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Year 10 Methods pathway Victorian Curriculum 2.0

© 2023 Mathspace Pty Ltd Copyright Notice This Work is copyright. All rights are reserved. Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. For permission to use material from this text or product, please email hello@mathspace.com For our full digital offering, visit mathspace.co Mathspace Pty Ltd Suite 1102, 418a Elizabeth St Surry Hills NSW The Victorian Curriculum F-10 content elements are © VCAA, reproduced by permission. The VCAA does not endorse or make any warranties regarding this resource. The Victorian Curriculum F-10 and related content can be accessed directly at the VCAA website. Title: Mathspace Victorian Curriculum - Year 10 Methods pathway - 2024 edition Print Run: xx ISBN: xx Editors and lead authors: Erin Gallagher, Elisha Templin, Sarah Butterworth, Julia Reid, Anjelica Sieger, Jacqueline Cichello, Stephen George, Grace Peake, Jonathan Kim Sing, Toni Smythe, Dot Yung Writing and development team: Adam Humphries, Ma. Lourdes Ona, Bernadette Barillo, Sheramie Castillo, Abegail Fontanilla, Shaira Llanita, Lesley Lopez, Neil Lopez, Kerby Mendoza, Regine Jara, Albert Jara, Rolly Apdo, Michaela Belen Images and Design team: Alex Duckmanton, Chastine Marquez, Keith Gimeno

Contents

1 2

Indices and surds

1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

4 12 19 25 32 39 43 52 63

Algebraic fractions 2.01 2.02 2.03 2.04 2.05

3 4

Index laws Negative indices Optional: Simplify surds Optional: Add and subtract surds Optional: Multiply and divide surds Optional: Binomial expansions with surds Optional: Rationalise the denominator Optional: Fractional indices Chapter 1 review

Add and subtract fractions with numeric denominators Add and subtract fractions with algebraic denominators Optional: Add and subtract further algebraic fractions Multiply and divide algebraic fractions with numeric denominators Optional: Multiply and divide algebraic fractions with algebraic denominators Chapter 2 review

68 73 78 84 89 94

Expand and factorise

3.01 3.02 3.03 3.04 3.05 3.06

98 103 110 116 121 127 132

Factorise algebraic expressions Expand binomial products Factorise binomial products Factorise monic quadratic trinomials Factorise non-monic quadratic trinomials Complete the square Chapter 3 review

Linear equations 4.A 4.B 4.01 4.02 4.03 4.04 4.05

134

Linear graphs Equations of more than 3 steps Equations involving fractions Rearrange equations Parallel and perpendicular lines Linear models Graph linear inequalities Chapter 4 review

136 144 150 160 172 188

Contents mathspace.co

iii

5 6

Quadratics 192 5.01 5.02 5.03 5.04 5.05 5.06 5.07

194 202 224 235 250 256 268

280

Intersections and simultaneous equations 284 6.01 6.02 6.03 6.04 6.05 6.06

7 8

Quadratic equations Quadratic graphs Solve by factorisation Solve by completing the square The quadratic formula The discriminant Optional: Further quadratic equations Investigation: Derive the quadratic formula Investigation: Use bisection to solve quadratics Chapter 5 review

Solve graphically Simultaneous inequalities Solve using substitution Solve using elimination Applications of simultaneous equations Investigation: Intersections of curves Optional: Intersections of curves Chapter 6 review

Introduction to functions (Optional) 7.01 7.02 7.03 7.04

Optional: Define relations and functions Optional: Domain and range Optional: Graphs of xn Optional: Transformation of functions Chapter 7 review

Further trigonometry (Optional) 8.01 8.02 8.03 8.04

Optional: Angles with exact values Optional: The unit circle Optional: Graphs of sine and cosine Optional: Trigonometric equations Chapter 8 review

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

286 294 304 314 320

333

338 340 351 363 373 385

388 390 397 416 425 431

Exponentials 434 9.01 9.02 9.03 9.04 9.05 9.06

10.01 10.02 10.03

507 522

Logarithms Order of magnitude and logarithmic scales Optional: Other bases Investigation: Logarithms Optional: Logarithm laws Optional: Connect exponentials and logarithms Optional: Solve exponential equations with logarithms Chapter 10 review

528 537 546 550 555 562 576 582

Circles 586 11.01 11.02 11.03

12 13

436 446 460 484 493

Logarithms 526

10.04 10.05 10.06

Exponential relationships Characteristics of exponential graphs Optional: Transformations of exponentials Exponential equations Compound interest Investigation: Half life Exponential models Chapter 9 review

Investigation: Circles Circles at origin Translations of circles Investigation: Equation of any circle Optional: Transform circle equations Chapter 11 review

588 595 605 615

Hyperbolas 618 12.01 12.02 12.03

Investigation: Hyperbolas Direct and indirect proportion Hyperbolas Optional: Transformations of hyperbolas Chapter 12 review

620 631 643 656

Trigonometry 660 13.01 13.02 13.03 13.04 13.05 13.06

Pythagoras’ theorem Trigonometric ratios Angles of elevation and depression Bearings Optional: Sine and cosine rules Optional: Pythagoras in 3D

662 671 685 695 711

Contents mathspace.co

13.07

16 17 vi

730

Measurement 736 14.01 14.02 14.03 14.04

14.05 14.06 14.07

Optional: Trigonometry in 3D Chapter 13 review

Surface area of prisms and cylinders Volume of prisms and cylinders Surface area and volume of composite solids Estimation and accuracy Investigation: Rainfall collection Investigation: Instruments and methods Investigation: Scale models Optional: Spheres Optional: Cones Optional: Pyramids Chapter 14 review

738 748 758 770

779

Probability 784 15.A 15.01 15.02 15.03 15.04

Review Venn diagrams and two-way tables Independent and dependent events Conditional probability Mixed probability concepts Optional: Geometric probability Investigation: Simulation Chapter 15 review

Univariate statistics 16.A 16.01 16.02 16.03 16.04 16.05 16.06 16.07

Representations of data Quartiles Boxplots Cumulative frequency Compare data sets Samples and populations Optional: Standard deviation Optional: Comparisons using standard deviation Chapter 16 review

Bivariate statistics 17.01 17.02 17.03 17.04 17.05 17.06 17.07

Two-way tables Associations between categorical variables Bivariate numerical data Associations between numerical variables Line of good fit Interpolation and extrapolation Misrepresentations of data

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

786 800 812

827

830 832 845 859 871 887

897

904 906 917 935 945 959 970 983

17.08

998

Networks 18.01 18.02 18.03 18.04

Investigation: Misrepresentations in the media Investigation: Conduct a statistical investigation Optional: Correlation coefficient Chapter 17 review

Introduction to networks Network representations Paths Investigation: Traversing networks Planar graphs Chapter 17 review

Geometry and proofs 19.01 19.02 19.03 19.04 19.05 19.06 19.07 19.08 19.09

Deductive reasoning Proofs with parallel lines Congruent triangles Similar triangles Further geometric proofs Investigation: Shortest path Optional: Circles and chords Optional: Angles in circles Optional: Tangents Investigation: Algorithms in geometry Optional: Transformations on the plane using algorithms Chapter 19 review

Functions and polynomials (Optional)

Counting Methods (Optional)

20.01 20.02 20.03 20.04 20.05

21.01 21.02 21.03 21.04 21.05

Optional: Operate with polynomials Optional: Polynomial division Optional: Factor and remainder theorem Optional: Graph polynomials in factored forms Optional: Rates of change Chapter 20 review

Optional: Counting principle Optional: Factorial notation Optional: Permutations Optional: Combinations Optional: Applications of counting methods to probability Chapter 21 review

Contents mathspace.co

vii

1 Indices and surds Big ideas • Understanding and applying index laws enable us to manipulate and simplify expressions involving powers, facilitating problem-solving and exploring mathematical concepts. • Exploring irrational numbers, surds and their operations reveals the underlying principles of manipulating expressions involving roots. • Mathematical expressions can be transformed and represented in different forms using principles of equivalence and algebraic properties.

Curriculum Outcomes VC2M10A01 Factorise algebraic expressions by taking out a common algebraic factor VC2M10A02 Simplify algebraic products and quotients using exponent laws VC2M10AN01 Define rational and irrational numbers and perform operations with surds and fractional indices VC2M10AN02 Perform operations on numbers involving fractional exponents and surds

Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

4 12 19 25 32 39 43 52 63

A “surd” is a term used to describe an irrational number. The name originally meant “deaf” or “unheard” because they cannot be written exactly using fractions and decimals.

1.01 Index laws After this lesson, you will be able to… • apply the multiplication law to simplify expressions. • apply the division law to simplify expressions. • apply the power of a power law to simplify expressions. • apply the zero power law to simplify expressions. • solve problems using a combination of index laws.

Multiply terms with indices For the expression a5 × a3, since the factors have like bases we can use the multiplication law. a5 × a3 = a5 + 3 = a8 We can remind ourselves of the multiplication law by writing out the original expression in expanded form.

a× a× a× a× a × a× a× a For any base number a and any indices, also called exponents, m and n the multiplication law is: am × an = am+n When multiplying terms with a common base: • Keep the same base. • Find the sum of the indices.

Example 1 Simplify 6x2 × 4x6 using index laws.

Create a strategy Use the multiplication law.

Apply the idea 6x2 × 4x6 = 6 × 4 × x2+6 8

= 24x

Multiply the coeﬃcients and add the indices of x Evaluate and simplify

Reﬂect and check Notice that the coeﬃcients are multiplied separately when we use the multiplication law on the powers.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary The multiplication law: am × an = am+n

Divide terms with indices Similarly, we can avoid having to write each expression in expanded form by using the division law, which is also known as the quotient law.

a × a ×a× a× a a×a

To simplify

we can use the fact that

to get:

For any non-zero base number a, and any indices m and n we get,

When dividing terms with a common base: • Keep the same base. • Find the difference between the powers. We can also write the division law in the form: am ÷ an = am−n

Example 2 Simplify

Create a strategy Use the division law.

Apply the idea Subtract the indices of x

Evaluate the subtraction

Idea summary The division law:

where a ≠ 0.

1.01 Index laws mathspace.co

Power of a power law For any base a, and indices m and n, we also have the power of a power law: (am)n = am×n When simplifying a term with a power that itself has a power: • Keep the same base. • Find the product of the powers. Brackets are used to show which powers are applied to any coefficient. Be careful when raising negative numbers to a power. A negative number raised to an odd power gives a negative result, but raised to an even power gives a positive result. For example: (−2x2)3 = −8x6

Example 3 Simplify (−2x2)2

Create a strategy Use the power of a power law.

Apply the idea (−2x2)2 = (−2)2 x2×2 = 4x

Multiply the powers of the variable Evaluate the multiplication and coeﬃcient

Idea summary The power of a power law: (am)n = am×n

Zero power law We also have the zero power law which says that for any non-zero base a, a0 = 1 That is, taking the zeroth power of any number will always result in 1.

Example 4 Simplify 18a0

Create a strategy Use the zero power law.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea Notice that only a is raised to the power of 0, but not the coeﬃcient 18. 18a0 = 18 × a0

Separate the two parts of the term

= 18 × 1

Evaluate a0

= 18

Evaluate

Idea summary The zero power law: a0 = 1 where a ≠ 0.

Mixed operations Now we are going to look at questions that can be solved using a combination of these rules. It’s important to remember the order of operations when solving such questions. • The multiplication law: am × an = am+n • The division law: am ÷ an = am−n, a ≠ 0 • The power of a power law: (am)n = am×n • The zero power law: a0 = 1, a ≠ 0 We may also come across expressions such as (am × bn)p, and we can use the power of a power law on each variable to see that: (am × bn)p = am×p × bn×p

Example 5 Simplify (x5 y4)4

Create a strategy Use the power of a power law.

Apply the idea (x5 y4)4 = (x5)4 × ( y4)4

Raise each variable to the power of 4

× y Multiply the powers

y16 Simplify

=x =x

1.01 Index laws mathspace.co

Example 6 Simplify: a

Create a strategy We can use the zero power law, followed by the power of a power law, on the numerator. Then we can use the division law to simplify the expression, since there will be a power of x in both the numerator and denominator.

Apply the idea Use the zero power law

Use the power of a power law

Simplify the numerator

Use the division law

Simplify the power

Create a strategy Use the division law then the power of a power law.

Apply the idea Use the division law

Simplify

Use the power of a power law

Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary Some expressions require a combination of the power laws to simplify them. It’s also important to remember the order of operations when simplifying these expressions. The multiplication law: am × an = am+n The division law: am ÷ an = am−n, an ≠ 0 The power of a power law: (am)n = am×n The zero power law: a0 = 1, a ≠ 0

Practice questions Choose your question path Explorer 3, 4, 5abeghjk, 6aceghk, 7bdf, 8acegi, 9abe, 10abde, 11abfins, 12abh, 14, 15ace, 16–18

Adventurer 1–4, 5aeil, 6achk, 7bdf, 8bdhj, 9bde, 10begh, 11bhjmq, 12cdj, 14, 15, 17, 18, 20

Trailblazer 1–4, 5acdl, 6bhjl, 7ade, 8ceil, 9ceh, 10d-fh, 11eklqt, 12dfj, 13, 15, 17, 18, 19, 21

Understanding 1

When simplifying a division of terms that are powers with the same base, what operation is performed on the indices?

Rewrite this expression in expanded form and demonstrate how the division law for indices works.

Consider the expression ( p3)2. a

Are these expressions equivalent to ( p3)2: i

( p × p × p) × ( p × p × p)

p3 × p3

iii

( p × p × p)2

p3 × p2

( p × p × p) × ( p × p)

Determine whether these equations are true or false: i

( p3)2 = p3+2

( p3)2 = p3×2

Complete the equation: ( p3)2 = p⬚

1.01 Index laws mathspace.co

Complete each statement: a

x⬚ ÷ x3 = x7

(b8 × b2) ⬚ = b20

( y⬚ × y17)1 = y17

(3m2 × ⬚)4 = 81m24

am a n

5z5 × 3z8 × z9

Fluency Ex 1

Simplify: a

6y2 × y5

u2 × u6 × u3

i Ex 2

4y × 6y × 3y

a10 ÷ a5

Ex 4

4 2

3 4

5g h × 8g h

6bc × 8b a × 0

a3 × −2a8

6y7 × 6y5 × 6y3

8m2n4 × 7m6n

c m5 ÷ m2

d j11 ÷ j4

9y9 × 8 (−y)8 × 7y7

4y4 × (−2y6) × 4y5

2s2 t × (−5r2 s3) × 3r3 t4

(−4y7) × (−6y2) × (−4y2)

(−20b4) ÷ 2b4 ( y4)3

9r8 ÷ (4r2)

Simplify:

(−9y ) ÷ (−3y )

Simplify: a

(j4)6

(w2)11

(5h7)2

(xy2)5

(−x9)4

(−5x4)3

i 9

2w4 × w10 × w7

e 8

Ex 3

3y6 × 4y

Simplify: a

(3a2)4

Simplify: a

(w9 v6)4

(k5)3 × (k2)8

6 (r6)5 × 4 (r4)7

(x9 y)7 × (xy2)4

10 (r7)5 × (2rs)4

(a5)9 × (b6)10

( p10)6 × ( p4)3

((x3)4)5

(g0)7

(j8)0

(7st)0

6k0 ÷ (4x2)0

Simplify: a

24a0

10r0 + (2q3)0

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 5

Simplify: a

(x7 y2)4

p5 × ( p4)3

(4y4)3

m8 ÷ m2 × m5

p 9 ÷ p5 ÷ p2

(−2x3)4

(11x4)0 − (12x8)0

(15x3)0 + 150 − 15y0

((x2)6)5

m (4u5 v2)2

(3a2 b5 c)4

(3y5)2 × (5y2)3

Ex 6

(4h ) + 18 ÷ (3g )

(c10)11 ÷ (c8)3

(10x2 × 10x5)0 − 10x0

(m12)9 ÷ (m4)2

Simplify:

(−240u32) ÷ (−8u9) ÷ (−5u12)

Reasoning 13

When simplifying the expression

, Clarice’s ﬁrst line of work was a3 + a2.

Is Clarice’s work correct? Explain. 14

When simplifying the expression

• Indira’s first line of work was a3 × a4 • Chad’s first line of work was a3 × a2 • Ernestino’s first line of work was Who is correct? Explain.

1.01 Index laws mathspace.co

Explain why each statement is incorrect, then give the correct solution: a

3s6 × 3s2 = 3s12

c e

(2x3)5 = 2x15

(4a)0 = 4a

Problem-solving 16

A scientist is modelling the growth of a bacteria colony. The colony doubles in size every hour. If she starts with a single bacterium, how many bacteria will there be after 6 hours? Express your answer using indices.

Will any values of x make 3x be equal to x3? Use examples to justify.

Find the value of a and b in the equation:

Determine whether each statement is true or false. a c

−xn = (−x)n for all x, only if n is odd n

−xn = (−x)n for all x always

−x = (−x) for all x, only if n is even

Rewrite 16p with a prime base.

Simplify: a

1.02 Negative indices After this lesson, you will be able to… • apply the negative index law to simplify expressions. • extend the concept of negative indices to include fractional bases.

Negative indices By simplifying

using the division law, we subtract the powers and get a−2. By using expansion

we see this is the same as

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

a×a×a 1 = a × a × a × a× a a× a This gives us the negative index law: for any non-zero base a, base to a negative power: • Take the reciprocal of the expression. • The power becomes positive.

. That is, when raising a

This also means that:

Example 1 Fill in the box to make the statement true:

Create a strategy Use the negative index law.

Apply the idea Write the denominator as a power of 5

Apply the negative index law. −2 should be placed in the box.

Example 2 Simplify

Create a strategy Use the division law.

Apply the idea Subtract the powers and divide the coeﬃcients

Simplify the power

Apply the negative index law

1.02 Negative indices mathspace.co

Example 3 Simplify 20m6 ÷ 5m13 × 9m2

Create a strategy Use the division law then the multiplication law.

Apply the idea Use the division law

Multiply the coefficients then use the multiplication law

Use the negative index law

Idea summary The negative index law: where a ≠ 0.

Fractional bases with negative indices When raising a fractional base to a negative power we can combine the individual rules we have seen so far.

Exploration We may see expressions in the form

We can simplify these expressions by starting with

Power law

Rewrite as division

Use negative index law

Multiply and take the reciprocal

Simplify the product

How could we rewrite

Are there any values that a or b cannot have?

using positive indices?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 4 Simplify: a

Create a strategy Find the reciprocal of the fraction inside the bracket then change the power to positive.

Apply the idea Find the reciprocal of the fraction then change the power to positive

Apply the power to the fraction

Evaluate the powers

Create a strategy Use the negative index law, remembering we take the reciprocal of the fraction.

Apply the idea Take the reciprocal of the fraction and change the power to positive

Use

Use (a × b)n = an × bn

Simplify using index laws

Use the negative index law again to end up with positive indices

1.02 Negative indices mathspace.co

Idea summary Negative index law with fractions:

where a, b ≠ 0.

Practice questions Choose your question path Explorer 1–3, 4b, 5a-g,l, 6abdfj, 7ac, 8ac, 10ae, 11acf, 12bd, 13, 15, 19, 20

Adventurer 1–3, 4a, 5acegik, 6egknp, 7bd, 8dg, 9adh, 10ceh, 11cgko, 12ce, 13, 15, 17–20, 22

Trailblazer 1–3, 4c, 5cdghji, 6fhlmo, 7cd, 8gh, 9bceh, 10d-f,h, 11dklp, 12ef, 14–23

Understanding 1

True or false: a

Is 4−2 less than or greater than 1?

Write

a−4 = −a4

102

103

104

using a negative index?

Fluency 4

Complete each table: a

2−4

10−4

2−3

10−3

2−2

10−2

2−1

10−1

100

101

100 c

3−4

3−3

3−2

3−1

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 1

Fill in each box. Evaluate if possible: a

Rewrite with positive indices: a

a−9

p−2

3x−4

8p−3

2x−8 y3

m 3a4 b−5 c−7

4−2 k−3 l7

Ex 3

m−5 n−4 p4

5y9 × 4y−3 2h

−4

× 4h

7a4 × 4a−2 3y

−2

−5

× 2y

5x4 × (−3x−8) 2

−5

−4y × (−4y )

3y−2 × 4y−3

(5mp)2 × mp−2

(3y2)−2

Evaluate: a

2−2 × 24

2 × 3−2

10 ÷ 2−1 + 3

20 × 2−2 + 6

3 + 4

3 × 4−2 + 5 × 2−4

102 × 5−2

(7 + 32) × 2−2

−1

6y7 × 2y−5 × 5y3

Simplify: a

Ex 4

p−2 q3

Simplify: a

Rewrite without fractions: a

Ex 2

7x−9

2y6 × 4y7 × 4y−5 −2

× 5x × 4x

12x−9 × 2x4 × 3x−2

4y3 × 3y8 ÷ 2y−1

8y10 ÷ 2y−4 ÷ y3

40x−2 ÷ 5x8 ÷ 4x−9

14x−8 ÷ 2x × 3x6

(3p−4)−2

−5

Simplify: a

(2m)−3

(4m−6)4

1.02 Negative indices mathspace.co

Simplify: a

(3x−4 × 2x2)2

(10x3 y−4)2

(4x3)−1 × ( y−2)−3

(10x3)3 × (2x4 y3)−2

Reasoning 13

Oreste and Donovan are writing an equivalent expression for Oreste writes down

as a single term.

and Donovan writes down x−10.

Who is correct and why? 14

We can evaluate 03 = 0, but not 0−3. Explain why 0−3 is undeﬁned.

A student wanted to express 5a−1 without negative indices and wrote their answer is incorrect, and write the correct answer.

For x > 1, what happens when we raise x to a very large negative index? That is, what happens to x−a as a gets closer to 0? Explain your answer.

. Explain why

For variable p, calculate the following expressions for p = −2, 0, 4, 10: • 2p • 4p • 2−p + 4p • 2−p × 4p

What do you predict will happen to the value of each expression as p increases signiﬁcantly?

Problem-solving 18

Write

Complete:

Solve for n:

at least three different ways.

(x3 y−5)n = x−12y20

Why is 2−2 < 2 but

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(a−5 b3)n = a15 b−9

? Give another example where this pattern applies.

It is estimated there were 7 × 108 jellyﬁsh in the largest bloom ever observed. If each jellyﬁsh can consume 5 grams of plankton in a day, how long would it take the bloom to consume 10 tonnes (1 tonne = 1 000 000 g) of plankton?

In electrical engineering, the formula used for calculating the capacitance of a capacitor is , where C is the capacitance measured in farads, Q is the charge on the capacitor measured in couloumbs and V is the voltage across the capacitor measured in volts. Use this formula to answer the following questions. a

A capacitor has a charge (Q) of 15 × 10−6 C (coulombs) and a voltage (V ) of 5 volts. What is the capacitance (C) of the capacitor?

1 microcoulomb is equivalent to 1 × 10−6 couloumbs. A capacitor has a charge of 20 microcoulumbs, and a voltage of 4 volts, what is the capacitance of the capacitor? Remember to ﬁrst convert the charge to coulombs.

A capacitor has a capacitance (C) of 10 × 10−4 F ( farads), and a charge of 30 microcoulombs. What voltage was applied across the capacitor?

1.03 Optional: Simplify surds After this lesson, you will be able to… • understand the difference between rational and irrational numbers. • define and recognise surds. • distinguish between exact values and approximations. • simplify surds.

Rational and irrational numbers A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. This might also look like a terminating or recurring decimal. An irrational number is a number which cannot be written as a fraction of two integers. These also cannot be written as terminating or recurring decimals. A particular type of irrational number is a surd. A surd is a root which cannot be simpliﬁed to a rational number. root symbol

index

radicand

surd

• The radicand of a square root is not a perfect square, like • The radicand of a cube root is not a perfect cube, like • The radicand of a fourth root is not a perfect fourth power, like

1.03 Optional: Simplify surds mathspace.co

A number has an exact value, such as fractions and roots. Numbers also have approximations. These are other numbers which are close but not equal to the exact value. We usually ﬁnd approximations by rounding the decimal representation. For example, if

is the exact value, then 0.667 is one possible approximation.

Example 1 Is

rational or irrational?

Create a strategy Put the number in your calculator.

Apply the idea When

we get 3.608 826 080 139. This value goes on forever with no pattern, so it is irrational.

Also 47 is not a cube number - there is no interger which gives 47 when cubed, since 33 = 27 and is irrational. 43 = 64. So this further conﬁrms that

Reﬂect and check For this to be rational, the radicand would need to be a perfect cube. The ﬁrst four perfect cubes are: • 13 = 1 • 23 = 8 • 33 = 27 • 43 = 64 27 < 47 < 64, so 47 isn’t a perfect cube, and

is irrational.

Idea summary A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers. A surd is a root which is irrational. A number has an exact value, such as a fraction or root. Numbers also have approximations. We usually ﬁnd approximations by rounding the decimal representation of the exact value.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Simplify surds Square root and cube root expressions can sometimes be written in simpler forms by simplifying any factors which are perfect squares, cubes, etc. For simplifying square roots, we can use the properties:

Combining these two, we can see that:

The same operations that can be applied when simplifying numerical radicals can be applied when simplifying radicals with algebraic expressions.

Exploration We can use prime factors to help us split a number into a product of a perfect square and a remainder. If we wanted to simplify

Using the factor tree, we can see that 60 = 22 × 3 × 5, so there is a perfect square factor of 22 = 4.

60 2

, we want to see if 60 has any factors which are perfect squares.

30 10

3 2

How can you identify if a number has a perfect square factor after drawing a factor tree?

Is it possible to get two different factor trees for the same number?

Is it possible to get two different prime factorisations?

Can every surd be simpliﬁed?

How would you rewrite 180 as a product of a perfect square and a remainder?

What would happen if we didn’t use the largest perfect square?

In general, we can use the following facts to simplify surds, for a, b ≥ 0 and m, n positive integers,

1.03 Optional: Simplify surds mathspace.co

Example 2 Simplify

Create a strategy Find the highest perfect square that is a factor.

Apply the idea We can ﬁnd that 12 = 4 × 3 where 4 is a perfect square. Write the surd as a product of its factors

Evaluate

Simplify

Reﬂect and check We can double check that we found the largest perfect square to factor out by looking at the radicand after simplifying. Since 3 has no factors other than 1 that are perfect squares we know this is fully simpliﬁed.

Example 3 Simplify

Create a strategy Use

to split up

into two factors.

Apply the idea

Write the surd as a product of its factors

Rewrite 8 as 23

Simplify

Evaluate the product

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary We can simplify surds by writing the radicand as a product of the largest possible perfect square or cube and a remainder. These identities are useful when simplifying expressions involving surds:

Practice questions Choose your question path Explorer 2–4, 6, 7acfepv, 8bef, 9ac, 10, 12, 16, 17

Adventurer 2–4, 5bd, 6cd, 7bfjnrv, 8afg, 9bc, 11–13, 15, 16, 18ace

Trailblazer 1–4, 5bc, 6d, 7dhlptx, 8dgh, 9cd, 11, 13–17, 18aceg

Understanding 1

Which statements are true and which are false? a

All surds are irrational.

All square roots are surds.

The cube root of a perfect square is always rational.

Some square roots are rational and some are irrational.

The cube root of a perfect cube is a surd.

Are these surds? b

a 3

Fill in the boxes:

Fluency Ex 1

Is each number rational, irrational, or neither? a

∞

1.03 Optional: Simplify surds mathspace.co

Rewrite each root as the product of the highest perfect square or cube factor and a whole number: a

Simplify: a

Ex 2

Simplify, and then determine if the value is rational or irrational: a

Simplify:

Simplify: a

Reasoning 10

Consider a

an exact value?

A calculator states that

is 17.029 386 366. Is this still exact? Explain.

During a physics assignment, Jordan tried to evaluate the cube root of 27 on their scientiﬁc calculator. They ended up with an answer of 19 683. Identify and correct Jordan’s error.

Are these expressions written in their simplest surd form? Explain your reasoning. b

a 13

Heiko is asked to simplify

. He notes that 80 = 4 × 20, and simpliﬁes to

Has he fully simpliﬁed the surd? If yes, explain how you know. If not, further simplify Heiko’s answer.

Use another pair of factors for simplifying

. Explain your choice.

If p is a rational number and p < 0, which statements are true and which are false? Explain your answer. a

is negative

is rational

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

is positive

Problem-solving 15

The volume V of a regular tetrahedron with edge length a is given by:

Find the volume of a tetrahedron with a side length of 2 cm.

Rearrange the formula to isolate a.

Now, ﬁnd the side length of a tetrahedon with a volume of 72 cm3.

Write the expression

Simplify

Simplify:

as a single surd.

1.04 Optional: Add and subtract surds After this lesson, you will be able to… • understand what “like surds” are and how to identify them. • add surds. • subtract surds. • solve problems involving the addition and subtraction of surds.

Like surds We know that we can simplify 5y + 2y to give 7y as 5y and 2y are like terms, so we can collect them together. Any surd which the same index and radicand is a like surd. For example, surds.

and

are like

1.04 Optional: Add and subtract surds mathspace.co

Exploration Sometimes terms might not look like like terms, but they can be simpliﬁed to be added or subtracted. and

appear to be unlike terms, but if we simplify:

So these are like terms and just as 2p + 3p = 5p.

can be rewritten as

. This would simplify to

Can all surds be simpliﬁed?

What are two surds that don’t appear to be like, but can be simpliﬁed to be like?

If two surds are fully simpliﬁed and have different radicands, can we say that they are not like surds or could they be simplified differently to be like surds?

Example 1 Are each pair “like surds”? a

and

Create a strategy Simplify each surd and compare their radicands.

Apply the idea Express 80 using the largest possible square factor

Simplify Express 45 using the largest possible square factor

Simplify

These are like surds as they are both of the form

Reﬂect and check When simplifying

, if 4 was identiﬁed as a perfect square factor to give

it would not be fully simpliﬁed as there would still be a factor of 4 in the radicand.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

and

Create a strategy Simplify each surd and compare the radicands.

Apply the idea Express 18 using the largest possible square factor

Simplify Express 27 using the largest possible square factor

Simplify

These are unlike surds as they have different radicands.

Idea summary We can determine if two surds are like surds by simplifying them and comparing their radicands.

Add and subtract surds We want to add and subtract surds. Similar to working with algebraic terms, we need like surds in order to add or subtract. In particular,

. For example: Simplify the roots

Evaluate the addition

Now incorrectly simplifying the same expression:

Incorrectly combining like terms

Evaluate the addition

Simplify the root

If a and b are different values, we can not simplify the expression

. This can be extended to adding or

But if they have the same value, then subtracting multiples of surds, such as

We can add and subtract like surds by adding or subtracting their coeﬃcients:

The same ideas can be used with cube roots, fourth roots, etc.

1.04 Optional: Add and subtract surds mathspace.co

Example 2 Simplify

Create a strategy Add the coeﬃcient of like surds.

Apply the idea Since both terms have irrational part

, they are like surds and we can add 10 and 14 together.

Example 3 Simplify

Create a strategy Group the like surds and then add the coeﬃcients of like surds.

Apply the idea Group like surds

Simplify like surds

Example 4 Simplify

Create a strategy Simplify each surd before adding.

Apply the idea We can see that 45 = 9 × 5 where 9 is a perfect square, and 80 = 16 × 5 where 16 is a perfect square. Write the surds as a products of their factors

Evaluate

Add like surds

and

Reﬂect and check To ﬁnd the largest perfect square factor, we can use a factor tree or trial and error. To use trial and error, we can divide 45 and 80 by perfect squares such as 4, 9, 16, 25, 36, … on the calculator and use the largest number with an integer remainder.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 5 Simplify

Create a strategy Simplify the surd and then add the like surds.

Apply the idea Write the surds as a products of their factors

Simplify the cube root using 83 = 512

Subtract the like surds

Idea summary We can add and subtract like surds:

The radicand, a, may be numerical or involve variables. Before we add and subtract surds, we make sure that all terms are already in their simplest form.

Practice questions Choose your question path Explorer 1, 2, 3aceg, 4adfjln, 5bc, 6ace, 7bd, 9, 10, 12, 13

Adventurer 1, 2, 3bdfh, 4aceim, 5ac, 6adfi, 7aegi, 9–11, 13, 14

Trailblazer 1, 2ae, 3bdfh, 4fijn, 5cg, 6ghkl, 7dfhj, 8, 10, 11, 13–15

Understanding 1

Fill in the blanks: a b

Ex 1

To add or subtract two surds they must be ⬚.

For a surd to be in its simplest form, the radicand must not have any factors that are a ⬚.

Are each pair “like surds” or not? a

and

and and

and

1.04 Optional: Add and subtract surds mathspace.co

Are these equations true or false? a

Fluency Ex 2

Ex 3

Simplify:

Simplify: a

g Ex 4

Ex 5

Simplify: a

Simplify:

Reasoning 8

Explain why we simplify surds.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For

Simplify 4x + 5y + x − 2y.

How does 4x + 5y + x − 2y relate to

Simplify

For

. :

Choose some values of k and m to show that the equation is not generally true.

Are there values of k and m that would make the equation true?

Apply the same reasoning to

. What are your conclusions?

Identify and explain the error in each equation: a

Problem-solving 12

Find the perimeter of the triangle in simpliﬁed surd form. 80

125

An athlete is participating in a race around the track shown in the diagram, which has dimensions in kilometres. a

Determine the exact distance of one lap around this track.

If the track distance is rounded to the nearest kilometre, determine by how much this approximation would overestimate the distance. Give your answer to the nearest metre.

A rectangle has length and width the nearest whole number to each surd.

4− 7

4+ 7

. Katie estimates the perimeter by using

What was Katie’s answer?

Is Katie’s answer an overestimate or an underestimate? Explain.

Write an expression for the exact perimeter.

What is the percentage difference between Katie’s estimate and the true value? Round your answer to two decimal places.

Emma found a piece of paper on which another student had written the ﬁrst four terms of a sequence. It read . a

Emma wanted to continue the sequence. What would the next three terms be?

The sum of the ﬁrst n integers is sequence as an exact value.

If the last term of the sequence that Emma wrote was terms in the sequence as an exact value.

. Find the sum of the ﬁrst seven terms of this , ﬁnd the sum of all the

1.04 Optional: Add and subtract surds mathspace.co

1.05 Optional: Multiply and divide surds After this lesson, you will be able to… • multiply surds. • divide surds. • solve problems involving multiplication and division of surds.

Multiply surds We have seen that we can simplify surds using the property

If we wanted to multiply two surds together, we could combine them back in the same way. That is This is sometimes known as the multiplication law of surds. Unlike adding and subtracting surds, these do not have to be like terms before multiplying. If we want to ﬁnd the product of surds that have a coeﬃcient, we can use the properties of multiplication to rearrange the expression.

Example 1 Simplify

Create a strategy Use the rule

Apply the idea

Apply the rule Evaluate 19 × 17

Example 2 Simplify

Create a strategy Multiply the integer terms together then the surds together.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea Rearrange the terms

Perform the multiplication

We can simplify our answer since 363 = 121 × 3, where 121 is a perfect square.

Write the surd as a product of its factors

Evaluate

Simplify

Example 3 Simplify

Create a strategy Use the distributive law: A (B − C) = AB − AC

Apply the idea

Apply the distributive law

Multiply the surds and integers

Evaluate the products

Idea summary To multiply terms involving surds, use the rule:

Divide surds Similar to multiplying surds, we can divide surds by writing the values underneath the same symbol. That is

This is sometimes know as the division law of surds. If we want to ﬁnd the quotient of surds that have coeﬃcients, we can divide the integer terms and the surd terms separately.

1.05 Optional: Multiply and divide surds mathspace.co

Example 4 Simplify

Create a strategy Use the rule:

Apply the idea Apply the rule

Perform the division

Example 5 Simplify

Create a strategy Divide the surd parts and integer parts separately.

Apply the idea

Separate the integer and surd parts

Write the surds as one surd

Evaluate the divisions

Evaluate the square root

Evaluate

Reﬂect and check Similar to the multiplication example above, we can approach this question by ﬁrst simplifying . We can rewrite the original expression: Simplify

Separate the integer and surd parts

Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary To divide terms involving surds, use the rule:

Practice questions Choose your question path Explorer 1, 2abfgi, 3, 4abgh, 5acfg, 6acg, 8abdfgit, 11, 13, 16, 18, 20, 22a

Adventurer 1, 2bdfhj, 3ac, 4cgh, 5cfik, 6cdk, 7bf, 8adegk, 9abe, 11, 13, 17, 18, 20, 21, 22c

Trailblazer 1, 2g-j, 3bd, 4bef, 5chjk, 6dhl, 7adjl, 8lmqv, 9aceg, 10, 12, 14, 15, 17, 19, 21, 23

Understanding 1

Is each expression deﬁned or undeﬁned in the real numbers? a

Which of these statements are true and which are false? Assume all variables are positive. a

Complete these statements using this example:

Fluency 4

Simplify: a

h 1.05 Optional: Multiply and divide surds mathspace.co

Ex 1, 2

Ex 3

Ex 4, 5

Simplify: a

Simplify:

Reasoning 10

Identify whether the resulting values are always a surd, sometimes a surd, or never a surd. Justify your answers with examples. a

The quotient of two surds.

The quotient of a surd and an integer.

The quotient of two integers.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Eliora claimed that if a rectangle has sides whose lengths are surds, that the area must also be a surd. Is Eliora correct? Provide an example to justify your conclusion.

The cross-sectional area of a prism can be found by dividing the volume by the height. If both the volume and the height are surds, is it possible for the height to not be a surd? Provide an example to justify your conclusion.

Pythagoras’ theorem states that c2 = a2 + b2, where c is the length of the hypotenuse of a right-angled triangle, and a and b are the lengths of the shorter sides. b

Give an example where a and b are surds, but c is not a surd.

Give an example where a and b are surds, and c is also a surd.

Suppose that a and b are different prime numbers. a

Explain why

is a surd, but

Explain why

is a surd?

is not.

ISO paper, for example A4 paper, always has a length that is times the width.

A3 A0

A4 A4

Give an example where a and b are not surds, but c is a surd.

Scarlett is starting a new standard series of paper where the length is times the width. The smallest size in the series has a width of . Each subsequent member times the lengths of of the series has lengths that are the previous size. Explain why all of the paper in this series would have an area which is a surd.

Problem-solving 16

Find the exact area of these rectangles. Give your answers as simpliﬁed surds. a

10 m

125 cm

2 20 cm

1.05 Optional: Multiply and divide surds mathspace.co

Find the area of the trapezium in simpliﬁed surd form:

The body surface area of a person in square metres can be approximately modelled by , where A is the surface area, h is the height of the person in cm, and w is the weight of the person in kg. a

Use the model to ﬁnd the surface area of a person who is 164 cm tall and weighs 63 kg. Leave your answer in exact form.

Find the approximate surface area of the person, to the nearest hundredth of a square metre.

What are the possible height and weight ranges for a person for a person with a surface area of 2 m2?

Find the exact perpendicular height of a triangle whose area is measures .

A rectangle has a side length of

and a width of

and whose base .

Find the exact area of the rectangle.

State the area of the rectangle to the nearest metre.

Find an approximation for the area of the rectangle by rounding the length and width to the nearest metre before multiplying.

Is there a signiﬁcant difference between the estimated area found by rounding before and after the area calculation? Calculate the difference as a percentage of the exact area.

A big rectangular carpet is metres long and metres wide. A small rectangle, 4 metres long and metres wide, is cut out of the carpet. Find the area of the carpet that is remaining in simpliﬁed surd form.

Ciera needs to perform each of these calculations, where uses x rounded to one decimal place as an approximation.

. In her calculations Ciera

Determine how much each of her calculations differ from the exact value. Give your answer rounded to 2 decimal places. a 23

10x

x2 + 5

50x − 2x2

Vaughan needs 40 lengths of rope, each

long. Vaughan rounds the individual

rope length required up to the nearest metre and then multiplies by 40 to obtain an approximate amount to purchase from the hardware store. Determine how much less Vaughan’s estimate would be if he hadn’t used rounding for each rope length.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

1.06 Optional: Binomial expansions with surds After this lesson, you will be able to… • understand how to apply the distributive law to multiply binomials. • apply the method of multiplying surds to binomial expressions. • expand and simplify binomial expressions involving surds. • identify and apply the special cases of expanding binomials, such as perfect squares and the difference of two squares.

Binomial expansions with surds We can multiply surds using the property , and we can multiply two binomial expressions using the property (A + B) (C + D) = AC + AD + BC + BD. A, B, C and D can be any type of terms - they don’t need to be only variables or integers. We can now combine the two concepts together allowing us to expand and simplify binomial expressions involving surds. For example:

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.06 to answer these questions. 1.

Give two examples where the simplified expansion has four terms. What similarities do you notice? What differences?

Where did the rational terms in the expansion come from?

Where did the irrational (surd) terms in the expansion come from?

When did terms in the product need to be simplified?

Can you come up with a product where the simplified expansion only has two terms?

Can you come up with a product where the simplified expansion is purely rational?

1.06 Optional: Binomial expansions with surds mathspace.co

Example 1 Expand and simplify

Create a strategy Expand using the distributive law: (A − B) (C − D) = A (C − D) − B (C − D)

Apply the idea Apply the distributive law

Expand both brackets

Example 2 Expand and simplify

Create a strategy Expand using the rule for the difference of two squares: (A + B) (A − B) = A2 − B2

Apply the idea Apply the rule

Evaluate the indices

Evaluate

Reﬂect and check If we hadn’t noticed that this was a difference of two squares expression, we could have performed the multiplication as usual and then simpliﬁed to get the same result:

Example 3 Expand and simplify

Create a strategy Use the binomial expansion (A − B)2 = A2 − 2AB + B2 to expand the brackets.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea

Apply the rule

Evaluate the indices and product

Perform the multiplication

Add the integer terms

Idea summary We can expand the product of two binomial expressions using the rule: (A + B) (C + D) = AC + AD + BC + BD There are two special cases of expanding binomials: • •

(A + B)2 = A2 + 2AB + B2 (called a perfect square) (A + B) (A − B) = A2 − B2 (called a difference of two squares)

Practice questions Choose your question path Explorer 1ac, 2bc, 3ac, 5abg, 6a-c, 7bc, 10, 13, 15

Adventurer 1bd, 2bd, 3bf, 4bd, 5b-d, 6aeg, 7abf, 8bc, 9, 10, 13–15

Trailblazer 1d, 2ad, 3ef, 4ad, 5cef, 6bfh, 7ceh, 8ad, 9, 11, 12, 14, 15

Understanding 1

Expand and simplify: a

(a + b) (c + d)

(x + 3) (x − 2)

(x − 5) (x + 5)

(x + 2)2

Simplify: a

When expanded and fully simpliﬁed, will these expressions have all rational terms, all surd terms, or a mix of rational and surds? a

d 1.06 Optional: Binomial expansions with surds mathspace.co

Fluency Ex 1

Expand and simplify:

Ex 2

Ex 3

Expand and simplify: a

Fill in the boxes for each equation. a

Reasoning 9

Without expanding, explain whether or not expanded.

Explain why when expanding and simplifying there would be two terms and there would be four terms, provided c ≠ a and d ≠ b. for

For:

will have an

term when it is

Find values of a, b, c, and d, so that the expansion has four terms, none of which are like terms.

Find values of a, b, c, and d, so that the expansion simpliﬁes to two terms.

Can a fully simpliﬁed product of two binomials involving surds ever have three terms? If so, give an example. If not, explain why not.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Problem-solving 12

Expand and simplify

What value of m can be substituted into rational?

. so that the expression is

For the given right triangle: a

Find c, the length of the hypotenuse of the triangle.

Find the exact perimeter of the triangle.

Find the exact area of the triangle.

10 − 7

10 + 7

For the given isosceles triangle: a

Find h, the height of the isosceles triangle.

Find the exact perimeter of the isosceles triangle.

Find the exact area of the isosceles triangle.

5 2

5 −1

For the given trapezium: a

Find the exact area of the trapezium.

Find the exact perimeter of the trapezium.

2 5 17 + 5 2 17

1.07 Optional: Rationalise the denominator After this lesson, you will be able to… • understand the process and importance of rationalising the denominator. • apply the technique of squaring a surd to rationalise denominators. • use conjugates to rationalise binomial denominators. • apply these techniques to rationalise denominators in solving problems.

Rationalise the denominator with perfect squares Fractions with surds in the denominator are harder to calculate with. For these fractions we can rationalise the denominator, which is a method used to rewrite the expression without surds in the denominator.

1.07 Optional: Rationalise the denominator mathspace.co

Exploration For the expression

What do you think will happen if we multiply the denominator by

Multiply the denominator by . Does the result have the same value as the original expression? Use your calculator to verify your answer.

What number can we multiply by that does not change the value of the original number?

Using your answer, what fraction should we multiply

that will not change the value of

the expression, but will eliminate the surd from the denominator? 5.

Multiply

by your answer to the previous question. Does the result have the same value

as the original expression? Use your calculator to verify your answer.

When an expression has only one term in the denominator, we can rationalise the denominator by multiplying the numerator and denominator by the surd in the denominator. To rationalise the denominator of the form

we want to multiply it by the fraction

Since we are multiplying the numerator and denominator by the same number, it is the same as multiplying by 1, which does not change the value of the expression.

Example 1 Express the fraction in simplest surd form with a rational denominator:

Create a strategy Multiply the numerator and denominator by

Apply the idea Multiply by

Evaluate the multiplication

Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 2 Simplify

Create a strategy Check if we can simplify the fraction ﬁrst before rationalising.

Apply the idea Notice that 3 is a factor of both 39 and 6

Separate into two fractions

Rewrite the surds

Evaluate the divisions

Reﬂect and check Rationalising the denominator would have given us the same result, but it requires more steps: Multiply by

Evaluate the multiplication

Simplify the surds

Evaluate the division

Idea summary To rationalise the denominator of the form

we want to multiply it by the fraction

If possible, we should simplify the surds before rationalising.

1.07 Optional: Rationalise the denominator mathspace.co

Binomial denominators Exploration For these surds: • • • • 1.

Calculate the product of every possible combination of two surds. That is, , and so on.

When multiplied together, which pairs result in a rational number?

How are the two expressions related?

For the expression

, what should we multiply by to eliminate the surd from the

denominator? Justify your answer. When there is a binomial expression in the denominator, multiplying by

will not eliminate

the surd. Instead, we can rationalise the denominator by multiplying top and bottom by the conjugate of the denominator. For any binomial expression A + B, we can ﬁnd a conjugate, A − B, by changing the sign of the second term. For example, the conjugate of

A binomial and its conjugate are called a conjugate pair. The product of a conjugate pair takes the familiar form of the difference of two squares: (A + B) (A − B) = A2 − B2 The expression A2 − B2 will be rational even if either term A or B are surds. For example:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 Express the fraction in simplest surd form with a rational denominator:

Create a strategy Multiply the numerator and denominator by the conjugate of

Apply the idea The conjugate of

is Multiply the top and bottom by the conjugate

Use difference of 2 squares in the denominator

Evaluate the indices

Evaluate the denominator

Factorise the numerator

Reflect and check Factorising the numerator is not necessary to answer this question, but doing so helps us check if the fraction can be simplified further.

Example 4 Evaluate, expressing your answer in simplest surd form with a rational denominator:

Create a strategy To subtract fractions, the denominators must be the same. We will rationalise the denominators before subtracting.

1.07 Optional: Rationalise the denominator mathspace.co

Apply the idea Rationalise the denominators

Evaluate the multiplication

Find a common denominator

Evaluate the multiplication

Evaluate the subtraction

Reﬂect and check Rationalising the denominator after subtracting will result in the same answer, but may require more steps. Find a common denominator

Evaluate the multiplication

Evaluate the subtraction

Rationalise the denominator

Evaluate the multiplication

Simplify the surds

Evaluate the multiplication

Idea summary For any binomial expression A + B, we can ﬁnd a conjugate A − B by changing the sign of the second term. A binomial and its conjugate are called a conjugate pair. To rationalise the denominator of the form

, we multiply the numerator and

denominator by the conjugate of the denominator. That is, we multiply it by the fraction :

Where possible we should simplify the expression before rationalising.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1, 3ac, 4ac, 5, 6aceg, 7acegik, 8acegh, 9ace, 10ace, 11a-d, 12a-c, 13, 15–17, 19

Adventurer 1, 3bc, 4bd, 5, 6bdfh, 7acegik, 8bdfhj, 9bdfi, 10bdf, 11c-e,h, 12b-d, 13, 15–18, 20

Trailblazer 1, 2, 3bd, 4gh, 5, 6bdfh, 7bdfhjl, 9adgk, 10dhl, 11c-e,kl, 12bfhjl, 14, 15, 17, 18, 20–22

Understanding 1

Simplify

Evaluate

Why is

equal to

Determine whether each statement is true or false: a

The product of a conjugate pair will result in a rational number.

A conjugate pair is a fraction with same numerator and denominator.

A fraction with the same numerator and denominator simplifies to 1.

A product of conjugate pairs is similar to a difference of squares.

Simplify: a

For each surd, what is the simplest surd we could multiply it by to produce an integer result? a

Simplify

Evaluate

1.07 Optional: Rationalise the denominator mathspace.co

Does each product show a conjugate pair or not? a

Fluency 7

Ex 1

Ex 2

Fi nd the conjugate: a

Rationalise the denominator and simplify: a

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 3

Ex 4

Rationalise the denominator and simplify: a

Evaluate in simplest surd form with a rational denominator: a

Reasoning 13

For x2 − y2 a

Explain why this expression is known as a difference of perfect squares.

Factorise the expression.

Explain how this relationship is useful in rationalising the denominator of an expression like

, if x and y are surds.

Explain why the rationalised form of

Xander says that to rationalise Justify your answer.

will contain a term of we can multiply it by

. Is he correct or incorrect?

Problem-solving 16

A rectangle has a length of

Pythagoras’ theorem states that in a right-angled triangle, c2 = a2 + b2, where c is the length of the hypotenuse and a and b are the lengths of the two shorter sides.

and width of

The shorter sides of a right-angled triangle are hypotenuse of this right-angled triangle.

. Find the area.

and

. Find the length of the

1.07 Optional: Rationalise the denominator mathspace.co

The ﬁrst term of a sequence is

. Subsequent terms of the sequence are found by , so the second term would be

multiplying the current term by so on. a

Find a term of the sequence that is rational.

Is the term found in part(a) the only term that would be rational?

A rectangular garden has an area of 18 m2 and width of garden in simplest form with a rational denominator.

Rewrite

Find the mean of

. Find the length of the

where x, y, and z are real numbers.

in the form of and

, and

, giving the answer in simpliﬁed rational form. Assume that

x is positive. 22

Evaluate

, giving the answer in simpliﬁed rational form.

1.08 Optional: Fractional indices After this lesson, you will be able to… • rewrite surds in index form. • evaluate expressions with fractional indices. • apply understanding of fractional indices to simplify fractions using negative indices.

Fractional indices with numerator of 1 We want to look at expressions of the form

Exploration Let’s explore different types of indices, starting by looking at the equation: a+b=1 • Find a combination where a and b are integers. • Find a combination where one of a or b is zero. • Find a combination where both a and b are positive.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

• Find a combination where both a and b are fractions. • Find a combination where a and b are equal. Next let’s look at 16a × 16b = 161 1.

What must be true about a and b?

Use your calculator to evaluate 16a and 16b for each combination of a and b found previously.

What do you notice about when a and b are fractions?

What do you notice about when a and b are equal?

Let’s formalise our exploration to help us work with fractional indices, starting by writing different form.

in a

First, using the multiplication index law, we have:

The definition of a square root is a value which multiplies by itself to give the original number. We know that Comparing these two statements we see that

We can confirm this by using the power of a power rule:

In a similar way we can look at the value

. Using the multiplication law:

From the definition of a cube root: Once again, by comparison we see that

These results can be generalised for any positive denominator. That is,

1.08 Optional: Fractional indices mathspace.co

Example 1 For a Rewrite in surd form.

Create a strategy Use the rule

Apply the idea

Reﬂect and check We can also write this as , but writing the root without an index is usual for a square root.

b Evaluate

Create a strategy

Apply the idea

Take the square root of 36

From part a

Use that 36 = 62

Simplify

Example 2 Write

with a fractional index.

Create a strategy Use the rule

Apply the idea .

Idea summary Fractional index law:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fractional indices with other numerators Next we look at expressions of the form

We know the power of a power law: Applying this index law, and the fractional index law that we’ve just seen:

In a similar way, we can see:

For example:

or:

More generally, the fractional index law states:

When solving problems with fractional indices, it doesn’t matter whether you start with the powers or the roots (although it might be easier to do it one way for any particular expression). Let’s simplify

in two ways:

Starting with the root:

Now let’s start with the powers:

You can see that we get the same answer both ways, but starting with the powers led to working with much larger numbers. 1.08 Optional: Fractional indices mathspace.co

Example 3 Rewrite each expression in index form: a

Create a strategy Use the rule

Apply the idea

Create a strategy Use the rule

Apply the idea

Example 4 Evaluate: a

Create a strategy Use the rule

Apply the idea Apply the rule

33 = 27

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Create a strategy Use the rule

Apply the idea Apply the rule

Solve inside the brackets, note that

22 = 4

Example 5 Write as a simpliﬁed fraction: a

Create a strategy Use the rules

and

Apply the idea Apply the rules

Evaluate the cube roots

Evaluate the squares

Reﬂect and check If we had evaluated the squares before the cube roots, the numbers would have become very large and would likely require a calculator.

1.08 Optional: Fractional indices mathspace.co

Create a strategy Use the rules

and

Apply the idea Use the negative index law

Write as roots

Evaluate the square roots

Idea summary When simplifying expressions with fractional indices, we may need to use many index laws to fully simplify. Fractional index law: The multiplication law: am × an = am+n The division law: am ÷ an = am−n, an ≠ 0 The power of a power law: (am)n = am×n Powers of fractions: The zero power law: a0 = 1, a ≠ 0 Negative index law: Negative index law with fractions

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1–4, 5abefijop, 6abd, 7ace, 8ac, 9abd, 10ab, 11ab, 12ace, 13ac, 14abe, 15eik, 16a-c, 19, 22, 23

Adventurer 1–4, 5cd,g-j,lo, 6ace, 7bdf, 8bd, 9bce, 10bc, 11bc, 12bdf, 13ac, 14abe, 15ejk, 16acd, 18, 19, 22, 23, 26achkmp

Trailblazer 1–4, 5ghjlprt, 6beh 7f-h, 8bd, 9dfh, 10fg, 11dgl, 12cdh, 13bd, 14beh, 15ikl, 16e-h, 17, 20, 21, 24, 25, 26m-p, 27, 28

Understanding 1

Complete the statement:

Complete the expressions: a

Which expressions are equivalent to a

160.75

e 4

c ?

Evaluate on your calculator, rounding to two decimal places, if necessary. b

80.5

Write in surd form

Evaluate

320.2

Fluency Ex 1

For each expression:

1.08 Optional: Fractional indices mathspace.co

Ex 2

Ex 3

Evaluate: a

Express with a fractional index: a

Express in surd form: a

Ex 4

Evaluate:

Express with a fractional index: a

Simplify:

Evaluate: a

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 5

Evaluate: a

Evaluate:

Express in the form xk, where k is rational: a

Reasoning 17

Explain circumstances where it is easier to evaluate

by writing it as

, rather than

. Use a numeric example in your explanation. 18

Without calculating, which is greater:

Given

Blair attempted to fully simplify an expression:

? Explain your reasoning.

, explain how this expression was simpliﬁed.

Rewrite the radical in index form

Rewrite the indices with a common denominator

Add the indices

Multiply the indices using power of a power law

Rewrite from index form into surd form

1.08 Optional: Fractional indices mathspace.co

Identify any errors that Blair has made and explain what he has done wrong.

Correct any errors.

Phoebe attempted to fully simplify an expression with positive variables: Use the power of a power law

Simplify numeric coeﬃcients

2 3

Multiply numeric and algebraic parts

Rewrite from index form into radical form

4 a

Identify any errors that Phoebe has made and explain what she has done wrong.

Correct any errors.

Problem-solving 22

A rectangle has a length of

and a width of

. Find the area of the rectangle in:

Fractional index form

A cube has sides of length

. Find the volume of the cube.

Using six different digits from 1 to 9, ﬁll in the boxes to make a true statement.

Using each digit from 1 to 6 once, ﬁll in the boxes to make the largest possible product.

Simplify each expression in index form. Assume all variables represent positive numbers.

Surd form

Simplify: a

Simplify:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Chapter 1 review 1

Simplify: a

x3 × 6x2

a4 × −3a5

Find the value of a and b in the equation:

Simplify:

4y10 × 5y−4

(4a)−3

5x4 × (−3x−8)

3y−2 × 4y−3

(6x−2 × 2x3)4

(Optional) Simplify fully: a

(Optional) Simplify fully, given the variable represents a positive value:

(Optional) Simplify fully: a

(5mp)2 × mp−2

(Optional) Simplify fully, then determine if the value is rational or irrational: a

Fill in the blanks: a

Simplify, giving your answers with positive exponents: a

(−2x3)4

Simplify, giving your answers with positive exponents: a

6m3 n7 × 9m6 n

Fill in each box. Evaluate if possible. a

−2y7 × 2y2 × −2y6 d

Simplify: a

(Optional) Assuming all variables represent positive values, simplify fully: a

Chapter 1 review mathspace.co

(Optional) Find the perimeter of the triangle in simpliﬁed surd form. 12

(Optional) Simplify fully: a

(Optional) Expand and fully simplify: a

(Optional) For this shape: a

Find the exact perimeter of the ﬁgure.

Find the exact area of the ﬁgure.

( 5 + 7 ) cm ( 5 + 7 ) cm

(Optional) Rationalise the denominator of the given expressions. Express your answer in simplest surd form: a

(Optional) A rectangular garden has an area of 21 m2 and width of of the garden in simplest form with a rational denominator.

(Optional) The initial term of a sequence is

. Find the length

. Consecutive terms of the sequence

are generated by multiplying the current term by

, thus the second term would be

, and so forth.

Identify a term of the sequence that is rational.

Is the term found in part(a) the sole term that would be rational?

(Optional) Write with a fractional exponent: a

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) For each expression: i

Write in surd form. ii

(Optional) Evaluate: a

Evaluate.

(Optional) Evaluate: a

Chapter 1 review mathspace.co

2 Algebraic fractions Big ideas Mastering the basics of algebraic operations and understanding common denominators are essential for simplifying and solving algebraic fractions.

Curriculum Outcomes VC2M10A03 Apply the 4 operations to simple algebraic fractions with numerical or single variable denominators VC2M10AA03 Simplify combinations of linear expressions with rational coefficients and the solution of related equations

Chapter outline 2.01 2.02 2.03 2.04 2.05

68 73 78 84 89 94

Did you know that the word “fraction” comes from the Latin word “fractus,” meaning “broken”?

2.01 Add and subtract fractions with numeric denominators After this lesson, you will be able to… • find common denominators for fractions with algebraic numerators and numeric denominators. • add and subtract fractions with algebraic numerators by finding a common denominator. • simplify resulting fractions by collecting like terms or dividing by common factors apply the techniques of adding and subtracting algebraic fractions in problemsolving scenarios.

Add and subtract fractions with numeric denominators An algebraic fraction is a fraction with a pronumeral in the numerator or denominator. Adding and subtracting algebraic fractions with numeric denominators is very similar to adding and subtracting purely numeric fractions. We ﬁrst need to ﬁnd equivalent fractions with a common denominator, then we add or subtract the numerators. We may be able to simplify our answer by collecting like terms, and/or dividing the numerator and denominator by common factors.

Exploration Notice that

Write another equivalent fraction with a denominator of 12.

Write a fraction equivalent to

What would be the lowest common denominator for

with a denominator of 15. Explain how you got this. and

Consider this expression, where a and b are numbers: To add these fractions, we need to find a common denominator. The most efficient way to do this is to use lowest common multiple (LCM) of the denominators. In this case, a common multiple of a and b is ab. We need to rewrite the fractions with a common denominator of ab, then add the numerators. The first fraction needs to have both numerator and denominator multiplied by b, and the second fraction must have both multiplied by a: Now, we can add the numerators to get the final result: 68

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 Simplify:

Create a strategy The fractions already have a common denominator, so we can subtract the numerators. Simplify by collecting like terms and dividing the numerator and denominator by any common factors.

Apply the idea

Subtract the numerators

Collect like terms

Divide numerator and denominator by 3

Reﬂect and check To check our answer, substitute x = 1 into both the original expression and the simpliﬁed expression. We could use any value of x, but 1 is the simplest. Substitute x = 1 into the original expression

Multiply and subtract

Evaluate Substitute x = 1 into the simpliﬁed expression Both the original expression and the simpliﬁed expression gave the same result, suggesting that the answer is correct.

Example 2 Simplify:

Create a strategy Rewrite the fractions with a common denominator, then add the numerators.

Apply the idea The lowest common multiple (LCM) of 3 and 5 is 15. Rewrite the fractions to have a common denominator

Evaluate the products

Add the numerators

Collect like terms

2.01 Add and subtract fractions with numeric denominators mathspace.co

Reﬂect and check In this case, the lowest common denominator is the product of the two denominators. The product will always be a common denominator, but may not be the lowest common denominator. For example with

, the LCM is 30, but the product of the denominators is 90. If we used 90

as the denominator, we would have to work with larger numbers than if we used 30.

Example 3 Simplify: 8x −

Create a strategy Rewrite the ﬁrst term as a fraction with the same denominator as the second term, then subtract the numerators.

Apply the idea Write 8x as a fraction

Rewrite the ﬁrst fraction to have a denominator of 3

Evaluate the products

Subtract the numerators

Collect like terms

Reﬂect and check To check our answer, substitute x = 1 into both the original expression and the simpliﬁed expression. Substitute x = 1 into the original expression

Evaluate the products and write the fraction as a mixed number

Evaluate the difference

Substitute x = 1 into the simpliﬁed expression

Evaluate the product and write the fraction as a mixed number

Both the original expression and the simpliﬁed expression gave the same result, suggesting that the answer is correct.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary To add or subtract fractions where the numerators are algebraic terms: 1. Rewrite the fractions with a common denominator (preferably the lowest common denominator) 2. Add or subtract the numerators 3. Simplify by collecting like terms and dividing the numerator and denominator by any common factors.

Practice questions Choose your question path Explorer 1–4, 5acgj, 6bdfk, 7a-c, 8ad, 9, 12, 13ac, 14a, 15, 17

Adventurer 1–4, 5acdfhjl, 6bcfkl, 7acfg, 8bdfg, 9–12, 13ac, 14ab, 15, 16a, 17

Trailblazer 1–4, 5cdfhj-l, 6bchl, 7acf-h, 8bdfh, 10–12, 13acd, 14–17

Understanding 1

Evaluate and simplify: b

a 2

Find the lowest common denominator for these pairs of algebraic fractions: a

and

Are these statements true or false? simpliﬁes to

a c

The expression

simpliﬁes to

is equivalent to 2z

The expression

Find the lowest common denominator.

Simplify

is equivalent to 3a

Fluency 4

For a

Ex 1

and

Simplify: a

2.01 Add and subtract fractions with numeric denominators mathspace.co

i Ex 2

Ex 3

Simplify:

Simplify each expression, by ﬁrst simplifying the fractions (where possible). a

Reasoning 9

Using mathematical vocabulary, explain how you can complete the equation.

Complete the equation:

Why is it important to simplify fractions before adding or subtracting them in algebraic equations?

Can we add or subtract fractions with different denominators directly? Explain your answer.

Problem-solving 13

Fully simplify these expressions: a

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Write an expression for the perimeter of the triangle shown, and simplify this expression. x 5

3x 10

Given y1 =

and y2 = − ,

Find y3 if y3 = y1 + y2

Sketch the graph of y3 on the Cartesian plane.

The fuel tank in Dario’s car holds p litres of petrol. After ﬁlling the tank, Dario uses tank driving to his friend’s house, then another

of the

of the tank driving to the beach. Write an

expression for number of litres of fuel left in the tank when Dario arrives at the beach, and simplify this expression.

2.02 Add and subtract fractions with algebraic denominators After this lesson, you will be able to… • determine the lowest common multiple (LCM) of two algebraic terms by considering the coefficients and pronumerals. • apply the knowledge of finding the LCM to add or subtract fractions that have algebraic denominators. • simplify fractions with algebraic denominators.

Add and subtract fractions with algebraic denominators Just like with other types of fractions, we need to ﬁnd a common denominator to add or subtract fractions with algebraic denominators. Once we have a common denominator, we can add or subtract the numerators of the fractions and simplify the result if possible. When at least one of the denominators is algebraic, then we will have an algebraic common denominator. Remember that the best common denominator to use is the lowest common denominator, which is the lowest common multiple (LCM) of the denominators. The LCM of two algebraic terms will be the LCM of the coeﬃcients, multiplied by the different pronumerals. For example, the LCM of 12 and 8 is 24, so the LCM of 12x and 8y is 24xy.

2.02 Add and subtract fractions with algebraic denominators mathspace.co

Exploration For 60x and 24y, let’s look at how the prime factors of the coeﬃcients can help us ﬁnd the lowest common multiple. 24

60 6 2

10 3

60 = 22 ⋅ 3 ⋅ 5

24 = 23 ⋅ 3

6 2

Between 60 and 24, what is the highest power of each prime factor (2, 3, and 5)?

What is the lowest common multiple of 60 and 24?

What is the lowest common multiple of 60x and 24y?

What is the product of 60x and 24y?

If simplifying

, which would be the easiest common denominator to use, the

lowest common multiple of 60x and 24y, or the product of 60x and 24y? Explain.

Example 1 Simplify

Create a strategy The fractions already have a common denominator, so we can add the numerators. Simplify by dividing the numerator and denominator by any common factors.

Apply the idea Add the numerators

Evaluate the sum

Divide numerator and denominator by 7

Reﬂect and check When working with fractions (algebraic or numeric), it is always best to check our solution for any common factors. If there are common factors, we should simplify the fraction further as we did in this example.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 2 Simplify

Create a strategy Rewrite the fractions with a common denominator, then subtract the numerators.

Apply the idea The lowest common multiple of 8y and 12y is 24y.

Multiply to get a common denominator

Simplify the products

Subtract the numerators

Collect like terms

Example 3 Write

as a single fraction.

Create a strategy Find a common denominator and combine the numerators of the fractions.

Apply the idea The lowest common multiple of 3y and 6x is 6xy. Multiply to get a common denominator

Simplify the products

Add the numerators

Reﬂect and check Notice in this case that the terms in the numerators are not like terms, so we cannot add them and our answer is fully simpliﬁed.

2.02 Add and subtract fractions with algebraic denominators mathspace.co

Idea summary To add or subtract fractions where the denominators are algebraic terms, we need to ﬁnd a common denominator. We can then add or subtract the numerators and simplify the result if possible. When the denominators are algebraic terms, the lowest common denominator will be the lowest common multiple of the coeﬃcients, multiplied by the different pronumerals.

Practice questions Choose your question path Explorer 1–3, 5a-d, 6a-cg, 7abh, 8abd, 9acg, 10ac, 12, 15e, 16

Adventurer 1–4, 5abe, 6dfh, 7bce-g, 8cef, 9b-e, 10bd, 11, 13, 14, 15bd

Trailblazer 1–4, 5e-h, 6ei-l, 7d-g, 8f-h, 9befh, 10, 13, 14, 15a, 16, 17

Understanding 1

Are these statements true or false? a

To add or subtract algebraic fractions, we need a common denominator.

When adding or subtracting algebraic fractions, we add the numerators and add the denominators.

If at least one of the denominators is an algebraic term, then the lowest common denominator of two fractions will be an algebraic term.

Evaluate and simplify: b

a 3

Find the lowest common multiple of each pair of algebraic terms. a

15a and 24a

12x and 15x

5c and 25c

Are these expressions true or false? simpliﬁes to

The expression

simpliﬁes to

is equivalent to

The expression

is equivalent to

Fluency Ex 1

6y and 14x

Simplify:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 2

Ex 3

Simplify: a

Simplify:

Write as a single fraction: a

Reasoning 11

If x does not equal 0, is

Explain why it is important to ﬁnd a common denominator when adding or subtracting algebraic fractions with different denominators.

This expression was simpliﬁed incorrectly:

Explain the mistake.

the same as

? Explain your reasoning.

Give the correct solution.

Problem-solving 14

Complete the equation:

2.02 Add and subtract fractions with algebraic denominators mathspace.co

Simplify: a

A rectangle has a length of rectangle.

To get her driver’s licence, Attia took a theory test and a practical test. The theory test had t points in total and the practical test had twice as many points as the theory test. Attia scored x points in the theory test and y points in the practical. a

Write an expression for Attia’s percentage score in the: i

and a width of . Find an expression for the perimeter of the

Theory test

Practical test

Attia will get her licence if the average of her two scores is greater than 70%. Use your answers to part (a) to write this as an inequality, simplifying the algebraic expression to a single fraction.

2.03 Optional: Add and subtract further algebraic fractions After this lesson, you will be able to… • add or subtract the binomial numerators while keeping the denominator constant. • simplify algebraic fractions to a single expression.

Add and subtract algebraic fractions Adding and subtracting algebraic fractions with binomial numerators and numerical denominators is similar to adding and subtracting numerical fractions. The key is to have the same denominator for the fractions. If the denominators are different, we need to ﬁnd the common multiple and adjust the fractions accordingly. Once the denominators are the same, we add or subtract the numerators, and the denominator remains the same.

Example 1 Simplify

as a single fraction.

Create a strategy Since the denominators are the same, add the numerators.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea Add the numerators

Simplify the numerator

Example 2 Simplify

as a single fraction.

Create a strategy Since the denominators are the same, subtract the numerators.

Apply the idea Subtract the numerators

Distribute negative sign in (3x + 2)

Simplify the numerator

Example 3 Simplify

as a single fraction.

Create a strategy Find a common multiple of 3 and 4. Then, adjust the fractions so they have the same denominator and add the numerators.

Apply the idea Adjust the fractions to have the same denominator

Distribute 4 in (x − 3) and 3 in (2x + 5) in the numerator

Simplify the numerator

2.03 Optional: Add and subtract further algebraic fractions mathspace.co

Example 4 Simplify

as a single fraction.

Create a strategy Find the common multiple of 8 and 6. Then, adjust the fractions so they have the same denominator and subtract the numerators.

Apply the idea Adjust the fractions to have the same denominator

Distribute 3 in (3x − 6) and −4 in (2x − 4) in the numerator

Simplify the numerator

Idea summary To add and subtract algebraic fractions with binomial numerators and numerical denominators, the denominators of the fractions must be the same. We then add or subtract the numerators, and the denominator remains the same.

Practice questions Choose your question path Explorer 1–3, 6aef, 8ace, 9a-c, 10ab, 11ac, 12ab, 13b, 15, 16ab, 20ad, 24a

Adventurer 1, 2, 4, 5, 6aceg, 7, 8aceg, 9b-f, 10bdfg, 11bcfg, 12be-g, 13bde, 14ac, 15, 16abd, 17, 19, 20ab, 21bc, 24ab

Trailblazer 1, 3–5, 6ac-fh, 7, 8cd-h, 9bdf-h, 10bdf-h, 11begh, 12be-h, 13d-h, 14acd, 15–19, 21, 22, 24, 25

Understanding 1

Which statements are true and which are false? a

Adding algebraic fractions is similar to adding numerical fractions.

The denominators of algebraic fractions need to be different before you can add or subtract them.

When adding or subtracting algebraic fractions, the numerator is always added or subtracted, not the denominator.

When adding or subtracting algebraic fractions, we need to factorise the numerator.

When the denominators of algebraic fractions are different, we need to ﬁnd the common multiple of the denominator.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Tom needs to change this fraction to have a denominator of 12.

He is aware that the denominator needs to be multiplied by 3. Which is the correct numerator of the new fraction? A 3

3x + 2

3x + 6

x+6

x+2

Jenny is trying to add these algebraic fractions:

She has to ﬁnd the common multiple of 3 and 6. Then, adjust the fractions so they have the same denominator. Which is the correct numerator of the ﬁrst fraction after the adjustment? A 4

x + 10

x+4

2x + 10

2x + 8

Mike needs to solve this algebraic fractions:

He has to ﬁnd the common multiple of 5 and 7. Then, adjust the fractions so they have the same denominator. Which is the correct numerator of the second fraction after the adjustment? A 5

7x + 49

5x + 35

7x + 42

5x + 30

Match the 3 pairs of equivalent fractions. a

iii

Fluency 6

Simplify:

Are these expressions equal to a

8 Ex 1

Simplify:

2.03 Optional: Add and subtract further algebraic fractions mathspace.co

9 Ex 2

12 Ex 3

13 Ex 4

Simplify: a

Simplify:

Simplify: a

Reasoning 15

If a =

Are these calculations correct? Explain how you know.

and b =

, explain how you would ﬁnd the value of a + b.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Two students, Joanne and Edmon tried to solve the problem:

Joanne’s working:

Edmon’s working:

Who is correct? Explain why.

What mistake did the other student make?

Jonathan and Jefferson were given algebraic fractions:

Jonathan says that b + c = d. Jefferson says that d − c = b. Who is correct? Explain your reasoning.

Problem-solving 19

Consider the algebraic fraction:

Simplify the expression.

If x = 4, what is the value of the expression?

Given a = a

a+b

,b=

,c= b

a−b

and d =

, express each expression in simplest form: c

a−d

c+b

2.03 Optional: Add and subtract further algebraic fractions mathspace.co

Given p = a

,q=

s−r

,r= b

and s =

r−q

, express each expression in simplest form. c

s−q+r

p−q−r

Given a =

,b=

,c=

, and d =

, solve for x if a − b + c − d = 2.

Given p =

,q=

,r=

, and s =

, solve for x if p + q − r + s = 1.

Simplify: a

If 3x − 2y = 7, then ﬁnd:

2.04 Multiply and divide algebraic fractions with numeric denominators After this lesson, you will be able to… • apply the multiplication and division rules to algebraic fractions with numeric denominators. • use indices to simplify numerators and denominators in algebraic fractions. • identify and cancel common factors in the numerator and denominator.

Multiply fractions with numeric denominators We have multiplied numerical fractions in the past. When multiplying fractions, we multiply the numerators and multiply the denominators

For example:

When it comes to working with algebraic fractions with numeric denominators and applying the four operations, the process is exactly the same as when we worked with numeric fractions. Let’s have a look at a simple example of multiplying two algebraic fractions with numerical denominators.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

We may need to use indices to simplify the numerators. Remember that: a × a = a2 and in general am × an = am+n We should check if the ﬁnal answer can be simpliﬁed by cancelling out a common factor in the numerator and denominator using that

= 1, a ≠ 0.

Example 1 Simplify:

Create a strategy Multiply the numerators together and multiply the denominators together.

Apply the idea Multiply the numerators and denominators

Evaluate

Example 2 Simplify:

Create a strategy Multiply the numerators together and multiply the denominators together.

Apply the idea Multiply the numerators and denominators

Evaluate

Factorise 12 out

Simplify

2.04 Multiply and divide algebraic fractions with numeric denominators mathspace.co

Reﬂect and check We could also have removed any common factors before we performed the multiplication: Factorise the terms

Remove the common factors

Simplify

Idea summary The product of two fractions is the product of their numerators divided by the product of their denominators. For B, D ≠ 0:

We may need to use indices to simplify the numerators. Remember that: a × a = a2 and in general am × an = am+n Before multiplying two fractions together, look for common factors that we can remove ﬁrst. This will make the resulting multiplication easier in most cases, as there will be less factors to deal with. We can use that

= 1.

Divide fractions with numeric denominators When we divide fractions, we take the reciprocal of the second fraction and multiply by it. Sometimes we say ﬂip or invert when we take the reciprocal. Multiply by the reciprocal

Evaluate

The process for dividing is the same as when we divided numerical fractions, but we may need to use indices.

Example 3 Simplify:

Create a strategy Multiply the ﬁrst fraction by the reciprocal of the second fraction.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea

Multiply by the reciprocal

Remove the common factor

Reﬂect and check Notice that if common factors are not removed, we would end up with very large numbers which would then need to simplify.

Multiply by the reciprocal

Multiply the numerators and denominators

Simplify

Idea summary To divide by a fraction, multiply by the reciprocal. For B, C, D ≠ 0:

Simplify if possible by removing common numerical or algebraic factors.

Practice questions Choose your question path Explorer 1, 2, 3ab, 4abfg, 5acfg, 6a-cfj, 7, 10a, 11ac, 12a

Adventurer 1, 2, 3bc, 4b-e, 5bdefh, 6ad-gj, 9, 10bc, 11bc, 12c, 13

Trailblazer 1, 2, 3bd, 4ah, 5bei-l, 6hikl, 8, 9, 10d, 11acd, 12cd, 14

Understanding 1

Evaluate: a

Are these statements true or false? a To multiply two algebraic fractions, we multiply the numerators and multiply the denominators. b When multiplying or dividing algebraic fractions with numeric denominators, the denominators must be the same before performing the operation. c To multiply or divide algebraic fractions, you must ﬁrst ﬁnd a common denominator. d The order of operations still applies when multiplying or dividing algebraic fractions with numeric denominators. 2.04 Multiply and divide algebraic fractions with numeric denominators mathspace.co

Complete each simpliﬁcation. a

Fluency Ex 1

5 Ex 2

6 Ex 3

Simplify: a

Simplify:

Reasoning 7

The following expression was simpliﬁed incorrectly:

a 8

Explain the mistake.

Give the correct solution.

Is each statement always true, sometimes true, or never true? Justify your answer. a

The quotient of two algebraic fractions is an numeric fraction.

The product of two numeric fractions is an algebraic fraction.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Amarion and Dean both simpliﬁed

differently than the others in their group. Their

ﬁrst two steps are shown. Who could go on to be correct? Explain your answer. Amarion:

Dean:

Problem-solving 10

Simplify fully: a

. Find an expression

Simplify fully: a

and

. Find an expression for

2.05 Optional: Multiply and divide algebraic fractions with algebraic denominators After this lesson, you will be able to… • apply multiplication to the numerators and the denominators of algebraic fractions. • apply division by using the reciprocal of the second fraction. • simplify resulting algebraic fractions by cancelling out common factors.

2.05 Optional: Multiply and divide algebraic fractions with algebraic denominators mathspace.co

Multiply and divide algebraic fractions with algebraic denominators In the same way as when multiplying algebraic fractions with numeric denominators, we apply all the same fractional and algebraic thinking to algebraic denominators.

Example 1 Simplify as a single fraction:

Create a strategy Remember to simplify a product of algebraic fraction we should cancel out common factors across numerators and denominators.

Apply the idea Simplify common numerical factors across numerators and denominators Simplify common algebraic factors across numerators and denominators

Multiply the numerators and multiply the denominators

Simplify

Example 2 Simplify as a single fraction:

Create a strategy Dividing by a fraction is the same as multiplying by its reciprocal.

Apply the idea Take the reciprocal of the second fraction

Simplify common factors across numerators and denominators

Multiply the numerators and multiply the denominators

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary To multiply algebraic fractions, multiply the numerators together and the denominators together. To divide algebraic fractions, multiply the ﬁrst fraction by the reciprocal of the second fraction. Always simplify the resulting fraction by canceling any common factors in the numerators and denominators.

Practice questions Choose your question path Explorer 1–3, 4a-c, 5, 6, 7a-eg, 8b, 9abdg, 10a, 11ab, 13a, 15a, 16

Adventurer 1–5, 6, 7acegi, 8, 9ac-eg, 10ace, 11abd, 12, 13a, 14, 15ac, 15, 16, 18

Trailblazer 1–5, 6, 7adgi-l, 8, 9ace-h, 10ad-f, 11–14, 15acd, 15–19

Understanding 1

When dividing algebraic fractions, you should: A

Multiply the ﬁrst fraction by the second fraction

Multiply the ﬁrst fraction by the reciprocal of the second fraction

Add the ﬁrst fraction to the second fraction

Subtract the second fraction from the ﬁrst fraction

Are these statements true or false? a

To simplify a product of algebraic fractions, you can cancel common factors in the numerators and denominators before multiplying.

To simplify a quotient of algebraic fractions, you can divide across the numerators and across the denominators.

Simplify: a

a 2 × a3

(x3 × x4)2

2.05 Optional: Multiply and divide algebraic fractions with algebraic denominators mathspace.co

Fluency Ex 1

Simplify as a single fraction: a

Ex 2

Simplify as a single fraction: a

Reasoning 12

The product of two fractions is 1. If one of the fractions is

Complete the equations: a

Show that the given expression simpliﬁes to 1.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

, what is the other fraction?

Problem-solving 15

Simplify: a

Simplify:

Simplify

, what is the value of A?

where

2.05 Optional: Multiply and divide algebraic fractions with algebraic denominators mathspace.co

Chapter 2 review 1

At the end of last month, a dam was

full. Over the current month,

was used up by the town. Meanwhile, rain filled dam is full at end of this month? 2

of the dam’s capacity

of the dam’s capacity. What fraction of the

Simplify: a

The fuel tank in Lannie’s car holds q litres of petrol. After filling the tank, Lannie uses the tank driving to her cousin’s house, then another

of the tank driving to the mall. Write

an expression for number of litres of fuel left in the tank when Lannie arrives at the mall, and simplify this expression. 4

(Optional) Two students, Waheed and Azize tried to solve the problem:

Waheed’s working:

What mistake did the other student make?

a 5

Who is correct? Explain why.

Simplify: a

Simplify:

(Optional) Simplify: a

Azize’s working:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Simplify: a

(Optional) Complete the equations:

a 12

Simplify fully: a

(Optional)Simplify as a single fraction: a

(Optional) Show that the given expression simplifies to 1.

(Optional) Show that

(Optional) Simplify as a single fraction: a c

Chapter 2 review mathspace.co

3 Expand and factorise Big ideas • Algebraic expressions can be manipulated and transformed through different ways allowing for a deeper understanding of their structure and relationships. • Algebraic fractions can be manipulated by ensuring common denominators, performing operations and simplifying through factorisation.

Curriculum Outcomes VC2M10A01 Factorise algebraic expressions by taking out a common algebraic factor VC2M10A04 Expand binomial products and factorise monic quadratic expressions using a variety of strategies

Chapter outline 3.01 3.02 3.03 3.04 3.05 3.06

Factorise algebraic expressions Expand binomial products Factorise binomial products Factorise monic quadratic trinomials Factorise non-monic quadratic trinomials Complete the square Chapter 3 review

98 103 110 116 121 127 132

Engineers use algebraic factorization and expansion to simplify complex equations representing the motion of robotic arms, spacecraft, and drones.

3.01 Factorise algebraic expressions After this lesson, you will be able to… • identify the highest common factor in algebraic expressions. • factorise the highest common factor out from algebraic expressions. • apply the distributive law to factorise algebraic expressions with common factors.

Highest common factor Any factorising problem should start with an inspection of the terms to see if there is a common factor. If there are common factors, other than 1, these should be factored out before any other work is done.

Example 1 Find the highest common factor for the following pairs of terms: a 9ab and 12ac

Create a strategy Break down the terms into factors. Note, the HCF of 9 and 12 is 3.

Apply the idea 9ab = 3 × 3 × a × b

Factorise 9ab

12ac = 3 × 4 × a × c

Factorise 12ac

HCF = 3 × a Select the common factors = 3a Simplify

b 16c3d2 and 20c2d3

Create a strategy Break down the terms into factors. Note, the HCF of 16 and 20 is 4.

Apply the idea 16c3d2 = 4 × 4 × c × c × c × d × d

Factorise 16c3d2

20c2d3 = 4 × 5 × c × c × d × d × d

Factorise 20c2d3

HCF = 4 × c × c × d × d 2 2

= 4c d

Select the common factors Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

c 4(x − 3) and y(x − 3)

Create a strategy Note that a common factor can be an expression in brackets.

Apply the idea 4(x − 3) = 4 × (x − 3)

Factorise 4(x − 3)

y(x − 3) = y × (x − 3)

Factorise y(x − 3)

HCF = (x − 3) Select the common factor

Idea summary The ﬁrst step in any factorising problem is to identify and factor out the highest common factor. It can help to break each term down into factors and compare.

Factorise algebraic expressions Consider 9ab + 12ac We can write each term in factors: 3×3×a×b+3×4×a×c We can see that the common factors are 3 and a, so we factor 3a out, and what is left goes into a bracket. 9ab + 12ac = 3a(3b + 4c) We can view this as factorising using the law, AB + AC = A(B + C) A is the highest common factor, which is taken out. This extends to three or more terms. AB + AC + AD = A(B + C + D)

Example 2 Factorise the expression by taking out the highest common factor: 8 + 16x − 4x2

Create a strategy The terms have a highest common factor of 4, which we can factor out using AB + AC + AD = A (B + C + D)

3.01 Factorise algebraic expressions mathspace.co

Apply the idea 8 + 16x − 4x2 = 4 × 2 + 4 × 4x − 4 × x2 2

= 4(2 + 4x − x )

Write the terms as products of 4 Factorise

Example 3 Factorise the expression 4(x − 3) − y(x − 3).

Create a strategy Factorise using the law, AB + AC = A (B + C). Here the common factor is the binomial expression (x − 3).

Apply the idea 4(x − 3) − y(x − 3) = 4 × (x − 3) − y × (x − 3) Write the terms as products of (x − 3) = (x − 3) (4 − y)

Factorise

Example 4 Factorise the expression 2t2k7 + 18t9k9.

Create a strategy Factorise using the law, AB + AC = A (B + C).

Apply the idea The coeﬃcients have a common factor of 2, and both terms have factors of t2 and k7, so we can factorise 2t2k7 out. 2t2k7 + 18t9k9 = 2t2k7 × 1 + 2t2k7 (9t7k2) = 2t2k7 (1 + 9t7k2)

Write the terms as products of 2t2 k7 Factorise

Idea summary We can use the distributive law to factorise an algebraic expression like AB + AC. This means writing the expression with any common factors between the terms taken outside of the brackets: AB + AC = A(B + C) Factorising is the reverse of expanding.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1bdfg, 2, 3, 4acegi, 5bef, 6, 7ace, 8, 9, 11, 12, 14

Adventurer 1aceh, 2bc, 3, 4bfj, 5cgh, 6, 7bdf, 8, 9, 11, 13, 14

Trailblazer 1ah, 2d, 3, 4egi, 5fhj, 6, 7d-f, 9, 10, 11, 13–15

Understanding Ex 1

Find the highest common algebraic factor of each pair of terms. a

20x and 15x2n

16abnx and 12abmy

6abmn and 8bnxy

16a2bx3y6 and 12b4m2n6x

12a7bm3y2 and 9ab5n2x3

x(x + 1) + 3(x + 1)

2x(4 − x) − (4 − x)

5y3 + 15y2 + 10y = 5y( y2 + ⬚ + ⬚)

Fill in the boxes to complete the factorisation: a c

15n2m and 8nm2

2x3 + 4x2 + 2x = ⬚(x2 + 2x + 1) 2

ax − 3ax + 6a = ⬚(x − 3x + 6)

4kx2 + 12kx + 16k = 4k(⬚ + ⬚ + 4)

Find the highest common factor of the terms: 12p2r2t3, 9prt3, and 18p3r3.

Fluency Ex 2

Factorise: b

2x3 − 16x2 + 2x

2ax + 2ax + 4ax

5xy2 + 10xy + 15x

4b3 − 16b2 + 20b

6qr2 − 36qr − 12q

3xyz2 + 21xyz + 9xz

−4abc2 + 8abc + 4ac

10rstu2 + 70rstu − 40rtu

i Ex 3

3ax2 + 6ax − 6a 3

−6efgh − 30efgh − 12ehg

Factorise: a

2y2(x + 2) − 3(x + 2)

3m2(n + 2) + 5(n + 2)

4m2(n − 3) + 6(n − 3)

5y3(z + 5) + 7(z + 5)

2p q(r + 4) + 5q(r + 4)

6a2b(c − 2) − 4b(c − 2)

( y + 4) ( y + 7) + x( y + 7)

2f(g + h) + (g + h)2

4(7 − x2) − 2y(7 − x2) + 5(x2 − 7)

3(8 − c2) − 5d(8 − c2) − 4(c2 − 8)

(3m − 2) (4m + 7) − (2m + 3) (4m + 7)

(6z − 5) (2z + 4) + (4z + 2) (2z + 4)

A farmer wants to fence a rectangular field with an area of 48xy − 24y. If the length of the field is 24y, find the dimension of the width of the field.

3.01 Factorise algebraic expressions mathspace.co

101

Ex 4

Factorise: a

20j − 15jk − 55j2k 2

3p2q2 + 4p4q4 + 5p6q6

2yz − 10xy + 12xy z

pqr + p2q2r + p3q3r

−hf − hj + hg

25m − 15mn − 10m2n

Reasoning 8

12x4 + 18x3 − 24x2 can be factorised into the form 6x(2x3 + 3x2 − 4x). Has the expression been fully factorised? Explain your answer.

Xander was asked to factorise the expression 35x2y + 10xy2 − 5xy. He stated that the highest common factor was 5xy so the factored form of the expression is 5xy(7x + 2y). Identify and correct Xander’s error.

Explain why the highest common factor of the variables in any expression has the least possible index of any of the terms.

When factorising an algebraic expression, explain how you can check your answer.

Problem-solving 12

A rectangle’s length is three times its width. Its length can be represented by the expression: 3x2 + 6x + 3

Factorise this expression.

Interpret what the factors represent in the context of this problem.

A farmer wants to create a set of adjacent fields which all have the same width. He plans to create the smallest field in the shape of a square, with the largest field 9 times the size of the smallest field, and the middle field to have a length that is 5 units more than the smallest field. a

Write expressions for the area of each ﬁeld.

Use factorising to determine the dimensions of the entire ﬁeld.

A rectangle has area 4x2 + 8x cm2 and length 4x cm. For what values of x is it a square? For what values is it just a rectangle?

Using the digits 0 to 9 at most once time each, place a digit in each box to create a true statement. ⬚x + ⬚⬚ = ⬚(⬚x + ⬚⬚)

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

3.02 Expand binomial products After this lesson, you will be able to… • understand and apply the rule for expanding binomial products. • visualise the expansion of binomial products using area models. • apply the formulae for expanding perfect squares to relevant problems. • understand and apply the difference of two squares identity. • solve problems requiring the expansion of binomial products.

Expansion of binomial products A C

+ D

x x

x × x = x2

5 × x = 5x

+ 3

Consider the expression (A + B) (C + D). We can represent it visually using this area model. This diagram shows that (A + B) (C + D) = AC + AD + BC + BD. In other words, the area of the whole shape is the sum of the smaller areas.

Let’s look at using this model to expand (x + 3) (x + 5). Here we can see that: (x + 3) (x + 5) = x2 + 3x + 5x + 15 We can simplify this to:

3 × x = 3x

5 × 3 = 15

x2 + 8x + 15

Example 1 Expand and simplify: (x + 10) (x + 5)

Create a strategy Use the rule (A + B) (C + D) = AC + AD + BC + BD.

Apply the idea (x + 10) (x + 5) = x × x + x × 10 + 5 × x + 10 × 5 Apply the rule = x2 + 10x + 5x + 50 2

= x + 15x + 50

Simplify the products Combine like terms

3.02 Expand binomial products mathspace.co

103

Reﬂect and check After using the rule we can then simplify the expression using any of the algebraic rules that we have learned.

Idea summary We can expand the product of two binomial expressions using the rule: (A + B) (C + D) = AC + AD + BC + BD

Expanding perfect squares Exploration Consider the expansions of the binomials of the form (a + b) (a + b) = (a + b)2: • (x + 3) (x + 3) = x² + 6x + 9 • (x + 5) (x + 5) = x2 + 10x + 25 • (x + 6)2 = x² + 12x + 36 • (2r + 3s)2 = 4r2 + 12rs + 9s2 • (5s - 3)2 = 25s² - 30s + 9 1.

What do you notice about the linear coefficient of the product?

What do you notice about the constant of the product?

Is there a general rule for this type of product?

A perfect square is something of the form (A + B)2. We can use the same area model to determine the expansion.

A A

+ B

Here we can see that the expansion is: (A + B)2 = A2 + 2AB + B2 In a similar way: (A − B)2 = A2 − 2AB + B2 We can use perfect square expansions whenever we have to expand a perfect square.

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Example 2 Expand and simplify: (x + 2)2

Create a strategy Use the perfect square expansion: (A + B)2 = A2 + 2AB + B2

Apply the idea (x + 2)2 = x2 + 2 × 2x + 22

Use perfect squares

= x + 4x + 4

Evaluate

Idea summary A special case of expanding binomials is for perfect squares: • •

(A + B)2 = A2 + 2AB + B2 (A − B)2 = A2 − 2AB + B2

Difference of two squares Exploration Consider the expansions of the binomials of the form (a + b) (a - b): • (x + 3) (x - 3) = x² - 9 • (x - 5) (x + 5) = x5 - 25 • (x + 6) (x - 6) = x² - 36 • (2r + 3s) (2r - 3s) = 4r2 - 9s2 • (5s - 3) (5s + 3) = 25s² - 9 1.

What do you notice about the linear coefficient of the product?

What do you notice about the constant of the product?

Is there a general rule for this type of product?

The difference of two squares identity is very useful in many contexts in Mathematics. Consider expanding (A − B) (A + B). Using the distributive law, (A − B) (A + B) = A2 + AB − AB − B2 If we consider the middle terms, AB − AB = 0, the right hand simpliﬁes and we have (A − B) (A + B) = A2 − B2 The right hand side, A2 − B2, is a difference of two squares, which is where the identity gets its name. 3.02 Expand binomial products mathspace.co

105

Example 3 Expand and simplify: (m + 3) (m − 3)

Create a strategy Expand using the identity for difference of two squares: (A − B) (A + B) = A2 − B2

Apply the idea We are given: A = m and B = 3. (m + 3) (m − 3) = m2 − 32

Apply the identity

= m – 9

Evaluate the square

Idea summary A special cases of expanding binomials is the difference of two squares: (A + B) (A − B) = A2 − B2

Practice questions Choose your question path Explorer 1ac, 2ac, 3ac, 4ac, 5abdg, 6a-d, 8bc, 9dceg, 10ab, 11ac, 12aceg, 14a, 15adg, 16, 17, 20, 22–24

Adventurer 1cd, 2bd, 3bd, 4bd, 5cg, 6cegi, 7ac, 8ad, 9a-d, 10ad, 11be, 12acegk, 13a, 14bc, 15egh, 16, 19, 20, 22–25

Trailblazer 1d, 2d, 3d, 4d, 5dh, 6dfhj, 7bd, 8ad, 9abeh, 10bdfh, 11cf, 12acdhjl, 13cd, 14cdf, 15e-g, 16, 18–21, 25–27

Understanding 1

Multiply the terms. a

(4y) (5y)

(−3z) (7z)

(x + 2) (x + 1) = x2 + ⬚ + 2 2

(x − 3) (x + 2) = x2 − ⬚x − 6

(x + 2) (x − 5) = x − 3x − ⬚

(x + 3)2 = x2 + ⬚ + 9

Fill in the missing boxes. a c

106

(5p) (−2p)

Fill in the missing boxes. a

(2x) (3x)

(x + ⬚)2 = x2 + ⬚x + 36

(x − 3) (x − 4) = x2 − ⬚x + ⬚ (x − 3)2 = x2 − ⬚x + ⬚

(3x − 8)2 = 9x2 − ⬚ + 64

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fill in the missing boxes. a c

(x + 5) (x − 5) = x2 − ⬚

(x − ⬚) (x + ⬚) = x ⬚64

(x + 2) (x − 2) = ⬚2 − 4

(x + 14) (x − ⬚) = x2 − ⬚

Fluency Ex 1

Expand and simplify: a

(x − 1) (x − 8)

(x + 1) (x + 4)

(x − 3) (x − 6)

(x − 4) (x + 3)

(x − 7) (x − 5)

(x − 5) (x − 1)

(x + 5) (x − 6)

(x − 9) (x + 2)

(3x − 2) (4x + 5)

(4x + 1) (x − 2)

(3x − 4) (4x + 5)

3(2x−5) (x + 4)

−3(3x + 2) (2x − 1)

Expand and simplify: a

(2x + 3) (3x − 4)

(5x − 1) (x + 2)

2(x + 3) (x−2)

5(2x + 1) (x − 2)

2(2x + 1) (3x − 4)

Expand and simplify: b

a 8

c 9

(3 + 4)2 = 32 + 42 2

(2y + 3) = 4y + 12y + 9

( y − 3)2 = y2 − 9y + 9

( y − 7)2 = y2 − 21y + 49

(a + b)2

(a − b)2

(x + 7)2

( y − 5)2

(c + 6)2

(e + 5)2

(7 − z)2

(−6 − a)2

(2y − 5)2

(2y − 3)2

(x2 + 1)2

Expand and simplify: (3x + 7)2

Ex 3

Expand and simplify:

Are the expansions correct or not? If not, write the correct expansion. a

Ex 2

Expand and simplify: a

2(x + 5)2

3( y − 3)2

8(q + 3)2

3x(7x − 5y)2

5(m − 5)2

(u + 5) (u − 5)

6(n + 6)2

Expand and simplify: a

(a + b) (a − b)

(x − 7) (x + 7)

(8 − m) (m + 8)

(12 − p) (12 + p)

(7 − 6p) (7 + 6p)

(3x − 8) (3x + 8)

(8y + 9) (8y − 9)

(−9y − 1) (−9y + 1)

(−7y − 8) (−7y + 8)

(9x − 2y) (9x + 2y)

(0.2x + 3) (0.2x − 3)

3.02 Expand binomial products mathspace.co

107

Expand and simplify: a

3x(5x − 6y) (5x + 6y)

3(9x − 8y) (9x + 8y)

a + (9 − a) (9 + a) − 10

[(5t + 9) − 6r] [(5t + 9) + 6r]

6(8x − 9y) (8x + 9y)

5x(2x − 9y) (2x + 9y)

Expand and simplify: a

(5a + 4)2

(x + 1) (x − 8)

(x + 3) (x + 7)

(a + 3)2

(2x − 0.9) (2x + 0.9)

16(4x + 3y)2

e g 16

( y2 + 9) ( y2 − 9)

Using the diagram shown: a

Find the area of the smaller squares A, B, C and D.

Add the smaller areas together to get the total area.

Find the area of the large square by multiplying the side lengths (2y + 3).

Reasoning 17

Is (3x + 4)2 equal to 9x2 + 16? Explain your answer.

Show and explain how 36 × 24 may be multiplied using the structure of a difference of two squares.

Given two expressions (x + 3) (x + 3) and 13 × 13, perform the multiplication and describe the similarities between them.

A student wrote (a + b)2 = a2 + b2. Explain why they are incorrect and write the correct answer.

Consider the expression

108

Explain how to expand and simplify the expression.

Is the result a trinomial?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Problem-solving 22

The length of a rectangular ﬁeld is (x + 10) metres and the width of the ﬁeld is (x − 2) metres. a

Find the simpliﬁed expression for the perimeter of the ﬁeld.

Find the simpliﬁed expression for the area of the ﬁeld.

If x = 6, ﬁnd the length, width, perimeter, and area of the ﬁeld.

A square of side lengths measuring (x − 1) centimetres has each side enlarged by a factor of 2. Write an expression, in expanded form, for the area of the new square.

A cube has side length x + 3: a

Find the surface area of the cube in expanded form.

Find the volume of the cube in expanded form.

x+3

Form an expression in expanded form for the shaded area in the given ﬁgures. a

5x − 7

8x − 5y

8x + 5y 5x + 7

If the smallest of three consecutive integers is n, ﬁnd the product of the three integers in terms of n. Write your answer in expanded form.

A rectangular prism has side lengths y, 12 − y, and y + 8. a b

Write an expression for the volume of the prism in expanded form.

y y+8

Determine which of these are viable options for the length y and explain why or why not in relation to both the side lengths and volume. i

y = −10

y = 20

y=0

iii

y=2

12 − y

y = 12

By trial and error or by using technology, determine the maximum volume of the prism.

3.02 Expand binomial products mathspace.co

109

3.03 Factorise binomial products After this lesson, you will be able to… • understand the process of factorising binomial products. • factorise expressions by identifying and grouping common factors. • understand and apply perfect square factorisation. • understand and apply the factorisation of the difference of two squares. • apply the knowledge of factorising binomial products to solve problems.

Factorise by grouping We’ve seen the distributive law: A(B + C) = AB + AC We have also seen how the distributive law applies when the ﬁrst term is a binomial expression: (A + B) (C + D) = A(C + D) + B(C + D) This means that if we have an expression such as: y(x + 4) + 5(x + 4) we can write it as a product of two brackets: ( y + 5) (x + 4)

Example 1 Fully factorise 5(a + b) + v(a + b)

Create a strategy Identify the common factor between the two terms, and then factorise using the rule: AB + AC = A (B + C)

Apply the idea We can see that (a + b) is a common factor of both terms. 5 (a + b) + v (a + b) = (5 + v) (a + b) Use the rule for factoring

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Example 2 Factorise this expression by grouping in pairs: 2x + xz − 40y − 20yz

Create a strategy To factorise an expression with four terms, group the expression in pairs so that each pair has a common factor.

Apply the idea We can then factorise both pairs, since x is a common factor of (2x + xz) and 20y is a common factor of (40y − 20yz). 2x + xz − 40y − 20yz = x (2 + z) − 20y (2 + z) = (2 + z) (x − 20y)

Factorise by x and 20y Simplify

Idea summary We can use the rule (A + B) (C + D) = A(C + D) + B(C + D) = AC + AD + BC + BD. to factorise expressions in terms of the product of two binomial expressions.

Factorise using perfect squares We’ve seen how to expand perfect squares and used a model to visualise it. (A + B)2 = A2 + 2AB + B2 A A

+ B

When factorising an expression of the type A2 + 2AB + B2: • Look for a common factor to both terms first as this can simplify the expression for the remaining steps. • Identify values for A and B such that satisfy this form. • Write in factorised form using the rule (A + B)2 = A2 + 2AB + B2.

In our work in quadratics, the first term A, is often the variable x, so for the expression x2 + 4x + 4, we would identify A = x and B = 2, because 22 = 4. The middle term 2AB holds true as 2 × x × 2 = 4x.

3.03 Factorise binomial products mathspace.co

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Example 3 Factorise x2 − 12x + 36

Create a strategy Using the perfect squares rule (A + B)2 = A2 + 2AB + B2.

Apply the idea We can use the perfect squares rule because half of −12 squared is (−6)2 = 36. Using the perfect squares rule, A = x and B = −6. x2 − 12x + 36 = (x − 6)2 Use the rule

Reﬂect and check We can check the answer by expanding it again. It’s always worth checking if an expression ﬁts the pattern A2 + 2AB + B2, because we can use this special rule. We call this perfect square factorisation.

Idea summary We can factorise the product of two binomial expressions using the rule (A + B)2 = A2 + 2AB + B2 (called a perfect square)

Factorise using the difference of two squares In the previous section we saw how to expand the difference of two squares. We ended up with: A2 − B2 = (A − B) (A + B) When factorising an expression of the type A2 − B2: • Look for a common factor to both terms first as this can simplify the expression for the remaining steps. • Identify the terms that are the squares A and B. • Write in factorised form using the rule A2 − B2 = (A − B) (A + B).

Example 4 Factorise: 121m2 – 64

Create a strategy Factorise using the difference of two squares, A2 − B2 = (A + B) (A − B)

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Apply the idea We can tell that this is a difference of two squares expression because there are 2 terms that are both perfect squares, and the sign between them is a minus (−). Since (11m)2 = 121m2 and 82 = 64, we can use A2 − B2 = (A + B) (A − B), where A = 11m and B = 8. 121m2 − 64 = (11m + 8) (11m − 8)

Factorise

Idea summary We can factorise expressions of the form A2 − B2 using the rule: (A + B) (A − B) = A2 − B2 (called the difference of two squares)

Practice questions Choose your question path Explorer 1ac, 2adf, 3, 5ab, 6abeg, 7a-c,e, 8aceg, 9a-d, 10abdik, 11cj, 13, 15, 16, 17, 19ac, 20

Adventurer 1bc, 2e-g, 4, 5bdeg, 6b-d,h, 7aceh, 8aceg, 9aceg, 10aceil, 11cdg, 12, 15–17, 19bd, 20, 21

Trailblazer 1cd, 2f-h, 4, 5bcfh, 6cdfh, 7c,e-g, 8eghj, 9dfhj, 10dfhjl, 11fhkm, 12, 14, 15, 18, 19d, 20–22

Understanding 1

Identify the highest common factor: a

2a + 18b

10xy + 25x2

32(c + 5) − x(c + 5)

6d4 + 8d2 ⬚ = 42

Fill in the box to make the equation true: a

81 =⬚2

100 =⬚2

49 =⬚2

⬚ =122

24 = ⬚

One linear factor of x2 − 16 is x − 4. What is the other factor?

One linear factor of n2 + 12n + 36 is n + 6. What is the other factor?

Identify if these can be factored as a perfect square: a

k2 + 4

b4 + 4b2 + 4

a2 − 9a + 81

c g

100 − 10c + c2

25 = ⬚2 81 = ⬚4

12 − m + m2

k4 + 12k2 + 36

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Are these expressions a difference of two squares? a

a2 – 16

b2 – 9b – 81

4w2 – 25a2

16x4 – 81

100 + 25y2

1 − 16d2

–9z2 + 36

Fluency Ex 1

Ex 2

Ex 3

Ex 4

Factorise: a

5(a + b) + v(a + b)

x( y − z) − w( y − z)

5y(4w + 3x) − z(4w + 3x)

2y(2x2 + 3z) − (2x2 + 3z)

3f(g + h) + (g + h)2

8y( y − 4) + 3(4 − y)

(2c − d) (c + 5d) − 3(d − 2c)

3x2(x + 4y) − 5y(x + 4y)

Factorise: a

8x + xz − 16y − 2yz

24 + 3y + 8x + xy

7xy + wx + 7yz + wz

x2 + 2x + 5x + 10

2mp + 6 + 3p + 4m

x − 3x + 8x − 24 3

a + 8a + a + 8

3df 2 − 11ydf + 3rzf − 11yrz

4x + 24yz + 32xy + 3z

2x + 18yz + 12xy + 3z

Factorise: a

x2 + 6x + 9

6c2 − 24c + 24

64 + 16e + e2

49 − 14p + p2

5q + 10qt + 5t

u2 − 2uq + q2

b2 − 2br + r2

16w2 − 40w + 25

64r2 + 48r + 9

Factorise: a

x2 − y2

e i 11

f 25m2 − 49

81x2 − 16y2

v2 − 1

121 − v2

16 − 9y2

x2y2 − 49

7x2 − 63

45t2 − 20

Factorise: a

3x2y − 6y4 + 9x2y3

16x2y2 + 24xyz + 9z2

121c2 − 49d2

3x2y + x2z + 3y2 + yz

2 4

8a b + 12bc

(b + 4)2 − (3 − b)2

14h3p2r + 21h5p2 + 49hp4

16w2 − 64wt2 + 6wt − 24t3

16q + 24q r + 9r

48df 2 − 3dh2

128t3 − 18s4t

18j2 + 60jk + 50k2

a2 + 2ab + b2 − c2

m 4x2y + 28x2 − 4y − 28

114

n2 − 25

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reasoning 12

For each expression: i

What strategy or combination of strategies would you use to factorise? Explain your choice.

Factorise the expression fully.

x2 + 2x + 1

n3 − 121n

2x2 − 6x + 5x − 15

9x2 − 12xy + 4y2

(x + 16)2 − y2

4x3 + 16x2 − x − 4

Does the sum of two squares, a2 + b2, equal (a + b) (a + b)? Explain your reasoning.

What are the similarities and differences between a perfect square trinomial and a difference of two squares binomial?

Explain why the trinomial 4a2−72a + 9 cannot be factorised using perfect squares.

Identify and correct the error in each statement. a

x2 + 4x + 3 = (x + 2) (x + 2)

x2 − 4x + 4 = (x + 2) (x + 2)

a4 − 18a2 + 81

452 − 422

Problem-solving 17

Factorise: a

x4 + 8x2 + 16

Factorise: x4 − 1.

Using the difference of two squares, evaluate: a

652 − 632

992 − 972

582 − 522

A square has area t2 + 14t + 49 units2: a

Find the side length of the square.

Find the perimeter of the square.

Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a perfect square trinomial.

x2 + ⬚⬚x + ⬚⬚

The area of a quilt can be expressed as x2 + 6x + 4x + 24. a

Label the area model so that it represents the area of the quilt.

Express the area of the quilt in factorised form.

Area I

Area II

Area III

Area IV

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3.04 Factorise monic quadratic trinomials After this lesson, you will be able to… • understand factorising a monic quadratic trinomial. • find two numbers that satisfy the conditions of the trinomial’s coefficients. • factorise monic quadratic trinomials.

Factorise monic quadratic trinomials Quadratics have the general form ax2 + bx + c. When a = 1, we call this a monic quadratic. Let’s look at how we can approach trying to factorise an expression in the form x2 + bx + c. If we can ﬁnd numbers p and q so that b = p + q and c = pq then we can write x2 + bx + c = x2 + ( p + q)x + pq 2

Since b = p + q and c = pq

= x + px + qx + pq

By expanding the middle term we have four terms

= x(x + p) + q(x + p)

Factorise the ﬁrst two and the last two terms

= (x + q) (x + p)

Finish the factorisation

We can see that we have just used AC + AD + BC + BD = (A + B) (C + D) so once we have identiﬁed p and q, we can write (x + p) (x + q)

Example 1 Factorise: x2 + 6x + 8

Create a strategy Find two numbers p and q that add to 6 and multiply to 8, then factorise using (x + p) (x + q).

Apply the idea Notice that 6 and 8 are positive, then both p and q must be positive. We need factors of 8 that add to 6, which are 2 and 4. x2 + 6x + 8 = (x + 2) (x + 4)

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Example 2 Factorise: x2 − 17x + 60

Create a strategy Find two numbers that add to −17 and multiply to give 60, then factorise.

Apply the idea We need to choose factors of 60 that have sum of −17. The factors are −5 and −12. x2 − 17x + 60 = (x − 5) (x − 12)

Factorise

Example 3 Factorise: 3x2 + 6x − 45

Create a strategy There is a common factor of 3 to factor out. Factorise the remaining quadratic.

Apply the idea 3x2 + 6x − 45 = 3 (x2 + 2x − 15)

Factor out the common factor

If we now consider the trinomial in the bracket, we need to choose factors of −15 that have sum of 2. The factors are −3 and 5. 3x2 + 6x − 45 = 3 (x − 3) (x + 5)

Factorise the quadratic trinomial

Idea summary An expression of the form x2 + bx + c is a monic quadratic trinomial. To factorise expressions like this, we ﬁnd a pair of numbers p and q such that p + q = b and pq = c. Then the factorisation is x2 + bx + c = (x + p) (x + q).

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Practice questions Choose your question path Explorer 1, 2ac, 3, 4, 5ad, 6acegik, 7acegik, 8acegik, 9adeh, 10–12, 18, 19, 21

Adventurer 1, 2bd, 3, 4, 5cd, 6acegik, 7acegik, 8acegik, 9bfgk, 10, 12, 14, 17ac, 18, 19b, 20, 21

Trailblazer 1cd, 3d, 4, 5d, 6h-l, 7h-l, 8c,e-g, 9h-j,l, 12, 13, 15, 16, 17bd, 20–23

Understanding 1

Find a pair of numbers with the given sum and product. a

Sum: 19

Product: 90 2

Sum: −10

Sum: 3

x2 + 11x + 18

x2 + 16x + 64

x2 − 34x − 72

(m + 8) (⬚ + ⬚) = m2 + 18m + 80

x2 + x − 30

Complete the missing values in each equation. a c

For each of the quadratic expressions, list all factor pairs of the constant term. a

Sum: −2

Product: −63 Product: 24 Product: −88

(m + 4) (m + ⬚) = m2 + 10m + 24 2

( y − 14) ( y + ⬚) = y − 12y − 28

(x − 2) (⬚ − ⬚) = x2 − 6x + 8

Find the values of a and b that will make the equation true: y2 + ay + 35 = ( y + 5) ( y + b).

Fluency 5

Ex 1

Ex 2

118

Factorise: a

x2 + 3x + 2x + 6

x2 + 4x + 5x + 20

t2 + 7t + 8t + 56

y2 + 6y + 12y + 72

Factorise: a

x2 + 11x + 24

x2 + 8x + 16

x2 + 17x + 72

x2 + 16x + 64

t2 + 7t + 10

t2 + 8t + 7

y2 + 9y + 20

x2 + 8x + 15

x2 + 7x + 12

y2 + 9y + 14

40 + 13x + x2

36 + 12x + x2

Factorise: a

x2 − 4x + 3

x2 − 17x + 60

x2 − 19x + 84

x2 − 12x + 35

x2 − 8x + 7

y2 − 10y + 16

t2 − 13t + 42

x2 − 20x + 51

y2 − 18y + 77

t2 − 20t + 96

x2 − 12x + 11

x2 − 18x + 45

Factorise: a

x2 − x − 6

x2 − x − 30

x2 + x − 20

x2 − 3x − 54

x2 − 3x − 70

x2 + 4x − 117

y2 − 13y − 48

t2 + 8t − 84

y2 − 12y − 45

x2 + 10x − 56

x2 + 10x − 24

x2 + 3x − 28

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 3

Factorise: a

2x2 + 14x + 20 2

4x2 + 24x + 32 2

4x2 − 4x − 288 2

3x2 − 21x + 30

5x + 5x − 30

−2x − 18x + 44

−4x + 12x + 40

−3x2 + 12x − 12

−8 − 6x − x2

−12 + 7x − x2

x3 + 7x2 + 12x

2x3 + 16x2 + 30x

Reasoning 10

Octavia and Rajeev are factorising −3x2 − 15x − 18. • Rajeev states that the first step in factorising the trinomial is to find two numbers that multiply to −18 (the constant term) and add to −15 (the coefficient of the middle term). • Octavia states that the first step is factorising the highest common factor, which is −3. Who is correct? Explain and factorise −3x2 − 15x − 18.

Omar and Dima both factorise the expression x2 + 24x + 144. Their work is shown. Omar: x2 + 24x + 144 = x2 + 2(x) (12) + (12)2 = (x + 12)2 Dima:

⬚ + ⬚ = 24

⬚ × ⬚ = 144

x + 24x + 144 = (x + 12) (x + 12) = (x + 12)2

Explain and compare each of their strategies. 12

Identify and correct the error in each statement. a

x2 + 7x − 18 = (x + 2) (x + 9)

x2 − 11x + 30 = (x − 5) (x + 6)

Suppose we want to write the trinomial x2 + bx + c in the form (x + p) (x + q). For each condition, explain what must be true about the signs of p and q. a

c is negative.

c and b are both positive.

c is positive and b is negative. x

A square table of side length x has one of its dimensions decreased by 4. This can be expressed visually by the area model shown.

Determine if the following expressions are equivalent to the area x (x − 4). If so, explain why. If not, show what they are equivalent to. a

Area I + Area II

x2− Area II

Area I

x2− Area I

Area I

Area II

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Find two digits (1 to 9) to ﬁll in the blanks and create a trinomial in the form x2 + bx + c that a

Maximises the value of b.

(x + ⬚) (x + ⬚)

Minimises the value of c. (x + ⬚) (x + ⬚)

Minimises the value of b. (x + ⬚) (x − ⬚)

A square ﬁeld with side length x will have its length extended by a and its width extended by b. Both a and b are integers.

What values of a and b would maximise the length of the entire ﬁeld? Explain.

What values of a and b would minimise the difference in area of the bottom left and top right ﬁeld? Explain.

Problem-solving 17

Factorise: a c

120

(a + 3) + 3(a + 3) − 10

y4 − 5y2 + 4

(d − 1)2 − 12(d − 1) + 36

One expression for the area of the rectangle is m2 + 14m + 45. The rectangle is made up of four smaller rectangles. Use the diagram to express the area of the large rectangle in factorised form.

Find an expression for the total area of the rectangles in factorised form. a

x4 − x2 − 12

10m

The area of a patchwork artwork can be expressed as x2 + 8x + 15. a

Express the area of the quilt in factored form (x + p) (x + q).

What is the difference in the side lengths?

If both side lengths are tripled, what is the new expanded expression for the area of the artwork?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Factorise u2 − 11u + 24.

Factorise (x2 + 2x)2−11(x2 + 2x) + 24.

A square with side length y metres has c metres added to its length and d metres added to its width. The resulting rectangle has an area of ( y2 + 8y + 15) m2. Find the values of c and d.

Students decide to make rectangular appreciation cards for their family members. The total area of each card is equal to x2 + 2x − 3 cm2. If the shorter sides of a card are 6 cm in length and the longer sides are 10 cm, ﬁnd the value of x.

3.05 Factorise non-monic quadratic trinomials After this lesson, you will be able to… • understand the difference between monic and non-monic quadratic trinomials. • understand and apply the method of factorising non-monic quadratic trinomials by grouping. • apply the splitting of the middle term to factorise non-monic quadratic trinomials. • factorise non-monic quadratic trinomials.

Factorise non-monic quadratic trinomials A non-monic quadratic expression is an expression of the form: ax2 + bx + c where a ≠ 1. The general approach to factorising a quadratic trinomial of the form ax2 + bx + c: 1. Factor out any HCF. (If a is negative it is advised to factor out -1 at this stage). 2. Find two numbers, p and q that add to give b and multiply to give ac. 3. Rewrite the trinomial with four terms in the form ax2 + px + qx + c. 4. Factorise by grouping. 5. Check that the answer will not factorise further, and verify the factorised form by expanding. We will explore this approach in the following examples.

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Example 1 Factorise: 2x2 + 11x + 12

Create a strategy Find numbers that multiply to give 2 × 12 = 24 and add to give 11. Use those numbers to split the middle term and factorise by grouping.

Apply the idea 8 × 3 = 24 and 8 + 3 = 11, so split the middle term into 8x and 3x. 2x2 + 11x + 12 = 2x2 + 8x + 3x + 12

Split the middle term

= 2x(x + 4) + 3(x + 4)

Factorise the ﬁrst two and the last two terms

= (2x + 3) (x + 4)

Finish the factorisation

Reﬂect and check We can check our factorisation by expanding to make sure we get the original expression: (2x + 3) (x + 4) = 2x2 + 8x + 3x + 12 2

= 2x + 11x + 12

Expand the brackets Collect like terms

Example 2 Factorise: 6x2 − 26x + 24

Create a strategy There is a common factor of 2 to factor out. Look for numbers p and q to split the middle term, then factorise by grouping.

Apply the idea 6x2 − 26x + 24 = 2 (3x2 − 13x + 12) Take out the common factor of 2 The trinomial in the bracket is 3x2 − 13x + 12 so we need to ﬁnd numbers p and q so that pq = 36 and p + q = −13. p = −9 and q = −4 will do this. 6x2 − 26x + 24 = 2 (3x2 − 9x − 4x + 12)

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Use p = −9 and q = −4 to split the middle term

= 2 (3x(x − 3) − 4(x − 3))

Factorise the ﬁrst two and last two terms

= 2 (3x − 4) (x − 3)

Complete the factorisation by grouping

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reﬂect and check We can check our factorisation by expanding to make sure we get the original expression: 2 (3x − 4) (x − 3) = 2(3x2 − 13x + 12) 2

= 6x − 26x + 24

Complete the binomial expansion Do the remaining expansion

Example 3 Factorise the expression by ﬁrst taking out a common factor: −6x2 − 21x + 45

Create a strategy There is a common factor of −3 to factor out. Look for numbers p and q to split the middle term, then factorise by grouping.

Apply the idea −6x2 − 21x + 45 = −3(2x2 + 7x − 15)

Take out the common factor of −3

The trinomial in the bracket is 2x2 + 7x − 15 so we need to ﬁnd numbers p and q so that pq = −30 and p + q = 7. p = 10 and q = −3 will do this. −6x2 − 21x + 45 = −3(2x2 + 10x − 3x − 15)

Use p = 10 and q = −3 to split the middle term

= −3(2x(x + 5) − 3(x + 5))

Factorise the ﬁrst two and last two terms

= −3(2x − 3) (x + 5)

Complete the factorisation by grouping

Reﬂect and check We can check our factorisation by expanding to make sure we get the original expression: −3 (2x − 3) (x + 5) = −3(2x2 + 7x − 15) 2

= −6x − 21x + 45

Complete the binomial expansion Do the remaining expansion

Idea summary The steps in factorising a quadratic trinomial of the form ax2 + bx + c 1. Factor out any HCF. (If a is negative it is advised to factor out -1 at this stage). 2. Find two numbers, p and q that add to give b and multiply to give ac. 3. Rewrite the trinomial with four terms in the form ax2 + px + qx + c. 4. Factorise by grouping. 5. Check that the answer will not factorise further, and verify the factorised form by expanding.

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Practice questions Choose your question path Explorer 1–3, 4abf, 5a-d, 6aeh, 8, 9, 12, 13, 16, 17, 21

Adventurer 1–3, 4abfgi, 5bfgi, 6aeg, 8, 10, 11, 13, 14, 16–18, 21

Trailblazer 1–3, 4a-c,ij, 5c,h-j, 6efh, 7, 10, 11, 14, 15, 18–22

Understanding 1

To factorise −8x2 − 40x − 48, what is a possible ﬁrst step?

Fill in the boxes to complete each factorisation. a c

8x2 + 11x + 3 = (8x + 3) (x + ⬚)

2x2 − 19x + 45 = (⬚ − 9) (⬚ − 5)

b d

2x2 + 3x − 20 = (2x − 5) (x + ⬚)

10x2 + 29x + 21 = (⬚ + 3) (5x + ⬚)

Find an expression for the total area of the rectangle, 3x2 + 19x + 6, in factorised form.

3x2

18x

Fluency Ex 1

Factorise fully: a

Ex 2

2x2 − 11x − 40 2

64x − 48x + 9

3x − 25x + 28

12x2 + 7x − 10

6x2 + 41x − 56

4x2 + 40x + 100

12t2 − 13t − 4 2

81x2 + 72x + 16

8x − 19x + 6

8x2 − 21x − 9

10x2 + 5x − 30

8s2 + 6s − 54

Factorise fully: a

Ex 3

3y2 + 28y + 9

3x2 − 21x − 54 2

9x − 42x − 15

8x − 20x + 8

105x2 − 50x + 5

168x2 − 88x − 16

−6x2 + 25x − 14

24x − 24x − 18

72x2 − 6x − 6

−12x2 − 12x + 45

−6x2 + 22x + 40

−3x2 − 10x + 8

Factorise fully: a e

−6x2 + 5x + 6 2

−4x + 11x − 6

−9x + 24x − 16

−5x − 14x − 8

A rectangle has an area of 6x2 + 23x + 20. If the length and width are linear factors of 6x2 + 23x + 20, what are the dimensions of the rectangle in terms of x?

A ball is thrown from the top of a 140 m tall cliff, with an initial velocity of 50 m/s. The height of the ball after t seconds is approximated by the quadratic expression −10t2 + 50t + 140. Factorise this quadratic.

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The length of a rectangle is L = −56y + 11 and the width is W = 5y2. a

What is the perimeter of the rectangle in terms of y in expanded form?

What is the perimeter in fully factorised form?

The side length of a regular pentagon is S = 2x2 + 21x + 49. a

What is the perimeter of the pentagon in terms of x in expanded form?

What is the perimeter in fully factorised form?

A cube has a surface area of (6x2 + 36x + 54) square units, where x > 0. a

Factorise 6x2 + 36x + 54 fully.

What is an expression for the length of a side of the cube?

Reasoning 12

Consider each quadratic expression. The factorised form of the expression is (Ax + C) (Bx + D). Determine: i

the product of A and B

the signs for A and B: opposite or the same

iii

the product of C and D

the signs for C and D: opposite or the same

15x2 + 14x − 16

−45x2 + 29x − 4

Sheldon and Gabriella each factorised 16x2 + 48x + 36. Whose work is incorrect? Identify and explain the error. Sheldon: 2

Gabriella: 2

16x + 48x + 36 = 16x + 24x + 24x + 36

16x2 + 48x + 36 = (4x + 6)2

= 8x(2x + 3) + 12(2x + 3)

= 2 (2x + 3)2

= (8x + 12) (2x + 3) = 4(2x + 3) (2x + 3) = 4(2x + 3)2 14

Compare the process for factorising 23x2 + 438x + 19 and 24x2 + 74x + 45.

Consider these polynomials: 2x2 + x + 1, 3x2 − 2x − 4, 6x2 + 10x − 5, 12x2 − 12x + 1 a

What do the polynomials have in common?

How can you show that a polynomial where a ≠ 1 cannot be factorised?

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Problem-solving 16

Tara is practising diving. She springs up off a board 10 metres high, and after t seconds, her height in metres above the water is described by the quadratic: −5t2 + 5t + 10

Fully factorise the quadratic.

Substitute t = 2 into the factorised quadratic and ﬁnd the value of the expression.

Substitute t = 2 into the original quadratic and ﬁnd the value of the expression.

What is happening 2 seconds after Tara jumps off the board?

Consider the ﬁgure. a

Write an expression in expanded form for the area of the shaded region.

Write the expression for the area in factorised form.

The area of a rectangular ﬁeld is given by the general expression (6x2 + 17x + 12) m2 where x is a positive whole number. a

Find the length and width of the ﬁeld in terms of x.

Write an expression for the perimeter of the ﬁeld. Describe how to determine the expression for the perimeter.

If the perimeter of a particular ﬁeld is 114 metres, ﬁnd x.

A photographer wants to put a border around her photo. a

Why might the photographer use a variable to represent the width of the border?

What are the dimensions, length and width, of the total artwork if the total area of the photograph including the border is 4x2 + 24x + 35 square units?

What is the area of the photograph? Explain how you reached your solution.

Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a trinomial that can be factorised, and provide the factorised form: 4x2 + ⬚⬚x + ⬚.

Quadratic trinomials can be factorised using the identity:

Find the value of k that will make 16x2 − 24x + k a perfect square trinomial.

where m + n = b and mn = ac. Find the values m and n for the quadratic 4x2 − 14x + 12.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

3.06 Complete the square After this lesson, you will be able to… • understand how to complete the square using visual representation. • write quadratic trinomials in completed square form.

Complete the square We have seen the perfect square identity (A + B)2 = A2 + 2AB + B2 and we have seen how we can use this to factorise an expression such as x2 + 4x + 4 as (x + 2)2. All quadratics of the form y = ax2 + bx + c can be transformed into the form y = a(x − h)2 + k through the process of completing the square. This form is very useful for us in graphing, solving and identifying key features of quadratic which you will see in Chapter 4, but for now we will just focus on the algebraic skills associated with completing the square. From earlier in this chapter you will remember that Perfect squares : (x + a)2 = x2 + 2ax + a2

(x − a)2 = x2 − 2ax + a2

Difference of two squares : (x − a) (x + b) = x2 + b2 We also used a visual to help us understand the expansion of a perfect square. Let’s use the same ideas and look at how to complete the square, and indeed even how it got that name. Let’s start with the expression x2 − 4x + 23. We will start with the ﬁrst two terms and build a perfect square using our visual. We know that x × x is x2, so we can put that in.

x x

−2

−2x

To show the −4x, we will have two rectangles each with an area of −2x. This is now a square, with a side length of (x − 2).

−2 −2x

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−2

−2x

−2 −2x

To complete the square we will use −2 × −2 = 4. If the coefficient is 23, and we have used 4, there is 19 remaining.

So we can rewrite x2 − 4x + 23 = (x − 2)2 + 19. Algebraically the same example would look like this: x2 − 4x + 23 = x2 − 4x + (−2)2 + 23 − (−2)2 Add the square of half the coeﬃcient of x, but also subtract that same value so the expressions value is not altered = (x − 2)2 + 23 − (−2)2

Identify the perfect square we created by adding the square of half the coeﬃcient of x

= (x − 2)2 + 19

Simplify the constant term

Example 1 Using the method of completing the square, rewrite x2 + 4x in the form a(x − h)2 + k.

Create a strategy Form a perfect square from the terms and see what the required constant is.

Apply the idea Use x and half of the coeﬃcient of x to make a perfect square, (x + 2)2 (x + 2)2 = x2 + 4x + 4 2

Expand the perfect square

It follows that x + 4x = (x + 2) − 4

Example 2 Write 2x2 + 4x + 7 in the form a(x − h)2 + k.

Create a strategy Factorise 2 from the ﬁrst two terms. Complete the square inside the bracket, re-expand and simplify.

Apply the idea 2x2 + 4x + 7 = 2(x2 + 2x) + 7 2

= 2((x + 1) − 1) + 7

Complete the square on x2 + 2x

= 2(x + 1)2 − 2 + 7

Expand the bracket

= 2(x + 1) + 5

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Factor 2 out from the ﬁrst two terms

Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary Completing the square is a method that allows us to write a quadratic trinomial ax2 + bx + c in the form a(x + h)2 + k where

and

Factorise from completed square form We have met the difference of two squares identity: (A − B) (A + B) = A2 − B2 We can use this to factorise an expression such as x2 − 9 x2 − 9 = (x − 3) (x + 3) or an expression such as x2 − 5

If we have a completed square expression such as (x + 2)2 − 25, we can factorise as: (x + 2)2 − 25 = ((x + 2) − 5) ((x + 2) + 5)

Use difference of two squares

= (x + 2 − 5) (x + 2 + 5)

Remove the brackets

= (x − 3) (x + 7)

Simplify

Example 3 Factorise x2 + 4x + 3 using the method of completing the square.

Create a strategy Complete the square. Use the difference of two squares identity (A − B) (A + B) = A2 − B2

Apply the idea x2 + 4x + 3 = (x + 2)2 − 4 + 3 2

= (x + 2) – 1 2

Complete the square on the ﬁrst two terms Simplify

= (x + 2) − 1

Write as difference of 2 squares

= ((x + 2) + 1) ((x + 2) − 1)

Factorise by difference of 2 squares

= (x + 3) (x + 1)

Simplify each bracket

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Example 4 Factorise 5(x + 3)2 − 15.

Create a strategy Factor out the common factor of 5 Use the difference of two squares identity (A − B) (A + B) = A2 − B2

Apply the idea Factor out the common factor

Use the difference of two squares identity

Remove the unnecessary brackets

Idea summary If completing the square leaves us with an expression of the form (x + h)2 − k we can use the difference of two squares identity A2 − B2 = (A − B) (A + B) to factorise the completed square form to

Practice questions Choose your question path Explorer 1, 2, 3ac, 4, 5a-d, 6ac, 7abef, 8a-d, 9ac, 11, 12, 14af, 15, 17

Adventurer 1, 2, 3bd, 4, 5aceg, 6bc, 7c-f, 8bdfh, 9be, 11, 12, 14cd, 15–17

Trailblazer 1, 2, 3bd, 4, 5befh, 6cd, 7gikl, 8degj, 9cdg, 10, 13, 14ef, 16–18

Understanding 1

What is the squared value of half the coeﬃcient of x in each quadratic: a

130

x2 + 6x + 36

x2 + 8x + 12

x2 − 14x − 72

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x2 − 10x + 24

Identify if the correct value for ‘the square of half the coeﬃcient of x’ has been applied in each expression, ensuring that the expression value is not altered. a

x2 + 4x + 18 = x2 + 4x + 22 + 18 − 22

x2 − 10x − 21 = x2 − 10x + 52 − 21

x2 + 30x − 64 = x2 + 30x + 152 − 64 + 152

x2 − 24x + 80 = x2 − 24x + 122 + 80 − 122

Fluency 3

Complete each equation: a c

3x2 + 6x − 8 = ⬚(x2 + 2x) − 8

4x2 + 12x − 6 = ⬚(x2 + ⬚) − 6

5x2 − 10x + 4 = ⬚(x2 − 2x) + 4

Complete each solution for completing the square: a

2x2 − 10x + 1 = ⬚(x2 − 5x) + 1

6x2 + 24x − 7 = ⬚(x2 + ⬚) − 7 6x2 + 36x − 1 = ⬚(x2 + ⬚) − 1

= ⬚(x2 − 2x + ⬚) + 4 − ⬚

Ex 1

= ⬚(x − ⬚)2 − 1

x2 + 18x

x2 − 8x

x2 + 10x + 31

x2 + 14x + 47

x2 − 10x + 30

x2 − 18x + 77

x2 + 9x + 16

x2 − 7x + 15

(x + 4)2 − 9

(x + 5)2 − 49

(x − 1)2 – 3

Factorise fully:

Ex 4

(x − 3)2 − 1

Rewrite the following in the form a(x − h)2 + k by completing the square: a

Ex 3

= ⬚(x + ⬚)2 − ⬚

Rewrite each quadratic in the form a(x − h)2 + k using the method of completing the square:

a Ex 2

= ⬚(x2 + 6x + ⬚) − 1 − ⬚

2x2 + 8x − 7 2

5x2 + 20x − 9 2

4x2 − 16x + 3 2

10x2 + 20x − 11

7x − 14x + 1

9x + 54x − 2

3x + 33x + 88

5x2 + 5x + 1

4x2 − 11x + 7

3x2 + 9x + 8

3x2 − 27x + 58

2x2 − 7x + 6

m 4x2 + 20x + 16

4x2 + 20x + 28

Factorise fully using the method of completing the square: a

x2 + 6x + 4

x2 − 8x + 11

x2 + 24x + 63

x2 − 20x + 19

x2 + 42x + 185

x2 − 6x + 5

x2 − 28x + 115

x2 + 11x + 10

x2 − 11x + 30

(x + 3) (x + 19) − 17

Factorise fully: a

3(x − 1)2 − 12

2(x + 4)2 − 50

4(x + 2)2 − 1

−2(x − 9)2 + 32

−4(x + 2)2 + 16

9(x + 2)2 − 4

4(x + 3)2 − 5

2(x − 1)2 − 3

Reasoning 10

Explain why we cannot factorise x2 + 6x + 13.

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Alessandro is completing the square on x2 + 6x + 12. Here is his work:

x2 + 6x + 12 = (x2 + 6x + 9) + 9 + 12

= (x + 3)2 + 9 + 12

= (x + 3)2 + 21

What mistake has Alessandro made? 12

Beatrix is completing the square on 3x2 + 18x − 7. Here is her work:

3x2 + 18x − 7 = 3(x2 + 6x) − 7

= 3[(x + 3)2 − 9] − 7

= 3(x + 3)2 − 9 − 7

= 3(x + 3)2 − 16

What mistake has Beatrix made? 13

A function is written as y = 3(x − 4)2 + 2. Explain why the minimum value of y is 2. For what value of x does y take the value 2?

Factorise each quadratic expression in 2 different ways. Show your working. a

x2 − 18x + 81

t − 10t + 24

b f

4c2 − 12c + 9

y2 + 7y + 10

3a2 + 14a + 8

2x + 5x − 3

Problem-solving 15

A square has an area of x2 − 10x + 25 cm2. a

What is the length of one of the sides?

What are the restrictions on the value of x?

A cube has a surface area of 6x2 − 48x + 96. What is the length of one of the sides?

Complete the square for the function y = −x2 − 8x + 20.

What is the maximum value of y and at what value of x does this occur?

Using four of the digits 1 to 9 with no repeats, ﬁll in the blanks to complete the expression: x2 + ⬚x + ⬚ = (x + ⬚)2 − ⬚

Chapter 3 review 1

Find the highest common algebraic factor of each pair of terms. a

132

16a2 bx3 y6 and 12b4 m2 n6 x

5y2 + 10y + 15

3p2 q2 + 4p4 q4 + 5p6 q6

Factorise: a

20x and 15x2 n k2 − 8k 2

−4bc + 8bc + 4c

A student was asked to factorise the expression 35x2 y + 10xy2 − 5xy. They stated that the highest common factor is 5xy, so the factored form of the expression is 5xy (7x + 2y). Identify and correct the student’s error. Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Expand and simplify: a

(x + 1) (x + 3)

(x + 5) (x − 6)

(x − 9) (x + 4)

(4y + 3)2 = 16y2 + 24y + 9

(y − 3)2 = y2 − 9y + 9

(3x − 0.7) (3x + 0.7) d

Expand and simplify: a

Are the expansions correct or not? If not, write the correct expansion. a

(x − 2) (x − 8)

(m + 4) (m − 4)

(8 − 5p) (8 + 5p)

Expand and simplify: a

−2 (3x−4) (4x + 3)

A square of side lengths measuring (x − 4) centimetres has each side enlarged by a factor of 3. Write an expression, in expanded form, for the area of the new square.

Factorise:

k3 + 8k2 + k + 8

5r2 + 10rs + 5s2

35m2 − 49

5 (a + b) + x (a + b)

5x2 (x + 4y) − 5y (x + 4y)

Factorise: a

x2 − 10x + 25

x + 4x − 117

x2 + 16x + 64

1 − 2x + x2

x2 + 17x + 72

y − 13y − 48

x + 10x − 24

x2 + x − 20

6x2 + 41x − 56

81x2 − 90x + 25

−6x2 + 22x + 40

Factorise fully: a

12t2 − 13t − 4

A rectangle has an area of 20x2 + 39x + 18. If the length and width are linear factors of 20x2 + 39x + 18, what are the dimensions of the rectangle in terms of x?

The side length of the regular pentagon is S = 2x2 + 15x + 25. a

What is the perimeter of the pentagon in terms of x in expanded form?

What is the perimeter in fully factorised form?

Find the value of k that will make 4x2 − 12x + k a perfect square trinomial.

Rewrite each quadratic in the form a (x − h)2 + k using the method of completing the square:

x2 − 8x

10x + 20x − 11

b f

x2 − 4x + 6 2

2x + 8x − 7

c g

x2 + 10x + 21 2

4x + 20x + 28

x2 + 3x + 3

5x2 + 5x + 1

x2 − 11x + 30

Factorise fully using the method of completing the square: a

x2 + 6x + 4

Complete the square for the function y = 20 − 8x − x2.

What is the maximum value of y and at what value of x does this occur?

x2 + 14x + 33

x2 + 42x + 185

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4 Linear equations Big ideas • Graphs and models represent relationships between variables, with linear functions exhibiting a constant rate of change and non-linear functions demonstrating a varying rate of change. • Graphical representations of inequalities provide a visual tool to depict solution regions in the Cartesian plane. • Equations can be solved more easily by manipulating algebraic expressions and applying a series of organised steps and techniques. • Parallel and perpendicular lines have special rules that make them different from other lines.

Curriculum Outcomes VC2M10A05 Substitute values into formulas to determine an unknown and rearrange formulas to solve for a particular term VC2M10A07 Solve problems involving linear equations, including those derived from formulas VC2M10A08 Solve linear inequalities and graph their solutions on a number line VC2M10A10 Solve problems involving gradients of parallel and perpendicular lines VC2M10A12 Solve linear equations involving simple algebraic fractions

Chapter outline 4.01 4.02 4.03 4.04 4.05

Equations involving fractions Rearrange equations Parallel and perpendicular lines Linear models Graph linear inequalities Chapter 4 review

136 144 150 160 172 188

If you bought one less ice cream per week you how much would you save in a year? You can use a linear model to measure it!

4.01 Equations involving fractions After this lesson, you will be able to… • apply the three steps in solving equations with fractions. • use the lowest common multiple to clear fractions from equations. • apply inverse operations to isolate the variable and solve equations. • recognise when an algebraic denominator makes a solution invalid. • solve equations that have algebraic expressions in the denominators.

Equations with algebraic fractions To solve equations involving algebraic fractions, we can solve as we did with equations of three or more steps, but at some point we will clear the fractions. We can clear fractions by ﬁnding the lowest common multiple of the denominators and multiplying both sides by it. Using the lowest common multiple will eliminate the fractions and make it easier to solve the equation.

Exploration Let’s look at part of the process for solving this equation. 1

Start with the equation

Multiply by the lowest common multiple

Simplify the fraction

What do you notice about the coeﬃcients outside the brackets in the last step and the denominators in the ﬁrst step?

Will this pattern always hold? For example, does it work for

What will happen to the constant, 1, when solving

? ?

Here are some things to remember: • We often start with the inside brackets and work our way out. • The distributive law can still be used if there are more than two terms in the grouping symbol. • We must multiply the whole side of an equation by the lowest common multiple, not just the fractional terms. It is essential to simplify the equation and check your solution to ensure accuracy in your ﬁnal answer.

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Example 1 Solve:

Create a strategy We can start by expanding the brackets, and then clear the fraction. Once we have done that we can solve by using inverse operations.

Apply the idea Start with the equation

Expand the brackets

Multiply both sides by 8

Combine like terms on the left-hand side

Subtract 15x from both sides

Subtract 21 from both sides

Divide both sides by −14 and simplify

Reﬂect and check An alternative method for solving this problem involves ﬁrst multiplying through by 8, and then expanding the brackets.

Keep in mind that when using this strategy, we multiply each group, not each individual term.

Example 2 Solve:

Create a strategy First, we need to ﬁnd the lowest common multiple to clear the fraction. Then we can solve using inverse operations to isolate the variable.

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Apply the idea Start with the equation

Multiply both sides by the lowest common multiple, 10

Distribute the 10

Simplify the fractions

Expand the brackets

Add 2x to both sides

Subtract 50 from both sides

Divide both sides by 27

Reﬂect and check We can substitute the value of x to check the answer. Start with the equation

Substitute

Multiply both sides by the lowest common multiple, 10

Evaluate the brackets then expand

Multiply both sides by 27

Evaluate both sides

Since both sides are equal, we can conﬁrm that our answer is correct.

Example 3 Solve:

Create a strategy First, we need to ﬁnd the lowest common multiple to clear the fractions. Then we can solve using inverse operations to isolate the variable.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea Start with the equation

Multiply both sides by the lowest common multiple, 12

Distribute the 12

Simplify the fractions

Expand the brackets

Combine like terms

Subtract 73 from both sides and add x to both sides

Divide both sides by −7 and simplify

Reﬂect and check We can substitute the value of x to check the answer. Start with the equation

Substitute

Write each term as a fraction with a denominator of 12

Multiply both sides by 12 to clear them

Evaluate the brackets

Expand the brackets and simplify

Multiply both sides by 7

Evaluate both sides

Since both sides are equal, we can conﬁrm that our answer is correct.

Idea summary Solving equations with algebaric fractions involves simplifying the equation, clearing fractions by multiplying both sides by the lowest common multiple, and solving for the variable using inverse operations.

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Algebraic denominators If we have an algebraic expression in the denominator, then we can multiply both sides by the algebraic expression. When we simplify, we should get an expression which we can solve using previous strategies. Since we can’t divide by 0, the value which makes the denominator zero cannot be a possible solution. For example,

cannot have a solution of x = 2. We sometimes write x ≠ 2.

Example 4 Solve:

Create a strategy To solve this equation, we can multiply both sides by the lowest common multiple, which is 7(2x + 5), to clear the fractions. Then we can solve for the variable using inverse operations.

Apply the idea Start with the equation

Multiply both sides by 7(2x + 5)

Remove the common factor

Expand the brackets and evaluate

Subtract 30 from both sides

Divide both sides by 12 and simplify

Reﬂect and check We can use cross-multiplication as an alternative method for solving problems with fractions.

Start with the equation

Cross multiply to eliminate the fractions

Expand the brackets and evaluate

Subtract 30 from both sides

Divide both sides by 12 and simplify

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Idea summary If we have an algebraic denominator, the lowest common multiple will be an algebraic expression. We can multiply both sides of the equation by the lowest common multiple and then simplify and solve. Since we can’t divide by 0, the value which makes the denominator zero cannot be a possible solution.

Practice questions Choose your question path Explorer 1, 2, 4, 5a-dg, 6, 7a-d, 9, 12, 13, 16

Adventurer 1–6, 7a-f, 8a-ceg, 9, 10, 12–16

Trailblazer 1–4, 5gh, 6e-h, 7–12, 14–18

Understanding 1

What is the lowest common denominator of each set of fractions? b

a 2

, in solving for x, would this be a valid next step?

For the equation b

3x + 2 = 15

Fill in the blanks: a

Is each statement true or false? a

x = 1 is a solution for the equation

is a solution for the equation

is a solution for the equation 4x − 3 = 3

is a solution for the equation

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Fluency Ex 1

Ex 2

Ex 3

Solve: a

Solve:

Solve: a

Ex 4

Solve: a

Reasoning 9

Dylan attempted to solve the equation 1

8 × (3x + 6) = 4 − 2 × x

24x + 48 = 4 − 2x

26x = −44

142

. Here is his work:

Cross multiply Expand the brackets and simplify Add 2x to both sides and subtract 48 from both sides Divide both sides by 26 and simplify

Identify where Dylan has made an error and explain what it is.

Solve the equation correctly, showing the correct steps of work.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Complete the equation so it is true for all values of x.

Give an example of a linear equation with algebraic fractions that has no solutions. Justify your answer.

Faith looked at the equation

and noticed that the fraction could be simpliﬁed

since it had the same numerator and denominator. She then wrote down 1 = 1 as her next line of working and said that x = 1 was the solution to the equation. a

Does x = 3 satisfy the equation?

Does the equation have any other solutions?

What value is not a possible solution?

Problem-solving 13

A construction company has spent $15 000 000 to develop new cranes, and wants to limit the cost of development and production of a single crane to $4000. Given that the production cost of each crane is $2000, the cost for development and production of x cranes is given by 2000x + 15 000 000 dollars. a

Write an expression for the cost of one crane.

If the cost of one crane must equal $4000, ﬁnd x, the number of cranes that must be sold.

Fiona, a tennis player, has won 40 out of 55 matches in her career. Find x, the number of matches she must win in a row to raise her win percentage to 80%.

Solve for u:

If x = 2 and

At an annual charity event, the money raised was distributed in several different ways. One-third of the funds came from the silent auction, art sales comprised one-fourth of the total, and direct in-person donations made up one-ﬁfth of the revenue. The food booth raised $250. How much money was raised at the charity event? Round your answer to two decimal places.

, ﬁnd the value of k.

, ﬁnd the value of r.

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4.02 Rearrange equations After this lesson, you will be able to… • apply correct algebraic operations such as addition, subtraction, multiplication, and division to rearrange equations effectively. • solve real-world problems that require making different variables the subject.

Rearrange equations In mathematics and science, there are many formulas that are often given in a speciﬁc way by default.

Exploration Let’s look at the formula for speed:

s d t 1.

speed distance time

Use the formula to ﬁnd each of the following: • The speed at which a person travels if they drive 100 km in 75 min • The distance travelled when walking 5 kph for 30 min • The distance travelled by a car driving 85 kph for 5 hr • The time it will take to travel 1 000 km if you walk 80 m/min • How fast a plane is travelling if it can ﬂy 2 789 km in 6 hr • How long it will take to walk to the store 2 km away if you walk 100 m/min

Which ones took the most effort to solve?

If we always wanted to ﬁnd the distance given time and speed, how could we write the formula to make these calculations more eﬃcient?

By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, depending on which variable is unknown. To rearrange an equation and make a speciﬁc variable the subject, we need to perform various algebraic operations such as addition, subtraction, multiplication, and division. The key is to isolate the desired variable on one side of the equation while maintaining the balance of the equation. Here are some general steps to follow: • Identify the variable that you want to make the subject. • Perform algebraic operations on both sides of the equation to isolate the desired variable. Remember to use correct order of operations. • Keep track of the signs and the terms you are moving across the equal sign. • Simplify the equation as much as possible, and always double-check your steps to ensure accuracy. 144

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 Rearrange kx = m + nx to make x the subject.

Create a strategy Notice the like terms. Rearrange so that all the terms in x are on one side of the equation.

Apply the idea Write the equation

Subtract nx from both sides

Take out the common factor of x

Divide both sides by (k − n) to make x the subject

Reﬂect and check When the variable we want to make the subject appears more than once in the equation, we should get them all on the same side and factorise it out.

Example 2 Make M the subject of

Create a strategy We need to have one instance of M in the equation, so we ﬁrst need to clear the fraction and then factorise.

Apply the idea Start with the equation

Multiply both sides by (M + A)

Simplify

Expand the brackets

Subtract M from both sides

Subtract PA from both sides

Factorise each side of the equation

Divide both sides by (P − 1)

Simplify

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Reﬂect and check We can substitute values for A and P, to see if we get the same value solving for M in the original equation and using the rearranged equation. However, this does not guarantee we haven’t made any errors. It is worth checking every step for errors.

Example 3 Make R the subject of V = IR − E.

Create a strategy I is multiplied by R, then E is taken away to give V. Reverse these operations.

Apply the idea Write the equation

Adde E to both sides

Divide both sides by I

Write R as the subject

Idea summary We can rearrange equations and formulas to make different variables the subject. This can make it easier to do repeated calculations. Carefully using inverse operations, we can rearrange equations in a similar way to solving equations. It is very important to keep the equation balanced by doing operations on both sides. However, instead of operating on numbers, we are operating on pronumerals.

Practice questions Choose your question path Explorer 1–4, 5a-d, 6a-e, 7a-c, 8ab, 10, 12–15, 18, 20, 22, 23

Adventurer 1–7, 8abc, 9, 10, 12, 13, 14, 16, 18–21, 22, 23, 25

Understanding 1

146

Why do we rearrange equations?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 1–9, 11–13, 16, 17, 20, 21, 22–25

For the formula, Q = m × c × ΔT: • Q is the amount of heat absorbed in Joules, (J) • m is the mass in grams (g) • c is the heat capacity (J/g °C) • ΔT is the temperature change, (°C) We are given the amount of heat absorbed, the mass, and temperature change for ten different substances. Which variable should we make the subject to ﬁnd the unknown values more eﬃciently?

To make p the subject of R = pq − 4pt2, which option could be a ﬁrst step? A

Divide pq − 4pt2 by p

Multiply both sides of the equation by 4pt2

Factorise p from pq − 4pt2

Add p to one side of the equation

Is each statement true or false? a

When we rearrange a formula, we will get a numerical value.

When rearranging a formula, the formula may involve numerical and algebraic terms.

We use inverse operations to make a variable the subject.

When we rearrange a formula, we will eliminate all of the other pronumerals.

Fluency Ex 1

Make x the subject of these equations: a

y=x+z

y = 5 (8 + x)

3y = 2x + z

y = 5 (3 − x)

Make m the subject of these equations: a e

x = 8k (n + m)

x = −7m (p + q)

y = −5mx − 14

x = −7k (l − m)

8x3 = 4y − 12

9x2 + 3y = 12

Make k the subject of these equations: a

Ex 3

y = 3mx − 5

Make y the subject of these equations: a

Ex 2

b = −2am + 10

Make p the subject of the equation M = pq − 7pr2.

Make v2 the subject of the formula for kinetic energy,

The bend allowance for sheet metal is given by the formula

. .

Make T, the thickness of the sheet, the subject of the formula. 4.02 Rearrange equations mathspace.co

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Make r2 the subject of these formulas: .

The volume of a cone is given by the formula

The area of a circle is given by the formula A = π r2.

The area of a sector of a circle is given by the formula

Make the mentioned variables the subject in the following equations: a

E = mc2, make m the subject

A = P (1 + r)t, make P the subject

P = I2 R, make R the subject

V = π r2 h, make h the subject

, make h the subject

D = b2 − 4ac, make c the subject

, make z the subject

g = hx2 k, make h the subject

, make c the subject

d = uv4 − t, make u the subject

The formula for the force of an object is F = ma, where F is the force, m is the mass of the object, and a is the object’s acceleration. Write an equation that could be used to solve for the mass of the object.

The formula for the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width. Write an equation that could be used to ﬁnd the length of the rectangle.

Betsy knows the formula to convert Celsius into Fahrenheit is

. She wants to

convert a temperature from Fahrenheit to Celsius. Find an equation that she could use to do this conversion. 17

The displacement of an object is given by used to ﬁnd v0.

. Write an equation that could be

Reasoning 18

Describe the similarities and differences of solving 3x + y = z and 3x + 8 = 2. Explain your thinking.

Two students, Alice and Bob, are given the Pythagora’s theorem equation to rearrange: c2 = a2 + b2 Alice rearranges the equation to make a2 the subject, while Bob rearranges the equation to make b2 the subject. Their results are as follows: Alice: a2 = c2 − b2 Bob: b2 = c2 + a2 Determine which student, if any, is correct in their rearrangement of the equation and explain your reasoning.

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Pamela attempted to make r the subject of the equation

and followed these steps:

Multiplied both sides by R + r

Divided both sides by L

Subtract R from both sides of the equation

What was Pamela’s mistake? Explain and correct her error. 21

Ruﬁno has submitted the following work to write the equation for the mass of an object, m, given the kinetic energy, KE, and velocity, v:

Was Ruﬁno correct? If so, explain each step of his working. If not, ﬁx his error(s).

Problem-solving 22

The area, A, of a triangle is given by the formula and b is the length of its base.

, where h is the height of the triangle

Write an equation that could be used to solve for h.

Find the height of a triangle with a base of 12 cm and an area of 18 cm2.

The volume of a sphere is given by the formula

, where r is the radius of the sphere.

Rearrange the formula to make, r the subject of the formula.

Calculate the radius of a sphere with a volume of 91 cm3 to two decimal places.

When a certain amount of money, P, is invested, its future value is given by the formula: . a

Make P, the principal, the subject of the equation.

How much must be invested now, if it is to grow to $3000 in 10 years with interest of 4%

compounded annually? Round to the nearest whole dollar. 25

The volume of the cylinder that ﬁts exactly into a sphere is given by the formula:

Rearrange the formula so that r is the subject.

What is the radius of the sphere if the volume of the cylinder is 13.75π mm3 and the height of the cylinder is 5 mm? Round your answer to the nearest millimetre. 4.02 Rearrange equations mathspace.co

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4.03 Parallel and perpendicular lines After this lesson, you will be able to… • understand the characteristics of parallel and perpendicular lines. • apply the understanding of parallel and perpendicular lines to find equations of such lines. • interpret and solve problems involving parallel and perpendicular lines.

Parallel lines Lines that never intersect are called parallel lines.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.03 to answer these questions. 1.

What do you notice about the gradients of the two parallel lines?

What do we know about the gradients of vertical lines? Is it possible for two distinct, vertical lines to be parallel?

When two non-vertical lines are parallel, how will their equations be similar? How will they be different?

Parallel lines are lines that have the same gradient.

When two lines are parallel, they never cross and have no points in common.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

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Let’s look at how we can identify parallel lines given their equations. Equation form

Characteristic of parallel lines

y = mx + c

Parallel lines have the same m value.

Examples y = 2x – 1 y = 4 + 2x

For every straight line, there exists an inﬁnite number of lines parallel to it.

Here is the line y = x and two more lines in the same family of parallel lines: y = x + 3 and y = x - 4.

y=x+3

Notice that they have the same gradient (m-value) but different x and y-intercepts (c-values).

y=x

2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4 y = x − 4

The gradient of any vertical line is undefined. But by definition, any two vertical lines are parallel.

Example 1 Which two lines are parallel to y = −3x + 2? A y = 3x

−3y − x = 5

y = −10 − 3x

E y + 3x = 7

Create a strategy Choose the two options with the same gradient, m, as the given line.

Apply the idea In order for us to compare the gradients, we need to ﬁrst make all the expressions into the form of y = mx + c, so we will rearrange options C and E.

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For option C: Write the equation

Add x to both sides

Divide both sides by −3

Write in gradient-intercept form

For option E: y + 3x = 7 Write the equation y = −3x + 7 Evaluate The lines that have a gradient of m = −3 are y = −10 − 3x and y + 3x = 7. So the lines in options D and E are parallel to y = −3x + 2.

Example 2 Find the equation of the line that is parallel to

and passes through the point (9, −1).

Create a strategy First, ﬁnd the gradient of the given line. Next, substitute the gradient and the given coordinates into the equation y = mx + c.

Apply the idea The gradient of

. Since the new line is parallel to the given line, the gradient of

the new line is also . Equation of a line in gradient-intercept form

Substitute

Substitute x = 9 and y = −1

Evaluate the multiplication

Subtract 3 from both sides

Substituting the values of m and c into the gradient-intercept form of a line, the equation is:

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Reﬂect and check y −4 −3 −2 −1

−1

We can graph the given line with the line we found to see that they are parallel.

x 1

−2 −3 −4 −5 −6 −7 −8

Idea summary If two lines are parallel, they have the same gradient. If two lines have the same gradient, they are parallel. All vertical lines are parallel to each other.

Perpendicular lines Lines that meet at right angles (90°) are called perpendicular lines.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.03 to answer these questions. 1.

What do you notice about the gradients of the two perpendicular lines?

When two lines are perpendicular, what is the product of their gradients?

Is your answer to the previous question also true for a vertical and a horizontal line? Are vertical and horizontal lines perpendicular?

Perpendicular lines have gradients whose product is equal to −1. For two perpendicular lines, y = m1 x + c1 and y = m2 x + c2: m1 × m2 = −1 We say that m1 is the negative reciprocal of m2. This means the two gradients have opposite signs and are reciprocals of each other.

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Here are some examples of negative reciprocals:

If the gradient of the ﬁrst line is m1 then we can ﬁnd the gradient m2 of a perpendicular line using the formula:

Special case: a vertical line and a horizontal line are perpendicular but do not have gradients which have a product of −1. This is because the gradient of a vertical line is undeﬁned.

Example 3 Find the equation of a line that is perpendicular to y = 6x + 10, and has the same y-intercept. Give your answer in the form y = mx + c.

Create a strategy Find the gradient of the perpendicular line and the y-intercept of the given line, then substitute both values into the equation y = mx + c.

Apply the idea The gradient of y = 6x + 10 is 6, and the negative reciprocal of 6 is

, so the gradient of the

. The y-intercept is c = 10. So the equation of the line is:

perpendicular line is

Reﬂect and check y 15 12 9

We can check that the product of the gradients is equal to −1, and we can graph both lines to see they are perpendicular.

y = 6x + 10

1 y = − x + 10 6

6 3 x −6

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Idea summary Two lines are perpendicular if their gradients are negative reciprocals of each other. To test if lines are perpendicular, multiply the gradients together. If the result is −1, then the lines are perpendicular. Vertical and horizontal lines are perpendicular to each other, even though their gradients do not multiply to give −1.

Practice questions Choose your question path Explorer 1–8, 10a-d, 11, 12, 14, 15a-f, 16, 17a-d, 19, 20, 22a, 23a, 26, 27, 29, 31

Adventurer 1–8 10a-d, 12, 14, 15aceg, 16, 17, 19, 21, 22b, 23b, 24, 26, 27, 31

Trailblazer 1–7, 9, 10, 13, 15e-h, 17–21, 24, 25, 27, 28, 30, 32, 33

Understanding 1

The equations y = 2x, y = 2x + 5 and y = 2x − 7 have been graphed on the same Cartesian plane: a

What do all of the equations have in common?

What do all the lines have in common?

y 10 y = 2x 8 6 y = 2x + 5 4 2 −10 −8 −6 −4 −2 −2 −4 −6 −8 −10

y = 2x − 7

Which of these lines is parallel to the y-axis? A

2 4 6 8 10

y = −1

y = 2x − 1

y=x

x=5

If the equation of a line is y = mx + c, and m > 0, which of the statements is true? A

Any line parallel to this line will have a negative gradient.

Any line perpendicular to this line will have a positive gradient.

Any line perpendicular to this line will have a gradient of zero.

Any line perpendicular to this line will have a negative gradient.

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For the given graph: a

Which two lines are parallel?

Find two pairs of perpendicular lines.

y = 3x + 15

10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

Complete the statement:

The x-axis and the y-axis are a pair of ⬚ lines.

y y = 1.9x

x 2 4 6 8 10

1 y=− x 3 1 y = − x−9 3

If two non-vertical lines are perpendicular, what is the product of their gradients?

Fluency Ex 1

Which of these lines are parallel to y = 7x + 3? a

y = 7x − 3

y = 6x + 3

y = 7x + 4

y = 7x

y = 6x + 4

y = −7x + 3

y = 3x + 7

y = −3x − 7

Are these pairs of lines parallel? a

y = −3x − 2 and y = −3x + 9

y = −2x − 5 and y = −2x − 8

y = 7x + 8 and y = −5x + 8

y = 4x − 1 and y = 4x − 6

x = 4 and y = 5

y = 7x − 5 and y = −7x + 6

y = 3x − 5 and

y = −8x − 2 and 9x + 7

The line L1 passes through the point (9, −5) and is parallel to the line y = −5x + 2. a

Ex 2

156

Find the gradient of line L1.

Find the equation of line L1.

Find the equation of the line described. a

Parallel to the x-axis and passes through (−10, 2)

Parallel to the y-axis and passes through (−7, 2)

Parallel to the line y = −3x − 8 and crosses the y-axis at −4

Parallel to the line y = 8x − 3 and crosses the y-axis at 5

Parallel to the line y = −2x + 9 and passes through the point (−3, 1)

Given the lines L1 : 5x + 8y − 10 = 0 and L2 : a

Find the gradient of line L1.

Are the two lines parallel?

. b

Find the gradient of line L2.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Find the gradient of the line perpendicular to: a

A line with gradient 6

c 13

y = −3x

y=8−x

3y − x = 5

x + 3y = −10

y = 9x

y = −9x + 5

9y = −x + 2

Are these pairs of lines perpendicular? a

y = x + 1 and y = x − 1

y = 3x − 7 and y = −3x + 6

y = −3x + 6 and

and

and y = −(3x − 27)

y = 3x − 5 and

and

The line L1 is perpendicular to y = 5x − 4 and crosses the y-axis at 1. a

Ex 3

Are these lines perpendicular to y = 9x + 2? a

A line with gradient

Are these lines perpendicular to y = 3x + 2? a

Find the gradient of line L1.

Find the equation of line L1.

Find the equation of the line described. a

Perpendicular to the x-axis and passes through (−8, −1)

Perpendicular to the y-axis and passes through (−8, −8)

Perpendicular to

Perpendicular to y = −6x + 10 and has the same y-intercept

Perpendicular to

and goes through the point (0, 6)

and passes through (−8, 5)

Given the lines L1: 8x = −7y + 3 and L2: 7y − 8x − 7 = 0. a

Find the gradient of line L1.

Find the gradient of line L2.

Are the two lines perpendicular?

Reasoning 19

If every point on the line y = 2x + 2 is shifted 2 units up, ﬁnd the equation of the new line.

Every point on a particular line is shifted 6 units down. The resulting line has equation y = 2x − 2. Find the equation of the original line.

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A farmer wants to create an enclosure for his sheep. He puts fencing along these lines:

What is the shape of the enclosure? Justify your answer.

Problem-solving 22

Each pair of lines are parallel. Find the value of b. a

0 = 5x − 9y + 6

y = bx + 4

0 = bx + 3y − 2 23

Each pair of lines are perpendicular. Find the value of k. a

5x + 4y + 8 = 0

−6x + 6y − 8 = 0

kx + 9y − 8 = 0 kx + 10y + 8 = 0 24

A construction company is building a new road that runs parallel to an existing road. If the equation of the existing road is y = 0.5x + 3, ﬁnd the equation of the new road that passes through the point (2, 6).

The line L1 goes through (−9, 10) and (2, −1).

Find the gradient of line L1.

Find the equation of the line that passes through (10, 8) and is parallel to line L1.

The line L1 goes through (3, 2) and (−2, 4). Find the equation of the line that has a y-intercept of 1 and is parallel to line L1.

The line L1 goes through (1, 1) and (3, 13). a

Find the gradient of line L1.

Find the equation of the line that has a y-intercept of (0, −4) and is perpendicular to line L1.

The line L1 is perpendicular to y = 10x + 8 and passes through the point of intersection of the lines y = x + 5 and 8x − 10y + 60 = 0. a

Find the gradient of line L1.

Find the point of intersection of y = x + 5 and 8x − 10y + 60 = 0.

Find the equation of the perpendicular line L1.

The vertices of a quadrilateral are A (2, 2), B (6, 4), C (8, 10), and D (4, 8). a

Find the gradient of each side.

Is quadrilateral ABCD a parallelogram?

Is quadrilateral ABCD a rectangle?

The line y = 2x + 4 is perpendicular to the line cx − 4y + 7 = 0. Find the value of c.

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The 4 vertices of square ABCD have been plotted on a Cartesian plane.

y 7

Find the gradient of side AB.

Find the gradient of side BC.

Find the product of the gradients in parts (a) and (b).

Are sides AB and BC perpendicular?

6 A

5 4 D

2 C

1 1

A quadrilateral ABCD has been graphed on the Cartesian plane. a

Find the gradient of each side: i

iii

What type of quadrilateral is ABCD? Explain your answer.

y x

−1 1 2 3 4 5 6 7 8 9 −1 D −2 −3 −4 A −5 C −6 −7 −8 B −9

The points P (0, −2), Q (−3, −2), R (0, 4) and S (2, 2) are graphed on the Cartesian plane:

y 4 R

Find the gradient of PQ.

Find the gradient of RS.

Are PQ and RS parallel?

Are QR and PS parallel?

What type of quadrilateral is PQRS?

−4 −3 −2 −1 Q

−1

S x 1

−2 P −3 −4

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4.04 Linear models After this lesson, you will be able to… • understand the roles of independent and dependent variables in a linear model. • interpret linear models in real-world contexts, including understanding the significance of the gradient and vertical intercept. • explore the concept of negative gradients in the context of decreasing quantities. • understand the characteristics of linear graphs in real-world scenarios. • interpret linear relationships from a table of values. • solve real-world problems involving linear models.

Linear models In linear relationships we have two variables, independent and dependent variables which are x and y respectively. These will often represent physical quantities such as time, distance, cost, mass or temperature. Instead of being labelled x and y, they may be labelled with letters or names that better represent those quantities. For example, if we were modelling the rate of fuel being used in a car, the independent variable may represent the distance travelled, in kilometres, and the dependent variable may represent the volume of fuel, in litres, in the car’s fuel tank. Instead of x and y, we might use d and V as our variables. Each of these variables has units of measurement associated with them. Fuel consumption of a mid-sized Sports Utility Vehicle V (L) 70 60 50 40 30 20 10

d (km) 100

300

500

700

900

As we know already, the two key features of a linear function, or straight-line graph, are the gradient and the y-intercept. In linear modelling situations, the y-intercept is often referred to as the vertical intercept, because the vertical axis may be labelled with a variable other than y. The vertical intercept represents an initial value. In our example above, the vertical intercept is 63 litres. It represents the volume of fuel in a full tank, before the car began its journey.

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In a real-world context the gradient represents a rate of change. Using our example above, the gradient would represent the volume of fuel used per distance travelled. In other words, the gradient is a measure of the car’s fuel consumption. The graph above is decreasing as the fuel is being used, so it has a negative gradient. If we divide the ‘rise’ of −63 by the ‘run’ of 900 we get a gradient of −0.07. This means fuel is being consumed at a rate of 0.07 litres per kilometre (or 7 litres per 100 kilometres). Because most physical quantities like distance, volume or time do not contain negative values, the graphs of most linear models tend to exist only in the ﬁrst quadrant of the Cartesian plane (like the example above). This is not always the case though - for example, temperature is a physical quantity that can have negative values.

Idea summary In linear modelling situations: • • • •

The vertical intercept represents the initial value. The gradient represents the rate of change. The independent variable goes on the horizontal axis. The dependent variable goes on the vertical axis.

Linear relationship from a table of values Real world situations often involve the collection of data, which is commonly displayed in a table of values. The independent variable is always displayed in the ﬁrst row of the table, followed by the dependent variable, in the second row. We may ﬁrst want to decide whether the values in the table represent a linear relationship. To do this, we could plot the values on a cartesian plane. If we can draw a straight line through all of the points, then we have a linear relationship. Once we know that the relationship between the variables is linear, we can work out the gradient and vertical intercept, and express the relationship as a linear equation (or function).

Example 1 Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars: Number of litres (x) Cost of petrol ( y)

0 0

10 16.40

20 32.80

30 49.20

40 65.60

a Graph the linear relationship represented in the table.

Create a strategy Write the values in the table as points, plot the points on a Cartesian plane, and draw the line that passess through the points.

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Apply the idea y 70 60 50 40 30 20 10

x 10

20 30 40 50 60

b Write an equation linking the number of litres of petrol pumped (x) and the cost of the petrol ( y).

Create a strategy Find the gradient of the line and the y-value of the y-intercept, then write the equation of the line in the form y = mx + b.

Apply the idea Find the gradient of the line: Divide the change in y by the change in x

Evaluate

Since the y-intercept is at (0, 0), b = 0. The equation of the line is y = 1.64x

c How much does petrol cost per litre?

Create a strategy The gradient of the line represents the unit rate of the cost of petrol per litre.

Apply the idea The gradient of the line is

and represents the increase in the price of petrol per 1 litre.

In other words, the cost of petrol per litre is $1.64

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Reﬂect and check We can also ﬁnd the cost of petrol per litre by dividing the cost of the petrol ( y) by the number of litres (x). Divide 16.40 by 10

Evaluate

d How much would 47 litres of petrol cost at this unit price?

Create a strategy Using the equation from part (b), substitute x = 47.

Apply the idea y = 1.64x

Write the equation

= 1.64 × 47

Substitute x = 47

= $77.08

Evaluate

Idea summary The independent variable is always displayed in the ﬁrst row of the table, followed by the dependent variable, in the second row. From a table of values we can plot the graph and determine the equation for the relationship. We can use the relationship to determine the value of one of the variables given the value of the other.

Practice questions Choose your question path Explorer 1−8, 10, 12, 14, 16, 20, 22, 24, 27, 29, 32, 33

Adventurer 1−5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 30, 31, 33

Trailblazer 1−5, 7, 9, 11, 15, 19, 21, 25, 26, 30, 31, 34, 35

Understanding 1

Which of the equations represents a linear model? A

y = 2x + 3

y = x2 + 2x + 1

y = 3x3 + 2x + 1

f (x) = 2x+1

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Which of the statements is a characteristic of a linear relationship? A

The rate of change is variable.

The rate of change is constant.

The rate of change is exponential.

The rate of change is never zero.

Complete the sentences with ‘horizontal’ or ‘vertical’: a b c d

The independent variable is plotted on the ⬚ axis. The dependent variable is plotted on the ⬚ axis. The line y = 5 is a ⬚ line.

The line x = −2 is a ⬚ line.

The initial value is the starting value or the y-intercept. Find the inital value for these linear models: a

C = 250 + 7.55x

T = 12.5 + 1.5m

4x + 5y = 40

f (k) = 0.6k − 5

Fluency 5

The horizontal axis represents the time in hours and the vertical axis represents the distance travelled in kilometres. a

What is the gradient of the line?

What does the gradient represent? A

The initial value

The speed

Cost of petrol

Time travelled

Distance (km) 575 460 345 230 115 t

The following graphs represent the cost and revenue in dollars for x items produced and sold, respectively. C = 15x is the cost function, indicating each item costs $15 to make. R = 0.5x2 + 50 is the revenue function. a

Identify each function on the graph.

Explain the meaning of the intersection points of the graphs.

$ 600 500 400 300 200 100 Items (x)

164

Identify whether the situation is represented by a linear or non-linear function: a

Ross deposits $100 into a savings account each week.

Tammy’s parents record her height every year for the ﬁrst 12 years of her life.

Julia collected lollies from a pinata at a party and eats two lollies per day.

Janet starts running and doubles the distance she runs each day.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A dam used to supply water to the neighbouring town had this data recorded for its volume over a number of months: Month (M ) Volume in billions of litres (V )

1 112

Plot the points on the Cartesian plane.

Is this relationship linear?

80 16

140 22

160 24

Graph the linear relationship represented in the table. Determine the temperature when the crickets make 130 chirps each minute.

How many chirps per minute will the crickets make if the temperature is 27 °C?

How many chirps per minute will the crickets make if the temperature is 20 °C?

A diver starts at the surface of the water and begins to descend below the surface at a constant rate given by g (x) = 0.8x Complete the table to show the depth of the diver over 4 minutes: 0 0

Graph the linear relationship represented in the table.

Calculate the increase in depth each minute.

Calculate the depth of the diver after 24 minutes.

The time in minutes and the temperature in degrees Celsius (°C) are given in the table: Time (x) Temperature ( y °C)

Ex 1

110 19

Number of minutes passed (x) Depth of diver in metres g(x)

4 80

3 110

Chirping crickets can be used as an indication of how hot or cool it is outside. For a particular species this data was recorded: Number of chirps per minute (x) Temperature ( f (x) °C)

2 106

1 8

2 13

3 18

4 23

5 28

By how much is the temperature increasing each minute?

Find the initial temperature.

Form an equation relating x and y.

Graph the linear relationship represented in the table.

Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars: Number of litres (x) Cost of petrol C(x)

0 0

10 12

20 24

30 36

40 48

Graph the linear relationship represented in the table.

Write an equation linking the number of litres of petrol pumped, x, and the cost of the petrol C(x).

Find the cost of petrol per litre.

Calculate C(75), the cost of 75 L petrol.

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The table shows the temperature of a metal plate, in °C, after an amount of time in minutes: Time (x) Temperature f (x)

1 10

2 15

3 20

Graph the linear relationship represented in the table.

By how much is the temperature increasing each minute?

Find the initial temperature.

Form an equation relating x and f (x).

After Sally starts running, her heartbeat increases at a constant rate as shown in the table: 4 87

Complete the table. Graph the linear relationship represented in the table.

By how much is her heartbeat increasing each minute?

6 93

8 99

10 105

Emma is gathering data on the growth of her plant as shown in the table: 2 6

4 10

6 14

8 18

Graph the linear relationship represented in the table.

How tall was the plant when Emma started tracking the growth?

Determine the height of the plant on day 12.

On which day will the plant be 24 cm tall?

Write a linear equation to represent the height, h, as a function of the day, d.

When a plane is coming in to land, the pilot requires the plane to lose altitude at a rate of 5 m/s. At the beginning of a particular descent the plane starts at an altitude of 8500 m. The altitude of the plane at various times after starting the descent are shown in the table: 0 8500

200 7500

400 6500

1600 500

Graph the linear relationship represented in the table.

Determine the altitude of the plane 500 seconds after starting the descent.

How many seconds did the plane take to descend to the ground?

Write a linear equation to represent the altitude of the plane, A, as a function of the number of seconds after commencing descent, t.

Katrina is going to catch a taxi into the city. She knows that the taxi has a ﬂat fee of $0.30 and that the fare rises by $0.10 per kilometre travelled. The fares for particular distances are shown in the table: Distance travelled, d (km) Fare, F ($)

166

2 81

Descent (t seconds) Altitude (A)

0 75

Day (d) Height of plant (h cm)

5 30

Number of minutes passed (x) Heart rate ( y)

4 25

6 0.90

7 1.00

8 1.10

9 1.20

Graph the linear relationship represented in the table.

Write a linear equation to represent the fare, F, as a function of the journey’s length, d.

Calculate the fare for a 14 km journey.

Find the distance travelled for a trip that costs $0.60

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Justin is looking into the details of his mobile phone plan. He knows the costs for several call lengths as shown in the table: Length of call in minutes (t) Cost (C)

2 $1.00

6 $2.20

10 $3.40

14 $4.60

Graph the data in the table.

How much will it cost to make an 8 minute call?

Find the length of a call that costs $1.60

Determine the rule that connects the cost of a call, C, to the length of the call, t.

Beth’s income is based solely on the number of hours she works, and she is paid a ﬁxed hourly wage. She earns $750 for working 30 hours. a

Identify the two variables in the question and assign a variable name to each.

Sketch a graph that displays her income against her hours worked.

Form an equation relating the variables.

Calculate Beth’s income when she works 25 hours.

Calculate the number of hours that Beth must work to earn $125.

The cost for a business to operate, can be expressed in terms of the total number of hours it has operated for. The cost is $150 an hour. a

Identify the two variables in the question and assign a variable name to each.

Form an equation relating the two variables.

Find the total cost for the business to operate for 22 hours.

Find the number of hours that the business needs to operate to incur a total cost of $5400.

Reasoning 21

What does the gradient represent in these situations? a

Petrol used (L)

Distance (km)

Amount earned ($)

Hours worked (h)

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Heartbeat (b)

Altitude in metres (a)

Minutes running (m)

Number of seconds flying (f)

The graph shows the conversion between Country A and Country B’s currency:

Use the graph to convert 8 of currency A to currency B.

Use the graph to convert 2 of currency B to currency A.

Discuss the usefulness and limitations of this model.

14 10 8 6 4 2

B 1

A baseball is thrown vertically upward, and the velocity of the baseball, V in metres per second after T seconds is given by V = 120 − 32T. a

Complete the table of values: Time (T ) Velocity (V )

What does the negative value of V, when T = 4, mean in context?

Discuss the usefulness and limitations of this model.

A racing car starts the race with 250 litres of fuel. From there, it uses fuel at a rate of 5 litres per minute. a

Complete the table of values: Number of minutes passed, x Amount of fuel left in tank, y

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2 3 4 5 6 7 8 9 10

Determine an algebraic rule linking the number of minutes passed, x, and the amount of fuel left in the tank, y.

Describe how the amount of fuel in the car is changing over time.

Discuss the usefulness and limitations of this model.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A ball is rolled down a hill. The table shows the velocity, in m/s of the ball after a given number of seconds: Time in seconds, t Velocity, V

0 18

1 19.7

2 21.4

3 23.1

4 24.8

5 26.5

Find and interpret the gradient of the line.

Find and interpret the vertical intercept of the line.

Determine the rule that connects the velocity, V, to the time in seconds, t.

Find the velocity of the ball after 17 seconds, rounded to one decimal place.

Discuss the usefulness and limitations of this model.

The graph shows the temperature of a room after the heater has been turned on:

9 8

Find and interpret the gradient of the line.

Find and interpret the y-intercept.

Write an equation to represent the temperature of the room, y, as a function of time, t.

Find the temperature of the room after the heater has been turned on for 30 minutes.

Discuss the usefulness and limitations of this model.

c d e

Temp (°C)

5 3 1

t (mins) 1

Problem-solving 27

The graph shows the conversion between miles and kilometres:

Convert 10 miles to kilometres.

How many kilometres is 1 mile equal to?

12 10 8 6 4 2

mi 1

2 3 4 5 6 7

8 9 10

Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars: Number of litres (x) Cost of petrol ( y)

0 0

10 16.40

20 32.80

30 49.20

40 65.60

Calculate the cost of 47 L petrol.

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The graph shows the conversion between temperatures in Celsius and Fahrenheit: 50

°C

40 30 20 10 −20

−10

°F 10

100

−20 −30 −40 −50

Use the graph to convert as indicated:

40 °C to Fahrenheit

50 °F to Celsius

−20 °F to Celsius

The graph shows the conversion between temperatures in Celsius and Fahrenheit:

170

°F

Use the graph to convert 10 °C into Fahrenheit.

0 °C is 32 °F. For every 1 °C increase, by how much does the Fahrenheit temperature increase?

100 80 70 60

Would 80 °F be above or below normal body temperature? Normal body temperature is approximately 37 °C.

50 40 30

Write the rule for conversion between Celsius (C) and Fahrenheit (F).

Convert 35 °C into Fahrenheit.

20 10

°C 10 15 20 25 30 35 40

The number of ﬁsh in a river is approximated over a ﬁve year period. The results are shown in the table: Time in years (t)

Number of fish (F )

6000

5700

5400

5100

4800

4500

Find and interpret the gradient of the line.

What is the value of F when the line crosses the vertical axis?

Write an algebraic equation for the line relating t and F.

How many ﬁsh remain in the river after 9 years?

Determine the number of years it takes for there to be 1800 ﬁsh remaining in the river.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The table shows Peter’s earnings from ironing shirts: Shirts ironed Earnings ($)

0 4

2 8

4 12

6 16

8 20

Find and interpret the gradient of the line.

Find and interpret the y-intercept of the line.

What will be his total earnings if he irons 14 shirts?

How many shirts will he have to iron in order to earn $28?

The graph shows the conversion between miles and kilometres: a

If a car is driving 32 km/hr in a school zone, at what speed are they travelling in miles per hour?

A road sign says the speed limit is 128 km/hr. Calculate this speed limit in miles per hour.

22 20 18 16 14 12 10 8 6 4 2

km 4

The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side. a

Find and interpret the gradient of the line.

Find and interpret the y-intercept.

Write an equation to represent the amount of water remaining in the bucket, y, as a function of time, x.

Find the amount of water remaining in the bucket after 54 minutes.

32 Water remaining (L) 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 5

8 12 16 20 24 28 32 36

Time (mins)

20 25 30 35

The table shows the linear relationship between the number of plastic chairs manufactured and the total manufacturing cost: Number of plastic chairs Cost (dollars)

5 135

10 185

15 235

Find and interpret the gradient of the line.

Find and interpret the y-intercept.

Form an equation relating the number of plastic chairs and the cost.

Find the total cost of manufacturing 25 plastic chairs.

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4.05 Graph linear inequalities After this lesson, you will be able to… • graph linear inequalities on the cartesian plane. • apply test points to verify if they satisfy the given linear inequality.

Graph linear inequalities We can represent a linear relationship using a line on the Cartesian plane, for example, y = −2x − 5.

2 1 −5 −4 −3 −2 −1

−1

x 1

−2 −3 −4

The line indicates the set of all points on the Cartesian plane that make the equation true.

−5 −6

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.05 to answer these questions. 1.

What effect do the different inequality signs have on the area that is shaded?

What is the difference between < and ≤, or > and ≥?

Test other points using the equation and verify using the shaded region if the point is “in” or “out”.

There are two key considerations when plotting inequalities. 1. Is the region shaded above or below the line? 2. Is the line included (solid) or excluded (dotted)?

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The inequality y > 2x + 3 is satisfied for y-values greater than the expression 2x + 3. This corresponds to y-values ABOVE the line, which is the shaded region.

5 4 3 2 1 −4 −3 −2 −1

x 1

−1

The line corresponds to the equation y = 2x + 3. Since this is not included in the strict inequality y > 2x + 3, the points on the line are not included in the solution. We indicate this by using a dotted or dashed line.

−2

Compare this to the inequality y ≤ 2x + 3. This represents all of the y-values less than or equal to 2x + 3. This includes all of the points BELOW the line, but also the points ON the line itself. We show that the line is included in the inequality by using a solid line.

5 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2

Example 1 Determine whether or not each of the following points satisfies the inequality y < 2x − 5: a (1, −4)

Create a strategy Substitute x = 1 into the right side of the inequality, and compare the result to the value of y, which is −4.

Apply the idea y < 2x - 5

Start with the inequality

y<2×1-5

Substitute the value of x

y < -3

Evaluate

-4 < -3

Compare to -4

This result matches the inequality, and so this point satisﬁes the inequality.

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b (−3, −9)

Create a strategy Substitute x = −3 into the right side of the inequality, and compare the result to the value of y, which is −9.

Apply the idea y < 2x - 5

Start with the inequality

y < 2 × (-3) - 5

Substitute the value of x

y < -11

Evaluate

-9 > -11

Compare to -9

This result does not match the inequality, and so this point does not satisfy the inequality. c (3, 1)

Create a strategy Substitute x = 3 into the right side of the inequality, and compare the result to the value of y, which is 1.

Apply the idea y < 2x - 5

Start with the inequality

y<2×3-5

Substitute the value of x

y<1

Evaluate

1=1

Compare to 1

This result does not match the inequality, and so this point does not satisfy the inequality. The fact that the two values are the same tells us that this point lies on the boundary line y = 2x − 5, but due ro the inequality symbol < the boundary is not included in the inequality.

Example 2 Consider the inequality 3x − 4y ≥ 24. a Find the x-intercept of the boundaty line 3x − 4y = 24

Create a strategy The x-intercept occurs when y = 0.

Apply the idea 3x − 4y = 24 3x − 4 × 0 = 24

Start with the equation Substitute y = 0 into the equation

3x = 24

Simplify

x=8

Solve for x

The x-intercept for the boundary line is (8, 0).

174

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b Find the y-intercept of the boundary line 3x − 4y = 24.

Apply the idea 3x − 4y = 24 3 × 0 − 4y = 24

Start with the equation Substitute x = 0 into the equation

−4y = 24

Simplify

y = −6

Solve for y

The y-intercept for the boundary line is (0, −6).

c Does the origin (0, 0) satisfy the inequality?

Create a strategy Substitute x = 0 and y = 0 into the inequality and see if it is true.

Apply the idea 3x − 4y ≥ 24 Start with the inequality 3 × 0 − 4 × 0 ≥ 24 Substitute the values into inequality 0 ≥ 24 Evaluate the left-hand side The statement is false, so the origin does not satisfy the inequality.

d Graph the inequality.

Create a strategy Draw the boundary line using the intercepts found in parts (a) and (b). Decide which side to shade based on the answer to part (c) which states that the origin is not in the region.

Apply the idea 1 −1 −1 −2 −3 −4 −5 −6

y x 1

2 3 4 5 6 7 8 9

Since the inequality is ≥ we know the boundary line is included so is solid. The boundary line goes through (8, 0) and (0, −6). We know that (0, 0) is not in the required region, so we shade the opposite side.

−7 −8 −9

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e Do points on the line 3x − 4y = 24 satisfy the inequality?

Create a strategy Is the boundary included or excluded?

Apply the idea The inequality is 3x − 4y ≥ 24. The inequality symbol ≥ means “greater than or equal to”, so points that satisfy 3x − 4y = 24 are included. Points on the boundary line satisfy the inequality.

Example 3 A linear inequality has points (1, 4) and (−1, 10) on its boundary line. The inequality is satisﬁed by (5, 5). Points on the boundary line itself do not satisfy the inequality. a Find the gradient of the boundary line.

Create a strategy Use m =

Apply the idea Let (1, 4) = (x1, y1) and (−1, 10) = (x2, y2) Substitute the values into the gradient formula

Simplify numerator and denominator

Calculate

The gradient of the boundary line is −3

b Find the equation of the boundary line in the form y = mx + c.

Create a strategy Use the gradient found in part (a) and one of the points that lie on the boundary and substitute into y = mx + c.

Apply the idea y = mx + c

Start with y = mx + c

4 = −3 × 1 + c

Substitute m = −3 and (1, 4)

c=7

Evaluate c

The equation of the boundary line is y = −3x + 7, which may be expressed as y = 7 − 3x

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

c State the inequality that satisﬁes the given conditions.

Create a strategy We are told that the point (5, 5) satisﬁes the inequality. Substitute x = 5 into the expression for the line and then compare this to the value y = 5 to see which way the inequality should go. We are also told that points on the boundary line do not satisfy the inequality, so the relationship will either be < or > and not ≤ or ≥.

Apply the idea 7 − 3x = 7 − 3 × 5

Substitute 5 for x

= −8

Evaluate

5 > −8

Compare to the y-value of 5

So the inequality is y > 7 − 3x.

d Graph the inequality.

Create a strategy Draw the boundary line. The given points (1, 4) and (−1, 10) can be used to draw the boundary. The boundary line should be dashed. Shade the side that contains (5, 5).

Apply the idea 10 9 8 7 6 5 4 3 2 1 −1 −1

Since the inequality is > we know the boundary line will be dashed.

The boundary line goes through (−1, 10) and (1, 4). We know that (5, 5) is in the required region, so we shade that side.

x 1

2 3 4 5 6 7 8 9

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Example 4 Bernie sells his homemade pizza sauce in two sizes of containers: 200 mL, and 500 mL. One week he has 4 L of sauce to sell. a Let x be the number of 200 mL bottles he ﬁlls, and y be the number of 500 mL bottles he ﬁlls. Write an inequality, with y as the subject, to represent this situation.

Create a strategy The amount of sauce in 200 mL bottles, added to the amount of sauce in 500 mL bottles, must be less than, or equal to, the amount of sauce available, 4000 mL.

Apply the idea x of the 200 mL bottles contains 200x mL. y of the 500 mL bottles contains 500y mL. he amount in bottles must be less than or equal to the amount T available

Divide each term by the common factor of 100

Subtract 2x from each side of the inequality

Divide each term of the inequality by 5

b Graph the solutions.

Create a strategy The boundary line may be expressed as 2x + 5y = 40. We can find the intercepts for this line by setting x = 0 to find the y-intercept and y = 0 to find the x-intercept. We can then decide which side of the boundary line to shade.

Apply the idea To ﬁnd the y-intercept set x = 0 2 × 0 + 5y = 40

Substitute x = 0

5y = 40

Simplify

y=8

Divide both sides by 5

To ﬁnd the x-intercept set y = 0 2x + 5 × 0 = 40

178

Substitute y = 0

2x = 40

Simplify

x = 20

Divide both sides by 2

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The boundary line is drawn between the two intercepts.

The boundary is solid as the relationship is ≤.

A possible solution to the inequality is (0, 0), (which equates to ﬁlling no bottles), so we need to shade the side of the line that contains the origin.

6 5 4 3 2 1

x 2 4 6 8 10 12 14 16 18 20

c Which combinations of bottles of sauce could Bernie ﬁll? Select all that apply. A

No 200 mL bottles and eight 500 mL bottles

Four 200 mL bottles and ten 500 mL bottles

Ten 200 mL bottles and four 500 mL bottles

Twenty 200 mL bottles and no 500 mL bottles

Create a strategy Try each set of coordinates in the inequality.

Apply the idea We can express the inequality as 2x + 5y ≤ 40 A: x = 0, y = 8. 2 × 0 + 5 × 8 = 40 Substitute x = 0, y = 8 This satisﬁes the inequality, so A is a possible solution. B: x = 4, y = 10. 2 × 4 + 5 × 10 = 58

Substitute x = 4, y = 10

This does not satisfy the inequality, so B is not a possible solution. C: x = 10, y = 4. 2 × 10 + 5 × 4 = 40

Substitute x = 10, y = 4

This satisﬁes the inequality, so C is a possible solution. D: x = 20, y = 0. 2 × 20 + 5 × 0 = 40

Substitute x = 20, y = 0

This satisﬁes the inequality, so D is a possible solution.

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Idea summary The following are features of two variable inequalities: • • • •

A line is solid if the symbols are ≥ or ≤. A line is dotted if the symbols are > or <. If the symbols are ≥ and >, then the shaded region must be above the line. If the symbols are ≤ and <, then the shaded region must be below the line.

Practice questions Choose your question path Explorer 1, 3−5, 6ab, 7a, 8, 10, 11, 12, 15, 16ai, 18, 19ab, 20ab, 21, 22, 26, 29, 33, 35

Adventurer 1−5, 6bc, 7b, 8, 10, 11, 13, 15, 16af, 18, 19ac, 20ac, 22, 23, 27, 30, 32, 34, 36

Trailblazer 1−6, 7c, 9, 11, 14, 16dhi, 17, 19ce, 20ce, 22, 24, 25, 28, 31, 34, 37−39

Understanding 1

The two entertainment options for the school holidays are movies and ice-skating. • Movies, m, cost $12 each • Ice-skating, s, costs $21 per session The inequality 12m + 21s ≤ 150 represents the total amount of money you can spend. What does the point (0, 0) on this graph signify?

180

The point (0, 0) signiﬁes that there is no money spent on movies or ice-skating sessions.

The point (0, 0) signiﬁes that you spent all your budget on movies and ice-skating sessions.

The point (0, 0) signiﬁes that you attended only movies.

The point (0, 0) signiﬁes that you attended only ice-skating sessions.

Are these statements true or false? a

If a point lies in the shaded region of the graph of an inequality, it is not a solution to the inequality.

The point (0, 0) lies in the solution region of the inequality x + y > 0.

If an inequality has < or > symbol, the graph of the inequality will have a dashed line.

In the context of budgeting, the inequality 10m + 20s ≤ 100 could represent spending on movies (m) and snacks (s) with a total budget not exceeding $100.

The inequality 3x − 2y > 6 and 3x − 2y ≥ 6 will have exactly the same graph.

Changing 2x + 3y ≤ 20 to 2x + 3y ≤ 30 will shift the line upwards on the graph.

The inequality x ≥ 0 represents all points to the right of the y-axis, including the y-axis itself.

When graphing an inequality, we shade the region that contains the solutions to the inequality.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

What does this shaded region mean in the context of a problem? A

The shaded region represents the line of the graph.

The shaded region represents all possible combinations of x and y-values that make the inequality false.

The shaded region represents all possible combinations of x and y-values that make the inequality true.

The shaded region looks nice and has no mathematical significance.

Is the inequality represented using a solid or dashed line on the graph? a

y ≤ 6x − 3

y < 5x − 7

y ≥ 3x − 2

y>8

y ≥ 6x − 2

y < −9

Is the shaded region above or below the line? a

y ≤ 5x − 3

y < −2x − 7

Fluency Ex 1

Is the coordinate pair a solution to the inequality? a c

(6, 20); y ≥ 4x + 5 b (3, 5); 2y ≤ x

(3, 2); 3x + 2y ≥ 12 (−1, 7); y ≥ 2x2 − 4x

Does the point satisfy the inequality? a

x − y > 10 i

(−2, −9)

(15, 20)

iii

(16, 7)

(10, −6)

(−2, −3)

iii

(0, 0)

(2, 7)

(−2, 4)

iii

(−4, 30)

(2, −13)

y > 3x i

(12, 20)

y > −8x − 3 i

(6, −50)

For each graph: i

Find the equation of the boundary line.

Write the inequality that represents the shaded region.

7 6 5 4 3 2 (0, 2) 1

−4 −3 −2 −1

−1 −2 −3 −4 −5 −6 −7

3 (−3, 2)

(3, 2)

x 2

−4 −3 −2 −1

−1

x 1

−2 −3 (−3, −4) −4

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7 6 5 4 3 (−2, 1) 2 1

6 x 3

−4 −3 −2 −1

(0, 7)

x 1

y 4 (0, 4)

−1

(0, 0) 1 2

x 3

−4 −3 −2 −1

−1

−2

−3

−4

(3, 0) x 1

Write the inequality that describes the shaded region: a

6 5 4 3 2 1

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

182

−2 −4 −6 −8 −10

y (−2, 4)

−4 −3 −2 −1

4 2

(−1, 3)

−4 −3 −2 −1 −1 1 2 −2 −3 −4 −5 (0, −5) −6 −7

10 8

x 1

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

x 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

6 5 4 3 2 1

7 y 6 5 4 3 2 1

x 1

−3 −2 −1−1

2 3

x 1

2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

The line y = x − 5 is shown. a

Graph the solution to y > x − 5.

Graph the solution to y ≤ x − 5.

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Ex 2

x 1

2 3 4 5

For 3x + 4y = 12: a

Find the x-intercept.

Find the y-intercept.

Graph the line on a cartesian plane.

Write a related inequality that describes the region where (−18, 15) is located.

For 5x + 2y = 10: a

Find the x-intercept.

Find the y-intercept.

Graph the inequality 5x + 2y > 10.

Do the points on the line satisfy the inequality 5x + 2y > 10?

For y = −3x + 4: a

Find the x-intercept.

Find the y-intercept.

Graph the inequality y ≤ −3x + 4.

Do these points satisfy the inequality y ≤ −3x + 4? i

(3, 4)

(4, −8)

iii

(5, −5)

(2, 2)

Do the points on the line satisfy the inequality y ≤ −3x + 4?

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183

For y = −5x − 10: a

Find the x-intercept.

Find the y-intercept.

Graph the inequality y < −5x − 10.

Do these points satisfy the inequality y < −5x − 10? i

(−8, 3)

Graph the boundary line.

Does the point (0, 0) satify the inequality?

iii

Graph the inequality.

y≥x+5

184

(4, −3)

(−7, −4)

y > 2x − 4

2x + 3y < 12

6x + 3y > 12

y > 3x − 5

y < 4x − 7

y ≥ −3x + 5

Graph the inequality: y ≤ 4x − 3

Ex 3

iii

For each inequality:

(−7, 4)

−4x + y < 5

g −3x + y < 7

−2x + y < 9

3x + y < 11

iii

x = −3, y = 1

x = 4, y = 5

Which points are a solution to each inequality? i

(3, 5)

3x − 2y > −6

(1, 0)

For each set of conditions: i

Find the gradient of the boundary line.

Write the equation of the boundary line in the form y = mx + c.

iii

Write the inequality that satisﬁes the given conditions.

A linear inequality has solutions of (0, −3) and (−2, −7) on its boundary line and is not satisﬁed by (1, 4).

A linear inequality has solutions of (0, −4) and (−4, −20) on its boundary line and is satisﬁed by (−2, −8).

By using a graph of the inequality, find the maximum value of y with the given conditions. a

2x + y ≤ 10, when x = 3

3x + y ≤ 9, when x = 2

0.5x + y ≤ 7, when x = 4

, when x = 3

, when x = 4

By using a graph of the inequality, find the minimum value of y with the given conditions. a

x − 2y ≤ 6, when x = 10

2x − 3y ≤ 12, when x = 8

0.5x − y ≤ 5, when x = 6

, when x = 5

, when x = 4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reasoning 21

Explain the differences between the graphs of: a

4x + 5y > 20 and 4x + 5y ≥ 20

4x + 5y ≤ 20 and 4x + 5y ≥ 20

Can the inequality x + y > 10 have integer solutions where both x and y are less than 3? Explain your reasoning.

Can the inequality x − y > 10 have integer solutions where both x and y are less than 3? Explain your reasoning.

A gardening club wants to invest in new tools and plants. They can buy a set of tools for $12 and a bundle of plants for $21. They have a budget of $150. a

Given the inequality 12t + 21p ≤ 150 (where t represents the number of tool sets and p the number of plant bundles), explain why the club can’t buy more than 12 tool sets or more than 7 plant bundles.

Suppose the club decides to buy 5 sets of tools. Explain why the maximum number of plant bundles they can afford is 4.

Suppose the club decides to buy an equal number of tool sets and plant bundles. What is the maximum number of each they can afford? Explain your reasoning.

Suppose the budget for the gardening club was raised to $180. They decide to buy an equal number of tool sets and plant bundles. What is the maximum number of each they can afford?

A local bakery is gearing up for a grand celebration. They have the option to prepare a batch of cookies at a cost of $15 or a batch of cupcakes at $25. The oven can only accommodate 300 units of heat. a

Given the inequality 15c + 25p ≤ 300 (where c represents the number of cookie batches and p the number of cupcake batches), explain why the bakery can’t bake more than 20 cookie batches or more than 12 cupcake batches.

If the bakery decides to prepare 10 batches of cookies, explain why they can only afford to bake 8 batches of cupcakes at the most.

If the bakery chooses to bake an equal number of batches of cookies and cupcakes, what is the maximum number of each that can be baked? Explain your reasoning.

Suppose the bakery’s oven capacity was increased to 360. They decide to bake an equal number of cookie and cupcake batches. What is the maximum number of each they can bake?

Problem-solving 25

Applicants for a particular job are asked to sit a numeracy test and verbal reasoning test. Successful applicants must obtain a minimum combined score of 43 for both tests. Write an inequality relating the applicant’s score on the numeracy test, x, and the verbal reasoning test, y.

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Athletes applying for a speciﬁc sports scholarship must pass both a physical and a written test. The minimum combined score for both tests is 50. Write and graph an inequality relating the athlete’s score on the physical test, x, and the written test, y.

Rochelle is being careful with her spending so she can later purchase a house. She has allocated no more than $855 each month for both travel expenses and eating out. On average, every time she eats out it costs $19, and every travel journey costs $9. Write an inequality relating the number of times she eats out, x, and the number of her travel journeys each month, y.

In an oﬃce building, an elevator has a maximum carrying capacity of 5800 kilograms. This is based on an average female’s weight of 68 kilograms and an average male’s weight of 77 kilograms. Write an inequality relating the number of women, x, and the number of men, y, who can enter the elevator at one time.

Throughout university, Luigi works as a mentor getting paid $20 per hour, and as a barista getting paid $25 per hour. The number of hours he works in each job can vary from week to week, and he needs to be able to at least cover his weekly expenses of $360. Write an inequality relating the number of hours he works as a mentor, x, and barista, y.

Edward is setting up a monthly budget to save for a new car. He plans to spend no more than $720 each month on groceries and utility bills. On average, each grocery bill is $30, and each utility bill is $60. Write and graph an inequality relating the number of utility bills, x, and the number of grocery bills each month, y.

A school bus has a maximum carrying capacity of 7500 kilograms. This is based on an average student’s weight of 55 kilograms and an average teacher’s weight of 70 kilograms. Write an inequality relating the number of students, s, and the number of teachers, t, who can board the bus at one time.

Ex 4

186

Roxanne has set aside $14.00 in her shopping budget for fruit this week. Currently, oranges are on sale for $1.40 each, while apples are on sale for $1.75 each. a

Write an inequality relating the number of oranges, x, and the number of apples, y, that Roxanne buys.

Graph the solution to the inequality.

Can Roxanne buy these combinations of fruit? i

6 oranges and 7 apples

3 oranges and 4 apples

iii

7 oranges and 4 apples

1.5 oranges and 2.1 apples

A book seller makes a proﬁt of $8 on every book sold online and $5 on every book sold in- store. The company wants to make a proﬁt of at least $240 a day selling books through online and in-store sales. a

Write an inequality relating the number of books sold online, x, the number of books sold in-store, y, and the target proﬁt.

Graph the solution to the inequality.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Will the combination of books sold in-store and online be enough for the book seller to meet the target proﬁt? i

15.1 online and 29.1 in-store

25 online and 5 in-store

iii

13 online and 23 in-store

20 online and 20 in-store

Using the graph, ﬁnd the minimum number of total books ( from both online and in-store) that need to be sold to reach the target proﬁt.

In the new basketball season, William is looking to beat his personal best of scoring 102 points in total for the whole season (this does not include points from fouls). a

If he is to score x ‘two pointers’ and y ‘three pointers’ throughout the new season, write an inequality relating x, y and the number of points he hopes to score.

Graph the solution to the inequality.

In which given scenarios will William beat his personal best for the season? A

He scores 17.6 ‘two pointers’ and 27.9 ‘three pointers’

He scores 23 ‘two pointers’ and 24 ‘three pointers’

He scores 4 ‘two pointers’ and 27 ‘three pointers’

He scores 31 ‘two pointers’ and 8 ‘three pointers’

Tom has several bottles of Fruita and Greatorade that he wants to take to a picnic, but he only has enough room in his backpack for 12 bottles. a

Write an inequality relating number of bottles of Fruita, x, and the number of bottles of Greatorade, y, that Tom takes to the picnic.

Graph the solution to the inequality.

Can Tom take this combination of bottles to the picnic? i

7 bottles of Fruita and 8 bottles of Greatorade

3 bottles of Fruita and 1 bottles of Greatorade

iii

3.8 bottles of Fruita and 3.3 bottles of Greatorade

6 bottles of Fruita and 8 bottles of Greatorade

Airline passengers are told to be at the gate no less than 15 minutes before a ﬂight’s departure. It takes Gwen 30 minutes to get to the airport, and y minutes to go through check-in and security, depending on how busy the airport is. a

Write an inequality relating x and y, where x represents the number of minutes left until Gwen’s ﬂight departs. Express your ﬁnal answer with y as the subject.

Graph the solution to the inequality.

If the check-in and security process take 20 minutes, how many minutes before her ﬂight’s departure can Gwen leave her house at the latest?

Oprah is thinking about how to use her last arcade tokens. A game of Table Tennis costs 2 tokens and a game of Frog Days costs 5 tokens. She has a total of 30 tokens left. a

Write an inequality relating the number of times Oprah plays Table Tennis, x and the number of times she plays Frog Days, y.

Construct a graph of the region containing the points corresponding to all the different ways Oprah could spend her remaining tokens across the two games.

Interpret the value of the x-intercept of the boundary line in the given context.

Interpret the value of the y-intercept of the boundary line in the given context.

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Sean is thinking about how to use his remaining spending money for snacks. A pack of dried fruit costs $6 and a bag of mixed nuts costs $4. He has $48 remaining. a

Write an inequality relating the number of packs of dried fruit, x and the number of bags of mixed nuts, y that Sean buys.

Sketch the region containing the points corresponding to all the different ways Sean could spend his remaining spending money on these snacks.

Supposing that Sean does not need to spend all of it, how many different ways can Sean spend his remaining money on some combination of dried fruit and mixed nuts?

Monica works part-time as a tutor and also as a sales associate. The number of hours she works in each role varies from week to week, and she needs to earn at least a certain amount weekly to cover her expenses. The graph of the inequality relating the number of hours she works as a tutor, x, and as a sales associate, y, is shown.

45 40 35 30 25 20 15

Write the rule for the inequality.

Find Monica’s hourly rate of pay for each job.

Find the amount of money she needs weekly to cover her expenses.

Complete the table of values: x y

b 2

T F

For the equation y = 4x + 7: a

−1

A R

Sketch the line on a Cartesian plane.

For the linear equation

x 5 10 15 20 25 30 35 40 45

Chapter 4 review 1

Find the gradient.

Find the y-intercept.

Use the gradient and y-intercept to sketch the line.

Sketch a line that passes through the point (−2, 7) and has a gradient of −2.

Sketch the graph of y = −3x + 8 by ﬁnding any two points on the line.

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Javier tracked the number of apps on his phone over a number of months. Month (M ) Apps (A)

1 118

Is this relationship linear?

The time in minutes and the temperature in degrees Celsius (°C) are given in the table: 1 8

2 11

3 14

4 17

5 20

By how much is the temperature increasing each minute?

Find the initial temperature.

Form an equation relating x and y.

Graph the linear relationship represented in the table.

T F

Rice costs a certain amount per kilogram. The table shows the cost of various amounts of rice in dollars: 0 0

10 32

20 64

30 96

A R

Complete the table.

Graph the linear relationship represented in the table.

Find the cost of rice per kilogram.

Calculate C (24), the cost of 24 kg of rice.

Find the amount of rice that costs $44.80.

Solve: a c e

g 9

4 96

Plot the points on the number plane.

Number of kilograms (x) Cost of rice C(x)

3 104

Time (x) Temperature (y)

2 106

d f h

Solve: a

e g

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189

Make the mentioned variables the subject in the following equations: a

W = Fd, make F the subject

Q = n2L, make n the subject

Rearrange the formula to make, r the subject of the formula.

Calculate the radius of a cylinder with a volume of 200 cm3 and a height of 10 cm. Round your answer to two decimal places.

Are these lines parallel to y = 3x + 5?

y = 3x + 2

3x + y = −5

x+5

y = −4x + 5

The graph shows the conversion between metres and feet:

A R

If a plank of wood is 10 ft in length, how long is it in metres?

In softball, the ﬁrst base is 60 ft from home plate. Calculate this length in metres.

x−2

y = 3x

4y = x

8 7

6 5 4

3 2 1

ft 2 4 6 8 10 12 14 16 18 20 22

Is the coordinate pair a solution to the inequality? (2, 2) ; x + 4y > 11

(−1, 6) ; y ≤ 3x2 − 2x + 1

For each graph, write the inequality that represents the shaded region. a

b 4

4 3

1 −4 −3 −2 −1 −1

x 1

−2

190

T F

a 16

y = −3x + 5

Are these lines perpendicular to y = −4x + 1? a

, make H the subject

The volume of a cylinder is given by the formula V = π r2h, where r is the radius of the base and h is the height of the cylinder.

a 13

S = I (1 + rt), make I the subject

1 −4 −3 −2 −1 −1 −2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

For 3x + 5y = 15: a

Find the x-intercept.

Find the y-intercept.

Graph the line on a Cartesian plane.

Write a related inequality that describes the region where (−12, 9) is located.

For y ≤ −2x + 2: a

Graph the inequality y ≤ −2x + 2.

Do these points satisfy the inequality y ≤ −2x + 2? i

c 19

(3, −6)

iii

(4, −2)

(1, 2)

Do the points on the line satisfy the inequality y ≤ −2x + 2?

Graph the inequality: a

(2, 3)

T F

2x − 3y < 15

A linear inequality has solutions of (0, −3) and (7, −31) on its boundary line and is satisﬁed by (3, −11). a

Find the gradient of the boundary line.

Write the equation of the boundary line in the form y = mx + b.

Write the inequality that satisﬁes the given conditions.

A R

Throughout university, Ben works as a waiter, getting paid $14 per hour, and as a valet, getting paid $18 per hour. The number of hours he works in each job can vary from week to week, and he needs to be able to at least cover his weekly expenses of $210.

Write an inequality relating the number of hours he works as a waiter, x, and valet, y.

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5 Quadratics Big ideas Quadratic equations can be represented graphically and solved through various methods, and their properties can be determined using the discriminant.

Curriculum Outcomes VC2M10A05 Substitute values into formulas to determine an unknown and rearrange formulas to solve for a particular term VC2M10A13 Solve simple quadratic equations using a range of strategies, including null factor law VC2M10AA02 Devise and use algorithms and simulations to solve mathematical problems VC2M10AA07 Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts VC2M10AA10 Experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06 5.07

Quadratic equations Quadratic graphs Solve by factorisation Solve by completing the square The quadratic formula The discriminant Optional: Further quadratic equations Chapter 5 review

194 202 224 235 250 256 268 280

Developers use quadratics to control the speed and physics of race cars in video games!

5.01 Quadratic equations After this lesson, you will be able to… • identify the key features of quadratic graphs and how they relate to the solutions of the equation. • recognise that quadratic equations can have zero, one, or two solutions. • apply the null factor law to solve quadratic equations.

Quadratic graphs The standard form of quadratic function is y = ax2 + bx + c, where a ≠ 0. For example, the quadratic function y = 2x2 − 3x + 5 has the values a = 2, b = −3, c = 5. The simplest quadratic function is y = x2. The graph of a quadratic function is called a parabola. This parabola is concave up as it opens upwards. The point (0, 0) is the minimum point on the curve and is called the turning point or vertex.

y 7 6 5 4

y = x2

3 2 1

−2

−1

x 1

−1

y 1 −2

−1

x 1

If the value of a is negative, such as y = −x2, the graph opens downwards. It is called concave down and the turning point is a maximum.

−2 2

y = −x −3 −4 −5 −6 −7

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There are many other key features including the intercepts and axis of symmetry. axis of symmetry

y x x-intercepts Maximum x

y-intercept

y-intercept Minimum

Concave up with two x-intercepts

axis of symmetry

Concave down with no x-intercepts

The number of solutions to a quadratic equation can be zero, one or two. This is related to the number of x-intercepts on the graph.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.01 to answer these questions. 1.

If a graph is concave up with two x-intercepts, what does this tell us about the value of a and the location of the turning point?

If a graph is concave down with two x-intercepts, what does this tell us about the value of a and the location of the turning point?

If a graph has one x-intercepts, what does this tell us about the value of a and the location of the turning point?

If a graph has no x-intercepts, what does this tell us about the value of a and the location of the turning point?

The graph of y = x2 − 4 has a = 1 and turning point at (0, −4). How many solutions would be related quadratic equation, x2 – 4 = 0 have?

Example 1 For the parabola deﬁned by the equation y = −x2 + 3: a Is the parabola concave up or concave down?

Create a strategy For a parabola in the form y = ax2 + bx + c, concavity depends on the x2 coeﬃcient a: positive for concave up, negative for concave down.

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Apply the idea Looking at the equation, the coeﬃcient of x2 is −1. So the parabola is concave down. b What is the y-value of the y-intercept of the parabola?

Create a strategy To ﬁnd the value of the y-intercept, set the x-value equal to 0.

Apply the idea y = −x2 + 3 =3

Rewrite the equation Substitute x = 0

Idea summary A quadratic function is of the form y = ax2 + bx + c, where a ≠ 0. The graph is called a parabola. If a > 0 the graph is concave up as it opens upwards. If a < 0 the graph is concave down as it opens downwards. The minimum or maximum point on the curve is called the turning point or vertex. The turning point and whether it is concave up or down can tell us about the number of solutions.

Quadratic equations Quadratic equations can be written in the form ax2 + bx + c = 0. Quadratic equations can come in many forms such as: x2 = 16, (x − 5)2 = 25, 2x2 + 5x = 17 Quadratic equations can have zero, one or two solutions. For example: • x2 = −4 has no solutions as a squared number is always positive. The related quadratic graph will have no x-intercepts. y

• x2 = 0 has one solution only, x = 0. The related quadratic graph will have one x-intercept. y

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• x2 = 9, has two solutions, x = 3 and x = −3. We can also write this as x = ±3. The related quadratic graph will have two x-intercepts. y

The symbol ± means “plus or minus”. We can use this as a shorthand for both the positive and negative of a number.

Example 2 Solve for x: a x2 = 16

Create a strategy To ﬁnd x, we need to reverse the square operation.

Apply the idea x = ±4

Take the square root of both sides

Reﬂect and check Remember to always use the plus or minus sign (±) when dealing with perfect square answers.

b 81x2 − 64 = 0

Create a strategy Isolate x and reverse the square operation.

Apply the idea Add 64 to both sides

Divide both sides by 81

Take the square root of both sides

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Create a strategy Isolate x using inverse operations.

Apply the idea Add 3 to both sides

Multiply both sides by 16

Take the square root of both sides

Example 3 Solve (7 − x)2 = 81.

Create a strategy We can directly reverse the square operation on both sides, without expanding the binomial square on the right-hand side.

Apply the idea 7 − x = ±9

Take the square root of both sides

We need to further simplify both equations 7 − x = 9 and 7 − x = −9. 7−x=9

Write the first equation

−x = 2

Subtract 7 from both sides

x = −2

Divide both sides by −1

7 − x = −9

Write the second equation

−x = −16

Subtract 7 from both sides

x = 16

Divide both sides by −1

Idea summary Equations where the highest power is two are quadratic equations. They can be written in the form ax2 + bx + c = 0 and can have zero, one or two solutions. If we can rearrange a quadratic equation into the form x2 = k, then we can solve the equation by taking the positive and negative square roots. That is, x = ± . The symbol ± means “plus or minus”.

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The null factor law The null factor law states that if a product of two or more factors is equal to 0, then at least one of those factors must be equal to 0. For example, if ab = 0 then either a = 0 or b = 0 We can use this to solve quadratic equations. If we can rearrange a quadratic equation into the form (x − p) (x − q) = 0 then we know that either x − p = 0 or x − q = 0. We can solve the quadratic equation by solving each of these linear equations.

Example 4 Solve: a (x − 6) (x + 7) = 0

Create a strategy Use the null factor law and set both factors to 0, then solve the resulting linear equations.

Apply the idea x−6=0 x=6

Set the ﬁrst factor to 0 Add 6 to both sides

x+7=0

Set the second factor to 0

x = −7

Subtract 7 from both sides

So the solutions are x = 6 and x = −7.

b 4m (m + 5) = 0

Create a strategy Use the null factor law and set both factors to 0, then solve the resulting linear equations.

Apply the idea 4m = 0

Set the ﬁrst factor to 0

m=0

Divide both sides by 4

m+5=0

Set the second factor to 0

m = −5

Subtract 5 from both sides

So the solutions are m = 0 and m = −5.

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Idea summary The null factor law states that if a product of two or more factors is equal to 0, then at least one of those factors must be equal to 0. If we can rearrange a quadratic equation into the form (x − p) (x − q) = 0 then we know that either x − p = 0 or x − q = 0. We can solve the quadratic equation by solving each of these linear equations.

Practice questions Choose your question path Explorer 1–6, 7bceg, 8ace, 9afgk, 10, 11, 13, 15, 16, 17

Adventurer 2–6, 7acfh, 8cdf, 9abkl, 12–14, 16–18

Trailblazer 2–6, 7eikl, 8cgh, 9bjkl, 12–14, 16, 18–20

Understanding 1

Given that we know 5 × 5 = 25. What is

How many solutions can a quadratic equation have?

A quadratic equation can be written in the form ax2 + bx + c = 0. Write the values of a, b and c for these quadratic equations:

2x2 − 7x + 5 = 0

−3x2 + 8x = 7

−x2 = 4x − 3

x = 16

How can you use the null factor law to solve a quadratic equation in the form (x − a) (x − b) = 0?

The equation x2 − 144 = 0 has a positive integer solution of x = 12. Find its other solution.

Fluency Ex 1

Are the graphs of these quadratic equations concave up or concave down? a

Ex 2

200

y = −x2 + 4x − 2

y = 3x2 + 8x

y = −x (x + 2) + 5

Solve: a

a2 = 2

k2 = 25

x2 = 121

t2 = 294

x − 121 = 0

15 = x − 10

36 = 25y2

−3k2 = −12

81k2 + 8 = 24

−16 = −81x2

4y = 100

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 3

Solve: a e

Ex 4

(z + 3)2 = 49 2

(m − 7) = 81

(−3 + p)2 = 64

(7 − x) = 81

(x − 6)2 = 2 2

(8l + 9) = 256

81 = (2 − n)2

10 ( p2 − 7) = 930

Solve: a c e

j (j + 7) = 0

x (2x − 9) = 0

(4k − 9)2 = 0

(e − 4) (e − 2) = 0

(10x − 9) = 0 (−3 + 7x) = 0

(s + 6) (s + 7) = 0

(8x − 5) (3x − 7) = 0

(3w + 8) (5w − 7) = 0

(3x − 17) (2x − d) = 0

−5m (m + 5) = 0

Reasoning 10

If a quadratic equation is given in the form x2 = k, how can we ﬁnd the solutions for x?

Explain why the quadratic equation x2 − 9 = 0 has two solutions.

A quadratic equation has two solutions, x = 4 and x = −3. Explain how to write the equation in the form (x − a) (x − b) = 0.

Jade used the null factor law to solve the equation (x − 4) (x + 6) = 0 and found the solutions x = −4 and x = 6. Is Jade’s answer correct? Explain why.

Explain why the equation x2 + 4 = 0 has no solutions.

Problem-solving 15

The equation 4x2 + kx + 16 = 0 has one solution: x = 2. Find the value of the coeﬃcient k.

The equation ax2 − 32x − 80 = 0 has one solution: x = 4. Find the value of the coeﬃcient a.

The Widget and Trinket Emporium has released the forecast of its revenue over then next year. The revenue R (in dollars) at any point in time t (in months) is described by the equation: R = −(t − 12)2 + 4 a

Is the graph of R concave up or concave down?

Does the revenue reach a maximum or minimum point?

When will the revenue be zero?

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Neville needs a sheet of paper x cm by 13 cm for an origami giraffe. The local origami supply store only sells square sheets of paper. The lower portion of the image below shows the excess area, A, of paper that will remain after Neville cuts out the x cm by 13 cm piece. The excess area, in square centimetres, is given by the equation: A = x (x − 13) a

At what lengths of x will the excess area be zero?

Does the graph of A have a minimum or maximum turning point?

For what value of x will Neville be able to make an origami giraffe with the least amount of excess paper?

The equation px2 + qx − 20 = 0 has solutions coeﬃcients p and q.

Given ax2 − c = 0: Solve for x, in terms of a and c.

For what values of a and c, would there be: No solutions

One solutions

and x = 4. Find the value of the

iii

Two solutions

5.02 Quadratic graphs After this lesson, you will be able to… • understand the different forms of quadratic equations and how they relate to the graph of the equation. • identify the turning point and interpret its significance in the graph of a quadratic function. • understand the transformations of a parabola, including translations, dilations and reflections. • calculate the y-intercept and x-intercepts of a parabola and understand their significance in sketching the graph. • apply the knowledge of quadratic graphs to solve problems.

Transformations of parabolas We can graph quadratic functions using transformations that can be identified from the equation. Let’s explore the possible transformations and how they are represented in the equation and graph.

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.02 to answer these questions. 1.

What does changing the value of a do?

What does changing the value of k do?

What does changing the value of h do?

How is reflecting the graph shown in the equation?

What values of a and k give no x-intercepts? One x-intercept? Two x-intercepts?

A parabola can be vertically translated by increasing or decreasing the y-values by a constant number. A translation of y = x2 up by k units gives the equation y = x2 + k. This graph shows y = x2 translated vertically up by 2 to get y = x2 + 2, and down by 2 to get y = x2 − 2.

y 4 y = x2 + 2

3 2

y = x2

x −4 −3 −2 −1

y = x2 − 2

−1 −2

Similarly, a parabola can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must be squared together. That is, to translate y = x2 to the left by h units we get y = (x + h)2. This graph shows y = x2 translated horizontally left by 2 to get y = (x + 2)2 and right by 2 to get y = (x − 2)2.

y 6 5 y = x2

4 3 y = (x + 2)2 2

y = (x − 2)2

1 −4 −3 −2 −1

−1

x 1

−2

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A parabola can be dilated by multiplying every y-value by a constant number greater than 1. So to expand the parabola y = x2 by a scale factor of a we get y = ax2. We can compress a parabola by using a scale factor between 0 and 1. This graph shows y = x2 vertically expanded by a scale factor of 2 to get y = 2x2 and compressed by

y 6 5 y = 2x2 4

a scale factor of 2 to get

y = x2

3 2

1 −4 −3 −2 −1

x2 2

x 1

−1

−2

Finally, we can reflect a parabola across the x-axis by multiplying by −1. So to reflect y = x2 across the x-axis we get y = −x2. Notice that reflecting will change the concavity (in this case from concave up to concave down).

y 4 3 2 y = x2

1 −4 −3 −2 −1

−1

x 3

−2 −3 −4

y = −x2

Example 1 For the parabola y = −(x + 2)2 − 5: a Which successive transformations turn y = x2 into the equation y = −(x + 2)2 − 5? A

Reﬂection across the x-axis, horizontal shift 2 units to the left and vertical shift 5 units down.

Horizontal shift 2 units to the right, vertical shift 5 units down, reﬂection across the x-axis.

Reﬂection across the x-axis, vertical shift 2 units down and horizontal shift 5 units to the left.

Vertical shift 2 units up, horizontal shift 5 units to the left, reﬂection across the x-axis.

Create a strategy Look at which transformation each part of the equation represents.

Apply the idea y = (x + 2)2 shifts y = x2, 2 units to the left. y = −(x + 2)2 reﬂects y = (x + 2)2 about the x-axis.

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y = −(x + 2)2 − 5 shifts y = −(x + 2)2 down 5 units. So the answer is option A.

b State the coordinates of the turning point of the curve.

Create a strategy We can use the transformation from part (a).

Apply the idea y 2 1 −4 −3 −2 −1

x 1

−1

−2 −3

The turning point of y = −(x + 2)2 − 5 is the turning point of y = x2 reﬂected across the x-axis, horizontally shifted 2 units to the left and shifted vertically 5 units down. As we can see in the graph, the coordinates of the turning point of the curve are: (0 − 2, 0 − 5) = (−2, −5).

−4 −5 −6

c Plot the graph of the equation.

Create a strategy Use the transformations and turning point we found in the previous parts.

Apply the idea 3 2 1

−4 −3 −2 −1 1 2 3 4 −1 −2 −3 −4 −5 −6 −7 −8 y = −(x + 2)2 − 5

Starting with the top dashed curve, y = x2, we need to reﬂect the curve across the x-axis, shift the curve 2 units to the left, then shift this curve down 5 units to get the ﬁnal parabola.

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Idea summary The parabola y = x2 can be transformed by: •

Vertical translation by k units: y = x2 + k • If k > 0, like y = x2 + 3, then it is translated upwards. • If k < 0, like y = x2 − 1, then it is translated downwards.

•

Horizontal translation by h units: y = (x − h)2 • If h > 0, like y = (x − 2)2, then it is translated to the right. • If h < 0, like y = (x + 3), then it is translated to the left.

•

Dilated by a scale factor of a: y = ax2 • If a > 1, or a < −1, like y = 2x2, then it is stretched. • If −1 < a < 1 (with a ≠ 0), like y = 0.5x2, then it is compressed.

•

Vertically reﬂected across the x-axis: y = −x2

All of these transformations can be performed on a quadratic and are represented by the equation y = a (x − h)2 + k.

Quadratics in turning point form A quadratic function in turning point form is written as y = a(x − h)2 + k, where a, h, k are numbers. We can find the key features and then use these to graph the parabola. The turning point is (h, k). The axis of symmetry is x = h. To find the y-intercept, substitute x = 0 and evaluate. To find the x-intercepts, if required, substitute y = 0 and solve the equation using inverse operations.

Example 2 For the quadratic function y = −2 (x − 3)2 + 8: a Is the parabola concave up or down?

Create a strategy Look at the sign of a.

Apply the idea Since the quadratic function is y = −2 (x − 3)2 + 8 and a = −2 < 0, then the parabola is concave down.

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b Is the parabola more or less steep than y = x2?

Create a strategy Look at the value of a. Use the fact that as the value of a gets further from 0, the parabola becomes more steep and narrow.

Apply the idea Since we have that a = −2, and −2 is further away from zero than 1 is, then the parabola is more steep than y = x2.

c Find the coordinates of the turning point.

Create a strategy Use the fact that turning point of the quadratic of the form y = a(x − h)2 + k is (h, k).

Apply the idea The turning point of the quadratic function y = −2 (x − 3)2 + 8 is at (3, 8).

d Find the y-intercept.

Create a strategy Substitute x = 0 into the equation and then solve for y.

Apply the idea y = −2(x − 3)2 + 8 Write the equation = −2(0 − 3)2 + 8 Substitute x = 0 = −2(9) + 8

Evaluate the power

= −10

Evaluate

The y-intercept is at (0, −10).

e Sketch the graph of this function. Include the axis of symmetry.

Create a strategy Plot the intercepts and turning point and draw a curve through the points.

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Apply the idea Since the axis of symmetry occurs at x = h, then the axis of symmetry for y = −2(x − 3)2 + 8 is x = 3. 10 8 6 4 2

y (3, 8)

1 2 3 4 5 6 7 8 9 −2 −1 −2 −4 −6 −8 −10 (0, −10)

Reﬂect and check Graphing using as many key features as possible helps us to identify any errors we may have made when ﬁnding the key features.

Example 3 A parabola has the equation y = x2 + 4x − 1: a Express the equation of the parabola in the form y = (x − h)2 + k by completing the square.

Create a strategy Add and subtract

into the expression and factorise the perfect square part.

Apply the idea For x2 + 4x – 1 we need to add and subtract x2 + 4x − 1 = x2 + 4x + 4 − 4 – 1

Adding and subtracting 4

Factorise the perfect square part

Simplify the constant

= (x + 2) − 4 – 1 = (x + 2) – 5

b Find the y-intercept of the curve.

Create a strategy Substitute x = 0 into the equation of the parabola.

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Apply the idea y = (0)2 + 4(0) – 1

Substitute 0 to x

y = −1 Evaluate

Reﬂect and check We can check that we have correctly completed the square by ﬁnding the y-intercept using that form as well. y = (x + 2)2 – 5 2

Write the equation

y = (0 + 2) – 5

Substitute x = 0

y = 4 – 5

Evaluate the brackets and indices

y = −1 Evaluate the subtraction This is the same as when found using the standard form, so means completing the square was likely done correctly.

c Find the turning point of the parabola.

Create a strategy We can using the turning point form found in part (a).

Apply the idea The equation in turning point form is: y = (x + 2)2 − 5. We can write this as y = (x − (−2))2 − 5. The coordinates of the turning point are (−2, −5).

Reﬂect and check If we hadn’t found the equation in turning point form, we could have used the idea that the axis of .

symmetry is at

From the equation, y = x2 + 4x − 1, a = 1, b = 4. Substitute a and b

Evaluate

The equation of the axis of symmetry is x = −2, which is also the x-coordinate of the turning point. y = x2 + 4x – 1 2

y = (−2) + 4(−2) – 1

Write the equation Substitute x = −2

y = −5 Evaluate

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d Is the parabola concave up or down? A

Concave up

Concave down

Create a strategy Look at the coeﬃcient of x2.

Apply the idea A positive coefficient of x2 means the parabola is concave up while a negative coefficient of x2 means the parabola is concave down. Since there is a positive coefficient of x2 in the equation of the parabola y = x2 + 4x − 1, then the parabola is concave up. So, the correct answer is Option A.

e Plot the curve y = x2 + 4x − 1.

Create a strategy Plot the intercepts and turning point and draw a curve through the points.

Apply the idea 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4

Idea summary If an equation of a parabola can be factorised into the form y = a (x − h)2 + k then: • • •

The turning point form will be (h, k). The axis of symmetry will be x = h. The y-intercept, can be found by setting x = 0 and evaluating to get (0, a(h)2 + k).

These key features can be used to plot the parabola.

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Quadratics in factorised form A parabola will always have one y-intercept and can have zero, one or two x-intercepts. A quadratic function in factorised is written as y = a (x − p) (x − q). We can graph a quadratic in factorised form using these steps: 1. Find the y-intercept by substituting x = 0, this gives us y = apq. 2. Find the x-intercepts by setting y = 0 and using the null factor law. 0 = a (x − p) (x − q), so that x = p or x = q, and the two x-intercepts are ( p, 0) and (q, 0). 3. Find the axis of symmetry by ﬁnding the average of the two intercepts, 4. The x-value of the turning point is the axis of symmetry, so to ﬁnd the y-value, we can substitute the x-value into the original equation and evaluate for y. If the equation of a parabola factorises into the form y = a (x − p) (x − q) then it is possible to use these key features to plot the parabola.

Example 4 For the equation y = (2 − x) (x + 4): a State the y-value of the y-intercept.

Create a strategy Substitute x = 0 into the equation and then solve for y.

Apply the idea y = (2 − x) (x + 4) Write the equation y = (2 − 0) (0 + 4) Substitute x = 0 =8

Evaluate the product

b Determine the x-values of the x-intercepts.

Create a strategy Substitute y = 0 into the equation and then solve for x.

Apply the idea y = (2 − x) (x + 4)

Write the equation

0 = (2 − x) (x + 4)

Substitute y = 0

2 − x = 0 Equate the ﬁrst factor to 0 x = 2 Add x to both sides x+4=0 x = −4

Equate the second factor to 0 Subtract 4 from both sides

So the x-intercepts are x = 2, x = −4.

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c Determine the coordinates of the turning point of the parabola.

Create a strategy The x-coordinate of the turning point is half-way between the x-intercepts.

Apply the idea Find the average of the x-intercepts

Evaluate Substitute x = −1 into the equation Evaluate The coordinates of the turning point are (−1, 9).

d Plot the graph of the parabola.

Create a strategy Plot the intercepts and the turning point and draw a curve through the points.

Apply the idea y

9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

x 1

−1 −2 −3 −4

Idea summary If an equation of a parabola can be factorised into the form y = a (x − p) (x − q) then: • •

The y-intercept will be (0, apq). The x-intercepts will be ( p, 0) and (q, 0).

•

The x-value of the turning point is

. The y-value of the turning point will be

found by substituting in the x-value turning point into the equation. These points can be used to plot the parabola.

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Quadratics in standard form A quadratic function in standard form is written as ax2 + bx + c = 0, where a, b, and c are integers and a ≠ 0. If a > 0 the graph is concave up and the turning point is a minimum. If a < 0 the graph is concave down and the turning point is a maximum. To graph the parabola we can: 1. Find the axis of symmetry using

2. Find the turning point by substituting in the axis of symmetry for x and evaluating for y. 3. Find the y-intercept by setting x = 0 and evaluating to get y = c. If it is able to be factorised, we can use the null factor law to ﬁnd the x-intercepts. The turning point and y-intercept can provide a sketch of the graph. If other points are required they can be found by substituting x-values into the equation.

Example 5 For the equation y = 4x + x2: a Factorise the expression 4x + x2.

Create a strategy Take out the common factor of the two terms.

Apply the idea The common factor of the two terms is x. 4x + x2 = x(4 + x)

b Solve for the x-values of the x-intercepts of y = 4x + x2.

Create a strategy Substitute y = 0 into the equation and then solve for x.

Apply the idea y = 4x + x2 Write the equation y = x (4 + x) Factorise the quadratic equation 0 = x (4 + x) Substitute y = 0 x = 0 Equate the ﬁrst factor to 0 4+x=0

Equate the second factor to 0

x = −4 Subtract 4 from both sides So the x-intercepts are x = 0, x = −4.

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c Determine the y-value of the y-intercept of the graph.

Create a strategy Substitute x = 0 into the equation then solve for y.

Apply the idea y = 4x + x2

Write the equation 2

=4×0+0

Substitute x = 0

Evaluate

d Find the coordinates of the turning point.

Create a strategy and substitute the value of x into the equation of the parabola.

Use the equation

Apply the idea From the equation, y = 4x + x2, a = 1, b = 4. Substitute a and b Evaluate The equation of the axis of symmetry is x = −2, which is also the x-coordinate of the turning point. Write the equation y = 4x + x2 = 4(−2) + (−2)2 Substitute x = −2 = −4

Evaluate

The coordinates of the turning point are (−2, −4).

Reﬂect and check We could also have used the average of the x-intercepts to ﬁnd that

e Plot the graph of the parabola.

Create a strategy Plot the intercepts and turning point and draw a curve through the points.

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Apply the idea y 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Example 6 For the quadratic y = 3x2 − 6x + 8: a Find the axis of symmetry.

Create a strategy Use the formula

Apply the idea From the equation y = 3x2 − 6x + 8, we have a = 3 and b = −6. Substitute the values of a and b

Evaluate the numerator and denominator

Simplify the fraction

b Find the turning point of the parabola. Give your answer in the form (a, b).

Create a strategy Substitute x = 1 into y = 3x2 − 6x + 8 to ﬁnd the y-coordinate of the turning point.

Apply the idea y = 3(1)2 − 6(1) + 8 =5

Substitute 1 to x Evaluate

So, the turning point of the parabola is (1, 5).

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c What is the y-intercept of this quadratic function?

Create a strategy Substitute x = 0 into y = 3x2 − 6x + 8 to ﬁnd the y-intercept.

Apply the idea y = 3(0)2 − 6(0) + 8

Substitute 0 to x

Evaluate

So, the y-intercept is (0, 8).

d Plot the graph of the equation.

Create a strategy Plot the y-intercept and the turning point and draw a curve through the points.

Apply the idea y 11 10 9 8 (0, 8) 7 6 5 (1, 5) 4 3 2 1 −4 −3 −2 −1

−1

x 3

Idea summary A quadratic function in standard form is written as y = ax2 + bx + c, where a, b and c are integers, and a ≠ 0. To graph the parabola we can: 1. Find the axis of symmetry using

2. Find the turning point by substituting in the axis of symmetry for x and evaluating for y. 3. Find the y-intercept by setting x = 0 and evaluate to get y = c. If a > 0 the graph is concave up and the turning point is a minimum. If a < 0 the graph is concave down and the turning point is a maximum. If it is able to be factorised, we can use the null factor law to ﬁnd the x-intercepts.

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Practice questions Choose your question path Explorer 1–4, 5ab, 6ac, 7a, 8a-c, 9bd, 10ab, 11, 12, 13ab, 14a, 15aeg, 16ac, 17, 20, 23, 25–27

Adventurer 1–4, 5ac, 6ac, 7ab, 8ace, 9de, 10bc, 11, 12, 13bc, 14a, 15dh, 16bf, 17, 19, 20, 23, 25–28

Trailblazer 1–4, 5ad, 6bd, 7cd, 8d-f, 9ef, 10cd, 11, 12, 13cd, 14, 15j-l, 16dh, 18, 21, 22, 26–30

Understanding 1

Describe the basic shape of a parabola.

Given that a general quadratic is y = ax2 + bx + c: i

What does the sign of a tell us?

What does changing the value of c do?

Complete the following statement and ﬁll in the key features of a parabolic curve. The graph is concave ⬚.

y 10 8 6 4 2 –2

x 1

What feature is at the point (h, k) on the parabola deﬁned by y = a (x − h)2 + k?

What features are at the points ( p, 0) and (q, 0) on the parabola defined by y = a (x − p) (x − q)?

Fluency Ex 1

For each equation: i

Describe the transformations from y = x2.

What is the turning point?

iii

Sketch the graph of the parabola.

y = (x − 3)2 − 4

y = 3 (x + 1) + 4

y = −(x + 3)2 − 6

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Ex 2

For each quadratic function: i

Is the parabola concave up or down?

Is the parabola more or less steep than the parabola y = x2?

iii

Find the coordinates of the turning point.

Find the y-intercept.

Sketch the graph of this function. Include the axis of symmetry.

g (x) = 2 (x − 1)2 − 3

c Ex 3

g (x) = −3 (x + 5) − 4

g (x) = 3 (x + 1)2 + 2

y = −x2 + 4x − 3

y = x2 + x + 3

For each quadratic equation: i

Write in the form y = a (x − h)2 + k.

Find the coordinates of the turning point.

iii

Sketch the graph of the parabola.

y = x2 + 6x

c 8

y = −x − 8x − 18

Each of the graphs has an equation of the form, y = a(x − h)2 + k. Find the values of a, h and k for each graph and write the equation. a

2 −3

−2

−1

−2

x 1

−2

−1

y 12

−2

x 1

−4

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−2

2 −3

−1

−4

6 4 2

−6

−5 −4 −3 −2 −1 −2

−8

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

f 5

1 −5 −4 −3 −2 −1

−1

x 1

3 −3 −2 −1

−2

Ex 4

x 1

−1

−3

−2

−4

−3

For each quadratic function: i

Find the y-intercept.

Find the x-intercepts.

iii

Find the axis of symmetry.

Find the coordinates of the turning point.

Sketch the graph of the parabola.

f (x) = x (x + 4)

f (x) = x (4 − x)

f (x) = (x − 2) (x − 6)

f (x) = (x − 3) (x + 1)

f (x) = (5 − x) (x + 1)

f (x) = (1 − x) (3 − x)

A parabola has x-intercepts at x = 1 and x = −7. State the equation of the parabola in the form y = a (x − p) (x − q) for each of these cases, and make a sketch of the curve. a

The turning point is a minimum turning point and the y-intercept is at (0, −7)

The turning point is a maximum turning point and the y-intercept is at (0, 14)

The turning point is a minimum turning point and the y-intercept is at (0, −21)

The turning point is a maximum turning point and the y-intercept is at (0, 3.5)

The parabola y = x2 has been translated 2 units to the right and 6 units down. a

Find the equation of the parabola.

Sketch the graph of the parabola.

For the graph: a b

Find the equation of the parabola in factorised form.

What is the turning point?

12 8 6 4 2 −2 −1 −2

x 1

−4

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These parabolas each have x-intercepts at x = ± . What is the equation of each parabola in the form y = a (x − p) (x − q) and y = a (x − h)2 + k? a

−4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

Ex 5

−4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

9 8 7 6 5 4 3 2 1

x 1

−4 −3 −2 −1−1 −2 −3 −4 −5

For each quadratic function: i

Factorise to ﬁnd the x-intercepts.

Express the equation in the form y = a (x − h)2 + k by completing the square.

iii

Find the coordinates of the turning point.

Sketch the graph of the function.

y = x2 − 12x + 32

y = −2x2 − 8x + 10

For each quadratic: i

Describe the concavity of the parabola.

Factorise, and ﬁnd the x-intercepts.

iii

Find the y-intercept.

Find the axis of symmetry.

Find the coordinates of the turning vi point.

Sketch the graph and identify these key features on it.

y = x2 − 4x + 3

y = x2 − 36

y = 4 − x2

y = x2 + 4x + 3

y = x + 6x + 5

y = x2 − 8x + 15

y = −x2 + 2x + 24

y = −x2 − 2x + 8

y = x2 − 2x − 8

y = 8 + 2x − x2

y = 2x2 + 9x + 9

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7 6 5 4 3 2 1

−1 −2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1

x 1

−4 −3 −2 −1

7 6 5 4 3 2 1

y = x + 2x − 3

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 6

For each quadratic: i

What is the axis of symmetry?

What are the coordinates of the turning point?

iii

What is the y-intercept?

Sketch a graph of the function.

y = x2 − 4x + 1 2

y = x2 + 6x + 7

y = −4x − 4x + 1

y = −(x2 + 8x + 11)

y = x2 + 6x − 2

y = −x2 + 10x − 23

y = −x2 + 8x − 9

y = x2 + 2x − 11

Reasoning 17

Three different ways of expressing a quadratic equation are: y = a (x − p) (x − q), y = a (x − h)2 + k, and y = ax2 + bx + c. How can you ﬁnd the y-intercept for each form? Explain which form you would use to identify the y-intercept if you had all three available.

Given a quadratic of the form a (x − h)2 + k, what are the conditions on a, h, and k, so that: a

The turning point is a minimum turning point and the parabola has two x-intercepts?

The turning point is a maximum turning point and the parabola has one x-intercept?

The turning point is a minimum turning point and the parabola has no x-intercepts?

The turning point of a parabola is located at (2, 5). a

Determine the equation of the parabola if the y-intercept is at (0, 9).

The parabola is translated 2 units to the right and 7 units down. Determine the equation of the transformed parabola.

Explain the change that would be made to the equation in part (a) if the graph is translated a units to the right and b units upwards.

Determine whether each statement is true or false. Justify your answer. a

The turning point of a quadratic graph is always located at the origin (0, 0).

If the coeﬃcient of the x2 term in a quadratic equation is negative, the graph will open upwards.

All quadratic graphs have a line of symmetry.

The y-intercept of a quadratic graph is always equal to the constant term in the quadratic equation.

The roots of a quadratic equation can be determined by ﬁnding the x-intercepts of the corresponding quadratic graph.

For the functions: f (x) = (x − 2) (x − 4) and g(x) = 2 (x − 2) (x − 4) a

Which key features are the same for f (x) and g (x)?

Which key features are different for f (x) and g (x)?

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Marvin wants to understand how vertex form and standard form of a quadratic function are related. Consider the vertex form: y = a(x − h)2 + k a

Expand a(x − h)2 + k and write the equivalent equation to y = a(x − h)2 + k in standard form.

Determine the value of b in the equation from part (a).

Determine the value of c in the equation from part (a).

Explain how the equations are related.

Describe the relationship between the turning point of a quadratic graph and the coeﬃcients a and b of y = ax2 + bx + c.

Explain what possible differences there can be, if: a

Two different quadratic functions have the same intercepts.

Three different quadratic functions have the same turning point and share one common intercept.

Problem-solving 25

The height h, in metres, reached by a ball thrown in the air after t seconds is given by the equation: h = 10t − t2 a

Complete the table of values for h = 10t − t2: t

Graph the relationship h = 10t − t2.

Find the height of the ball after 5.5 seconds have passed.

Find the maximum height reached by the ball.

When an object is thrown into the air, its height above the ground is given by the equation: h = 163 + 38s − s2 where s is its horizontal distance from where it was thrown.

222

Graph the relationship h = 163 + 38s − s2.

Find s, how far horizontally the object is from where it was thrown at the point when it reaches its greatest height above the ground.

Find the maximum height reached by the object.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Over the summer, Susana and her friends build a bike ramp to launch themselves into the local lake. Susana decides that the shape of the ramp will be parabolic, and thinks that the parabola is given by the equation:

1 ( x + 2)2 + 2 4 x

where y is the height in metres above the ground, and x is −6 the horizontal distance in metres from the edge of the lake.

If the ramp starts 6 m back from the edge of the lake, how high is the start of the ramp?

At what height will the rider leave the ramp?

At what other distance x is the rider also at a height of 3 m?

Graph the shape of the ramp on a Cartesian plane.

The main cable on a suspension bridge is approximately parabolic in shape. A point (x, y) on the parabola represents the height, y m, above the roadway x m from the leftmost supporting pole. The lowest point of the main cable is 10 m above the roadway and occurs 150 m from the leftmost post. The main cable meets the supporting poles 90 m above the roadway. Determine an equation for the parabola formed by the main cable. y

(150, 10) x

Elise is the owner of a restaurant. Elise wants to install new wooden ﬂoors in several rooms. The rooms in the restaurant are square. The wood costs $6.25 per square metre. The cost of the ﬂooring in terms of its side length is shown by the quadratic function C (x) = 6.25x2. a

Determine how much Elise should expect to spend on ﬂooring if the room has side lengths of 12 m.

Determine how much the price would change if the side lengths decreased by 3 m.

Graph the quadratic model from part (a).

Describe what changes and what stays the same about the graph of the quadratic model if the cost per square metre decreases.

A quadratic function has the factorised form equation y = a (x − m) (x − 12). If the function has a turning point at the point (8, 15), show how we can ﬁnd the values of a and m.

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5.03 Solve by factorisation After this lesson, you will be able to… • solve quadratic equations by factorisation. • identify x-intercepts on a parabola and use them to solve a quadratic equation. • find the equation of a graph by identifying the intercepts and using them to write an equation.

Solve by factorisation We have seen how to solve quadratic equations using the null factor law. We can use these steps to solve factorisable quadratic equations: 1. Gather together all terms to one side of the equation and make it equal to 0, in the form ax2 + bx + c = 0. 2. Factorise the quadratic using any appropriate techniques, such as: • Finding the highest common factor • Using patterns like a2 − b2 = (a − b) (a + b) and a2 + 2ab + b2 = (a + b)2 • Splitting the middle term and using grouping • For x2 + bx + c, ﬁnding two numbers whose sum is b and whose product is c • Using the cross product method 3. Use the null factor law to split the quadratic equation into two linear equations and make each linear equation equal to 0. 4. Solve the linear equations to get the solutions to the original quadratic equation. 5. Check the solutions by substituting back into the original equation.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.03 to answer these questions. 1.

How are the x-intercepts related to the factorised equation?

If the factorised form exists, how many x-intercepts can the graph have?

If the factorised form does not exists, how many x-intercepts can the graph have?

Is is possible for a non-factorisable quadratic to have two solutions? Explain.

Is is possible for a factorisable quadratic to have no solutions? Explain.

What do you notice is common between the standard form and factorised form?

How can you use the graph to find the equation of a quadratic?

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Example 1 Solve the quadratic equations: a 3y − 15y2 = 0, which has two solutions. Solve for both values of y.

Create a strategy Factorise 3y − 15y2 and use the null factor law to solve.

Apply the idea 3y − 15y2 = 0 Write the equation 3y(1 − 5y) = 0 Factorise the quadratic expression 3y = 0 Set the ﬁrst factor equal to 0 y = 0 Divide both sides by 3 1 − 5y = 0 Set the second factor equal to 0 5y = 1 Add 1 to both sides Divide both sides by 5 So the solutions are y = 0 and

Reﬂect and check We can substitute y = 0 and

to check that these are the solutions of 3y − 15y2 = 0.

For y = 0: LHS = 3(0) − 15(0)2 =0

Substitute 0 for y Evaluate

= RHS For

: Substitute

for y

Evaluate the power Simplify

Evaluate the subtraction

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b x2 − 22x + 121 = 0

Create a strategy Factorise x2 − 22x + 121 and use the null factor law to solve.

Apply the idea x2 − 22x + 121 = 0

Write the equation

(x − 11) (x − 11) = 0

Factorise the quadratic expression

x − 11 = 0 x = 11

Set one of the factors equal to 0 Add 11 to both sides

Reﬂect and check Notice that x2 − 22x + 121 is in the form a2 + 2ab + b2, where a = x and b = −11. x2 − 22x + 121 = 0 2

Write the equation

Use the pattern

Factorise

(x) + 2 (x) (−11) + (−11) = 0 (x − 11) = 0 x − 11 = 0 x = 11

Set one of the factors equal to 0 Add 11 to both sides

Example 2 Solve x2 = 17x + 60 for x.

Create a strategy Move all terms to one side then factorise and use the null factor law to solve.

Apply the idea x2 = 17x + 60 2

Write the equation

x − 17x − 60 = 0

Move all the terms to one side

(x − 20) (x + 3) = 0

Factorise the quadratic expression

x − 20 = 0 x = 20 x + 3 = 0 x = −3

Set the ﬁrst factor equal to 0 Add 20 to both sides Set the second factor equal to 0 Subtract 3 from both sides

So the solutions are x = 20 and x = −3.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary We can solve most quadratic equations using these steps: 1. Gather together all terms to one side of the equation and make it equal to 0, in the form ax2 + bx + c = 0. 2. Factorise the quadratic expression using any appropriate techniques. 3. Use the null factor law to split the quadratic equation into two linear equations and make each linear equation equal to 0. 4. Solve the linear equations to get the solutions to the original quadratic equation. 5. Check the solutions by substituting back into the original equation.

Equations from graphs As well as drawing graphs from quadratic equations, we can also find the equation from the graph of a parabola. We have previously done this using the turning point, but it is not always easy to identify the coordinates of the turning point from the graph. Sometimes we will know other features, such as the x-intercepts, but not the turning point. To find the equation of a quadratic function, we can follow these steps: 1. Identify the x-values of the x-intercepts. These occur at the point where y = 0. 2. Substitute the x-values found in Step 1 for p and q in the equation y = a (x − p) (x − q). 3. Find the value of a by using another known point or by using second differences. Recall The first difference is the difference between consecutive y-values. The second difference is the difference between consecutive first differences. The second difference of any quadratic relationship is a non-zero constant. We can identify the first and second differences from a table of values or from a graph. 11 10 9 8 7 6 5 4 3 2 1 −4 −3 −2 −1

For our basic parabola, y = x2, where a = 1, the first differences are:

1-0=1 4-1=3 9-5=5

The second differences are: 3-1=2 3

5-3=2 x

1 1

In general,

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Example 3 Write an equation for the quadratic relation shown by the graph. y 4 3 2 1 x −4 −3 −2 −1

−1 −2

Create a strategy For this graph, we can easily read the coordinates of the x-intercepts, so we can write an equation using the factorised form of a quadratic function given by: y = a (x − p) (x − q) where p and q are the x-values of the x-intercepts.

Apply the idea The coordinates of the x-intercepts are (−1, 0) and (2, 0), so p = −1 and q = 2. To ﬁnd the value of a, we can use one of the points that passes through the curve of the parabola, such as the y-intercept, (0, −1). Write the factorised form of a quadratic equation Substitute p = −1 and q = 2 Substitute x = 0 and y = 1

Evaluate the brackets

Divide both sides by −2

Simplify The quadratic equation of the graph is:

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Reflect and check Instead of substituting a point to find the value of a, we can also look at the second differences. For this graph, the first differences are 1, 2, 3, …, so the second differences are all 1.

y 4

3 2 2

x −4 −3 −2 −1

−1

−2

Alternatively, we could have use the coordinates of points on the graph to create a table of values, then calculate the ﬁrst and second differences. x

−1

First Difference −1 − (−1) = 0 0 − (−1) = 1 2−0=2 5−2=3

Second Difference 1−0=1 2−1=1 3−2=1

Idea summary To ﬁnd the equation of a quadratic function, we can use these steps: 1. Identify the x-intercepts from the graph or otherwise. 2. Substitute the values for p and q found in Step 1 into y = a (x − p) (x − q). 3. Find the value of a by using another known point or by using second differences.

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Practice questions Choose your question path Explorer 1, 2abgh, 3, 4, 5ab, 6aceg, 7acegikmo, 8afgm, 9acg, 10c-e, 12, 13, 17a-c,e, 18, 19, 21

Adventurer 1, 2bdfh, 4, 5cd, 6bdfh, 7bdfjn, 8bdhn, 9cfh, 10bhkm, 11a, 12, 13, 17bdfh, 18, 19, 21

Trailblazer 2cde, 4, 5ad, 6fh, 7admp, 8bdfgh, 9hkn, 10hjmor, 11cd, 14–17egij, 18–22

Understanding 1

Which two options are solutions to the equation x (x − 7) = −10? A

230

x=5

x = −5

x=1

x=2

Find possible values of a and b: a

a + b = 5, ab = 6

a + b = −11, ab = 18

a + b = −1, ab = −2

a + b = −7, ab = 12

a + b = 0, ab = −9

a + b = 2, ab = −15

a + b = 8, ab = 16

a + b = −6, ab = 9

To factorise x2 + 11x + 24, we need to find two numbers: a

What should their product be?

What should their sum be?

What are the two numbers?

Factorise x2 + 11x + 24

One expression for the area of the rectangle below is m2 + 14m + 45. The rectangle is made up of four smaller rectangles. Use the diagram to express the area of the large rectangle in factorised form. m

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For the given graphs, ﬁnd the solution(s) to the equation. a

(x + 2) (x − 3) = 0

(x − 1) (x + 4) = 0

2 x

−6

−4

−2

x −6

−4

−2

−4

−6

(x − 3) (x + 3) = 0

−6

−4

x 2

−2

(x + 1) (x − 2) = 0

−6

−2

−4

−2 −2

−4

−6

−8

Fluency 6

Solve: a

Ex 1

x (x − 9) = 0

2m (m − 8) = 0

c (5c − 12) = 0

(k − 3) (k − 5) = 0

( y − 6) ( y + 11) = 0

(3x − 9) (2x − 5) = 0

5 (x + 5) (x − 5) = 0

(7a − 2)2 = 0

Solve: a

x2 − 64 = 0

x2 − 9 = 0

x + 12x = 0

x2 − 8x = 0

x2 + 5x + 4 = 0

x2 + 7x + 10 = 0

x − 8x + 15 = 0

x2 − 10x + 16 = 0

x2 − 2x − 24 = 0

x2 − x − 20 = 0

x2 + 3x − 10 = 0

x2 + 13x + 42 = 0

m x + 4x − 21 = 0

x2 − 14x + 40 = 0

x2 − 14x + 49 = 0

x2 − 24x + 144 = 0

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Solve: a

Ex 3

−3x2 − 3x + 396 = 0

−2x + 30x − 112 = 0

3x2 − 21x − 54 = 0

−5x2 + 10x + 40 = 0

4x2 + 24x + 20 = 0

3x2 − 30x + 75 = 0

2x − 32x = 0

Solve: a

m2 = 14m

m2 = m + 20

m2 = 7m − 6

m2 − 11m = −30

m2 − 27m = −182

−3m (m + 2) = 0

3 (n2 + 8) = −18n

2y − 6y2 = 0

2y − 10y2 = 0

15 − 11b − 12b2 = 0

6k2 = −7 + 13k

x2 = 24x − 144

m x = 81 10

c g Ex 2

10x2 − 10x − 120 = 0

Solve: a

−x2 + x + 6 = 0

x (x + 18) = −80

x (x + 2) = 48

x2 + 8x + 16 = 36

4x = 400

5x2 = −320 + 80x

x2 = 13x + 114

3x2 − 7x − 20 = 0

5x + 22x + 8 = 0

−4x2 + 25x − 36 = 0

2x2 − 3x − 20 = 0

−5x2 = −53x + 72

m −10x + x2 = 2 − 7x − x2

−8x + x2 = −6 − x − x2

( y − 3) = 2y + 2

( y + 1)2 = 4y + 4

6x2 − 27x + 28 = −3x2 + 3x + 3

(5x2 − 13x + 6) (2x2 − 13x + 20) = 0

For each graph, complete the following: i

What are the solutions of the related quadratic equation?

Determine the y-intercept(s).

iii

Write the equation in factored form.

18 16 14 12 10 8 6 4 2

12 10 8 6 4 2 −2 −1 −2

x 1

−4

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−2

−2 −4

x 2

2 −2

y 8

x 2

−4

−6 −8 −10

−8 −6 −4 −2 −2

−12

−4

−14

−6

−16

−8

x 2

Reasoning 12

In general, a quadratic function can have 0, 1, or 2 x-intercepts. If its equation can be factorised using rational numbers, how many x-intercepts will it have? Explain.

For x2 + kx + 24, where k is an integer:

Show that k = 10 is a possible value of k so that the expression can be factorised.

Find all of the integer values of k that allow the expression to be factorised over the integers.

For each integer value of k in part (b), state the solutions to x2 + kx + 24 = 0.

For x2 + kx + 28: a

Find the largest integer value of k that makes the expression factorisable. Explain.

Find the integer solutions to x2 + kx + 28 = 0 so that the difference between the two solutions is as large as possible.

For x2 + 4x + c: a

Find all positive integer values of c that allow x2 + 4x + c to be factorised as a product of rational linear factors. Explain your process.

Find all possible integer solutions to x2 + 4x + c = 0 for positive integer values of c.

Explain why it is not possible to list all negative integer values of c that allow x2 + 4x + c to be factorised into the product of rational linear factors.

Explain why there are always solutions to x2 + 4x + c = 0 for c < 0.

Explain why we can say that the solutions to (ax + b) (cx + d) = 0 are

and

but we cannot say anything about the solutions for (ax + b) (cx + d) = e, where e ≠ 0 without further work.

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Problem-solving 17

Solve: a

On the graph of y = x2 − 4, there are two points where y = 12. Find the x-coordinates of these two points.

Find the values of m and n that make both sides of the following equation equal: y2 + my + 60 = ( y + 10) ( y + n)

Software engineers are designing a self-serve checkout system for a supermarket. They notice that the traﬃc through the store during the day is described by the function: C = −t (t − 12) where C is the number of customers and t is the number of hours after the store opens. To meet the peak demand, the engineers allow for an extra checkout machine to automatically turn on when the number of customers ﬁrst reaches 27 people, and to automatically turn off when it next falls below 27 people. a

Find the times t when the number of customers is equal to 27 people.

After how many hours will the extra checkout machine turn on after opening?

How many hours will the extra machine be on for?

The height of a projectile above the ground is given by y = 10 + 3t − t2. Height is measured in metres and time in seconds. Find the time at which the particle hits the ground.

Write the equation of a quadratic in factorised form that has the following key features and draw a graph of the function:

234

Two x-intercepts that have the same values with opposite signs

One x-intercept that is a fraction and the other is a prime number

One unique x-intercept that is less than 1 but greater than 0

x-intercepts of x = 7, x = −2 and has a y-intercept at (0, 14)

x-intercepts of x = −1, x = −10 and has a y-intercept at (0, 30)

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

5.04 Solve by completing the square After this lesson, you will be able to… • solve quadratic equations by completing the square. • interpret the solutions of quadratic equations to sketch the graph of the equation.

Visual method - drawing a box To complete the square, the steps involved are: 1. Move the constant term to the end. 2. Create a box and complete the square. 3. Keep the expression balanced using subtraction. In general, we can start by setting up a box for x2 + bx + c with x2, bx, and c.

x x2

c bx

b 2

x x

b 2

b x 2

Then we can ﬁll in the relevant boxes with half of the bx term.

b x 2

b 2

x x

Finally, we can add and subtract

b x 2

c−

b 2

inside and outside

of the box to keep things balanced.

b 2

b x 2

b 2

This gives us that in general:

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Example 1 Write the expression x2 + 4x − 5 in the form (x + m)2 + n by completing the square.

Create a strategy Follow the steps on completing the square visually. 1. Move the constant term to the end. 2. Create a box and complete the square. 3. Keep the expression balanced using subtraction.

Apply the idea 1. Move the constant term to the end.

−5

2. Create a box writing x2 in the ﬁrst entry and 4x in the middle.

−5

x2 4x

3. Write the factors of x2 around it, and the factors of 4x on the right and below x2. x x

−5

4x 2x

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4. If we have x on top of the ﬁrst column where 2x is, the remaining factor

should be

written adjacent to 2x. Doing the same to the other x term gives also 2 which should be written on its top. x

−5

5. The remaining entry is the product of the constants outside which is 2 × 2 = 4. x

−5

4x 4

6. Subtracting the constants gives us −5 − 4 = −9. x

−5 −4 = −9

Each side of the box has x + 2 which when squared is (x + 2)2. We also have a constant of −9. x2 + 4x − 5 = (x + 2)2 − 9

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Idea summary The following are the steps of completing the square visually. 1. Move the constant term to the end. 2. Create a box and complete the square. 3. Keep the expression balanced using subtraction.

Algebraic method Let’s now look at how we can complete the square using algebraic steps. Completing the square algebraically: 1. Factor out the coeﬃcient of the x2 term, unless it is 1. 2. Halve the coeﬃcient of x and square it, add and subtract this value. 3. Factorise the terms that now form a perfect square. 4. Combine the constant and term left over from completing the square.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.04 to explore how the coefficient of x and the constant affects a quadratic equation when written through completing the square.

Example 2 Using the completing the square method, rewrite into the form y = (x + a)2 + b: y = x2 − 14x + 59

Create a strategy Use the completing the square method.

Apply the idea The coefficient of the x term is −14, and half of this is −7. y = x2 − 14x + 72 − 72 + 59 = (x − 7)2 − 72 + 59 2

= (x − 7) + 10

238

Add 72 − 72 Factorise the perfect square part Evaluate the constant

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 Solve x2 − 7x + 8 = 0.

Create a strategy Move the constant to the right-hand side and add

to both sides.

Apply the idea We have a coeﬃcient of -7 in the x term so x2 − 7x + 8 = 0

. Write the equation

x − 7x + 8 − 8 = 0 – 8 2

x − 7x = −8

Subtract 8 from both sides Evaluate Add

to both sides

Square the fractions

Factorise the perfect square part

Collect like terms Take the square root of both sides

Simplify the square root

Add

Evaluate

Simplify using a common denominator

So the solutions are

and

to both sides

Example 4 Solve 4x2 + 11x + 7 = 0.

Create a strategy Divide the equation by the x2 coeﬃcient to use the completing the square method.

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Apply the idea Write the equation

Divide both sides by 4

Evaluate

The coefficient of the x term is

, and so

Write the simpliﬁed equation

Subtract

Evaluate Add

from both sides

to both sides

Square the fractions

Factorise the perfect square part

Collect like terms

Take the square roots of both sides

Evaluate Subtract

from both sides

Evaluate

Simplify using a common denominator

We can see that we can still simplify each x value. Write the positive x value

Evaluate the subtraction

Evaluate

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Write the negative x value

Evaluate the subtraction

Simplify the fraction

Idea summary The following are the steps of completing the square algebraically. 1. Factor out the coeﬃcient of the x2 term, unless it is 1. 2. Halve the coeﬃcient of x and square it, add and subtract this value. 3. Factorise the terms that now form a perfect square. 4. Combine the constant and term left over from completing the square. We can then solve using inverse operations.

Use solutions to graph To sketch a quadratic in the form y = a(x − h)2 + k, follow these steps: 1. Find the coordinates of the turning point (h, k). When a is positive, the parabola has a minimum turning point, and when it is negative it has a maximum turning point. 2. Find the y-intercept by substituting x = 0. 3. Find the x-intercepts, if any, by substituting y = 0 and solving the equation. 4. Use the key features to graph the parabola. Going the other way, we can also ﬁnd the equation of a quadratic function from the graph using these steps. 1. Identify the turning point from the graph and ﬁll in the values of h and k in y = a (x − h)2 + k. 2. Substitute in any other point on the curve in for x and y to solve for the value of a. 3. Write the equation.

Example 5 Conside the quadratic equation

a Find the coordinates of the turning point.

Create a strategy Use the fact that the turning point of the quadratic in the form y = a(x − h)2 + k is (h, k).

Apply the idea The turning point of the quadratic function

is at (3, 2).

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b Find the y-intercept.

Create a strategy Substitute x = 0 into the equation and then evaluate using the order of operations to ﬁnd y.

Apply the idea

Write the equation

Substitute x = 0

Evaluate the power

Evaluate

c Find the x-intercept.

Create a strategy Substitute y = 0 into the equation and then solve for x.

Apply the idea Write the equation

Substitute y = 0

Subtract 2 from both sides Multiply both sides by −2

Take the square root of both sides of the equation

Evaluate the square root

3 ± 2 = x Add 3 to both sides x1 = 3 + 2

Find the ﬁrst intercept

x1 = 5

Evaluate

x2 = 3 - 2

Find the second intercept

x2 = 1

Evaluate

So, the x-intercepts are (1, 0) and (5, 0).

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d Sketch the graph.

Create a strategy Plot the turning points, intercepts, and draw a curve through the points.

Apply the idea y 4 3 2 1 −4 −3 −2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4

Example 6 Consider the graph shown. y 12 10 8 6 4 2 −7 −6 −5 −4 −3 −2 −1 −2

x 1

−4 −6

a Find the coordinates of the turning point.

Create a strategy Find the point on the parabola where the direction changes from downward to upward.

Apply the idea Based on the given graph, the coordinates of the turning point of the parabola are (−3, −6).

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b Find the equation in the form y = a (x − h)2 + k.

Create a strategy Use the coordinates of the turning point found in part (a) to ﬁll in h and k in the quadratic equation of the form y = a(x − h)2 + k. We can then use another point on the curve to ﬁnd the value of a.

Apply the idea Since the coordinates of the turning point of the parabola are (−3, −6), then the equation is y = a(x − (−3))2 − 6 which simpliﬁes to y = a(x + 3)2 − 6 To determine the value of a, we can use one of the points that passes through the curve of the parabola, such as (0, 12). Write the equation

Substitute the values of x and y

Add 6 to both sides

Evaluate the addition

Evaluate the power

Divide both sides by 9

Evaluate

So, the quadratic equation of the given graph is: y = 2(x + 3)2 − 6

Reﬂect and check We could have used any point we can identify from the graph, it does not need to be the y-intercept at (0, 12). For example, we could also have used (−5, 2), (−1, 2), or (−6, 12).

c Find the x-intercepts as exact values.

Create a strategy Substitute y = 0 into the equation and then solve for x.

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Apply the idea Write the equation

Substitute y = 0

Add 6 to both sides

Divide both sides by 2

Evaluate the division

Take the square root of both sides

Evaluate the square root

Subtract 3 from both sides

Find the ﬁrst intercept

Find the second intercept

So, the x-intercepts are

and

Idea summary To sketch a quadratic in the form a(x − h)2 + k, follow these steps 1. Find the coordinates of the turning point (h, k). When a is positive, the parabola has a minimum turning point and when it is negative it has a maximum turning point. 2. Find the y-intercept by substituting x = 0. 3. Find the x-intercepts, if any, by substituting y = 0 and solving the equation.

Practice questions Choose your question path Explorer 1–5, 7, 8a-d, 9ae, 10ac, 11a, 12af, 13bd, 14ab, 15, 18–20

Adventurer 1–5, 7, 8b-e, 9cd, 10ac, 11b, 12fg, 13ab, 14bc, 15, 18–20

Trailblazer 1–4, 6, 7, 8cdgh, 9ef, 10cd, 11b, 12de, 13ac, 14cd, 15–18, 20–22

Understanding 1

When solving x2 + 24x = 10, four different students showed their ﬁrst step of working. Identify if there is an error or not. a c

Asad: x2 + 24x + 144 = 10 2

Cordelia: x + 24x + 144 = 10 + 144

Blanche: x2 + 24x + 144 − 144 = 10

Dacia: x2 + 24x − 144 = 10

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What could be added to x2 + 10x in order to make it a perfect square trinomial?

What should be added to x2 − 8x in order to make it a perfect square trinomial?

Which value would complete the square for x2 + 6x?

For x2 − 3x = 6, What should be added to both sides of the equation in order to complete the square?

Complete the working out: Given

Subtract 2 from both sides

Complete the square

Evaluate and simplify

Factorise the left-hand side

Take the square root of both sides

Subtract ⬚ from both sides

Complete the working out:

Given

Factorising out a common factor

Divide both sides by 2

Subtract ⬚ from both sides

Evaluate and simplify

Factorise the left-hand side

Take the square root of both sides

Subtract ⬚ from both sides

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Complete the square

Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fluency Ex 1

Ex 2

Ex 3

Rewrite each expression in the form (x − h)2 + k using the box method: a

x2 − 10x

x2 + 12x + 40

x2 + 7x + 12

x2 − 2x + 46

x2 + 2x + 2

x2 − 14x + 56

x2 − 6x + 5

x2 + 3x + 6

x2 − 7x + 14

x2 + 5x − 1

x2 − 2x + 4

(4x − 4)2 = 4

2 (x − 3)2 = 8

(x + 2)2 = 20

x2 − 6x + 8 = 0

x2 − 9x + 8 = 0

x2 − 2x − 32 = 0

6x2 + 48x + 24 = 0 x2 + 11x + 5 = 0

Solve: a

(x + 6)2 = 4

(x + 5)2 − 2 = 15

Solve:

2x2 − 12x − 32 = 0

Solve, leaving your answers in exact form: a

x2 + 24x + 5 = 0

x2 − 7x + 8 = 0

2x2 − 5x − 4 = 0

2x2 − 8x − 28 = 0

−4x2 − 20x + 8 = 0

g Ex 5

x2 + 18x + 32 = 0

Solve: a

Rewrite in the form (x − h)2 + k:

a Ex 4

3x − 5x + 1 = 0

For each equation: i

Find the coordinates of the turning point.

Find the y-intercept.

iii

Find the x-intercepts.

Sketch the graph.

y = −3 (x − 3)2 + 9

y = 4 (x − 1)2 − 10

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Ex 6

For each graph: i

Find the coordinates of the turning point.

Find the equation in the form y = a (x − h)2 + k.

iii

Find the x-intercepts as exact values.

−1

x 1

2 3 4 5 6 7 8 9 10

−2 −3 −4 −5

1 x −2

−1

−6 −7 −8 −9

−2

y 5

2 1 −5

−4

−3

−2

−1

−2 −3 −4 −5

x −5

−4

−3

−2

−1

1 −1

Reasoning 15

Explain how to ﬁnd the number to add to x2 + bx to make it a perfect square trinomial.

Explain the relationship between solving equations by completing the square and solving using square roots.

Explain why we might use the method of completing the square to ﬁnd the exact solutions to some quadratic equations rather than by sketching a graph or factoring.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Tyler attempted to solve the quadratic equation x2 − 4x + 3 = 0 using the completing the square method. Tyler’s solution is shown below. Identify the errors in the solution and explain what Tyler should have done instead. Tyler’s solution: 1

Given

Subtract 3 on both sides

Complete the square

Evaluate and simplify

Factorise the left-hand side

Take the square root of both sides

Add 4 to both sides

Evaluate 4 + 1 and 4 − 1

Problem-solving 19

The revenue y (in millions of dollars) of a company x years after it ﬁrst started is modelled by y = 12.5x2 − 64x + 135 By completing the square, use this equation to predict the number of years x after which the revenue of the company will reach $1167 million.

A model submarine performs a parabolic dive so that its depth below the surface after covering a horizontal distance x was given by y = x2 − 20x All distances are in metres. By completing the square:

Find the maximum depth of the submarine beneath the surface.

Find the horizontal distance(s) travelled when the submarine is 10 m beneath the surface.

The bulbs for the headlights of a car are shaped like a parabola, to direct all light in the same direction. For a particular model of car, the bottom of the headlight can be modelled by the function:

y (0, 54)

where x and f (x) are measured in centimetres, and the y-intercept represents the top left point of the headlight. Determine the width and height of the headlight by ﬁrst rewriting the function in vertex form. Explain your reasoning. 22

Consider the quadratic equation ax2 − c = 0. Given the following conditions, state the real solution(s). If there are no real solutions state “No real solutions”. a

a and c are positive

a and c are negative

a and c have opposite signs

c is equal to 0

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5.05 The quadratic formula After this lesson, you will be able to… • understand what the quadratic formula is and how it is derived. • solve quadratic equations using the quadratic formula.

The quadratic formula The quadratic formula gives the solutions to any quadratic equation. It can be used to solve quadratic equations which cannot be solved by factorisation using rational numbers. Before we can use the quadratic formula, we have to rearrange the quadratic equation into the form ax2 + bx + c = 0, where a, b, and c are any number and a ≠ 0. Once the equation is in this form, the solutions are given by the quadratic formula:

a b c

The coeﬃcent of x2 The coeﬃcent of x The constant

Exploration We have seen that for an equation in the form ax2 + bx + c = 0, we can complete the square and then solve for x using inverse operations. What if instead of working with numbers, we solved for x in terms of variables? Here is the start of the process:

Explain what is happening in each step by ﬁlling in the blanks.

Complete the solution and solve for x in terms of a, b, and c. The result is the quadratic formula.

What would happen if we forgot to include both the positive and negative square root?

Recall that

is the axis of symmetry. Where do you see this in the quadratic formula?

Why do you think this is the axis of symmetry?

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Example 1 Solve 4x2 + 7x + 3 = 0 for x by using the quadratic formula or otherwise.

Create a strategy Use the quadratic formula.

Apply the idea Write the quadratic formula

Substitute a = 4, b = 7, c = 3

Simplify the expression inside the square root

Take the positive square root

Evaluate the numerator

Simplify

Take the negative square root

Evaluate the numerator

Simplify

So the solutions are

and x = −1.

Reflect and check We can also get the same solutions by factorising the equation. To do this, multiply the leading coefficient (4) by the constant term (3): 4 × 3 = 12. Then, two numbers whose product is 12 and sum is the middle coeﬃcient (7). These numbers are 3 and 4. So we can: 4x2 + 7x + 3 = 0 2

Write the equation

4x + 4x + 3x + 3 = 0

Split 7x

4x(x + 1) + 3(x + 1) = 0

Factor each pair of terms

(4x + 3) (x + 1) = 0

Factor out x + 1 Equate the first factor to 0

Subtract 3 from both sides

Divide both sides by 4 x + 1 = 0 x = −1

So the solutions are

Equate the second factor to 0 Subtract 1 from both sides

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Example 2 Solve: a 5x2 − 15x + 2 = 0

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula. Write the equation

Write the quadratic formula

Substitute a = 5, b = −15, c = 2

Simplify the expression inside the square root and

So the solutions are

b 10 − 6m + 2m2 = m2 + 8m + 9

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, and use the quadratic formula.

Apply the idea Write the equation

Subtract m2, 8m, 9 from both sides

Write the quadratic formula

Substitute a = 1, b = −14, c = 1

Simplify the expression inside the square root

Simplify the surd

Simplify the fraction

So the solutions are

252

and

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary For a quadratic equation of the form ax2 + bx + c = 0, the solutions are given by the quadratic formula:

where a, b, and c are any number and a ≠ 0.

Practice questions Choose your question path Explorer 1, 2a-d, 4, 6aegi, 7ace, 8adfi, 9, 10bd, 11, 15a, 16, 18

Adventurer 1, 2bdeg, 3, 4, 6bfjl, 7bdf, 8adij, 9, 10bd, 11, 15a, 17, 19, 20

Trailblazer 1, 2degh, 3, 5chkl, 7fhj, 8dghj, 10bfg, 12–14, 15b, 18, 20, 21

Understanding 1

Is each statement true or false? a

Any quadratic equation that can be solved by completing the square can also be solved by the quadratic formula.

Any quadratic equation that can be solved by factorising can also be solved by the quadratic formula.

The equation 3x2 + 3x − 7 = 0 can be solved using the quadratic formula.

The quadratic formula will always give 2 unique solutions.

For the equation 5x2 − x − 2 = 0, we would set a = 5, b = 1, and c = −2. , ﬁnd the values of a, b and c for each quadratic equation:

For a

x2 + 7x + 10 = 0

x2 − 3x − 4 = 0

2x + 9x = 0

4x2 + 3x = 5

8x2 − 3x = 0

−8x2 + 3x = 0

3x2 − 8x + 2 = 9x − 7

−2x + 9x + 5 = 0

Jeremy is using the quadratic formula for the equation 9y2 = 8y. He has correctly identified that b = 8, what are the values of a and c?

To solve x2 − x = 2, Emma substitutes b = −1 into the quadratic formula. Find the values of a and c that she must substitute into the quadratic formula.

Given the quadratic equation x2 − 4x + k = 0, where k is a constant, if one of the roots is , what is the other root?

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Fluency Ex 1

Solve: a

x2 + 5x + 6 = 0

2x2 + 6x − 8 = 0

4x2 − 10x + 4 = 0

2x2 + 7x + 3 = 0

4x2 − 17x − 15 = 0

5x2 + 7x − 6 = 0

5x2 − 13x − 6 = 0

5x2 + 21x − 20 = 0

k 7

Ex 2

x2 − 5x + 6 = 0

3x − 11x − 20 = 0

Solve, rounding your answers to three decimal places: a

x2 + 3x − 6 = 0

x2 + 7x − 3 = 0

4x2 + 7x + 2 = 0

5x2 + 9x + 2 = 0

7.1x + 5.3x − 1.5 = 0

1.8x2 + 5.2x − 2.3 = 0

2x2 = 14

3x2 + x − 30 = 0

2x (x − 4) = 3x + 1

3x (x + 4) = −3x + 4

Solve, leaving your answers in exact form: a

x2 − 7x + 9 = 0

x2 − 5x − 2 = 0

−2x2 − 15x − 4 = 0

3x2 + 9x − 4 = 0 −5x2 − 15x + 3 = 0

5x − 15x + 2 = 0

5x = (x − 5) (3x + 3)

3n2 = 2n2 − 2n + 7

12 − 8m + 2m2 = m2 + 12m + 15

Reasoning 9

For x (x + 9) = −20: a

Solve it by the method of factorisation.

Check your solution by solving it using the quadratic formula.

Solve each quadratic equation using two different methods and comment on the eﬃciency of both methods: a

x2 − 10 = 15

x2 + 24x + 63 = 0 f

x2 − 7x + 6 = 0

25y2 = 36

4x2 + 5x + 1 = 0

x2 − 7x = 0

5k2 − 17k + 13 = 0 h

x2 + 9x + 20 = 0

Harriet solve a quadratic using the quadratic formula and got the following result:

What can be concluded about the solutions of the equation?

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For the equation in terms of x: mx2 − 3x − 5 = 0, m ≠ 0

Given that it has two unique solutions, determine the possible values of m.

There is one value of m that must be eliminated from the range of solutions found in the previous part. Determine the solution and explain why it must be eliminated.

For b2 − 4ac, the part of the quadratic formula under the square root:

What type of number can we not take the square root of?

The value of b2 − 4ac determines the number of solutions that a quadratic equation has. i

How many solutions does a quadratic equation have if b2 − 4ac > 0? Explain.

How many solutions does a quadratic equation have if b2 − 4ac = 0? Explain.

iii

How many solutions does a quadratic equation have if b2 − 4ac < 0? Explain.

Problem-solving 14

A quadratic function has one of its roots as roots are conjugate pairs.

For each of the given solutions to a quadratic equation: i

Find the values of a, b and c.

Write down the quadratic equation that has these solutions. b

a 16

. Find the quadratic equation given that the

An object is launched from a height of 80 m with an initial velocity of 107 m/s. After x seconds, its height, h, is given by: h = −10x2 + 107x + 80 Find the number of seconds, x, after which the object is 30 m above the ground. Round your answer to one decimal place.

An object is launched from a height of 8 m with an initial velocity of k m/s. After t seconds, its height, h(t), is given by: h(t) = −10t2 + kt + 8 The object was in ﬂight for 4 seconds before reaching the ground. What was its initial velocity k?

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Yuri is playing cricket and hits a six with an initial velocity of 101 metres per second, from a height of 3 metres. After x seconds, its height (in metres) is given by y = −10x2 + 101x + 3

Determine the maximum height the ball reaches.

LaDeana is in the crowd and caught the ball in the stands from a height of 15 metres above the ground. Determine the number of seconds after Yuri hits the ball that LaDeana caught it. Round your answer to two decimal places.

The stopping distance of a car when the brakes are applied can be modelled by the equation in m/s.

where y is the stopping distance in metres and x is the initial speed

Determine the speed the car was travelling if its stopping distance was 100 m, rounding your answer to the nearest hundredth.

It takes 399 m to stop when travelling at the speed limit of 70 m/s. If it takes Ray 496 m to stop, determine how fast over the speed limit Ray was driving.

A rectangular park with dimensions 3 metres by 10 metres has a uniform pathway that is x metres wide running around it. a

If the total area including the pathway is 120 m2, show that the area of the pathway is given by the equation 4x2 + 26x − 90 = 0.

Find the width of the pathway.

The diagram shows a line divided in two parts, one of length a and the other of length b. is in the golden ratio if

a+b

. Show that if

, then x2 − x − 1 = 0.

Let

Find the solutions to this quadratic equation correct to three decimal places.

Which of the solutions is the golden ratio? Justify your answer.

5.06 The discriminant After this lesson, you will be able to… • define what the discriminant is. • identify how the discriminant determines the number and type of solutions. • analyse different cases of the discriminant: less than, equal to and greater than zero. • predict the nature of the roots of a quadratic equation based on the discriminant. • apply the knowledge of the discriminant to solve problems.

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The discriminant We have many ways to solve quadratic equations. Before we try to solve, we may want to know how many solutions or roots a quadratic equation will have. For ax2 + bx + c = 0, and a ≠ 0, the quadratic formula is shown. We will look at what happens if the value under the square root sign, the discriminant, takes on different values.

b2 − 4ac

discriminant

The discriminant determines the number and type of real solutions a quadratic function will have. The symbol Δ or D are often used to represent the discriminant b2 − 4ac.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.06 to answer these questions. 1.

How many roots can there be?

How does the number of roots relate to the value of the discriminant, D?

What types of roots can there be?

How does the type of root relate to the value of the discriminant, D?

How do the x-intercepts relate to the value of the discriminant, D?

We can have an idea of what the parabola will look like by observing the sign of a and the value of the discriminant. If b2 − 4ac < 0, then the value under the square root sign is negative. It is not possible to get real solutions if we square root a negative value. In this case there are no real roots.

y 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

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If b2 − 4ac = 0, then the square root is 0. The

y 4

quadratic equation becomes

equation for the axis of symmetry. In this case the

two roots are identical and the equation has only

1 −4 −3 −2 −1

, which is the

x 1

−1

one solution

−2 −3 −4

If b2 − 4ac > 0, then the square root will have two

y 4

values, one for and one for In this case there are two distinct real roots.

2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

When the discriminant is positive, we can also determine if the roots will be rational or irrational. We sometimes call this the nature of the roots. If the discriminant is a square number, for example: 1, 4, 9, 16, 25 …, and the coefficients of the quadratic are rational, we will have two rational roots. The square root of a square number is rational. If the discriminant is any other positive number, we will end up with surds, so two irrational roots.

Example 1 For 4x2 − 6x + 7 = 0: a Find the value of the discriminant.

Create a strategy Substitute a = 4, b = −6, and c = 7 into Δ = b2 − 4ac.

Apply the idea Δ = b2 − 4ac 2

258

Write the formula

= (−6) − 4 × 4 × 7

Substitute a = 4, b = −6, c = 7

= 36 – 112

Evaluate the index and multiplication

= −76

Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Using your answer from the previous part, determine the number of real solutions the equation has. A

Create a strategy If Δ is negative, the number of real solutions is 0.

Apply the idea Since Δ = −76, the number of real solutions is 0. The answer is option B.

Example 2 For x2 + 38x + 361 = 0 a Find the value of the discriminant.

Create a strategy Substitute a = 1, b = 38, and c = 361 into Δ = b2 − 4ac.

Apply the idea Δ = b2 − 4ac 2

Write the formula

= 38 − 4 × 1 × 361

Substitute a = 1, b = 38, c = 361

= 1444 – 1444

Evaluate the index and multiplication

Evaluate

b Using your answer from the previous part, detemine whether the solutions to the equation are rational or irrational.

Create a strategy The solutions are rational if Δ is a square number, while irrational if Δ is not a square number.

Apply the idea = 0, so there will be only one solution to x2 + 38x + 361 = 0 and it will be rational.

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Example 3 Identify the graph of the quadratic f (x) = ax2 + bx + c, where a > 0 and b2 − 4ac < 0. A y

x –10

–5

–10

–5

–10

C y

x –10

–5

x –10

–5

–10

Create a strategy Consider the signs of a and the discriminant b2 − 4ac.

Apply the idea Notice that a > 0, so it means that the parabola is concave up. Since b2 − 4ac < 0, then the parabola has no x-intercepts. This means that the graph of the quadratic is a parabola with concave up and has no x-intercepts. So, the correct answer is Option A.

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Example 4 Find an expression for the discriminant of x2 + 5x + p − 5 = 0.

Create a strategy Substitute a = 1, b = 5, and c = ( p − 5) into Δ = b2 − 4ac.

Apply the idea Δ = b2 − 4ac

Write the formula

= 5 − 4 × 1 × ( p − 5) Substitute a = 1, b = 5, c = ( p − 5) = 25 − 4 ( p − 5)

Evaluate the index and multiplication

= 25 − 4p + 20

Expand the brackets

= 45 − 4p

Simplify

Reﬂect and check Without knowing the value of p, we cannot make any conclusions about the number or nature of the roots. We may be asked to solve for a value or range of values of p, which gives a certain number of solutions.

Idea summary Discriminant of a quadratic Δ: • • •

b2 − 4ac < 0, 0 real solutions, the parabola has no x-intercepts b2 − 4ac = 0, 1 real solution, 2 equal real roots, the parabola touches the x-axis once b2 − 4ac > 0, 2 real solutions, 2 distinct real roots, the parabola passes through two different points on the x-axis.

For a quadratic with rational coeﬃcients, if the discriminant is a square number like 0, 1, 4, 9, 16 then the roots will be rational, otherwise they will be irrational.

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Practice questions Choose your question path Explorer 1–4, 5aceg, 6ace, 7ac, 8abc, 9ad, 10, 12, 15b, 16, 17, 19, 20a, 21

Adventurer 1–4, 5aceg, 6bdf, 7acd, 8fgj, 9acd, 11, 12, 14, 15a, 17, 19, 20b, 22

Trailblazer 1–4, 5bdfh, 6dkl, 7ghi, 8hjl, 9def, 11, 13, 14, 15c, 18, 20c, 22–24

Understanding 1

When graphing a particular parabola, Katrina used the quadratic formula and found that b2 − 4ac = −5. How many x-intercepts does the parabola have?

When graphing a particular parabola, Tony used the quadratic formula and found that b2 − 4ac = 0. How many x-intercepts does the parabola have?

The solutions of a quadratic equation are 9 and −9. What can be said about the value of b2 − 4ac?

Would the discriminant be positive, negative, or zero for these scenarios? a

No real solutions

One real solution

Two distinct solutions

Two distinct real roots

Two equal real roots

Non-real roots

No x-intercept

One x-intercept

Two x-intercepts

Fluency 5

For each equation in the form ax2 − bx + c = 0: i

State the values of a, b and c.

Calculate the discriminant.

x2 + 11x + 28 = 0

x2 − 5x + 6 = 0

4x2 − 7x − 15 = 0

x2 + 5x +

e g 6 Ex 1

262

−x − 5x − 8 = 0 2

−2x + 3x − 2 = 0

16x − 24x + 9 = 0

For each quadratic equation: i

Calculate the discriminant.

State the number of real solutions for the equation.

x2 + 6x + 9 = 0

3x2 − 5x + 7 = 0

x − 4 = 0

−x2 − 8x − 16 = 0

−2x2 + 5x − 6 = 0

−x2 + 6x + 1 = 0

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

7 Ex 2

Ex 3

−x2 + 25 = 0

2x2 − 2x = x − 1

3x2 − 5x = 3x + 1

4x2 − x = x2 − 5

−3x2 − 5x = −4x2 − 2

For each equation: i

Find the value of the discriminant.

Are the roots to the equation rational, irrational or non-real?

x2 + 22x + 121 = 0

x2 − 8x − 48 = 0

5x2 + 4x + 8 = 0

4x2 + 4x − 6 = 0

4x2 − 4x + 1 = 0

x2 − 16 = 0 2x2 − 9x = −2

9x − 42x = −49

x2 = −6x − 10

For each graph of a quadratic f (x) = ax2 + bx + c, with discriminant Δ = b2 − 4ac: i

Is a > 0 or a < 0?

2 −8 −6 −4 −2 −2

Is Δ > 0, Δ < 0 or Δ = 0?

x 2

−4

−6

−8

d 8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

2 −8 −6 −4 −2 −2

−4

−6

−8

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f 8

2 −8 −6 −4 −2 −2

−4

−4 −6

−8

h 8

−4

−6

−8

j 8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

x 4

−8 −6 −4 −2 −2

−6

−8 −6 −4 −2 −2

264

2 −8 −6 −4 −2 −2

−4

−6

−8

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

l 8

2 −8 −6 −4 −2 −2

−4

−6

−8

n 8

−4

−6

−8

p 8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

Ex 4

m y

−8 −6 −4 −2 −2

−4

−6

−8

Find an expression for the discriminant: a

mx2 + 3x − 2 = 0

x2 − 4nx − 2 = 0

x + 5x + p − 5 = 0

−px2 + 4x + p = 0

mx2 − 2mx + m − 3 = 0

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Reasoning 10

Explain how the discriminant determines the nature of the roots to a quadratic equation and the three possibilities.

Determine the range of values of the constant k such that the equation 3x2 + kx + 12 = 0 has no real solutions. Justify your answer.

Use an algebraic method to show that the graphs of the f (x) = 3x2 − x + 8 and g (x) = −x2 + 2x − 4 do not intersect.

A quadratic function with rational coeﬃcients has an axis of symmetry at x = 3 and an x-intercept which is units from the axis of symmetry.

Is this the only x-intercept? If so, explain why. If not, explain why not and state the other intercept.

What does this tell us about the value of the discriminant?

Is it possible for a quadratic equation to have one rational root and one irrational root? If so, give an example. If not, explain why not.

Problem-solving 15

Find the values of the variables that result in one real solution. a c

n for x2 − 8nx + 1296 = 0

k for 16x2 + 8x + k = 0

m for (m + 4) x + 2mx + 2 = 0

For x2 + 12x + k + 10 = 0: a

Find the values of k for which the equation has no real solutions.

What is the smallest possible integer value of k?

For what value(s) of k will the graph of y = 4x2 − 4x + k − 15 just touch the x-axis?

For what range of values of a will the quadratic equation ax2 − bx + c = 0, where b = 6 and c = 10, have two distinct real solutions?

Given that mx2 + 2x − 1 = 0 has two distinct solutions, determine the possible values of m.

For each equation: a

266

2x2 + 8x + k = 0

kx2 + 20x + 2 = 0

(k + 4) x + 10x + 3 = 0

Find the discriminant in terms of k.

Find the value of k so that the equation has equal solutions.

iii

Find the values of k so that the equation has real solutions.

Find the values of k so that the equation has no real solutions.

Find the values of k so that the equation has real and distinct solutions.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For the graph of the quadratic function, y = m − 9x − 3x2: a

Find the possible values of m, if the graph has no x-intercepts.

State the largest possible integer value of m.

A right-angled triangle with side lengths x units, x + p units and x + q units, ordered from shortest to longest is shown where no two sides of the triangle have the same length. a

Complete the statement:

p and q have lengths such that 0 < ⬚ < ⬚.

Write a quadratic equation that describes the relationship between the sides of the triangle in terms of x.

Find the discriminant of this quadratic equation.

State the number of real solutions to the quadratic equation. Explain your answer.

Find the value of x in terms of p when q = 2p.

x+q x+p

A farmer wants to build a rectangular fence around his field. The length of the fence is x metres and the width is x − 5 metres. The area of the field can be represented by the quadratic equation A (x) = x2 − 5x. The farmer wants to know the possible dimensions of the field that will give him an area of at least 50 square metres. a

Write the quadratic equation representing the area of the ﬁeld when it is at least 50 square metres.

Find the discriminant of the quadratic equation.

Determine the number of real solutions to the quadratic equation and explain your answer.

What is the possible length of the field that will provide an area of at least 50 m2?

Comment on why the answer to part c is not enough to understand the solutions to the problem in context?

A water tank has the shape of a parabolic container with the equation y = −ax2 + h, where y represents the height of the water in the tank, x represents the horizontal distance from the centre of the tank, a is a positive constant and h is the maximum height of the tank. The tank is ﬁlled with water up to a height of k, where 0 < k < h. The width of the water surface is w. a

Write the quadratic equation that represents the water surface in the tank in terms of x, a and k.

Find the discriminant of the quadratic equation representing the water surface in terms of a and k.

Determine the number of real solutions to the quadratic equation when the discriminant is positive. Explain your answer.

If the width of the water surface is w, ﬁnd the relationship between a, k, and w.

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5.07 Optional: Further quadratic equations After this lesson, you will be able to… • use the structure of a quadratic equation to identify ways to rearrange it. • identify the advantages and disadvantages of each method for solving quadratic equations. • choose appropriate methods for solving quadratic equations based on the structure of the quadratic expression. • create and solve quadratic equations for real-world contexts.

Further quadratic equations We have looked at a number of ways to solve quadratic equations. These include: • Factorising the quadratic expression and then using the null factor law. In particular, when the quadratic expression is non-monic, we can use techniques for non-monic quadratic trinomials. • Completing the square and then taking the square root of both sides of the equation. • Using the quadratic formula. Is the equation in standard form?

Yes

Is it easily factorisable? No

No Can we get to Yes a(x − h)2 = k easily?

Complete the square

Yes

Do I want to know the turning point as well?

No No Rearrange to standard form

268

Quadratic formula

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Yes

Factorise

Example 1 Solve: a 5x2 + 22x + 8 = 0

Create a strategy We can try to factorise the equation and use null factor law, but if it takes too long or doesn’t work, then we can use the quadratic formula.

Apply the idea 5x2 + 22x + 8 = 0 2

Write the equation

5x + 2x + 20x + 8 = 0

Split 22x

x(5x + 2) + 4(5x + 2) = 0

Factor each pair of terms

(5x + 2) (x + 4) = 0

Factor out 5x + 2 Equate the ﬁrst factor to 0

Subtract 2 from both sides

Divide both sides by 5

x+4=0 x = −4 So the solutions are

Equate the second factor to 0 Subtract 4 from both sides

and x = −4.

Reﬂect and check As an alternative approach to ﬁnding the solutions, we can use the quadratic formula. Write the quadratic formula

Substitute a = 5, b = 22, c = 8

Simplify the expression inside the square root Take the positive square root

Evaluate and simplify

Take the negative square root

Evaluate and simplify

So we get the same answers:

and x = −4

This involves more numerical operations, but will always work.

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b 3x2 + 5x + 2 = 0

Create a strategy We can factorise, complete the square, or use the quadratic formula. In this case, complete the square.

Apply the idea Write the equation

Divide both sides by 3

Subtract

from both sides

Add

to both sides

Write the left-hand side as a perfect square

Evaluate the right-hand side

Take the square root of both sides

Evaluate the square root

Subtract

Evaluate for each sign and simplify

from both sides

and x = −1.

So the solutions are

Reﬂect and check As an alternative approach to ﬁnding the solutions, we can factorise. 3x2 + 5x + 2 = 0

Write the equation

(3x + 2) (x + 1) = 0

Factorise Set the ﬁrst factor to 0 Solve for x

x+1=0 x = −1 So we get the same answers:

270

Set the second factor to 0 Solve for x and x = −1

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

c x2 − 7x + 9 = 0

Create a strategy We cannot factorise, so we can either use the quadratic formula or complete the square. In this case, use the quadratic formula.

Apply the idea Write the quadratic formula

Substitute a = 1, b = −7, c = 9

Evaluate

So the solutions are

and

Reﬂect and check We can use completing the square as an alternative approach to ﬁnding the solutions. x2 − 7x + 9 = 0 2

x − 7x = −9

Write the equation Subtract 9 from both sides Add

to both sides

Write the left-hand side as a perfect square

Evaluate the right-hand side Take the square root of both sides

Evaluate the square root

Add

Simplify

So we get the same solutions:

and

to both sides

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271

Example 2 Solve: a

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula.

Apply the idea

Rewrite the equation

Multiply both sides by 5x

Subtract 5x2 from both sides

Multiply both sides by −1 and rearrange Write the quadratic formula

Substitute a = 5, b = −11, c = −12

Simplify the expression inside the square root

Take the positive square root

Evaluate and simplify

Take the negative square root

Evaluate and simplify

So the solutions are x = 3 and

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, then factorise the equation and use the null factor law.

Apply the idea Rewrite the equation

Multiply both sides by x (x + 1)

Simplify 7x (x + 1)

Simplify the right-hand side

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Combine like terms and rearrange

Factorise Equate the ﬁrst factor to 0

Subtract 1 from both sides

Divide both sides by 3 Equate the ﬁrst factor to 0

Add 1 to both sides

Divide both sides by 2

So the solutions are

and

Example 3 A parabola is described by the function y = x2 + 4x − 5. a Find the x-values of the x-intercepts of this parabola.

Create a strategy Substitute y = 0 into the equation and then solve for x.

Apply the idea 0 = x2 + 4x − 5

Substitute y = 0

0 = (x + 5) (x − 1)

Factorise

x+5=0

Equate the first factor to 0

x = −5

Subtract 5 from both sides

x–1=0 x=1

Equate the second factor to 0 Add 1 to both sides

So the x-intercepts are x = −5 and x = 1.

b What is the y-value of the y-intercept for this parabola?

Create a strategy Substitute x = 0 into the equation and then solve for y.

Apply the idea y = 02 + 4 × 0 – 5 = −5

Substitute x = 0 Evaluate

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c What is the equation of the vertical axis of symmetry for this parabola?

Create a strategy To ﬁnd the x-value of the vertex or turning point, use the formula

Apply the idea Substitute a = 1, b = 4

Evaluate

Reﬂect and check We can ﬁnd the x-value of the vertex by calculating the average of the two x-intercepts in part (a). Get the average

Evaluate

d Find the y-coordinate of the vertex for this parabola.

Create a strategy Substitute the value x = −2 found in part (c) into the function.

Apply the idea y = (−2)2 + 4 × (−2) – 5 Substitute x = −2 = −9 Evaluate e Plot the graph of the equation.

Create a strategy Plot the vertex or turning point and intercepts found in the previous parts, then sketch the parabola through these points.

Apply the idea 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

274

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x 1

2 3

Idea summary We can solve quadratic equations in many ways. There are beneﬁts and drawbacks to each strategy. • • •

Factorising the quadratic expression and then using the null factor law. Completing the square then using inverse operations. Using the quadratic formula.

Once we have the solutions, we can plot the x-intercepts on the graph.

Practice questions Choose your question path Explorer 1–4, 5adgp, 6cel, 7–9, 10a, 11a, 12a, 14, 17, 19, 20, 21, 24

Adventurer 1–4, 5bcin, 6bejl, 9, 10a, 11c, 12b, 14, 17, 18, 21, 22, 25

Trailblazer 1–4, 5flnp, 6d,jkl, 10b, 11cd, 12d, 13, 16, 19, 21, 23, 25, 26

Understanding 1

Are the following statements true or false? a

Completing the square is a method used to solve linear equations.

Factorising is a method used to ﬁnd the roots of a quadratic equation by expressing it as a product of two binomials.

The quadratic formula can be used to ﬁnd real solutions of any quadratic equation.

A quadratic equation always has exactly two real solutions.

When completing the square for the quadratic equation x2 + 6x + 11 = 0, what constant should be added and subtracted to the equation?

Given that one factor of the quadratic equation x2 − 5x + 6 = 0 is (x − 2), what is the other factor? A

(x + 2)

(x + 3)

(x − 4)

(x − 3)

Which of the following is the correct quadratic formula for solving a quadratic equation of the form ax2 + bx + c = 0? A B C

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Fluency Ex 1

Solve: a

5x2 + 45x = 0

17x2 = 3x

81x2 − 49 = 0

4x2 − 32x + 60 = 0

6x2 + 23x − 4 = 0

−2x2 − 3x + 5 = 0

3x + 14x + 8 = 0

2x2 − 11x + 15 = 0

2x2 + 11x + 12 = 0

5x2 − 14x + 8 = 0

12 − 7b − 12b = 0

3x2 + 5x + 2 = 0

m 3x2 − 12x − 36 = 0

4x2 − 3x − 10 = 0

4x2 − 13x + 2 = 0

4k2 = −15 + 17k

k o Ex 2

Solve: a

4x2 = 2 + 8x

3x2 = 2 − 6x

(3x2 + 11x + 10) (2x2 + 11x + 12) = 0

The equation 4x2 + kx + 36 = 0 has one solution x = −3. Find the value of the coeﬃcient k.

If you substitute x = 6 into ax2 − 6x + 144, what must be the value of a for the expression to be equal to 0?

Using the graph:

276

What are the x-intercepts of the parabola?

What is the y-intercept for this curve?

What is the equation of the axis of symmetry for this parabola?

What are the coordinates of the turning point?

Are these statements true or false?

2 −6

−4

−2

y x 2

−4 −6 −8 −10

The y-value of the turning point is the same as the y-value of the y-intercept.

he turning point is the maximum value of T the graph.

iii

he turning point is the minimum value of the T graph.

The turning point lies on the axis of symmetry.

The x-value of the turning point is the average of the x-values of the two x-intercepts.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−12 −14 −16

For each graph: i

What are the x-intercepts of the parabola?

What is the y-intercept for this curve?

iii

What is the equation of the axis of symmetry for this parabola?

What are the coordinates of the turning point?

−2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9

Ex 3

2 1

y 25

x 1

15 10 5 x −4

−2

−5

For each equation: i

What is the y-intercept?

What are the x-intercept(s)?

iii

What is the equation of the axis of symmetry?

What are the coordinates of the turning point?

Sketch the graph of the parabola.

y = x2 + 2x

y = 2x2 − 8

y = −2x2 + 12x − 10

y = 9x2 − 54x + 72

For each equation: i

What is the y-intercept?

What are the coordinates of the turning point?

iii

What are the x-intercepts?

Sketch the graph of the parabola.

y = x2 + 3x − 2

y = x2 − 62x + 945

y = 2x2 − 9x + 2

y = 3x2 − 25x + 29

A parabola is of the form y = (x − h)2 + k. It has x-intercepts at (9, 0) and (10, 0). a

Determine the axis of symmetry of the curve.

Find the equation of the curve in the form y = (x − h)2 + k.

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Reasoning 14

Jeric is solving the quadratic equation x2 − 6x + 8 = 0 using factorising. Is there an error in his solution? If so, identify and correct it. If not, ﬁll in the reasoning for his steps. x2 − 6x + 8 = (x − 2) (x − 4) x = 2, 4

⬚ ⬚

Solve the quadratic equation x2 + 2x − 8 = 0 and verify the solution by substitution. Explain your process.

We can solve some quartic equations using substitution. a

Solve the equation x4 − 13x2 + 36 = 0 that is reducible to a quadratic by substituting u for x2. Show your solution. How many solutions do you get?

Apply the same technique to solve x4 − x2 − 20 = 0. How many solutions do you get? Explain.

Three students, Alice, Grace and Bob, are solving the quadratic equation x2 − 7x + 10 = 0. Alice uses factorising, Grace completes the square and Bob uses the quadratic formula. Will they arrive at the same solutions? Explain your reasoning.

Sarah attempted to solve the quadratic equation x2 + 15x + 54 = 0 using the factorising method. Identify the errors in Sarah’s solution and explain what Sarah should have done instead. Sarah’s solution: 1

x2 + 15x + 54 = 0

x + 15x + 54 = (x + 9) (x + 6) Factoring the quadratic equation

3 19

Given

x = 9, 6

Finding the solutions

Tom attempted to solve the quadratic equation x2 − 2x − 3 = 0 using the quadratic formula. Identify the errors in Tom’s solution and explain what Tom should have done instead. Tom’s solution:

278

Applying the quadratic formula

Substituting the values

Simplifying the expression

Simplifying the surd

Finding the solutions

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Problem-solving 20

A rectangle has one side 3 cm longer than the other side, and its area is 28 cm2. Determine the length of the shorter side of the rectangle.

The length of a rectangular mat is twice its width and the diagonal length is 55 cm. Find the width of the mat, rounded to the nearest centimetre.

Mae throws a stick vertically upwards. After t seconds, its height h metres above the ground is given by the formula h = 25t − 5t2.

At what time(s) will the stick be 30 m above the ground?

How long does the stick take to hit the ground?

Can the stick ever reach a height of 36 m? Explain.

A t-shirt is ﬁred straight up from a t-shirt cannon at ground level. After t seconds, its height above the ground is h metres, where h = −16t2 + 48t. a

For what values of t is the t-shirt 11 m above the ground?

How long was the t-shirt at least 11 m above the ground?

How long was the t-shirt at least 35 m above the ground?

Let m and n be the solutions to the quadratic equation value of m2 + n2, rounded to the nearest whole number.

The Widget and Trinket Emporium has released the forecast of its revenue over the next year. The revenue R (in dollars) at any point in time t (in months) is described by the equation

. Determine the

R = −(2t − 18)2 + 16 Find the times at which the revenue will be zero. 26

An interplanetary freight transport company has won a contract to supply the space station orbiting Mars. They will be shipping stackable containers, each carrying a fuel module and a water module, that must meet certain dimension restrictions. The design engineers have produced a sketch for the modules and container. Each component has a square cross-sectional area of x2 cm2, and the heights of both modules sum to the height of the container. a

Write an equation that equates the height of the container and the sum of the heights of the modules.

Find the possible values of x.

Find the tallest possible height of the container. Give your answer to two decimal places.

17 cm

Water Module

663 cm x

6426 cm x2

Fuel Module

Container

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Chapter 5 review 1

Are the graphs of these quadratic equations concave up or concave down? a

y = −x (x + 2) + 5

y = 9x2 + 4x

55 = x2 − 45

36 = (n − 2)2

Solve: a

y = x2 + 6x − 9

x2 = 81

A company’s revenue, R (in dollars), after t months is described by the equation: R = − (t − 12)2 + 36

280

x (2x − 16) = 0

Does the revenue reach a maximum or minimum point?

When will the revenue be zero?

T F

For each equation: i

Describe the transformations from y = x2.

What are the coordinates of the turning point?

iii

What are the coordinates of the y-intercept?

Sketch the graph of the parabola.

y = 2 (x − 4)2

A R b

y = − (x + 1)2 + 7

For the quadratic function y = (x − 1) (x + 3):

Find the y-intercept.

Find the x-intercepts.

Write the equation of the axis of symmetry.

Find the coordinates of the turning point.

Sketch the graph of the parabola.

For the quadratic equation y = x2 + 4x − 1: a

Write in the form y = a (x−h)2 + k.

Find the coordinates of the turning point.

Sketch the graph of the parabola.

Solve by factorisation: a

x2 − 12x = 0

x2 + 9x − 10 = 0

−x2 + 24x − 144 = 0

3x2 − 34x + 75 = 0

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For each graph: i

What are the solutions of the related quadratic equation?

Write the equation in factored form.

−6 −4 −2 −2 −4 x 1

−8

−10 −12

y = x2 − 14x + 40

y = 2x + 9x + 10

A R

Find the maximum altitude of the rocket above the ground.

Find the horizontal distance(s) travelled when the rocket is 85 m above the ground.

(Optional) Solve by using the quadratic formula, leaving your answer in exact form: a c

T F

y = −x2 + 6x − 9

A rocket is launched into the sky and its altitude in metres above the ground after travelling a horizontal distance x metres is given by

By completing the square:

x 2 4 6 8 10 12

−6

y = −x2 + 24x

Solve for y = 0 by completing the square: a

6 2

−4 −3 −2 −1 −2 −4

18 16 14 12 10 8 6 4 2

−2x2 − 15x − 2 = 0

3x2 − 11x − 20 = 0

12 − 8m + 2m2 = m2 + 12m + 15

(Optional) The solutions to a quadratic equation are:

Write down the quadratic equation that has these solutions.

What can be concluded about the solutions of the equation?

(Optional) For each quadratic equation: i

Calculate the discriminant.

State the number of real solutions for the equation.

x2 − 4 = 0

x2 + 6x + 9 = 0

−3x2 − 5x = −4x2 − 2

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(Optional) For each equation: i

Find the value of the discriminant.

Are the roots to the equation rational, irrational or non-real?

x2 + 22x + 121 = 0

2x2 + 2x − 9 = 0

5x2 + 4x + 8 = 0

(Optional) For each graph of a quadratic f (x) = ax2 + bx + c, with discriminant Δ = b2 − 4ac: i

Is a > 0 or a < 0?

Is Δ > 0, Δ < 0 or Δ = 0? y

2 −8 −6 −4 −2 −2

T F 2

x 2

−8 −6 −4 −2 −2

−4

A R

−8

y 8 6 4 2 −8 −6 −4 −2 −2 −4 −6

D 2

6 4 2

x 2

−4 −6 −8

For the equation in terms of x:

Given that it has two distinct solutions, determine the possible values of m.

282

−8

mx2 − 3x + 5 = 0

−6

−8 −6 −4 −2 −2

−8

x 4

−4

−6

(Optional) For the equation: y = 3x2 + 18x + 24 a

What are the coordinates of the y-intercept?

What are the coordinates of the x-intercept(s)?

What is the equation of the axis of symmetry?

What are the coordinates of the turning point?

Sketch the graph of the parabola.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) Let a and b be the solutions to the quadratic equation value of a2 + b2.

(Optional) A rock is thrown vertically upwards. After t seconds, its height h metres above the ground is given by the formula h = 3 + 14t − 5t2. a

How long does the rock takes to hit the ground?

Can the rock ever reach a height of 13 m?

. Find the

T F

A R

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and 6 Intersections simultaneous equations Big ideas Solutions to simultaneous equations and inequalities can be graphically and algebraically determined through various methods. There is often a more efficient method that can be used based on the structure of the equation.

Curriculum Outcomes VC2M10A08 Solve linear inequalities and graph their solutions on a number line VC2M10A09 Solve simultaneous linear equations, using algebraic and graphical techniques including using digital tools VC2M10A16 Solve equations graphically or using systematic numerical guess-check-and-refine with digital tools, with consideration of whether all solutions have been found VC2M10AA02 Devise and use algorithms and simulations to solve mathematical problems VC2M10AA09 Solve linear and non-linear simultaneous equations using graphing or systematic guesscheck-and-refine with digital tools VC2M10AA10 Experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

Chapter outline 6.01 6.02 6.03 6.04 6.05

Solve graphically Simultaneous inequalities Solve using substitution Solve using elimination Applications of simultaneous equations Chapter 6 review

286 294 304 314 320 333

Businesses use optimisation to improve efficiencies. In 1987 a large airline removed a single olive from each of its inflight salads, reducing costs by $ 55,000 a year!

6.01 Solve graphically After this lesson, you will be able to… • plot pairs of linear equations on a Cartesian plane. • identify the point of intersection of two graphs as the solution to the simultaneous equations. • recognise the significance of parallel lines and their impact on solutions. • apply graphical methods to real-life equations to find their solutions. • draw conclusions about the presence or absence of simultaneous solutions based on the properties of the graphed lines.

Solve graphically To ﬁnd the solution to a pair of linear equations, which we call simultaneous equations, we can graph both lines on a single Cartesian plane and find any points of intersection.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 6.01 to explore the number of solutions to simultaneous equations.

If two lines are parallel and are not the same line, they will never intersect.

Example 1 Consider the following linear equations: y = 5x − 7 y = −x + 5 a Sketch the two lines representing these equations on a Cartesian plane.

Create a strategy Use the gradient and y-intercept from the equations to graph the lines.

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Apply the idea 8

For the line y = 5x − 7, m = 5 and c = −7. Plot the y-intercept of (0, −7).

6 4 2

−4 −3 −2 −1 −2

Use the gradient of 5 to find another point, by moving 1 unit to the right and 5 units upwards, which is the point (1, −2). Draw a line through these two points.

−4 −6 −8

y = 5x − 7

For the line y = −x + 5, m = −1 and c = 5. Plot the y-intercept of (0, 5).

6 y = −x + 5 4 2

−5 −4 −3 −2 −1 −2

Use the gradient of -1 to find another point, by moving 1 unit to the right and 1 unit downwards, which is the point (1, 4). Draw a line through these two points.

1 2 3 4 5

−4 −6 −8

y = 5x − 7

b What is the point of intersection for the two graphs?

Create a strategy Use the graphs to identify the point of intersection.

Apply the idea 8

The point where the two lines meet is at (2, 3). This is the point of intersection.

6 y = −x + 5 4 2 −5 −4 −3 −2 −1 −2

x 1 2 3 4 5

−4 −6 −8

y = 5x − 7

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Idea summary We can solve any pair of simultaneous equations by plotting their graphs and ﬁnding the point of intersection. If two different lines have the same gradient, then they will be parallel and will have no point of intersection. So their equations will have no simultaneous solution.

Practice questions Choose your question path Explorer 1–3, 4bcd, 5ab, 6a-d, 8, 9, 11, 12ab, 13ab, 14, 15, 17, 18, 21

Adventurer 1–3, 4ace, 5bc, 6e-h, 7, 9, 11, 12ab, 13ab, 15–18, 20, 21, 23

Trailblazer 1–3, 4aef, 5bc, 6fgkl, 7, 10, 11, 12cd, 13cd, 16–19, 22–24

Understanding 1

What does the point of intersection of two straight lines on a graph represent?

If a pair of linear equations has no solutions, what would this look like graphically?

State the number of solutions for the simultaneous equations: a

y 6 5 4 3 2 1

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

288

x 1 2 3 4 5 6

6 5 4 3 2 1 −6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1 2 3 4 5 6

y 6 5 4 3 2 1

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6

y 6 5 4 3 2 1

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

f 4

1 2 3 4 5

3 2 1

x 1 2 3 4 5 6

−4 −3 −2 −1

x 1

−1

−2 −3 −4

Fluency 4

Identify the solution to the simultaneous equations: a

y 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

10 8 6 4 2 −10 −8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

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−3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7 −8 −9

2 4 6 8 10

5 4 3 2 1

3 2 x

−1−1 −2 −3 −4 −5 −6 −7

4 3 2 1

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

1 2 3 4 5 6 7 8 9

−9−8−7−6−5−4−3−2−1 −1

x 1 2 3 4

−2 −3 −4

Use the graph to solve the simultaneous equations: 5 4 3 2 1 −2

−1

x − 2y = 4

y=x−2 1

−1 −2 −3 −4 −5

x 8

y = −8 + 2x

y=x−2

y = −8 + 2x

y = −8 + 2x x − 2y = 4 x − 2y = 4 Ex 1

For each pair of linear equations: i

Sketch the lines on the same Cartesian plane.

Find the solution.

y = 4x − 4

y=x+9

y = 8 − 2x y = −x – 9

290

y=x+3

2x − 4y = −8

y = −x + 3 −4x + 2y = −8

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

y = 3x + 6

y = −x + 2 i

y = 1.5x

y = −x − 3

y = −1 j

y = 2.5x − 4

y = 4x − 3

y = 4 − 3x y = 7 − 2x l

y = 4 − 0.5x y = 12 − 1.5x 7

y = 3x + 2

A band plans to record a demo at a local studio. The cost of renting studio A is $250 plus $50 per hour. The cost of renting studio B is $50 plus $100 per hour. The cost, y, in dollars of renting the studios for x hours can be modelled by the simultaneous equations: • Studio A: y = 50x + 250 • Studio B: y = 100x + 50

Sketch the two lines on the same Cartesian plane.

What coordinates satisfy both equations?

What do the coordinates from part (b) mean in context?

Michael plans to start taking an aerobics class. Non-members pay $4 per class. Members pay a $10 one-time fee, but only have to pay $2 per class. The cost, y, of taking x classes can be modelled by the simultaneous equations: • Non-members: y = 4x • Members: y = 2x + 10

Sketch the two lines on the same Cartesian plane.

What coordinates satisfy both equations?

What do the coordinates from part (b) mean in context?

A company’s cost is represented by C(x) = 0.4x + 2015, y and its revenue is represented by R(x) = 3x, where x 2790 represents the number of items produced. a

What coordinates satisfy both equations?

2325

What do the coordinates from part (a) mean in context?

1860

C(x)

R(x)

1395 930 465

x 155 310 465 620 775 930 1085 1240

The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2n. The revenue, R, made from selling n toys is given by R = 4n. a

Sketch the graphs of cost and revenue on the same Cartesian plane.

What coordinates satisfy both equations?

What do the coordinates from part (a) mean in context?

Laura is going on maternity leave. Her living expenses are represented by y = 1200x. Her bank account balance is represented by y = 10 000 + 772.60x. For both equations, x is time in weeks. a

Sketch both equations on the same Cartesian plane using technology.

Estimate the solution to one decimal place.

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c 12

A pair of linear equations has no solutions. One of the equations is y = −3x − 2. Determine if the following can be the other equation: a

What does the point of intersection of the two lines mean in context?

y = −3x − 3

y = −3x + 2

A pair of simultaneous linear equations has inﬁnite solutions. One of the equations is y = 4x − 5. Determine if the following can be the other equation: a

y = 8x − 10

y = 4x + 5

3y = 12x − 15

2y = 8x − 10

Reasoning 14

Is it possible to have a pair of linear equations with exactly two solutions? Explain why or why not.

In a pair of linear equations, one equation is a scalar multiple of the other. For example, 2x + 4y = 6 and x + 2y = 3. What can you say about the solutions of the simultaneous equations?

The graph of y = 3x + 2 is shown.

10 8 6 4 2

If a second line intersects this line at the point (0, 2), can we determine the equation of the second line? Explain your answer.

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

The graph of y = 5x + 3 is shown.

A second line, y = mx + c, intersects this line at the one point (0, 3).

What is the value of c? Explain your answer.

What value of m is not possible? Explain your answer.

1 −2

−1

−1 −2

Consider the simultaneous equations: y = −2x − 5 y = −2x + 4

292

2 4 6 8 10

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

Without graphing, is there a solution? Explain your answer. 19

A company needs to ﬁnd the best way to distribute goods from multiple suppliers to multiple customers. The distribution amounts must satisfy the simultaneous linear equations: y = 3x + 15 y = 9x − 10 Explain the implications of having non-integer solutions in this context and discuss how the company might handle such solutions.

Problem-solving 20

A rectangle is to be 3 m longer than it is wide, with a total perimeter of 18 m. a

Let y represent the length of the rectangle and x represent the width. Create two equations that represent this information.

Sketch the two lines on the same Cartesian plane.

Find the length and width of the rectangle.

Rochelle and Mohamad are sister and brother. Rochelle’s age is 11 more than 4 times the age of Mohamad. The sum of their ages is 21. a

Write a pair of linear equations to represent the situation.

Graph the simultaneous equations.

How old are Rochelle and Mohamad?

Kang and Elvia are working on an assignment together. They have broken down the work into 9 parts of the same size. Kang works at 2 times the speed of Elvia but has 6 pieces of work to do for another subject. a

Write a pair of linear equations to represent the situation.

Graph the simultaneous equations.

Solve the simultaneous equations and interpret the result in terms of the context.

Write a scenario to represent the simultaneous equations and its solution. Explain what the solution means in terms of the scenario.

20 18 16 14 12 10 8 6 4 2

x 2 4 6 8 10 12 14 16 18 20 22 24

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Two equations, y1 and y2 represent the growth of two different house plants over time. Use the graph of y1 and y2 to support the claim that the two plants will never reach the same height on the same day. 25 20 15 10 5 −40 −30 −20 −10 −5 −10 y1 −15 −20 y2 −25

x 10 20 30

6.02 Simultaneous inequalities After this lesson, you will be able to… • define simultaneous inequalities. • find solutions to simultaneous inequalities and represent them graphically. • interpret and solve problems involving simultaneous inequalities in given contexts. • apply the concepts of simultaneous inequalities to real-world scenarios.

Simultaneous inequalities Simultaneous inequalities is a set of inequalities which have the same variables. The solution to simultaneous inequalities is the set containing any ordered pair that makes all of the inequalities in the system true. A solution can also be represented graphically as the region of the plane that satisﬁes all inequalities in the system.

294

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−2

x 1

−1

−2

−3

−4

The solution to a simultaneous inequalities in a given context is viable if the solution makes sense in the context, and is non-viable if it does not make sense.

Example 1 Select the inequalities that describe the shaded region.

A x + y ≤ 9, x ≥ 3

B x + y > 9, x < 3 C x + y > 9, x > 3

D x + y ≤ 9, x ≤ 3

6 3 x −3

−3

Create a strategy If the lines are dotted and the shaded region is above or on the right of each line then the inequality symbol is >.

Apply the idea The answer is option C.

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Example 2 Consider the lines y = 4x − 2 and y = 3x − 3. a Find the x-coordinate of the point at which the two lines intersect.

Create a strategy

Apply the idea 4x − 2 = 3x – 3 Form an equation by substitution of y

Form an equation by substitution of y and solve.

x − 2 = −3 x = −1

Subtract −3x from both sides Add 2 to both sides

b Find the y-coordinate of the point of intersection.

Create a strategy

Apply the idea

Substitute x = −1 into one of the equations.

We can use y = 4x − 2. y = 4x – 2 = 4 × (−1) – 2

Substitute x = −1

= −6

Evaluate

c Graph the region that satisiﬁes both y < 4x − 2 and y < 3x − 3.

Create a strategy • Plot the intersection point and the y-intercepts of each inequality. • The inequality symbols are both < so the lines must be dotted. • Since it is an “and” inequality then shade the region which satisfies both inequalities.

Apply the idea We found from parts (a) and (b) that the intersection point is at (−1, −6).

y 1 x −2

−1

1 −1

y < 4x − 2 has a y-intercept at (0, −2), while y < 3x − 3 has a y-intercept at (0, −3). The shaded region must be below both lines.

−2 (0, −2) −3 (0, −3) −4 −5 −6 (−1, −6) −7

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Example 3 Applicants for a particular university are asked to sit a quantitative reasoning test and verbal reasoning test. Successful applicants must obtain a minimum score of 14 on a quantitative reasoning test and a minimum combined score of 29 for both tests. a Write a simultaneous inequalities for this scenario, where x represents the quantitative reasoning test score and y represents the verbal reasoning test score.

Create a strategy

Apply the idea

The quantitative reasoning will show an inequality with 14 as the lowest possible solution. The total of the two scores must sum to 29 or more.

b Sketch the graph of the simultaneous inequalities.

Create a strategy

Apply the idea

Graph each inequality on the same Cartesian plane and shade the region that satisfies both.

y 30 25 20 15 10 5 x 5

c Does the solution

make sense in terms of the context? Explain your answer.

Apply the idea No. This point satisfies both inequalities, but a test score is likely a positive integer or a simple fraction such as

or .

Idea summary A simultaneous inequalities is a set of two or more inequalities with the same variables. The solution of simultaneous inequalities are the values of the variables that make all the inequalities true. It is shown as a shaded region on the Cartesian plane.

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Practice questions Choose your question path Explorer 1–3, 5ab, 6, 7ab, 8, 10, 11, 13, 14, 16, 18ab, 20, 22, 23, 25

Adventurer 1–3, 5, 7ac, 8, 10, 11, 13, 15, 16, 18ac, 20, 21, 23, 25

Trailblazer 1–5, 7ad, 9, 10, 12, 15, 17, 18cd, 19, 21, 23–26

Understanding 1

The graph shows the region for the simultaneous inequalities: y > − 3x + 6

7 6 5 4 3 2 1

y>4 Which points are located in the region?

(−3, 2)

(2, 5)

(−1, −6)

(3, 2)

(4, 4)

−3 −2 −1−1 −2 −3 −4 −5 −6 −7

The graph shows the region for the simultaneous inequalities: x+y≤4

7 6 5 4 3 2 1

x≥1 y≥0 What are the coordinates of the three vertices of the region?

1 2 3 4 5 6 7

−3 −2 −1−1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

Which ordered pair satisﬁes the simultaneous inequalities? 2x − y > −6 2x + 2y < 0 A

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(2, 6)

(−3, 6)

(1, −3)

(−6, 1)

Is the statement true or false? a

The region can only include individual points and cannot include lines.

The solution set to a linear inequality is a region of the graph and can include an infinite number of points.

All the points on the boundary line of a linear inequality are included in the solution if the inequality is “less than or equal to” or “greater than or equal to”.

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Fluency Ex 1

Write the simultaneous inequalities that describe the shaded region: a

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Ex 2

3 2 1

x −4 −3 −2 −1

1 2 3 4 5

−1

−2 −3 −4

3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

5 4 3 2 1

4 3

1 2 3

1 −4 −3 −2 −1

−1

−2 −3 −4

For 2x − y < 4 and 6x + 3y > 0: a

Sketch the boundary lines, 2x − y = 4 and 6x + 3y = 0, on the same set of axes.

Does the point (2, 2) satisﬁes the simultaneous inequalities?

Graph the solution set to the simultaneous inequalities 2x − y < 4 and 6x + 3y > 0.

For each pair of simultaneous inequalities: i

Find the x-coordinate of the point at which the two lines intersect.

Find the y-coordinate of the point of intersection.

iii

Graph the region.

x ≤ 5 and y < 3

y ≤ 3x − 4 and y > −4x + 1

3x + y > 5 and 3x + y < 7

y < 4x − 2 and y < 3x − 3

For y > x − 5, y > −x − 5, and y < 0: a

Graph the lines y = x − 5, y = −x − 5, and y = 0 on the same set of axes.

Does the point (1, −2) satisfy the simultaneous inequalities?

Shade the region described by the inequalities y > x − 5, y > −x − 5, and y < 0.

Calculate the area of the region.

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Ex 3

For x > −2, y > 1 and y ≤ −0.75x + 2.5: a

Graph the region which satisfies the inequalities.

Calculate the area of the region.

Applicants for a particular university are asked to sit a numeracy test and verbal reasoning test. Successful applicants must obtain a minimum score of 17 on the numeracy test and a minimum combined score of 37 for both tests. Let x and y represent an applicant’s score on the numeracy and verbal reasoning test respectively. a

It’s not possible to get a negative score therefore y ≥ 0 is one inequality. Write two more inequalities from the information in terms of x and y.

Graph the solution to the simultaneous inequalities.

Which points represent scores that would make the applicant successful? i

(14, 16)

(33, 18)

iii

(20, 13)

(17, 22)

If an applicant obtains a score of 24 in the numeracy test, ﬁnd the minimum integer score they need to obtain in the verbal reasoning test to be successful.

An airline is checking passengers into two flights, A and B, simultaneously. Due to passenger numbers, there must be at least 10 staff at check-in for flight A and at least 7 staff at check-in for flight B. Since there must be staff on hand for other services, the airline can only allocate at most 23 staff for check-in of both flights. Let x and y represent the number of staff attending check-in of flights A and B respectively.

Write three inequalities representing the possible values of x and y.

Graph the solution to the simultaneous inequalities.

If 14 staff are allocated to checking in passengers of ﬂight A, what is the maximum number of staff that can be allocated to checking in passengers of ﬂight B?

An old airforce plane used for sky diving can ﬁt no more than 32 divers at a time, and has a maximum carrying capacity of 3900 kilograms. When carrying a parachute and equipment an average female’s weight is 100 kilograms and an average male’s weight is 150 kilograms. Let x represent the number of women. Let y represent the number of men.

Write four inequalities in x and y that represent the number of men and women who can be on the plane at one time.

Graph the inequalities on the same axis.

Which combinations of men and women can be on the plane at the same time? A

25 women and 11 men

19 women and 12 men

25 women and 9 men

7 women and 10 men

A bicycle manufacturer employs a mechanic and a painter to construct two types of bikes: racing bikes and mountain bikes. The time each worker spends on each bike is shown (assume both workers can work on the same bike at the same time). Let x represent the number of racing bikes built and y represent the number of mountain bikes built.

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Racing Bike Mountain Bike

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Mechanic 5 hours 8 hours

Painter 10 hours 4 hours

Each worker can work at most 40 hours in a week.

Write an inequality relating x and y to the total time spent working by the mechanic.

Write an inequality relating x and y to the total time spent working by the painter.

Lamborgotti Motors produces two types of car: a sedan and a coupe. Each type of car requires assembly in two different factories. The time that each car is required to be in each factory is given in the table. Factory A Factory B Each factory can operate for 18 hours a day. Sedan 3 hours 4.5 hours Let x represent the number of sedans Coupe 2 hours 1.5 hours produced each day, and y represent the number of coupes produced each day.

Write an inequality for Factory A.

Write inequalities for x and y given the number of cars cannot be a negative number.

Graph the region deﬁned by the inequalities for Factory A and Factory B

Write an inequality for Factory B.

Mahindra produces two types of tractors: an All Purpose and a Harvester. Each type of tractor requires production in two different factories. The time that each tractor is required to be in each factory is given in the table. Each factory can operate for 18 hours a day. Let x represent the number of All Purpose tractors produced each day, and y represent the number of Harvesters produced each day.

All Purpose Harvesters

Factory A 4 hours 2 hours

Factory B 6 hours 1.5 hours

Write an inequality for Factory A.

Write inequalities for x and y given the number of tractors cannot be a negative number.

Graph the region deﬁned by the inequalities for Factory A and Factory B.

Write an inequality for Factory B.

Reasoning 16

Without graphing, determine whether the simultaneous inequalities have no solutions or infinitely many solutions. Explain your answer. a

y > 4 and y < 4

x ≤ −2 and x ≥ −2

9x − y < 7 and 9x − y > 7

2x − y ≤ 7 and 2x − y ≥ 7

Tracy has experimented with growing two different types of banana trees on her land in different proportions. Let x and y represent the decimal proportion of area Type A and Type B banana trees respectively. a

She has found that bananas grow best if Type A takes up at most 35% of the plantation area. Write two inequalities representing the possible values of x.

She has found that bananas grow best if Type B takes up at least 20% of the plantation area. Write one inequality representing the possible values of y.

The whole area must be planted with a combination of Type A and Type B. Write one inequality representing the possible values of x and y.

Graph the solution to the simultaneous inequalties.

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Do these points correspond to appropriate areas for Type A and Type B banana trees? i

(0.28, 0.68)

(0.58, 0.68)

iii

(0.13, 0.57)

(0.09, 0.11)

Is it possible for Type B trees to grow optimally if they take up 100% of the plantation area? Explain your answer.

A company produces cans of cat food and dog food. Each product passes through two machines in their production, and the amount of time each machine takes to process one pallet of pet food is shown in the table. Each machine can operate for 12 hours in a day. Let x represent the number of pallets of cat food processed each day, and y represent the number of pallets of dog food processed each day.

Cat Food Dog Food b

Machine A 2 hours 4 hours

Machine B 3 hours 1.5 hours

Write an inequality for Machine A.

Can the values of x and y be negative? Explain your answer. Write any additional inequalities if necessary.

Write an inequality for Machine B.

Graph the region deﬁned by the inequalities for Machine A and Machine B on a Cartesian plane.

Is it possible for a coordinate point to solve one part of the linear inequality pair but not the other? Explain with a diagram what this would look like.

A light bulb manufacturer produces two types of light bulbs: incandescent and ﬂuorescent. Let x represent the number of incandescent bulbs produced in a day, and y represent the number of ﬂuorescent bulbs produced in a day. a

If it’s not possible for a number of lightbulbs to be less than zero, write an inequality for x and y to represent this fact.

Up to 200 light bulbs can be produced in one day. Write an inequality to represent this.

The ﬁnal inequality representing constraints on the number of light bulbs that can be produced each day is:

Explain the meaning of this ﬁnal inequality in the context of the question.

Problem-solving 21

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There are x surfers competing for wildcard entry into a pro-surﬁng competition, of which Joanne is one. She needs her total score, y, from all judges to be at least 35 in order to be chosen. The total score possible from all judges is at most 50. a

x > 1 is one of the inequalities that can be formed for the situation. Write down the two other inequalities related to y to complete the system.

Graph the solution to the simultaneous inequalities.

If Joanne receives scores of 8.4, 7.3, 7.3, 7.8, 8.9 from the judges, will she be chosen as a wildcard entry?

Whether Joanne is chosen as a wildcard entry is dependent on the number of surfers competing for the spot. True or false?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Roald has 24 centimetres of leftover wood that he is trying to make a rectangular photo frame out of. Let x represent the length of the photo frame and let y represent the width. a

Write three inequalities to represent the situation.

Graph the solution to the simultaneous inequalities.

Are these possible dimensions of the photo frame?

A length of 6 cm and a width of 4 cm

A length of 7 cm and a width of 6 cm

iii

A length of 9 cm and a width of 8 cm

A length of 7 cm and a width of 4 cm

If Roald wanted to create a square frame, what are the dimensions of the largest possible frame?

Throughout university, Tom works as a mentor, earning $15 per hour, and as a ﬁtness instructor earning $18 per hour. The number of hours he works in each job can vary from week to week, but he never works more than 29 hours in total each week, and he needs to be able to at least cover his weekly expenses of $270. Let x represent the number of hours he works as a mentor and y represent the number of hours as ﬁtness instructor.

Write four inequalities representing the possible values of x and y.

Graph the solution to the simultaneous inequalities.

If Tom works 6 hours as a mentor in one week, ﬁnd the minimum number of hours y he can work as a ﬁtness instructor so that he can cover his expenses.

Is it possible for him to work the same number of hours in both jobs and still be able to cover his expenses?

A warehouse is stocked with two types of storage containers: square boxes and rectangular boxes. The square boxes cost $55 each and have a volume of 2 m3 and the rectangular boxes cost $65 each and have a volume of 7 m3. The warehouse has a total storage space of 200 m3 and there is a budget of $800 to purchase the containers. Let x represent the number of square boxes purchased, and y represent the number of rectangular boxes purchased.

If it’s not possible for a number of boxes to be less than zero, write two inequalities for x and y to represent this fact.

Write an inequality relating x and y for the total budget for the warehouse.

Write an inequality relating x and y for the total storage space of the warehouse.

Eileen is spending the week at a sports camp. She is able to attend coaching sessions for her two favourite sports: basketball and football. She can attend a maximum of 26 sessions and a minimum of 20 sessions per week. Eileen’s basketball coach has told her that she must attend at least 5 basketball sessions. Her football coach has told her that she must go to at least 8 more football sessions than basketball sessions. Let x represent the number of basketball sessions she attends, and y represent the number of football sessions she attends. a

From this information, write four inequalities.

Graph the region deﬁned by the four inequalities. 6.02 Simultaneous inequalities mathspace.co

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List the four vertices of the region.

The vertices of the region represent the maximum and minimum number of session Eileen can attend. What is the maximum number of basketball sessions that she can attend?

Edward is going to a music camp for one week. Edward plays drums and guitar and can attend lessons for both instruments. He can attend a maximum of 18 lessons and a minimum of 13 lessons per week. Edward’s guitar teacher has told him that he must not go to more than two times as many guitar lessons as drum lessons. His drum teacher has told him he must attend at least ﬁve drumm lessons. Let x represent the number of drum lessons he attends, and y represent the number of guitar lessons he attends. a

From this information, construct a set of ﬁve inequalities.

Graph the region deﬁned by the ﬁve inequalities.

List the ﬁve vertices of the region.

Using the ﬁve vertices determine the maximum number of guitar lessons that Edward can attend.

6.03 Solve using substitution After this lesson, you will be able to… • utilise the substitution method to solve for both variables in two simultaneous linear equations. • demonstrate how to rearrange equations to isolate a variable when equations are not ready for direct substitution and then apply the substitution method.

Substitute variables When solving simultaneous equations, we are looking for values for the variables which make both equations true. This will be the point of intersection of the graphs of the equations. This section focuses on solving for the point of intersection of two linear equations, but the same process can be applied to any two equations with two unknowns. We can rearrange the equations to suit our purpose as long as we always maintain the equivalence. Let’s look at some methods in the following examples.

Example 1 We want to solve the following simultaneous equations using the substitution method.

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y = x + 11

y = 3x + 19

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

a First solve for x.

Create a strategy Substitute the expression for y from equation 1, into the variable y in equation 2 and then solve for x.

Apply the idea x + 11 = 3x + 19

Substitute 1 into 2

11 = 2x + 19

Subtract x from both sides

−8 = 2x

Subtract 19 from both sides

x = −4

Divide both sides by 2

Using substitution we have found that x = −4.

b Solve for y.

Create a strategy Substitute the value of x into one of the equations and solve for y.

Apply the idea y = x + 11

y = −4 + 11

Substitute x = −4 into 1

y=7

Evaluate

Reﬂect and check To check that x = −4, y = 7 is also a solution to 2 , substitute the value of x into 2 and see if we get the same value for y: y = 3x + 19

y = 3(−4) + 19

Substitute x = −4 into 2

y=7

Evaluate

y = 7 is the same value, this confirms that x = −4, y = 7 is the solution. We can also write this as (−4, 7), the intersection point of the two lines.

Idea summary To solve simultaneous equations using the substitution method: • • •

Substitute the expression for one variable, usually y, into the other equation. Solve the new equation for the other variable, usually x. Substitute the result back into one of the equations to find the value of the other variable.

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Rearrange and substitute In the previous example, both equations were already written with y as the subject. As such, it was easy to substitute the expression for y from one equation into the other. When the equations are not written with one of the variables as the subject, we can choose any of the variables to isolate in one equation in order to substitute it into the other.

Example 2 Solve these simultaneous equations using the substitution method. 1 2 a Solve for q.

Create a strategy First, find an equation in terms of p using equation 1, then substitute the value of p into the equation 2.

Apply the idea Rearrange 1 to make p the subject. 1

Subtract 2q from both sides

Divide both sides by −7

We now have an equation for p, we will call this equation 3 .

Substitute the value of p into 2 and solve for q. 2

Substitute p

Expand the brackets

Rearrange and combine like terms

Simplify the fraction

Divide both sides by 4

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b Find the value of p.

Create a strategy Using

from part (a), substitute into equation 3.

Apply the idea 3

Substitute the value of q

Simplify the fraction

Create a common denominator

Add the fractions and simplify

The solution is

and

Reﬂect and check To check that

is also a solution to 1 , substitute the value of p into 1 and see if we

get the same value for q.

1 into 1

Substitute

Evaluate the multiplication

Add

Create a common denominator

Add the fractions and simplify

Divide both sides by 2

is the same value, this confirms that

to both sides

and

is the solution. We can also write this as

, the intersection point of the two lines.

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Example 3 A man is ﬁve times as old as his son. Four years ago, the man was nine times as old as his son. Find their present ages. a Use the statement “a man is five times as old as his son” to set up equation 1.

Create a strategy Let x represent the present age of the man, and y represent the present age of the son.

Apply the idea The man is ﬁve times as old as his son, so x is equal to 5 times y. x = 5y

b Use the statement “4 years ago the man was 9 times as old as his son” to set up equation 2. Write this with y the subject.

Create a strategy To ﬁnd their ages 4 years ago, we can subtract 4 from x and y.

Apply the idea We can express the age of the man from four years ago as x − 4 and we can express the age of the son as y − 4. At this time the age of the man is 9 times the age of the son, this will give us equation 2. 4 years ago the man is 9 times as old as the son Expand the brackets Add 36 to both sides Divide both sides by 9 2 2 is

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c Solve for y to ﬁnd the age of the son.

Create a strategy Substitute equation 1 into equation 2.

Apply the idea 2 Substitute 1 to 2 Multiply both sides by 9 Subtract 5y from both sides Divide both sides by 4 The son is 8 years old.

d Solve for x to ﬁnd the age of the man.

Create a strategy Substitute the value of y into equation 1.

Apply the idea x = 5y

x=5×8

Substitute y = 8

x = 40

Evaluate

The man is 40 years old.

Reflect and check Check the ages for the son (8) and man (40) against the initial statement. A man is 5 times as old as his son, this is true as 5 × 8 = 40. Four years ago, the man was 9 times as old as his son, this is true as 4 years ago the man would be 36, and the son would be 4 and 4 × 9 = 36.

Idea summary To solve simultaneous equations using the substitution method, we may need to rearrange one of the equations so that one variable is the subject before we can perform the substitution.

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Practice questions Choose your question path Explorer 1, 3, 5aefi, 7bc, 8, 10, 12–14, 17, 18, 20abde

Adventurer 1–3, 5bdjk, 7ab, 8, 10, 12, 13, 15, 17, 19, 20bdg

Trailblazer 1, 2, 4, 5k-n, 7ceg, 8, 9, 11–13, 15, 16, 19, 20cdfh, 21

Understanding 1

A group of students were solving the equations: x+y=6 y = 2x − 3 a

Tomi substituted x from the ﬁrst equation into the second.

Carla substituted y from the second equation into the ﬁrst.

Regie substituted y from the ﬁrst equation into the second.

Junard substituted x from the second equation into the ﬁrst.

Write the equation that each student would then have to solve and comment on the eﬃciency of each. 2

State whether these statements are true or false. a

The method of substitution involves isolating one variable in one equation and substituting this expression into the other equation.

The substitution method can be used to solve any simultaneous linear equations.

Once you solve one of the equations for one variable, you must substitute it back into the ﬁrst equation.

Solving simultaneous equations by substitution will always result in the same solution, regardless of which variable you solve for ﬁrst.

Simultaneous linear equations can have exactly two solutions, no solutions, or inﬁnitely many solutions.

Is (5, 2) a solution of the simultaneous equations? x+y=7 x−y=3

Is (4, 17) a solution of the simultaneous equations? y = 6x – 7 4x + 2y = 67

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Fluency Ex 1

Solve the simultaneous linear equations using substitution: a

y = 6x + 56

y = 2x + 24 e

y = 3x + 14 y = −3x + 9

y = −3x + 22

y = −6x + 59

x + y = 10 −2x + y = −13 j

y = 4x – 28

6x − y = 0 3x + 5y = 44 m 5x − 2y = −10

y = 5x + 37

y = −4x – 27

y = 7x + 38 5y = 35x + 265 2y = 10x + 54

x+y=2 i

y = −5x − 22

y = 2x + 13 x − y = −11

y = −3x − 23

y = 2x – 15

x + 4y = −37 3x − 4y = 20

2x + 3y = 48

y = −5x + 3 4x − 5y = 8 6

For these simultaneous equations: 3x − 7y = 4 −12x + 28y = −16

Ex 2

Rearrange 3x − 7y = 4 to make x the subject in terms of y.

Substitute your expression for x from part (a) into −12x + 28y = −16 and solve for y.

Solve the simultaneous equations using the substitution method: a

2x + 3y = 26

2x + 7y = 23

5x + 4y = 10

4x + 3y = 17

8x − 5y = 36 5x + 6y = 0 3x + 7y = −17 9x − 4y = 6 e

Ex 3

−7f + 2g = −0.4

−4a + 7b = 11.5

−2.1f + g = 0.36 −2a + 2.1b = 2.25

A mother is currently 10 times older than her son. In two years time, she will be 7 times older than her son. Let x and y be the present ages of the son and mother respectively. This can be modelled by the simultaneous equations: y = 10x y = 7x + 12 What are the current ages of the son and the mother?

Farzad bought some fresh produce. He picked up 2 apples and 3 kiwi fruit. The cost of Farzad’s shopping was $10.08. Eileen went to the same shop and bought 5 apples and 7 kiwi fruit. The cost of Eileen’s shopping was $24.08. Let a and k be the price of apples and kiwi fruit respectively. This can be modelled by the simultaneous equations: 2a + 3k = 10.08 5a + 7k = 24.08 a

Solve the simultaneous equations using the substitution method.

What is the cost of one apple and one kiwi fruit?

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The number of new jobs created in Venhurst can be modelled by the simultaneous equations y = 4x − 190 000 y = x + 560 000 where x is the number of jobs created in 2008 and y is the number of jobs created in 2013.

Solve the simultaneous equations using the substitution method.

What does the solution to the simultaneous equations mean in context?

The members in a group can be modelled by the simultaneous equations y=x+6 x + y = 22 where x and y are the number of adults and children in the group respectively. a

Solve for the values of x and y.

What does the solution to the simultaneous equations mean in context?

Reasoning 12

If you solve simultaneous equations by substitution and ﬁnd that the result is a false statement such as 0 = 5, what does that tell you about the equations?

What does it mean if you solve simultaneous equations by substitution and get a true statement, such as 0 = 0 or 5 = 5?

For these simultaneous equations: 3x + y = 7 2x − y = 4 Explain the steps you would take to solve the simultaneous equations and why you would do it in that way.

Do the equations y = 3x + 5 and y = 3(x + 5) have inﬁnitely many solutions? Explain your answer.

For these simultaneous linear equations: −6x − 2y = −28 2x + 16y = 40 4x − 2y = 12

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Find the values of x and y that satisfy the ﬁrst two equations.

Does this solution satisfy the third equation?

Is this the only solution to the simultaneous equations? Justify your answer.

Conﬁrm the results by graphing the three linear equations on the same Cartesian plane.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Problem-solving 17

A taxi driver charges a fee for the initial pick up and an additional amount per km travelled. A journey of 10 km costs $105, while a journey of 15 km costs $155. a

Find the amount of the ﬁxed charge and the charge per km.

How much does a person have to pay for travelling a distance of 25 km?

The coach of a tennis team buys 7 rackets and 6 balls for $380. Later, he buys 3 rackets and 5 balls for $175. Find the price of rackets and balls respectively.

Mikhail sells sausages and soft drinks. Each sausage costs $1.50 and each soft drink costs $0.50. At the end of the day, he made a total of $78.50 for a total of 87 sausages and soft drinks combined. How many sausages and soft drinks were sold?

Use the substitution method to ﬁnd the solution(s) for each pair of simultaneous equations. a

(x + 3)2 + ( y − 3)2 = 4

x = −5 c

y = 24

x + ( y + 7) = 7

y = 5x + 4 y = x2 + x + 8

y = −7 e

y = x2 − 5x

y = x2

(x + 1)2 + ( y + 4)2 = 25

y = −5x y = −3x − 2 g

y = (x − 5) (x − 2)

y = −x + 1 21

y = (x − 1)2 x − 3y = 1

The plan for a proposed tunnel through a mountain side is shown in the diagram: y

Entrance to the Tunnel

nel

ed Tun

Propos

Sea Level

The height of the mountain above sea level is given by the equation height of the tunnel above sea level is given by the equation the height above sea level in kilometres.

, and the , where y represents

Use the substitution method to solve for y, the heights above sea level at which cars will enter and exit the tunnel.

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6.04 Solve using elimination After this lesson, you will be able to… • apply the elimination method to solve equations.

Elimination method As long as maintain equivalence of both the left and right-hand sides of any given equation we can combine them in specific ways to make other equations that are still true. For example: if we know that:

a=b

c=d

Then we can combine these two equations to create others. We label these new equations in similar way, so that we can reference them in our working later. 1 + 2

a+c=b+d

1 − 2

a−c=b−d

We can also take multiples of equations before combining them with addition or subtraction, for example: y = x + 11 y = 3x + 19 3× 1 −4× 2

3a − 4c = 3b − 4d

× 1 +3× 2

1 2 5 6

Equations 3, 4, 5 and 6 are all different, but they all maintain the equivalence that was present in equations 1 and 2. To solve equations simultaneously, we can use specific combinations of equations to create situations where we can eliminate one of the variables. This is the elimination method.

Example 1 Use the elimination method to solve the pair of simultaneous equations: 3x + y = 10

−4x + y = −11

Create a strategy We can remove the variable y, by subtracting the equations.

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Apply the idea 3x + y = 10

−4x + y = −11

7x = 21

1 − 2

x=3

Divide both sides by 7

Now that we a value for x, substitute this into equation 1 and solve for y. 3x + y = 10

3(3) + y = 10

Substitute x = 3

9 + y = 10 y=1

Evaluate Subtract 9 from both sides

The solution to simultaneous equations is (3, 1).

Reflect and check Substitute the values for x and y into equation 2 to confirm. −4x + y = −11

−4(3) + 1 = −11

Substitute

−12 + 1 = −11

Evaluate

−11 = −11

True

Confirmed that (3, 1) also solves equation 2.

Example 2 Use the elimination method to solve the pair of simultaneous equations: 2x + y = 4

x − 4y = −34 2

Create a strategy We can remove the variable x, by doing 1 − 2 × 2 .

Apply the idea 2x + y = 4

x − 4y = −34

2x − 8y = −68

−2 × 2

9y = 72

1 −2 × 2

y=8

Divide both sides by 9

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Now that we a value for y, substitute this into equation 1 and solve for x. 2x + y = 4

2x + 8 = 4

Substitute y = 8

2x = −4

Subtract 8 from both sides

x = −2

Divide both sides by 2

The solution to simultaneous equations is (−2, 8).

Reflect and check Substitute the values for x and y into equation 2 to confirm. x − 4y = −34

−2 − 4(8) = −34

Substitute

−2 − 32 = −34

Evaluate

−34 = −34

True

Confirmed that (−2, 8) also solves equation 2.

Idea summary To solve equations simultaneously, using the elimination method we want to combine equations so the new equation has only one variable. If the coeﬃcient of the variable we want to eliminate in both equations has the same value but different sign, we can add the equations to eliminate the variable. If the coeﬃcient of the variable we want to eliminate in both equations has the same value and the same sign, we can subtract the equations to eliminate the variable.

Practice questions Choose your question path Explorer 1, 2, 4, 5, 6ab, 7a-f,m, 8, 11, 13, 16, 19ac

Adventurer 1–5, 6bc, 7acegikm, 9, 12, 13, 15, 17, 19ace

Understanding 1

316

The method of elimination involves: A

Substituting one variable into another equation

Graphing the equations

Adding or subtracting equations to eliminate a variable

Isolating a variable

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 1–4, 6bd, 7bdfhjln, 10, 12, 14, 15, 18, 19ef

Explain how you would begin the process to eliminate x from the following set of equations. 5x − 9y = 2 7x − 8y = 1

Comment on whether each statement is true of false. a

The elimination method is best used when the coeﬃcients of one variable are the same in both equations.

When using the elimination method, the solution represents where the lines of the equations intersect.

There are situations when the elimination method will not work.

Fluency 4

For the simultaneous linear equations:

What number should the second equation be multiplied by to eliminate y?

Eliminate y and ﬁnd the value of x that satisﬁes both equations.

Substitute the x-value to ﬁnd the value of y that satisﬁes both equations.

For the simultaneous equations: 3x + 7y = −24 4x − 5y = 11

What value can we multiply each equation by to eliminate the variable y?

Solve for x and y.

For each pair of simultaneous equations: i

Rewrite the equations as equivalent equations with lowest common multiple integer coefficients.

Solve for x and y.

0.2x + 0.3y = 0.5

0.5x + 0.4y = 0.2 c

Ex 1, 2

Use the elimination method to solve each pair of linear equations: a

3x + 6y = 2

2x + 5y = 44

3x + 3y = −7 6x − 5y = −28 c

8x + 3y = −11

2x − 5y = 1

−8x − 5y = 29 −3x − 5y = −39 6.04 Solve using elimination mathspace.co

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The train in an amusement park has 15 cabs that can hold a total of 56 people. Some cabs hold 4 people while some hold 6 people. Let x be the number of four-passenger cabs and y be the number of six-passenger cabs. This situation can be represented by the simultaneous equations: x + y = 15 4x + 6y = 56

Find the number of four-passenger cabs and six-passenger cabs using the elimination method.

In terms of the context, is the solution valid?

Nine pens and 2 rulers cost $26, while 3 pens and 4 rulers cost $22. Let x and y be the price of the pen and ruler respectively. This can be modelled by the simultaneous equations: 9x + 2y = 26 3x + 4y = 22

Solve the simultaneous equations using elimination.

Explain what the solution means in context.

Marjan spent $55.25 to purchase 9 ﬂowers. She bought rhododendrons which cost $6.45 each and chrysanthemums which cost $5.75 each. This can be modelled by the simultaneous equations R+C=9 6.45R + 5.75C = 55.25 where R is the number of rhododendrons and C is the number of chrysanthemums that Marjan bought. a

Solve the simultaneous equations using elimination.

Explain what the solution means in context.

Reasoning 11

Explain how you could use the elimination method to solve the simultaneous equations: 2x − 3y = 8 4x + 3y = 12

Without solving, how can you determine the number of solutions to these simultaneous equations using the elimination method? 3x − 2y = 6 −3x + 2y = −6

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Describe a situation where it would be more eﬃcient to use the elimination method rather than the substitution method to solve simultaneous equations and explain why.

Elias solved nonlinear simultaneous equations using elimination. His work is shown. Aria chose to solve by graphing, and she found two solutions. a

Explain the mistake Elias made that caused him to only ﬁnd one solution.

Find the second solution. y = 3x2 − 6 2

y = −x – 2

y = −3 Original equations

Add the equations, solve for y

−3 = 3x − 6

y = 3x – 6

3 = 3x2

3y = −3x2 – 6 Multiply equation 2 by 3

1 = x2

4y = −12

1=x

Substitute y, solve for x

The solution is (1, −3).

Problem-solving 15

In a game show, Nigel has the opportunity to answer 30 questions. For every question he answers correctly, he earns 6 points, but for every question he answers incorrectly, he loses 3 points. Let x represent the number of correct answers and y represent the number of incorrect answers. a

If Nigel’s total score was 90 points, write a pair of simultaneous equations that represent this problem.

Solve these equations to ﬁnd out the number of correct and incorrect answers Nigel provided.

For two numbers, x and y: • The sum of y and two times x is 27. • The difference between ﬁve times x and y is 43. Find the two numbers.

For two numbers, x and y: • The sum of the x and y is 10. • The difference between six times x and ﬁve times y is 5. Find the two numbers.

The proportion of the workforce of a particular country that are teenagers is modelled by 3.3x + y = 70.8, where x is the number of years since 2015 and y is the proportion as a percentage. The proportion of the workforce that are pensioners is modelled by 3.2x − y = −64.5. a

Estimate the year in which the proportion of the workforce that are teenagers and the proportion of the workforce that are pensioners are the same.

Estimate the percentage of the workforce that are teenagers or pensioners during the estimated year in part (a).

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Use the elimination method to solve the simultaneous equations. a

x2 − y = −2

y = x2

−x + y = 4 c

x2 + y = 5

y = 3x2 − 2

x2 + y2 = 25

−x + y = −1 e

x2 + y2 = 100

(x + 3)2 + ( y + 1)2 = 25

2x + 10y = 68 (x + 3)2 − y = 6

6.05 Applications of simultaneous equations After this lesson, you will be able to… • define variables that represent unknown quantities to translate information into at least two equations. • apply simultaneous equations to real-life scenarios. • define and understand the concept of the break-even point. • perform break-even analysis using graphical representations.

Applications of simultaneous equations Our ﬁrst step is to deﬁne the variables that represent the quantities in the problem. Then translate the pieces of information about the quantities into equations. Lastly, by using either the substitution method, elimination method or a graphical method, we solve the equations simultaneously.

Example 1 When comparing some test results Christa noticed that the sum of her Geography and Science test score was 172, and that their difference was 18. Given that her Geography score is x and her Science score is y, and that she scored higher in the Geography test: a Use the sum of the test scores to form an equation. We will refer to this as equation 1.

Create a strategy Write an equation in terms of x and y.

Apply the idea The sum of her two test scores is 172, where x and y are the scores. Equation 1:

320

x + y = 172

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b Use the difference of the test scores to form an equation. We will refer to this as equation 2.

Create a strategy

Apply the idea

Write an equation in terms of x and y.

The difference between the scores is 18, where x and y are the scores. Since x > y we can subtract y from x. x − y = 18

Equation 2:

c Use these two equations to ﬁnd her Geography score.

Create a strategy We can use the elimination method to solve for x.

Apply the idea Notice that the coeﬃcient of y in both equations is the same, but with opposite sign. So we can eliminate the variable y by adding the two equations together. +

x x

+ -

y y

2x x

= =

172 18

= =

190 95

So her Geography score is 95.

d Find her Science score.

Create a strategy Substitute the x-value into one of the equations to solve for y.

Apply the idea x + y = 172

Write equation 1

95 + y = 172

Substitute the value of x

95 + y − 95 = 172 – 95 Subtract 95 from both sides y = 77

Simplify

So her Science score is 77.

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Example 2 Consider the following phone plans: • GO SMALL plan: This plan has a $20 monthly base charge and charges 90c per minute for all calls. • GO MEDIUM plan: This plan has a $26 monthly base charge and then charges 70c per minute for all calls. a Complete the table of values for various total monthly call times for the two plans: Call time (in minutes) 5 10 15 20

Total cost for GO SMALL plan (S)

Total cost for GO MEDIUM plan (M )

Create a strategy Add the monthly base charge to the number of minutes times the rate per minute.

Apply the idea For the call time of 5 minutes we can ﬁnd the cost for the GO SMALL plan by: GO SMALL cost = 20 + 0.9 × 5 = $24.50

Add the base charge to the rate times the minutes Evaluate

We can ﬁnd the cost for the GO MEDIUM plan by: GO MEDIUM cost = 26 + 0.7 × 5 = $29.50

Add the base charge to the rate times the minutes Evaluate

Continuing in this way for the rest of the call times gives us this table: Call time (in minutes) 5 10 15 20

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Total cost for GO SMALL plan (S) 24.50 29.00 33.50 38.00

Total cost for GO MEDIUM plan (M ) 29.50 33.00 36.50 40.00

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Sketch a graph for the two plans.

Create a strategy Use the points from the table to graph both plans.

Apply the idea 50

Cost

45 40 35 30 25 20

M S

15 10 5

To include the points from the table the vertical or cost axis needs to go over 40, and the horizontal or call time axis needs to go over 20. We will go a bit further than this along both axes, in order to see more detail in the graph. We will also label the lines S and M. To graph the lines we can use two points for each plan from the table. For the GO SMALL plan we can use (10, 29) and (20, 38). For the GO MEDIUM plan we can use (10, 33) and (20, 40).

Call time 5 10 15 20 25 30 35

c Using the graph, determine how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.

Create a strategy Use the point of intersection.

Apply the idea 50

Cost

The point of intersection is at (30, 47). This means that for a call time of 30 minutes, both plans will cost 47.

45 40 35 30 25 20

M S

15 10 5

Call time 5 10 15 20 25 30 35

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Idea summary To solve a problem with two unknown quantities: • • •

Deﬁne variables to represent the quantities. Translate the information into two equations. Use the substitution method, elimination method or graphical method to solve the equations simultaneously.

Break-even point It’s important for businesses to make a profit. To find this point, we use break-even analysis. Break-even analysis looks at the graphs of cost and revenue together to determine where they intersect. This point of intersection is the break-even point where revenue equals expenses. Here is an example of a break-even analysis for a single day of operation for the company Lovely Lemonade. Lovely Lemonade’s break even analysis 1200 Profit

Money (dollars)

1000

Break even point

800 600 400

Loss

200 1000

2000

3000

4000

5000

Number of drinks (units) Revenue

Expenses

The expenses line (red) starts off higher than the revenue line (green). For a low number of units sold, Lovely Lemonade will lose money. For example, if they only sell 1000 units, the green line tells us they earn $200, the red line tells us they spend $600, and so overall they make $200 - $600 = -$400. The negative sign means overall they lose $400. If they sell 5000 units the graph tells us that they will earn $1100 but only spend $1000. Overall they make $1100 - $1000 = $100, and the positive sign means a profit of $100. The important point is when the two lines intersect - this is the break-even point. In this example we can see that selling 4000 units means they don’t make any profit, but they don’t lose any money either - both their revenue and their expenses are $900. We can also use break-even analysis to compare two different pricing plans for the same product or service to see which one offers the best deal for a particular situation. Here’s an example comparison between two energy plans provided by rival companies, Thorgate and Callisto: 324

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Thorgate vs Callisto pricing comparison 18 Callisto cheaper

Price (dollars)

15 12

Break even point

9 6 Thorgate cheaper

3 10

Energy per day (kWh) Thorgate

Callisto

In this scenario the break-even point at (40, 10) tells us that both plans charge $10 for 40kWh. For energy amounts less than the break-even point we can see that Thorgate is cheaper, and for energy amounts more than the break-even point Callisto is cheaper. Knowing how much energy someone plans on using can then determine the best plan for them.

Example 3 The cost for a furniture manufacturer to make an armchair is $500 per armchair plus a ﬁxed setup cost of $3500. The armchairs will sell for $650 each. a Write an expression to represent the cost of manufacturing x armchairs.

Create a strategy Add the ﬁxed cost and the variable cost.

Apply the idea The ﬁxed cost is $3500. The variable cost is $500 per armchair, which is $500 × x. Cost = 3500 + 500x

Add the ﬁxed and variable costs

b Write an expression to represent the revenue generated from the sale of x armchairs.

Create a strategy The amount of revenue generated is equal to the number of pieces sold multiplied by the price at which they sell.

Apply the idea Revenue = 650x

Multiply the selling price and the number of armchairs sold

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c Find the break-even point.

Create a strategy Equate the expressions for the cost and the revenue and solve for x.

Apply the idea At the break-even point, the cost is equal to the revenue, so we equate their expressions. Equate the revenue and cost

Subtract 500x from both sides

Simplify

Divide both sides by 150

Evaluate

So the break-even point is when x = 24 since we can only sell whole numbers of armchairs.

Example 4 David decides to start his own yoga class. The cost and revenue from running the class have been graphed: 54

Money (dollars, $)

48 42

Cost

36 30 24 18 12

Revenue

x 1 2 3 4 5 6 7 8 9 10 11 Number of students

a Find the revenue that David makes for each student.

Create a strategy Use the revenue line on the graph.

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Apply the idea 54

For 1 student, x = 1. We move from x = 1 on the x-axis up to the revenue line, then across to the y-axis to get y = 6.

Money (dollars, $)

48 42

So the revenue per student is $6.

36 30 24 18 Revenue

12 6

x 1 2 3 4 5 6 7 8 9 10 11 Number of students

b How many students must attend his class so that David can cover the costs of running the class?

Create a strategy

Apply the idea

The number of students is given by the intersection point of the revenue and cost curves.

The point of intersection is (6, 36). So if David get 6 students, he can cover the cost of running the class.

c How much proﬁt does David make if there are 9 students in his class?

Create a strategy Find the difference between the revenue and the cost.

Apply the idea 54 48 Money (dollars, $)

To ﬁnd the revenue and cost for 9 students, we ﬁnd x = 9 on the x-axis and move up to the two lines and then across to the y-axis from each.

Cost

We can see that cost is $45 and the revenue is $54.

36 30 24 18 12

Revenue

x 1 2 3 4 5 6 7 8 9 10 11 Number of students

Profit = 54 – 45 = $9

Subtract the cost from the revenue Evaluate

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Idea summary The break-even point is where the cost and revenue are equal. To ﬁnd the break-even point, we can either: • •

Look at the graphs of cost and revenue together to determine where they intersect. Equate the expressions for revenue and cost and solve for the independent variable.

Practice questions Choose your question path Explorer 1–5, 7–9, 11, 13, 14, 16, 17ac, 18ad, 19, 22, 24

Adventurer 1–4, 6–8, 10, 11, 13, 15, 17bd, 18ad, 20, 22, 24–26

Trailblazer 1–4, 7, 8, 11–13, 15, 17ab, 18cd, 21–23, 26–28

Understanding 1

Each symbol in the diagram has a value. The sum of each row is shown:

Find the value of one triangle. Find the value of one square.

b c

Find the value of one circle.

What number should replace the question mark for the total of that column?

4 ?

Each item in the diagram has a price. The total price of each row of items is given:

$15

Find the price of:

$66

One football

One jersey

One pair of boots

Three students in a class bought some new equipment from the school stationery shop. The items they each bought and their total prices are indicated:

$52

= $27

= $18

= $27

Find the price of:

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One pencil set

One work book

One calculator

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Each item in the diagram has a price. The total price of each row of items is given: Find the price of one pair of boots.

= $31

= $61

Fluency 5

A family owns two businesses that made a combined profit of $6 million in the previous financial year, with Business B making 2 times as much profit as Business A. Let x and y be the profits (in millions) of Business A and Business B respectively.

Write a pair of linear equations to represent this situation.

Find Business A’s proﬁt.

Find Business B’s proﬁt.

Two twin soccer players, Arden and Benton, are having a contest over who will score the most goals in the season. The twins have scored a combined total of 34 goals so far over the season and Arden has scored 6 more goals than Benton. Let x and y be the number of goals scored by Arden and Benton respectively.

Ex 1

Write a pair of linear equations to model this situation.

Find the number of goals Arden has scored this season.

Find the number of goals Benton has scored this season.

When comparing some test results Akira noticed that the sum of his Geography score and Science score was 184, and that their difference was 22. He also noticed that he scored higher for the Geography test. Let x and y be his Geography score and his Science score respectively.

Ex 2

Write two equations in terms of x and y from the information given.

Find his Geography score.

Find his Science score.

For a phone company, customers can choose between two phone plans: • GO SMALL plan: $24 monthly base charge, and 80c per minute for all calls • GO MEDIUM plan: $32 monthly base charge, and 60c per minute for all calls a

Complete the table of values for various total monthly call times for the two plans: Call time (in minutes) 20 30 40 50

Total cost for GO SMALL plan

Total cost for GO MEDIUM plan

Find how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.

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Christa has $4000 to invest, and wants to split it up between two accounts: • Account A earns 7% annual interest. • Account B earns 8% annual interest. Her target is to earn $303 total interest from the two accounts in one year. Let x and y be the amounts, in dollars, that she invests in accounts A and B respectively.

Ex 3

Write a pair of linear equations to represent this situation.

Solve for y, the amount Christa invests in account B.

Solve for x, the amount Christa invests in account A.

The length of a rectangle is 12 units more than the width, and the perimeter of the rectangle is 56 units. Let y be the width and x be the length of the rectangle. a

Write a pair of linear equations to represent this situation.

Solve the simultaneous equations.

Toby’s piggy bank contains only 5c and 10c coins. It contains 70 coins with a total value of $3.85. a

Set up two equations by letting x and y be the number of 5c and 10c coins respectively.

Solve the simultaneous equations.

Qin invested $16 000 in total in two stocks, A and B. In one year, the investment in stock A made a 14% return, while the investment in stock B fell by 6%. The total annual interest from both stocks was $700. a

Set up two equations by letting x and y be the amounts that she invested in stocks A and B respectively.

Solve the simultaneous equations.

Explain what the solution means in context.

Kerry currently pays $50 a month for her internet service. She is planning to switch to a ﬁbre optic cable service. a

Write an equation for the total cost T of Kerry’s current internet service over a period of n months.

For the ﬁbre optic cable service, Kerry pays a one-off amount of $1200 for the installation costs and then a monthly fee of $25. Write an equation of the total cost T of Kerry’s new internet service over n months.

Determine how many months it will take for Kerry to break-even on her new internet service.

There are 36 members in a group, and the men outnumber the women by 16. Find the number of women and men in the group.

Consider the diagram of the rectangle: a

Solve for x and y.

Find the length of the rectangle.

Find the width of the rectangle.

3y + 384

8x + 60 D

B 15y + 48

64x + 12

Maria and Buzz both walk from their houses to the bus stop every morning. Maria walks 1.5 kilometres further, and together they walk 3.3 kilometres. Find the distance Buzz walks.

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Ex 4

A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. The cost of employing people to make the clothing is modelled by y = 800 + 60x and the cost of using machinery is modelled by y = 3200 + 20x, where y is the cost and x is the number of t-shirts to be made. a

Sketch a graph that depicts the two cost functions.

For what value of x will the cost of making t-shirts be the same, whether they are made by people or by machines?

For what values of x will it be more cost eﬃcient to use machines to manufacture the t-shirts?

For what values of x will it be more cost eﬃcient to employ people to manufacture the t-shirts?

Reasoning 18

For the simultaneous equations, explain which strategy you would use and why: a

y = 3x – 1

2x + y = 7 19

y = 2x + 3

y = 4x + 2

y = −x + 5 3x − 2y = 6

2x + 3y = 5 x−y=1

For each of the simultaneous equations: i

State whether the substitution method or the elimination method would be more eﬃcient for solving.

Explain why you chose the method in part (i).

y = 6x + 56

7x − 4y = 15

−5x + 16y = 82

y = 2x + 24 7x + 5y = 60 25x − 4y = 122 20

y = −3x − 23 x + 4y = −37

Pensacola high school hosts an annual badminton tournament. Student tickets cost $3.00 and general admission tickets cost $5.50. a

If 350 tickets were sold for a total of $1450, how many of each type of ticket were sold?

Explain which method you used to solve and why.

At Soul Food Express you can buy two orders of oxtail stew and a slice of sweet potato pie for $38.49, or you can get ﬁve orders of oxtail stew and three slices of sweet potato pie for $99.22. a

How much is the cost of an order of oxtail stew and a slice of sweet potato pie?

Explain which method you used to solve and why.

Barry is considering opening a restaurant. He has calculated his ﬁxed costs at $10 000 per month. He expects to generate a revenue of $12 per meal, but the variable cost for each meal is $15. a

Write equations for the total cost C and total revenue R for selling x meals.

Can Barry reach the break-even point? Explain.

How does this impact Barry’s restaurant idea and what changes can he make for it to succeed?

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Problem-solving 23

For the simultaneous equations: y = −4x2 + 8x y = 2x + c a

Determine the value of c that will result in one unique solution to the simultaneous equations.

Determine the values of c that will result in two real solutions to the simultaneous equations.

A bacteria population doubles every day, following the equation y = 2x. A scientist predicts that the population will increase by 3 times the number of days, following the linear equation y = 3x. At what point will the scientist’s prediction first match the actual population?

A circular garden has a radius of 1 unit and its centre is at the point (2, 1). A straight path is built through the garden, following the equation y = x − 1. Find the points where the path intersects the garden’s boundary.

When $2700 is deposited, a bank offers two types of savings accounts: • Account A: After depositing $2700 the account earns 2.9% simple interest p.a. • Account B: After depositing $1650 the account earns 2.8% compound interest p.a. After how many years will Account B have a greater balance than Account A?

A chef is experimenting with a new recipe that requires a speciﬁc ratio of ingredients. If 2 is added to the numerator and denominator it becomes

and if 3 is subtracted from the

numerator and denominator it becomes . Find the original ratio of the ingredients. 28

An engineer is designing a playground that will be placed at the centre of a hyperbolicshaped park. The engineer wants to place a fence along a straight line that intersects the park at two points. The situation can be modelled by the simultaneous equations: • Park: • Fence: Find the points where the fence intersects the park. Round your answers to two decimal places.

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Chapter 6 review 1

For each pair of linear equations: i

Sketch the lines on the same Cartesian plane.

Find the solution.

x = −4

2y = 5x − 12

y = −3x − 9 y = x − 3 2

Use the graph to solve the simultaneous equations: y = −3x −13

y = −x − 3

−8

−7

5 4 3 2 1

−6

−5

−4

−3

−2

T F

−1

A R

x + 2y = −6

y = −x − 3

y = −3x − 13 x + 2y = −6 3

−1 −2 −3 −4 −5

y = −3x − 13

x + 2y = −6

A boutique’s expenses are represented by E (x) = x + 1220, and its sales revenue by S (x) = 1.95x, where x is the number of dresses sold.

3500 3000

What coordinates satisfy both equations?

2500

What do the coordinates from part (a) mean in context?

2000 1500

E (x) S (x)

1000 500 x 256 512 768 1024 1280 1536

A pair of linear equations has inﬁnitely many solutions. One of the equations is y = 2x + 5. Which of these is the other equation? A

y = 2x + 7

−y = −2x − 5

Consider the simultaneous equations:

y = −3x − 4

y = −3x + 7

Without graphing, is there a solution? Explain your answer.

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A triangular garden has a base and two equal-length sides. One of the equal sides is 2 m longer than the base. The garden has a total perimeter of 22 m. a

Let y represent the length of one of the equal sides and x represent the base. Create two equations that represent this information.

Sketch the two lines on the same coordinate plane.

Find the length of one of the equal sides and the base of the triangle.

(Optional) For each pair of equations: i

Use technology to graph the curves on the same Cartesian plane.

Estimate the solution(s).

x2 + y2 = 16

y = (x − 4) (x − 3)

y = x − 2 8

T F

(Optional) Identify the solution(s) to the simultaneous equations. a

−8 −6 −4 −2 −2

A R

−6 −8

−10

y 5

4 3

2 1 −5 −4 −3 −2 −1

−1

x 1

−2 −3

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2 1

−5 −4 −3 −2 −1 −1

−4

1 2 3 4 5

−2 −3

−4 −5

8 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2

x 1

2 3 4 5

(Optional) The parabolas y = c − bx2 and y = x2 − x + c intersect at the point (2, 1). a

Find the value of c.

Find the value of b.

Sketch the graphs of the parabolas on the same Cartesian plane.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) The revenue and cost functions for a new smartwatch, where y is the amount of money (in hundreds of dollars) and x is the number of units made or sold (in hundreds) are shown: • Revenue function: y = 3x2 − 2 • Cost function: y = x2 − 3x + 25 a

Use technology to graph the functions on the same Cartesian plane.

Write the solution(s) as an ordered pair.

Describe the number of units that should be produced and the corresponding revenue to achieve a break-even point for this product.

(Optional) A line intersects a circle at exactly one point and is perpendicular to the radius of the circle. Determine the equation of the line if the equation of the circle is (x − 6)2 + (y − 8)2 = 50 and the point of intersection is (1, 13).

For the simultaneous inequalities y ≤ 2x − 1 and y > x − 4:

Find the x-coordinate of the point at which the two lines intersect.

Find the y-coordinate of the point of intersection.

Graph the feasible region.

A R

Write the simultaneous inequalities that describe the shaded region.

D 14

T F

8 7 6 5 4 3 2 1

−4 −3 −2 −1−1 −2 −3 −4 −5 −6

x 1

2 3 4 5

For y > x − 4, y ≥ −x − 4, and y < 1: a

Graph the lines y = x − 4, y = −x − 4, and y = 1 on the same set of axes.

Does the point (2, 0) satisfy the simultaneous inequalities?

Shade the region described by the inequalities y > x − 4, y ≥ −x − 4, and y < 1.

Calculate the area of the feasible region.

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ChocDelight produces two types of chocolates: milk chocolate and dark chocolate. Each type of chocolate is made in different machines. The time that each chocolate requires in each machine is detailed in the table. Each machine can operate for 24 hours a day. Let x represent the number of milk chocolates produced each day, and y represent the number of dark chocolates produced each day.

Write an inequality for Machine A.

Write an inequality for Machine B.

Write inequalities for x and y, given the number of chocolates cannot be a negative number.

Graph the region deﬁned by the inequalities for Machine A and Machine B.

T F

Write three inequalities to represent the situation.

Graph the solution to the simultaneous inequalities.

Are these possible dimensions of the bookmark?

A R

A length of 4 cm and a width of 3 cm

A length of 5 cm and a width of 4 cm

iii

A length of 6 cm and a width of 5 cm

A length of 5 cm and a width of 2 cm

If Mirela wanted to create a square bookmark, what are the dimensions of the largest possible bookmark?

Solve the simultaneous linear equations using substitution: a

y = 4x + 40

y = −2x – 11

y = 3x + 30 4x + 2y = −22 18

Machine B 4 hours 6 hours

Mirela has 16 centimetres of leftover cardstock paper that she wants to use to create a rectangular bookmark. Let x and y represent the length and width of the bookmark respectively.

Milk Chocolate Dark Chocolate

Machine A 5 hours 3 hours

2a + b = 23

−3f + 2g = 6

b = −2a + 35 2f + 5g = 53

Use the substitution method to ﬁnd the solution(s) for each pair of simultaneous equations. a

(x + 2)2 + (y − 4)2 = 10

y = 4x + 2

x = −1 y = −x2 + x + 6 19

Solve the simultaneous linear equations using elimination: a

2x + 3y = 8 x − 3y = −14

3x − 6y = 48

x + 2y = 10

x − y = −22.5

Use the elimination method to ﬁnd the solution(s) for each pair of simultaneous equations. a

x2 − y = 9

x2 + y2 = 80

−x + y = 3

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0.1x + 0.2y = 0.3

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A couple runs two restaurants that made a combined profit of $10 000 in the previous financial year, with Fine Diners making 3 times as much profit as Casual Eats. Let x and y be the profits of Casual Eats and Fine Diners respectively.

Write a pair of linear equations to represent this situation.

Find Casual Eats’ proﬁt.

Find Fine Diners’ proﬁt.

Alfred invested $20 000 in total in two bonds, C and D. In one year, the investment in bond C had a 10% return, while the investment in bond D rose by only 4%. The total annual interest from both bonds was $1340. a

Set up two equations by letting x and y be the amounts that he invested in bonds C and D respectively.

Solve the simultaneous equations.

Explain what the solution means in context.

T F

Juma and Jian both cycle from their homes to the gym every evening. Juma cycles 2.2 kilometres further, and together they cycle 5.8 kilometres. Find the distance Jian cycles.

A R

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to functions 7 Introduction (Optional) Big ideas Functions and their properties play a crucial role in understanding and analysing relationships between variables.

Curriculum Outcomes VC2M10AA06 Apply understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation VC2M10AA08 Use function notation to describe the relationship between dependent and independent variables in modelling contexts

Chapter outline 7.01 7.02 7.03 7.04

Optional: Define relations and functions Optional: Domain and range Optional: Graphs of xn Optional: Transformation of functions Chapter 7 review

340 351 363 373 385

Theme parks use functions to analyse visitor patterns and improve experiences!

7.01 Optional: Define relations and functions After this lesson, you will be able to… • understand what a relation is and how it can be represented. • classify relations. • understand what a function is. • write a function equation using function notation. • apply the vertical line test to determine if a graph is a function.

Functions and relations A relation is a relationship between sets of information. For example, consider the names of students in a class and their heights. If given a height such as 162 cm, multiple students might have that height. This data can be expressed as a relation. Relations can be represented using tables, rules, graphs, ordered pairs, or diagrams mapping elements of two different sets. Relations can be classiﬁed by how sets of inputs and outputs are related. The following diagrams show the four types of relations for two sets of x and y-values (inputs and outputs): x 1 2 3

2 4 6

−1 1 0 0 1 4 2 many-to-one

one-to-one

One input relates to one output.

Two or more inputs related to one output.

For example, the relationship y = 2x.

For example, y = x2.

−3 0 −2 0 4 9 2 3 one-to-many One input relates to two or more outputs. An example might be the parabola in y expressed by .

1 2 3

5 7 9

many-to-many Two or more inputs are related to two or more outputs.

A function is a special type of relation where each input corresponds to only one output. This means functions can be one-to-one or many-to-one mappings. This means that for every input, there’s only one output. Functions are often described using dependent and independent variables. For example, in a vending machine selling juice, the cost (dependent variable) depends on the number of juice bottles purchased (independent variable).

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A relation can be shown with a one-to-many or many-to-many mapping. When we are writing in function notation, instead of writing “y =”, we write “f (x) =”. This can be interpreted as “f is a function of the variable x” and read as “f of x”. Common letters to use for general functions are lower case f, g and h. However, any letter can be used and we can use variables that have meaning in context. For example:

f (x) = 3x2 f is the function name f (x) is the function output, or y-value 3x2 is what the function does to the input (x) We can show substitution by replacing the input with a particular value. For example, instead of saying “substitute x = 3 into y = 2x + 1”, we can define f (x) = 2x + 1, then say f (3) means to substitute in x = 3. We say f (3) as “f of 3”. A benefit of using function notation is that we can: • Name functions without needing to use the whole equation • Show substitution easily • Confirm it is a function and not a relation To use function notation, the equation must represent a function and be of the form y =.

Example 1 Does the following set of points describe a function? a {(2, 5), (7, −3), (5, 2), (−4, −9)}

Create a strategy Identify if the sets of points represent a many-to-one, one-to-many, many-to-many or one-to-one relationship.

Apply the idea Let’s analyse the given set of points: {(2, 5), (7, −3), (5, 2), (−4, −9)} For a set of points to be a function, each x-value must have a unique corresponding y-value. • For x = 2, there is only one y-value which is (5). • For x = 7, there is only one y-value which is (−3). • For x = 5, there is only one y-value which is (2). • For x = −4, there is only one y-value which is (−9). Since this demonstrates a one-to-one mapping, it is a function.

b {(2, 5), (2, 7), (−3, −4), (−9, 13)}

Create a strategy If the same input has different outputs, then it is not a function. If different inputs have the same outputs, it can still be a function.

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Apply the idea Let’s analyse the given set of points: {(2, 5), (2, 7), (−3, −4), (−9, 13)}. For a set of points to be a function, each x-value must have a unique corresponding y-value. Let’s check: • For x = 2, there is more than one y-value, which are (5) and (7). • For x = −3, there is only one y-value, which is (−4). • For x = −9, there is only one y-value, which is (13). Since there are x-values with multiple y-values, it is a one-to-many relation. This set of points is not a function.

Example 2 The pairs of values in the table represent a relation between x and y. Do they represent a function? x y

−4 −4

−3 −3

−2 −2

−1 −1

0 0

1 −1

2 −2

3 −3

4 −4

Create a strategy To determine whether the pairs of values in the table represent a function, we need to apply the deﬁnition of a function.

Apply the idea Let’s analyse the given table of values. For the pairs of values to represent a function, each unique x-value must have only one corresponding y-value. The table shows a many-to-one relation between x-values and y-values, as multiple x-values share the same y-value. For example, x = −3 and x = 3 both have a single y-value, which is −3. Given the many-to-one relation between x and y-values, the pairs of values in the table represent a function.

Example 3 Consider the function f (x) = 4x + 4. a Find f (2).

Create a strategy Substitute the x-value into the expression.

Apply the idea f (2) = 4(2) + 4

342

Substitute x = 2

=8+4

Evaluate the multiplication

= 12

Evaluate the sum

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Find f (−5).

Apply the idea f (−5) = 4(−5) + 4

Substitute x = −5

= −20 + 4

Evaluate the multiplication

= 16

Evaluate the sum

Reﬂect and check f (−5) = 16 means that (−5, 16) is a point on the graph of f (x) = 4x + 4.

Idea summary A relation is an association between elements of two sets. It can be represented in various ways, including tables of values and graphs. A function is a special type of relation where each input has only one output. We can deﬁne relations as being: • • • •

One-to-one: one x-value maps to one y-value. This relationship is a function. Many-to-one: many x-values correspond to one unique y-value. This relations is a function. Many-to-many: multiple x-values map to multiple y-values. One-to-many: one x-value may map to multiple y-values.

Vertical line test To determine if a graph represents a function, we use the vertical line test. If a vertical line intersects a graph at more than one point, then the graph does not represent a function. This is because a function is a one-to-one or many-to-one relationship.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.01 to answer these questions. 1.

What type of mapping is the relation?

Is it a function?

How do you know what type of mapping it is and if it is a function or not?

We can see that many-to-one and one-to-one mappings are functions as they would pass the vertical line test (VLT). However, many-to-many and one-to-many are only relations as they would not pass the VLT.

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Relations

y x

many-to-many Functions y

one-to-many y x

many-to-one

one-to-one

Example 4 Do the graphs describe a function? a 4

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy To determine whether the graph represents a relation or a function, we need to apply the vertical line test.

Apply the idea The vertical line test states that for a graph to represent a function, any vertical line drawn on the graph should intersect the graph only once. In this case, when we draw vertical lines on the graph of each line intersects the graph at exactly one point. This mean that this graph is a function.

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b 4

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Create a strategy To determine whether the graph represents a relation or a function, we need to apply the vertical line test.

Apply the idea When we draw vertical lines on the graph, no lines intersect the graph at more than one point. Since the vertical line test is satisﬁed for all points on the graph, the second graph represents a function.

Reﬂect and check This is a function, but it is not one-to-one as two points may have the same y-values for different x-values.

7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

Create a strategy To determine whether the graph represents a relation or a function, we need to apply the vertical line test.

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Apply the idea When we draw vertical lines on the graph, the lines intersect the graph at more than one point. Since the vertical line test is not satisﬁed for all points on the graph, the graph does not represent a function.

Idea summary The vertical line test is a graphical method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then it’s not a function.

Practice questions Choose your question path Explorer 1a-g, 3, 6, 7a, d, 10, 13ab, 14, 17a,c,e,g,i, 18, 20, 21, 22, 25, 29, 32

Adventurer 1, 2, 4, 6, 7, 8, 11, 13, 15, 16ab, 17b,d,f,h,j, 20, 21, 23, 27, 30, 32

Trailblazer 1, 2, 5, 6, 9, 12, 13, 15, 16, 17a,d,e,g,j, 19, 20, 21, 24, 26, 28, 30–33

Understanding 1

Are the statements true or false? a

When working with a function, substituting a certain value of x into the equation gives only one value of y for that value of x.

A horizontal line can intersect the graph of a function at more than one point.

All relations are functions.

Some relations are functions.

All functions are relations.

No functions are relations.

No relations are functions.

A relation always passes the vertical line test.

The graph of every straight line is a function.

The graph of every non-vertical straight line is a function.

There is no straight line graph that is a function.

The graph of every straight line is a relation.

m The graph of every non-horizontal straight line is a function. 2

Imagine you have a machine that represents a function. If you feed an item into the machine, it gives you an output. Can you get different outputs for the same input? Why or why not?

For a function f, what does f (3) represent?

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Suppose that (7, −6) is an ordered pair that satisﬁes the function g. Write this situation using function notation.

Suppose that the value of the function f is 22 when x = 7. Write this situation using function notation.

Fluency Ex 1

Ex 2

Does each set of points describe a function? a

{(1, 6), (2, 12), (3, 18), (4, 24)}

{(10, 2), (2, 4), (3, 5), (10, 3)}

{(−2, 4), (−1, 4), (0, 5), (1, 3)}

{(−2, −3), (−1, 0), (0, 2), (1, 4), (1, −1)}

Is each table a function? a

x y

−4 −4

−3 −3

−2 −2

−1 −1

0 0

x y

−2 1

−2 2

0 3

1 4

2 3

2 4

3 3

4 4

5 5

x y

−9 10

−7 10

−6 10

−5 10

−3 10

−2 10

3 10

5 10

10 10

1 1

2 2 12

3 3 16

4 4 18

For which of these values of k does the relation {(2, 1), (7, k), (5, 8), (3, 9)} not represent a function? A

k=7

k=8

k=9

o such value of N k exists

Find a value of k such that the relation {(6, 2), (8, 5), (1, 7), (k, 4)} does not represent a function.

Given the points in the table. i

Plot the points on the Cartesian plane.

Is this relation a function?

x y

−4 4

−3 3

0 2

−1 1

0 0

1 1

2 2

3 3

x y

−3 0

−2 1

1 2

2 3

3 4

4 3

5 4

6 5

4 4

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A relation is deﬁned as y = 1 if x is positive and y = −1 if x is 0 or negative. a

Complete the table. x y

−4

−3

−2

−1

Plot the points on the Cartesian plane.

Is this relation a function?

Given these ordered pairs: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}

Ex 3

Plot the ordered pairs on the Cartesian plane.

Is this relation a function?

Which ordered pair would need to be removed from the set so that the remaining ordered pairs represent a function?

Do the equations describe a function? a

y = 9x

y = x2 + 2

y2 = x

y = −x3

3x + 5y = 12

f (−4)

y2 + x2 = 25

5x − 5y + 1 = 0

If f (x) = 3x − 1, ﬁnd: a

f (3)

Consider the function f (x) = 2x2 − 2x + 5. Evaluate

For each of the following functions, evaluate the function outputs when:

x = 2

x = −3

f (x) = 4x + 1

g(x) = (x + 1) (x + 2)

c Ex 4

h(x) = x − 3x − 2

Is this graph a function? a

b 4

4 3

1 −4 −3 −2 −1 −1

348

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

1 −4 −3 −2 −1 −1

−2

−3

−4

−4 −3 −2 −1 −1

−2

−3

−4

4 3

2 1 −4 −3 −2 −1 −1

−2 −3 −4

4 3

2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

4 y 3 2 1 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11

x 1

−4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

x 1

7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

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CheapCalls Mobile charges $1.10 a minute plus a connection fee of 70c for an international call. a

Complete the table. Call length (minutes) International call cost (dollars)

b 19

Is this relation a function?

A particular store is offering one free T-shirt for every two T-shirts purchased, where each T-shirt costs $22. a

Complete the table. Number of T-shirts (x) Total cost

b 20

Is this relation a function?

The model P (t) = 3500 × 2t describes the population of fish in a lake after t years, where t = 0 is the initial time when the population of fish was first recorded. Find and interpret the value of P (2).

Reasoning 21

Given the ordered pairs (a, b), (c, d), (e, f ), if this set represents a function, what can you infer about the values of a, c, and e?

Sharon works at a pizza shop and makes $15 per hour. Her wages can be represented by the function W (t) = 15t, where t represents the number of hours that she works. a

Explain what the expression W (19) represents.

Explain what the equation W (t) = 165 represents.

Find her wages after working 26 hours.

A bookstore offers a yearly membership for $120 which includes unlimited book rentals. Additionally, the bookstore provides an option for non-members to rent a book for $5 per book. Is the relation between the number of books rented and the total rental cost a function? Explain why or why not.

Imagine a cinema where each seat corresponds to a unique ticket number. If the seating chart represents a function, what can you infer about any two people having tickets with the same number?

John lists his pizza topping preferences: • For Margherita, he likes olives. • For Pepperoni, he likes jalapeños. • For Vegetarian, he likes bell peppers. Does this represent a function? Justify your answer.

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You have a coffee machine that gives speciﬁc types of coffee for each button pressed. If pressing the “Espresso” button always gives an espresso shot and pressing the “Latte” button always gives a latte, can this machine’s operation be described by a function? Why or why not?

In a small town, every family owns at least one car, but multiple families can own the same type of car. Is the relation between families and the type of car they own a function? Explain.

The local bakery uses a set of ingredients (flour, sugar, eggs) to make a set of pastries (cakes, cookies, muffins). If each pastry type requires all three ingredients but in different quantities, explain whether the relation of ingredients to pastries is a function or not.

Problem-solving 29

Consider the functions f (x) = 7x + 2 and g(x) = 15 − 2x. Find the value of the following: a

f (1) + g(2)

f (−2) − g(7)

f (0) × g(0)

For each of the following functions, evaluate the function outputs when x = a + 1. a

f (x) = 4x + 1

h(x) = x2 − 3x − 2

g(x) = (x + 1) (x + 2)

Consider the function f (x) = x2 + 2x + k. a

Simplify the expression for f (2).

If f (2) = 15, ﬁnd the value of k.

You have two sets of numbers. Set A: 1, 2, 3 and Set B: 4, 5, 6. Design a relation between these sets such that it is a function. Then add one more ordered pair to make this relation no longer a function.

Given the relation (2, 4), (3, 9), (4, 16), (5, 25), determine if this represents a function. If yes, brieﬂy explain the real-world scenario it could depict.

7.02 Optional: Domain and range After this lesson, you will be able to… • use algebraic methods to determine the domain and range. • utilise graphs to identify domain and range. • distinguish between the domain and range in different types of functions.

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Domain and range of a function The domain is the set of all allowable input values (usually x-values) for which the function is deﬁned. A function is deﬁned for all values of x except where: • The denominator of a fraction is zero. • There is a negative value under an even root. • A logarithmic function has an argument which is negative or zero. The domain may also be found from the graph of a function. The domain is all x-values for which the function has a corresponding y-value.

Domain −3

−2

−1

1 0 −1 −2 −3 −4 −5

x 2

In the example shown, the domain is the interval of values −3 < x ≤ 1. Notice that −3 is not included in the domain, which is indicated by the unﬁlled point.

The range is the set of all possible output values (usually y-values) as x varies over the domain of the function. When considering the range of a function, we must remember that: • Even powers and even roots always result in non-negative outputs. • The minimum and maximum values of a function will limit its range. Just like the domain, the range can be determined either algebraically or from the graph of the function. 2

1 −3

−2

−1

0 −1 −2 −3 −4 −5

x 2

In the example shown, the range is the interval of values −4 ≤ y ≤ 0. Notice that both endpoints are included in the range, since the function reaches a height of y = 0 at the origin.

Range

The domain and range of a function can vary depending on the type of function. For example, the domain and range of a linear function are all real numbers. However, the range of a quadratic function is limited by the location of the vertex. The table gives the domain and range for some common types of functions:

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Function type Linear Quadratic

Equation y = mx + c y = ax2 + bx + c

Rational

Domain All real numbers All real numbers

Range All real numbers If a is positive, the range is y ≥ (vertex’s y-value) If a is negative, the range is y ≤ (vertex’s y-value) Typically needs more in-depth analysis. Watch for horizontal asymptotes for clues. If n is even, range is y ≥ 0 If n is odd, range is all real numbers.

All real numbers except where p (x) = 0 If n is even, domain is x ≥ 0 If n is odd, domain is all real numbers. y = ax where a > 0 All real numbers and a ≠ 1 x>0 y = loga(x) where a > 0 and a ≠ 1

Root

Exponential Logarithmic

y>0 All real numbers

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.02 to answer these questions. 1.

Is the lower bound of the domain always the minimum value of the function?

Does the upper bound of the domain always occur at the right-most point?

Explain how to find the range from a graph.

Example 1 State the domain of y =

Create a strategy Identify value(s) of x for which the function is undeﬁned, and write an inequality that excludes these value(s).

Apply the idea The function is undeﬁned when the denominator is equal to zero. x−3≠0 x≠3

Write an inequality for where the function is deﬁned Add three to both sides to solve for x

So the domain is x ≠ 3.

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Reﬂect and check The domain of x ≠ 3 means that a y-value can be calculated for any real x-value except x = 3. We can check our answer by graphing the function using technology:

4 3 2

The graph shows that there is a vertical asymptote at x = 3. This means that the function is undeﬁned at x = 3.

1 −1

x 1

−1

−2

1 x−3

−3 −4

Example 2 The graph of f (x) =

is shown.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

a What is the domain of the function? Express as an inequality.

Create a strategy

Apply the idea

Use the graph to identify values of x for which the function is deﬁned.

We can see that the graph starts at x = −1 and extends indeﬁnitely to the right. 4

3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

So, the domain is x ≥ −1.

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x 1

Domain

b Is there a value in the domain that can produce a y-value of −2?

Create a strategy Find the x-value that, when substituted into the function, produces a value of −2. Remember, a square root always produces non-negative values.

Apply the idea The square root function always produces non-negative outputs. It’s impossible for the function to produce a value of −2 since that’s a negative value. So, there’s no x-value in the domain that will produce an output of −2 for the function

Example 3 State the range of y = x2.

Create a strategy Think of the graph of this function.

Apply the idea

Reflect and check

The graph of the function is an upward-facing parabola with its vertex (turning point) at the origin (0, 0). The minimum value of y is 0, and y increases to positive inﬁnity.

Alternatively, we can use the fact that even powers always result in non-negative outputs: x2 ≥ 0 which means that y ≥ 0.

y = x2

3 Range

2 1

−4 −3 −2 −1

−1

x 1

−2 −3 −4

So the range is y > 0.

Example 4 State the domain and range of y = 3x.

Create a strategy Consider the graph of the function.

7.02 Optional: Domain and range mathspace.co

355

Apply the idea

Reﬂect and check As shown in the table given earlier in the lesson, all exponential functions have a domain of all real numbers, and a range y > 0.

y 75

25 x −4 −3 −2 −1

On the x-axis, the graph extends from negative infinity to positive infinity. So, the domain is all real numbers. The y-value is always positive. As x approaches negative infinity, y approaches 0 (but never equals 0). As x approaches positive infinity, the function approaches infinity so, the range is y > 0.

Idea summary The domain is all acceptable input values for a function. Certain conditions, like zero denominators or negative values under even roots, limit the domain of functions. The range is the set of possible output values (usually y-values) as x varies over the domain of the function.

Practice questions Choose your question path Explorer 1−3, 4ab, 5, 6bc, 7b,d,g, 8a,d, 9, 11, 12, 15a,d,f, 16, 19, 20, 22, 23, 26, 28, 29

Adventurer 1−3, 4, 5, 6, 7a,d,h, 8a,c, 10, 11a, 13, 14b,d,f,h, 15b,c,e,g, 16, 17, 19, 20, 21, 24, 26, 27, 28, 29

Trailblazer 1−3, 4, 5, 7, 8cd, 11b, 14adg, 15, 18, 19, 20, 21, 25, 26, 27, 28, 29

Understanding 1

356

If a function’s equation does not imply any restrictions on x, what can we say about its domain?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

What can affect the domain of a function?

True or false: a

The domain of a rational function is the set of all real numbers except for those that make the denominator 1.

If the graph of a function lies entirely above the x-axis, then its range is only positive y-values.

Fluency Ex 1

State the domain of each function. a

Ex 2

The function

is shown. 4

3 2 1

−1 −1

−2 −3 −4

What is the domain of the function?

Is there a value in the domain that can produce a y-value of −2?

For each of the given graphs, state the domain. a

1 −4 −3 −2 −1

−1

x 1

1 −7 −6 −5 −4 −3 −2 −1 −1

−2

−3

−4

x 1

7.02 Optional: Domain and range mathspace.co

357

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

3 2 1 x

−4 −3 −2 −1 −1

2 4 6 8 10

−2 −3 −4 −5

For the given graphs: i

State the domain.

−4 −3 −2 −1

State the range.

1 −1

−7 −6 −5 −4 −3 −2 −1 −1

−2

x 1

−2

−3

−4

1 −7 −6 −5 −4 −3 −2 −1 −1

1 2 3

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−2

358

x 1

f 4

4 3

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2

−3

−4

10 8 6 4 2

x 1

−1

−2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

x 1

6 4 2

−12−10−8−6−4−2 −2

2 4 6 8 10

x 2 4 6 8 10 12

−4 −6 −8

For the given functions: i

Plot the graph of the function.

What is the domain of f (x)?

iii

What is the range of f (x)?

f (x) = −x + 1

f (x) = 3x − 1

Given the graph of the function y = f (x): a

What is the absolute maximum of the graph?

What is the range of the function?

What is the domain of the function?

f (x) = −4

14 12 10 8 6 4 2 −9−8−7−6−5−4−3−2 −1 −2

x 1 2 3

7.02 Optional: Domain and range mathspace.co

359

Given the graph of the function y = f (x): a

What is the absolute minimum of the graph?

What is the range of the function?

What is the domain of the function?

80 70 60 50 40 30 20 10 −8−6−4−2

For the quadratic function that passes through the points shown in the table of values: i

Does the function have a maximum or minimum point?

What is the minimum value of the function?

iii

What is the range of the function?

x y

1 17

x y

−5.5 −12

6 7

−4.5 −4

−4 −3

−3.5 −4

7 17 −3 −7

−2.5 −12

0 ⬚

1 ⬚

2 ⬚

3 ⬚

4 ⬚

Graph the function.

Is there a value in the domain that can produce a y-value of −9?

Given the parabola deﬁned by the equation y = x2 + 5: a

Is the parabola concave up or concave down?

What is the y-intercept of the parabola?

What is the minimum y-value of the parabola?

What is the range of the parabola?

State the range of the functions: y = x2 + 1 y = 2x

y = x2 − 7

y=x +9

y = −x2

y = 4 − x2

y = −x − 5

y = (x + 1)2

y = log2 x

State the domain and range of the functions: b

a e

360

5 1

e 15

−5 −7

4 −1

Ex 4

3 1

Complete the table of values. x y

Ex 3

2 7

Given the function f (x) = (x − 2)2, for all real x. a

x 2 4 6 8 10 12 14 16 18

y = 2x

c y = 3x + 4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

y = log(x − 1)

The function

has been graphed.

What is the domain of the function?

What is the range of the function?

4 2 x −6 −4 −2

−2 −4 −6

Given the function y = x − 1. a

Complete the table of values. −2 ⬚

x y

−1 ⬚

0 ⬚

1 ⬚

2 ⬚

Sketch the graph of the function.

Is there a value in the domain that can produce a y-value of

State the domain and range for: a

y = x3 + 2x2 − 1

Reasoning 19

If the range of a function is given as all non-negative values, does that imply that the function’s graph is always above the x-axis? Justify your answer.

Explain why the domain of the function

A factory produces a number of items, y, as a function of hours worked, x, given by y = 100 − x2. Why can’t the factory work more than 10 hours?

A theme park’s popularity y is modelled by the function y = 2x, where x is the number of years since its opening. What can we say about the popularity of the park over time?

is all real numbers.

Problem-solving 23

The function f is used to determine the area of a square given its side length. a

Which of the following values is not part of the domain of the function? A

−8

6.5

For n ≥ 0, what is the area function for a side length of n?

Plot the graph of the function f.

A whale jumps above the surface of the ocean and then dives back under. Its height above the surface can be sketched on a set of axes where x represents time and y represents distance above or below the surface. State the domain of the graph. 7.02 Optional: Domain and range mathspace.co

361

A function f (n) describes the maximum height of a ball dropped from a certain height after the nth bounce. What is the domain of the function?

The graph of a function is shown.

What is the range of the function?

150

A ball is thrown from an apartment window in a high-rise building. The height of the ball above ground, y, over time, x, can be modelled using the function shown in the graph. If the ball is thrown at x = 0, what is the range of the function in this context?

125 100 75 50 25 −8 −6 −4 −2 −25

x 2 4 6 8 10

A comet is travelling through the solar system. It passes by the Earth before curving around the sun and heading back out into deep space. The distance of the comet from Earth, y, is tracked by satellite telescopes and expressed as a function of time, x, since the comet was ﬁrst spotted.

What is the domain of this function?

The satellites detect that, at its closest point, the comet was a distance of 0.14 AU (astronomical units) from Earth. What is the range of the function?

The number of bees in a colony is initially measured and left to form a hive. The number of bees in the hive is measured each day over the course of one week. The function H (x) = 30 (3)x is found to model the number of bees, H, after x days.

What is the domain of the function?

What is the initial number of bees in the colony?

An industrial process involves heating a material from 25 °C to 80 °C in order to remove impurities. Using the standard approach, the temperature of the material over time is given by the function T = −1t (t − 16) + 25. A graph of the temperature proﬁle is shown, where temperature T is given in degrees Celsius (°C) and time t is given in minutes. 90

T°C

80 70 60 50 40 30 20 10

t (minutes) 1

362

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

What is the domain of this function?

What is the range of this function?

A new heating process is being developed that uses more energy but takes less time.

This regime is based on a different quadratic function, and the table shows the theoretical value of the temperature at different points in time using the new function. t T

0 25

1 45

3 80

4 96

30 96

31 80

32 63

33 45

34 25

35 4

Over what domain does this new function have the same range as the standard process? d

Which of the two domains found in part (c) should be used for the new heating process?

7.03 Optional: Graphs of xn After this lesson, you will be able to… • Identify cubic functions and explain their characteristics. • complete tables of values for given cubic functions to understand their behaviour. • sketch graphs of cubic functions and polynomial curves using data points and equations. • understand and describe the end behaviour of polynomial functions with even and odd degrees. • explore the effects of the degree and leading coefficient on a polynomial curve’s shape.

Cubic function We have seen linear and quadratic functions that had an x and x2 term respectively. We call the highest power the degree of a polynomial function. For example: Equation y=2 y = 3x y = −9x2 y = 6x3

Degree 0 1 2 3

The cubic function, represented by y = kx3 + c, is a type of polynomial function where the highest degree is 3. The graph of a cubic function can have different shapes based on the values of k and c.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.03 to answer these questions. 1.

Describe the shape of y = x3, including what’s happenning at the origin.

Describe what happens to the graph of y = ax3 for a > 1, a < 0 and 0 < a < 1.

Describe the effect of c on the graph of y = ax3 + c.

7.03 Optional: Graphs of xn mathspace.co

363

The cubic function, y = x3, has a point of inflection at (0, 0) where the steepness of the curve flattens out before beginning to get steeper again. The shape of the basic curve y = x3 is always increasing. We can reflect, dilate, and translate this basic function.

Example 1 Sketch the graph of y = 3x3 − 5.

Create a strategy Let’s explore the shape by understanding its transformations.

Apply the idea We will start with the graph of the basic curve y = x3. 8

y = x3

6 4 2

−4 −3 −2 −1 −1

−2 −3 −4

We vertically dilate the graph by a factor of 3, which makes it steeper. It is vertically translated down by 5 units. This means the point of inﬂection will be at (0, −5). 8

y = x3

6 4 2 −4 −3 −2 −1 −1 −2 −3

x 1

y = 3x3 − 5

−4

Reﬂect and check This graph is narrower and shifted down from the basic graph of y = x3.

364

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary The cubic function, y = x3, has a point of inflection at (0, 0) where the steepness of the curve flattens out before beginning to get steeper again. The shape of the basic curve y = x3 is always increasing. We can reflect, dilate, and translate this basic function.

Graphs of xn The shape of the general polynomial y = kxn changes based on the degree, n. We can identify similarities and differences based on the degree.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.03 to answer these questions. 1.

Determine the end behaviour of each function based on the degree and leading coefficient. Fill in the blanks in the ‘End behaviour’ column of the table. Degree

Leading Coefficient

Even

Positive

Even

Negative

Odd

Positive

Odd

Negative

End Behaviour As x → −∞, f (x) → ⬚ As x → +∞, f (x) → ⬚ As x → −∞, f (x) → ⬚ As x → +∞, f (x) → ⬚ As x → −∞, f (x) → ⬚ As x → +∞, f (x) → ⬚ As x → −∞, f (x) → ⬚ As x → +∞, f (x) → ⬚

Graph of the function ⬚ to the left and ⬚ to the right ⬚ to the left and ⬚ to the right ⬚ to the left and ⬚ to the right ⬚ to the left and ⬚ to the right

After finding the end behaviour, state whether the graph rises or falls to the left and to the right. Fill in the blanks with “rises” or “falls” in the ‘Graph of the function’ column of the table from the previous question.

What patterns do you notice in the end behaviour of functions with an even degree versus functions with an odd degree?

What happens when k = −1?

For a function in the form y = kxn, where n is a positive integer, we call k the leading coeﬃcient and n the degree. Based on the degree and leading coeﬃcient, we can identify some trends on the end behaviour of the function, where the function values are increasing/decreasing, and where the function values are positive/negative.

7.03 Optional: Graphs of xn mathspace.co

365

Positive leading coeﬃcient, even degree, for example: y = 3x6

Negative leading coeﬃcient, even degree, for example: y = −2x8

Positive leading coeﬃcient, odd degree, for example: y = 8x5

Negative leading coeﬃcient, odd degree, for example: y = −7x9

Between these ends, y = kxn will have a turning point if n is even and a point of inflection of n is odd. The higher the power of n, the flatter the point of inflection or turning point will be. y

Odd degree and n = 3

366

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Odd degree and n = 9

Even degree and n = 2

Even degree and n = 10

Example 2 Consider the function y = x7. a Describe the end behaviour of the function.

Create a strategy To describe the end behaviour of a polynomial function, we look at the degree of the polynomial and the sign of the leading coeﬃcient.

Apply the idea y

The given function is y = x7, which has an odd degree (7) and a positive leading coeﬃcient (1). The end behaviour is: • As x → −∞, y → −∞ • As x → +∞, y → +∞

b Sketch the graph of the function.

Create a strategy We can sketch the graph of the function by sketching a curve that reflects the described end behaviour and the properties of an odd-degree polynomial with a positive leading coefficient. Since n = 7, the point of inflection will be fairly flat.

7.03 Optional: Graphs of xn mathspace.co

367

Apply the idea

Reﬂect and check 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

We can also graph using technology.

x 1 2 3 4 5

Idea summary The degree of a polynomial and the leading coeﬃcient determine the end behaviour. •

If the degree is even, the end behaviour tends toward the same direction • Positive leading coeﬃcient: both ends tend toward +∞. • Negative leading coeﬃcient: both ends tend toward −∞.

•

If the degree is odd, the end behaviour tends toward opposite directions • Positive leading coeﬃcient: the left side falls toward −∞ and the right side rises toward +∞. • Negative leading coeﬃcient: the left side rises toward +∞ and the right side falls toward −∞.

Practice questions Choose your question path Explorer 2, 3, 5, 6a-d, 8, 9, 10, 11a,c,e,g, 14, 15, 16, 17, 19, 20a-c

Adventurer 1, 2, 3, 5, 7a-c, 9, 10, 11b,d,f,h, 12a,c,e,g, 13a,c,e,g, 15, 16, 17, 19, 20a-c, 21

Understanding 1

For the general equation y = kxn what does n represent?

For y = x5, is the function increasing or decreasing?

368

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 1, 2, 3, 4, 5, 6e-h, 7d-f, 9, 10, 12b,d,f,h, 13b,d,f,h, 15, 16, 17, 18, 19, 20, 21

Is each statement true or false for functions of the form y = kxn? a

This graph could have a positive leading coeﬃcient and an odd degree.

This graph could have a negative leading coeﬃcient and an even degree.

If k = −3, then an even degreed function would open upwards.

If k = 2 and n = 9, then x → ∞ as y → −∞.

If n = 9, then there would be a point of infection at (0, 0).

The graph of y = 3x6 would have a ﬂatter turning point than y = 3x2.

Fluency Ex 1

Sketch the graph of y = 2x4.

For each of the given graphs: i

Is the degree odd or even?

What are the intervals where the function is increasing and decreasing?

iii

Describe the end behaviour of the function.

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

3 2 1

x −4 −3 −2 −1

1 2 3 4 5

−1

x 1

−2 −3 −4

What is the end behaviour of the graph of f (x)? a

f (x) = x6

f (x) = −x7

f (x) = −x8

f (x) = 2x5

f (x) = −3x4

f (x) = 4x3

f (x) = −5x3

f (x) = 3x2

7.03 Optional: Graphs of xn mathspace.co

369

For each of these graphs of the function y = f (x): i

Is the degree odd or even?

Determine the behaviour of the function as x → ∞.

iii

Determine the behaviour of the function as x → −∞.

−5 −4 −3 −2 −1 −2 −4 −6 −8 −10

8 6

x 1 2 3 4 5

4 2 x −4

1 2 3 4 5

−4 −6 −8

3 2 1 x 1

−1

370

1 2 3 4 5

−2

−5 −4 −3 −2 −1

Ex 2

−1

−2

10 8 6 4 2

−3 −2

y 10

−5 −4 −3 −2 −1 −2 −4 −6 −8 −10

10 8 6 4 2

Consider the function y = x5. a

Describe the end behaviour of the function.

Sketch the graph of the function.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

Consider the functions y = x2 and y = x4. a

Sketch the graphs of the functions on the same Cartesian plane.

For even powered functions in the form y = xn, desrcibe what you notice as n increases.

The function f (x) = x3 is transformed into the function g (x) = kx3. a

Determine the values of k that will g (x) result in a graph of f (x) that is wider and reﬂected across the x-axis.

Choose an appropriate value for k based on your previous answer, and sketch the graph of the functions f (x) = x3 and g (x) = kx3 on the same Cartesian plane.

Sketch each graph using technology. a

y = 2x2

y = −4x3

y = −0.5x7

y = 8x8

y = 7x5

y = −14x10

y = −2x5 + 6

y = −4x10 + 1

y = −7(x − 6)5

y = 14(x + 2)10

Sketch each graph using technology. a

y = 2x2 + 5

y = 5x4 − 4

y = −0.25x7 + 1

y = 12x8 − 5

y = −7x3 − 3

Sketch each graph using technology. a e

y = −9x4

y = −3(x + 1)2

y = 4(x − 3)3

y = −8(x − 7)8

y = 9(x + 9)4

Sketch the graph of these functions on the same Cartesian plane: y = x3 and y = x3 − 4

The given graph models the revenue earned over the ﬁrst year of a small business launching. 40 000

Revenue in $ ( y)

35 000 30 000 25 000 20 000 15 000 10 000 5000

Time in months (x) 1 2 3 4 5 6 7 8 9 10 11 12

Identify when the revenue is increasing.

Identify when the revenue is negative.

Predict the revenue after one year.

Predict when the revenue reaches $25 000.

7.03 Optional: Graphs of xn mathspace.co

371

Reasoning 16

The given graph shows the population of crows at a local park over 15 weeks. a

Estimate when the number of crows is decreasing.

Number of crows ( y)

80 70 60

Predict the number of crows after 10 weeks.

Predict when the number of crows will reach 70.

Describe what happens as the number of weeks increases indeﬁnitely, as x → ∞. Explain if this is reasonable.

50 40 30 20 10

Time in weeks (x) 2

Explain how the end behaviour of a function is determined by the leading term.

Florin is using the following graph to describe information about the equation of the polynomial function. Florin states that there is not enough information to determine anything about the equation of the function. Determine whether Florin is correct and explain your reasoning.

y 8 6 4 2

−8 −6 −4 −2 −2

2 4 6 8

−4 −6 −8

Problem-solving 19

The volume of a sphere has the formula

The graph relating r and V is shown. a

Complete the table of values, rounded to two decimal places. r V

372

A sphere has radius measuring 3.5 m. Determine the exact interval that the volume of the sphere falls within. According to the graph, what is the radius of a sphere of volume 288π m3?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

324π

V (cubic metres)

288π 252π 216π 180π 144π 108π 72π 36π

r (metres) 1

A cube has side length 5 cm and a mass of 375 g. The mass of the cube is directly proportional to the cube of its side length. a

Let k be the constant of proportionality for the relationship between the side length, x, and the mass, m, of a cube, such that m = kx3. Solve for k.

Now state the equation relating the mass and side length of the cube.

Sketch a graph of this equation.

Find the mass of a cube with side length 6.5 cm, to the nearest gram.

A cube has a mass of 576 g. Determine what whole number value its side length is closest to.

The mass in grams, M, of a cube of cork is given by the formula M = 0.28l3, where l is the side length of the cube in centimetres. a

What is the mass of a cubic centimetre of cork to the nearest 0.01 of a gram?

Complete the table of values, to the nearest 0.01 of a gram if necessary. Side length(cm) Mass (g)

Sketch the graph of the function.

7.04 Optional: Transformation of functions After this lesson, you will be able to… • define what a transformation of a function is and identify the different types of transformations. • understand and explain the concepts of shifts, stretches, compressions, and reflections. • analyse the effects of transformations on the graph of a function. • apply the knowledge of transformations to change the position or shape of a function’s graph. • identify and describe the transformations applied to a given function.

Transformation of functions A transformation of a function is a change in the position or shape of its graph. Transformations can be categorised as translations (shifting the graph up, down, left, or right), dilations (stretching or compressing the graph), or reflections (creating a mirror image of the graph across a line). In this lesson, we will explore different types of transformations and how they affect the graphs of various functions. A translation is a transformation in which every point in a figure is moved in the same direction and by the same distance. A dilation is a transformation in which every point in a figure is scaled by the same factor.

7.04 Optional: Transformation of functions mathspace.co

373

A reﬂection is a transformation that produces the mirror image of a ﬁgure across a line. Let’s consider the quadratic function y = a(x − h)2 + k. This function represents a parabola with vertex (h, k) and a vertical stretch or compression factor of ‘a’.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.04 to answer these questions. 1.

What happens to the graph when c is positive or negative?

What happens to the graph when b is positive or negative?

What happens to the graph when k is negative?

What happens to the graph when a is negative?

What happens graph when |k| > 1?

What happens to the graph when 0 < |k| < 1?

What happens to the graph when 0 < |a| < 1?

What happens to the graph when |a| > 1?

As we have experimented with applets and summarised the effects of a, h and k for each type of function, did you notice that each variable always had the same effect? We found that: • a dilates the graph by a factor of a from the x-axis. • When a < 0 the graph was reﬂected over the x-axis. • h translates the graph h units horizontally. • k translates the graph k units vertically. These transformations work in the same way for a general function f (x). Let’s review and summarise them. Transformation −f (x) af (x) f (x) + k f (x − h) f (−x)

Effect on graph reflects f (x) across the x-axis dilates f (x) by a factor of a from the x-axis translates f (x) by k units vertically translates f (x) by h units horizontally reflects f (x) across the y-axis

Effect on coordinates multiplies y-coordinate by −1 multiplies y-coordinate by a adds k to each y-coordinate adds h to each x-coordinate multiplies x-coordinate by −1

f (ax)

dilates f (x) by a factor of

divides x-coordinate by a

374

from the y-axis

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 If the graph of y = −x2 is translated horizontally 5 units to the right and translated vertically 6 units upwards, what is its new equation?

Create a strategy A horizontal translation involves moving to the left or right, where right is the positive direction and left is the negative direction. A vertical translation involves moving to up or down, where up is the positive direction and down is the negative direction.

Apply the idea When a graph of y = f (x) is translated horizontally by h units to the right, its new equation is y = f (x − h). When a graph of y = f (x) is translated vertically by k units upwards, its new equation is y = f (x) + k. The new equation is y = −(x − 5)2 + 6.

Example 2 Determine the coordinates of the corresponding point after the given transformations: a Point A is (5, 2). The transformation is y = f (x + 4) − 2.

Create a strategy

Apply the idea

Start inside the bracket and work outwards and for each transformation consider if it is affecting the x-values or y-values.

First x + 4 means 4 is subtracted from the x-value. Then the −2 means subtract 2 from every y-value. Apply each transformation to the coordinates in order. The new coordinates are (1, 0).

b Point A is (−3, 2). The transformation is

Apply the idea First x + 2 means 2 is subtracted from the x-value. Then the

means divide every y-value by 3.

Apply each transformation to the coordinates in order. The new coordinates are

7.04 Optional: Transformation of functions mathspace.co

375

Idea summary Here is a summary table demonstrating how transformations apply to a general function f (x). Transformation −f (x) af (x) f (x) + k f (x − h) f (−x)

Effect on coordinates multiplies y-coordinate by −1 multiplies y-coordinate by a adds k to each y-coordinate adds h to each x-coordinate multiplies x-coordinate by −1

f (ax)

dilates f (x) by a factor of

divides x-coordinate by a

from the y-axis

Practice questions Choose your question path Explorer 1, 2, 4d-f, 5, 7a, 9a, 10, 11a, 12, 13a, 14a-d, 15c, 18, 19, 20, 21, 22, 26, 28, 29

Adventurer 1, 2, 4, 5, 6a,c, 7ab, 9, 11ad, 12, 13b, 14b-f, 15b, 16, 18, 19b,d, 20, 21, 24, 26, 28, 30

Trailblazer 1, 2, 3, 4, 5, 6, 7cd, 8, 9, 11bc, 13cd, 14e,f,g, 15a, 17, 18, 19a,c, 20, 21, 24, 26, 27, 28, 31, 32

Understanding 1

Which transformation represents a reﬂection across the y-axis? A

376

y = −f (x)

y = f (x) + 2

y = 2f (x)

y = 0.5f (x)

y = f (0.5x)

y = f (x) + 0.5

y = 2f (x)

y = x3

Which function remains unchanged after reﬂecting across the y-axis? A

Which transformation corresponds to a vertical compression by a factor of 0.5? A

y = f (−x)

y = x2

y=x

y = −x

State whether the following statements are true or false. a

A transformation in the form y = f (ax) where a > 1 will horizontally compress the graph.

If y = f (x) is transformed to y = f (x) − 3, the resulting graph will move upwards by 3 units.

A transformation of the form of the graph.

The transformation y = 3 + f (x) will shift the graph of y = f (x) three units downwards.

For the transformation y = f (x) − 4, the maximum and minimum values of the function (if they exist) will also decrease by 4.

y = −f (−x) will reﬂect the graph of y = f (x) ﬁrst across the y-axis and then across the x-axis.

where a > 1 will result in a horizontal compression

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fluency 5

For each equation, describe the type of transformation from f (x) to g (x): a

g (x) = f (x + 4)

g (x) = 6f (x)

g (x) = −f (x)

If the x-intercepts of the graph of y = f (x) are (−5, 0) and (6, 0), determine the x-intercepts of: a

g (x) = f (x) − 2

y = f (x + 4)

y = f (x − 4)

y = 3f (x)

y = f (−x)

For each table: i

Describe the type of transformation from f (x) to g (x).

Write an equation for g (x) in terms of f (x) in the form g (x) = f (x) + k or g (x) = af (x) depending on the transformation.

f (x) 2 5 8 11 g (x) 4 7 10 13

14 16

f (x) g (x)

1 3

2 6

3 9

4 12

5 15

f (x) −2 −1 0 1 g (x) −4 −3 −2 −1

2 0

f (x) g (x)

4 −2

6 −3

8 −4

10 −5

12 −6

Ex 1

The graph of is translated horizontally 6 units to the right and vertically 5 units upwards. State the new equation after the transformations are applied.

Ex 2

Point A lies on the graph of y = f (x). Determine the coordinates of the corresponding point after the given transformation:

Point A is (−3, −1). The transformation is y = f (x − 5) − 2.

Point A is (0, 3). The transformation is

Point A is (1, −2). The transformation is y = f (3x) − 2.

Point A is (12, 3). The transformation is y = 6f (6x).

Consider the graph of the function f (x). Select the transformed function g (x) deﬁned by adding 2 to the x-values. A

g (x) = (x + 2)3

g (x) = (x − 2)3 3

g (x) = x + 2

g (x) = x3 − 2

5 4 3 2 1 −5 −4 −3 −2 −1 −1 y = f (x) −2 −3 −4 −5

x 1 2 3 4 5

7.04 Optional: Transformation of functions mathspace.co

377

For each function, describe the transformation applied to f (x) that is required to produce the graph shown: a

f (x) = x

f (x) = ∣x∣

y 4

1 −4 −3 −2 −1

−1

x 1

4 −5 −4 −3 −2 −1

−2

−3

−2 −3

1 −1

−4 −3 −2 −1

−1

−4

f (x) = x – 1

x 1

−2

−3

−4 −3 −2 −1

−4

x 1

−1

For what values of a will the transformation from f (x) to g (x) = af (x) involve a reﬂection across the x-axis?

Consider these graphs: i

Describe the transformation applied to f (x) to produce g (x).

Write an equation for g (x) in terms of f (x).

4 3

g (x)

−1

−2

378

f (x)

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

f (x)

x 1

g (x)

1 −4 −3 −2 −1

x 1

y 10

g (x)

8 g (x)

f (x)

2 x

−8 −6 −4 −2 −2

x −8 −6 −4 −2 −2

Find the resulting equation when y = x2 undergoes these transformations. a

A dilation by a factor of 10

A dilation by a factor of 5 and a reﬂection across the x-axis

A vertical translation of 10 units up

A horizontal translation of 2 units to the right

A reﬂection about the x-axis and a vertical translation of 7 units down

A horizontal translation of 4 units to the left and dilation by a factor of 8

A horizontal translation of 6 units to the right and a vertical translation of 5 units upwards

Consider the graph of f (x) and the table of values for g (x). i

Describe the transformation applied to f (x) to produce g (x).

Write an equation for g (x) in terms of f (x).

y 4 3

g (x)

−1 1

0 0

1 −1

2 0

−3

−2

−1

3 1

f (x)

−4 −3 −2 −1

−1

x 1

−2 −3 −4

f (x) −4 −3 −2 −1

9 8 7 6 5 4 3 2 1

−1

g (x)

x 1

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y 4

x g (x)

−4 12

−2 8

0 4

2 0

4 −4

f (x)

−4 −3 −2 −1

−1

x 1

−2 −3 −4

Consider the graph of the function y = f (x):

Graph y = g (x) where g (x) = f (−x).

Describe the transformation from f (x) to g (x).

8 6 4 2 x −10 −8 −6 −4 −2 −2

Consider the graph of the function y = f (x):

y 8

Graph y = g (x) where

Describe the transformation from f (x) to g (x):

6 4 2 x −6 −4 −2

2 −2

Consider the table of values of f (x) and the graph of g (x) = af (x): a b

Identify the type of transformation needed to get from f (x) to g (x): Determine the value of a. x f (x)

−3 3

−2 2

−1 1

0 0

1 1

2 2

3 3

3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 −5

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x 1

y = g (x)

For each graph that has been transformed from y = x2: i

Determine the vertical translation.

Determine the horizontal translation.

iii

Has there been a reﬂection across the x-axis?

Has the quadratic had a change in dilation from the standard y = x2 curve?

2 x

−6 −4 −2

x −6 −4 −2

−2

−4

−6

2 x

−6 −4 −2

x −6 −4 −2

−2

−4

−6

Reasoning 20

For each type of transformation, explain whether or not these key features change. If they change, explain how. • Intervals where the function is increasing or decreasing • Intervals where the function values are positive, negative, or zero • End behaviour a

Horizontal translations

Vertical stretches and compressions

Reﬂections across the x-axis

7.04 Optional: Transformation of functions mathspace.co

381

Stavros has a bank account with $100. Every week, Stavros deposits an additional $50 into the account. A graph of Stavros’ bank balance reveals that the growth of his account is linear. Suppose that Stavros was instead depositing an additional $100 every week. Explain whether or not this could be represented by a vertical stretch of the current graph.

Balance 500 400 300 200 100 1

The functions f (x) and g (x) = f (x + k) have been graphed on the Cartesian plane. a

Determine the value of k.

Explain what k represents on this context.

Weeks 7

y 8 6 4

f (x)

g (x)

2 x −4 −2

−2 −4

Oreste draws the graph of g (x) = −f (x), as shown. a

Describe the error Oreste made.

Describe the type of graphs that f (x) could represent for Oreste’s error to still give him the correct answer.

3 g (x)

f (x)

−4 −3 −2 −1

−1

x 1

−2 −3 −4

382

Maddox and Paulina are going to a lake 2 kilometres from their home to go ice skating. Their mother told them that if the ice is not thick enough they must turn around and come straight home. It took them 20 minutes to walk to the lake at a constant rate and when they arrived Maddox could tell the ice was not thick enough so they immediately turned around and went home which took them an additional 20 minutes. a

Graph Paulina and Maddox’s distance from home over their entire journey.

If Paulina and Maddox had walked twice as fast, describe how your graph would change.

Describe the transformations if instead you were graphing Paulina and Maddox’s distance from the lake.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Consider the graphs of y = f (x) and y = g (x) shown. a b

Describe a series of transformations to get from f (x) to g (x). Phoebe thinks she has two different sets of transformations to get from f (x) to g (x), but she isn’t sure which one is correct.

Can you ﬁnd a second series of transformations that is different to your answer in part (a)?

6 4 2 −8 −6 −4 −2 −2 −4

x 2

g (x)

f (x) −6 −8

Here are the graphs of f (x) = x3 and g (x) =

(x − 3)3 + 5:

Describe how the graphs are different and how they are similar.

Describe the transformations from f (x) to g(x).

10 9 8 7 6 5 4 3 2 1

−3 −2 −1 −1 −2

f (x)

g (x) x

1 2 3 4 5 6 7 8

Yasushi and Peter are looking at the graphs of a few types of functions. a

Yasushi claims that a translation of a units to the right and a translation of b units upwards can be performed in either order and still get the same result.

Determine whether or not Yasushi is correct. Support your answer with an example or counter-example. b

Peter thinks that the same might be true for other types of transformations, and that the order in which transformations is applied doesn’t matter.

Determine whether or not Peter is correct. Support your answer with an example or counter-example.

Problem-solving 28

A classroom’s noise level is modelled by y = f (x), where x is the number of students. With a new seating arrangement, the noise is predicted to be y = 2f (x − 5). If 20 students originally resulted in a noise level of 50 dB, what will be the noise level with 25 students in the new seating arrangement?

A café owner models his daily coffee sales with the function y = j(x). After a marketing campaign, the sales increase by a factor of 1.5. If he originally sold 100 cups at noon, how many cups will he sell at noon after the campaign?

7.04 Optional: Transformation of functions mathspace.co

383

A farmer uses the function y = n(x) to model the amount of sunlight his crops receive where x is the number of hours after 12 noon. Due to a change in the season, the sunlight pattern is compressed horizontally by a factor of 2. If his crops originally received maximum sunlight at 6 p.m., at what time will they now receive maximum sunlight?

A scientist is modelling the light intensity of a star over time using the function y = p(t). She notices that after a certain event, the light intensity behaves as y = 2p(0.5t + 3) − 4. If the original peak intensity was at time t = 10 hours, at what time t will the new peak intensity occur?

The effectiveness of a drug over time is modelled by y = v(x). After a reformulation, its effectiveness is modelled by y = v(1.5x − 2) − 3. If the original drug reached its peak effectiveness in 6 hours, how long will it take for the reformulated drug to reach its peak effectiveness?

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Chapter 7 review 1

(Optional) Does each set of points describe a function? a

{(2, 5), (7, −3), (5, 2), (−4, −9)}

{(2, 5), (2, 7), (−3, −4), (−9, 13)}

{(1, 5), (1, 1), (7, −2), (−5, −10)}

{(1, 5), (7, −2), (−5, −10), (13, −13)}

{(1, 5), (1, 7), (−2, −5), (−5, −10)}

{(−1, −9), (0, 0), (1, 9), (2, 18)}

{(−2, 4), (−1, 1), (2, 4), (6, 36)}

(Optional) If f (x) = −6x + 4, ﬁnd: a

f (0)

x y

(Optional) Is each table a function? a

f (4)

x −2 −1 0 1 y 0 5 10 15

2 20

−4 −3 0 4 3 2

0 0

1 1

2 2

3 3

4 4

(Optional) Is each graph a function? a

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2

−3

−3 −4

1 −1

x 1

−1

−4

−4 −3 −2 −1

−2

x −4 −3 −2 −1

−2

−1 1

−1

x 1

−2

−3

−4

(Optional) State the domain of each function. a

Chapter 7 review mathspace.co

385

(Optional) Consider the function shown.

What is the domain of the function?

At what x-values does the function reach a minimum?

10 8 6 4 2

−5 −4 −3 −2 −1 −2

2 3 4 5

−4 −6

(Optional) Determine the domain and range of these functions: a

b 8

x −6 −4 −2

−8 −6 −4 −2 −2

−2

x 2

−6

−8

d 6

x −6 −4 −2

−2

−4

−6 −5 −4 −3 −2 −1 −1

−6

−2

y = x2

y = x2 + 4

y = −x2

(Optional) For the function y = 3x3: a

Complete the table of values: −1

−0.5

0.5

y = 3x b

386

(Optional) State the range of each function. a

−4

Sketch the graph of y = 3x3

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

y = (x − 1)2 − 2

(Optional) Consider the function y = x6. a

Describe the end behaviour of the function.

Draw the graph of the function.

(Optional) Velin is using the graph to describe information about the equation of the polynomial function. Velin states that there is not enough information to determine anything about the equation of the function. Determine whether Velin is correct and explain your reasoning.

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

(Optional) Describe the transformations applied to f (x) to derive the equation y = 2f (x) + 4.

(Optional) Consider the equation y = 4 − x2.

Describe of the transformation(s) applied to y = x2 to achieve the given function.

Draw a graph of y = x2 and the transformed function on the same Cartesian plane.

(Optional) Consider the equation y = 53−x − 1. a

Describe the transformation(s) applied to y = 5x to achieve the given function.

Draw a graph of y = 5x and the transformed function on the same Cartesian plane.

(Optional) Consider the table of values of f (x) and the graph of g (x) = kf (x): x f (x)

−3 3

−2 2

−1 1

0 0

1 1

2 2

3 3

Determine the type(s) of transformation needed to get from f (x) to g (x).

Determine the value of k.

y 2 1 x −6

−3

−1 y = g (x) −2 −3 −4

(Optional) Write an equation for each description and describe how the domain and range changed from the parent function: a

The parent function y = −3x is horizontally translated by 4 units left and vertically translated by 1 unit up.

The parent function y = ∣x∣ is vertically translated by 6 units up and horizontally translated by 3 units right.

Chapter 7 review mathspace.co

387

trigonometry 8 Further (Optional) Big ideas • The symmetry of the unit circle helps to define the properties of trigonometric functions. • Trigonometric functions are periodic functions that repeat over a constant interval. • Equations involving trigonometric functions require solving for the angles that satisfy a given trigonometric expression.

Curriculum Outcomes VC2M10ASP03 Use the unit circle to define the simple trigonometric functions of y = sin(x), y = cos(x) and y = tan(x) as functions of a real variable, and graph them with and without the use of digital tools VC2M10ASP04 Solve simple trigonometric equations

Chapter outline 8.01 8.02 8.03 8.04

Optional: Angles with exact values Optional: The unit circle Optional: Graphs of sine and cosine Optional: Trigonometric equations Chapter 8 review

390 397 416 425 431

Trigonometry helps predict tides, allowing sailors to navigate safely and surfers to catch the perfect wave!

8.01 Optional: Angles with exact values After this lesson, you will be able to… • calculate the exact values of trigonometric ratios. • use the angles with exact values to solve problems.

Exact value triangles The figure shown is a right-angled isosceles triangle, with two equal sides of length 1 unit. Using Pythagoras’ theorem, we can work out that the hypotenuse is

45° 1

units long.

The angles in a triangle add up to 180° and the base angles in an isosceles triangle are equal, so we can also work out that these two base angles are 45°.

45° 1

We can then determine exact values for the following trigonometric ratios:

We can also find some exact values by starting with an equilateral triangle:

60° 2

60°

To ﬁnd the exact ratios of 30 and 60 degree angles, we need to start with an equilateral triangle with side lengths of 2 units. Remember all the angles in an equilateral triangle are 60°.

60° 2

30° 30°

390

60°

We then draw a line that divides the triangle in half, into two congruent right-angled triangles. The base line is cut into two 1 unit lengths, and the opposite angle is cut into two 30° angles.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Now let’s focus on just one half of this triangle. 30° 2

We can calculate the length of the centre line to be using Pythagoras’ theorem. We can then use our trigonometric ratios to determine the exact values.

60° 1

Now, not every isosceles right-angled triangle has sides measuring 1, 1 and , but no matter how large or small it is, the two base angles will always be 45° angles and therefore the ratios of the sides will always be the same. This also applies to the triangle with 60° and 30° angles. Any pair of similar triangles have the same side ratios, and so any triangle with the angles we have just seen will have the same exact value trigonometric ratios.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.01 to answer these questions. 1.

What do you notice about sin θ and cos(90 − θ )?

What do you notice about cos θ and sin(90 − θ )?

Notice that: • sin θ = cos(90° − θ ) • cos θ = sin(90° − θ ) These angles are called complementary angles.

Example 1 Use the exact value triangles in the diagram to answer the following: 30° 2

45°

60° 1

1 45° 1

a What is the exact value of cos 45°?

Create a strategy Use the triangle with an angle of 45° in it.

8.01 Optional: Angles with exact values mathspace.co

391

Apply the idea Using the triangle on the right, we can ﬁnd cos 45°:

Use the cosine ratio

Substitute the values

b What is the exact value of cos 60°?

Create a strategy Use the triangle with an angle of 60° in it.

Apply the idea Using the triangle on the left, we can ﬁnd cos 60°:

Use the cosine ratio

Substitute the values

Example 2 θ is an angle in a right-angled triangle where

Create a strategy Use the exact value triangle with a side length of

Apply the idea Using this exact value triangle we can see that So θ = 30°

30° 2

60° 1

392

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary We have found the following exact values for the angles 30°, 45° and 60°: sin

cos

tan

30° 45°

60° We also have some exact values that don’t describe physical triangles, at 0° and 90°. 0° 90°

sin 0 1

cos 1 0

tan 0 Undefined

Practice questions Choose your question path Explorer 1–7, 9, 11–13, 15, 16bce, 17, 18, 21–23

Adventurer 1, 2, 4–7, 9, 11, 13, 15, 16ace, 17–19, 22–24

Trailblazer 1, 2, 4, 5, 7, 8, 10, 12, 14, 16bdef, 17, 19, 20, 23–26

Understanding 1

hich three angles between 0° and 90° have trigonometric ratios that can be identiﬁed W using the exact value triangles?

Which four angles on the axes of the unit circle between 0° and 360° have trigonometric ratios that can be found without using a calculator?

Identify the complementary angle: a

30°

72°

42°

x°

Use Pythagoras’ theorem to ﬁnd the hypotenuse of this right triangle. 45° 1 45° 1

8.01 Optional: Angles with exact values mathspace.co

393

Copy and complete: a

30° 2

b c d

60°

tan 60° = ⬚

Fluency Ex 1

Find the exact value of the ratio: a

sin 30°

sin 60°

sin 45°

cos 45°

cos 60°

cos 30°

tan 60°

tan 45°

tan 30°

30° 2

45°

60° 1

For this equilateral triangle with side lengths of 2 cm. Find the exact value of: a

Perpendicular height of the triangle.

Angle x.

x°

sin 60°

cos 60°

iii

tan 60°

sin x

cos x

iii

tan x 60°

2 cm

Find the exact value of the variable: a

b 60°

30°

3 30°

60° B

14 60°

d 10

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

30°

Find the exact side length of an equilateral triangle with a perpendicular height of

Find the exact length of side a.

Find the exact length of side b.

b 60° 7

Find the exact value of x.

45°

Find the exact value of y. 4 2

60° y

3 3

Find m.

Find t.

6 10

30° t Ex 2

θ is an angle in a right-angled triangle. Find the value of θ : a

Find the value of θ.

and b

Find the value of θ.

Find the value of tan θ.

θ is an angle in a right-angled triangle where a

and b

Find the value of tan θ.

Find θ : a

8 3 θ 12

4 2 4

8.01 Optional: Angles with exact values mathspace.co

395

Evaluate the expressions, leaving your answer in exact and rationalised form:

sin 45° cos 30°

sin 45° + cos 30°

sin 45° cos 60° + tan 45°

sin 90° cos 0° + sin 270° cos 180°

sin2 90° + cos2 90°

Reasoning 17

Use exact values to show that sin2 45° + cos2 45° = 1.

Use the triangle to show that sin 30° = cos 60°.

sin θ is equal to the cosine of which angle?

sin(90° − θ ) is equal to the cosine of which angle?

What is the relationship between tan(90° − θ ) and tan θ?

30° 2

60° 1

Use

and exact values to explain why the tangent function has an undeﬁned

value at 90° and 270°. 20

for any angle θ, except 90°, in a right-angled triangle.

Prove that

Problem-solving 21

A right-angled triangle has one angle of 30° and a hypotenuse of length 10. Find the exact lengths of the other two sides and the area of the triangle.

Find the exact value of the angle θ if

A right-angled triangle has one angle of 45° and an adjacent side of length exact lengths of the other two sides and the area of the triangle.

Find the exact value of the expression

A right-angled triangle has one angle of 60° and an opposite side of length exact lengths of the other two sides and the area of the triangle.

If sin

396

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

and

, ﬁnd the exact value of cos θ and tan θ.

. . Find the

. Find the

8.02 Optional: The unit circle After this lesson, you will be able to… • divide the unit circle into four quadrants and describe the unique trigonometric properties of each. • identify and use reference angles to find trigonometric ratios in any quadrant. • recognise and apply the signs of trigonometric ratios in different quadrants. • apply the knowledge of the unit circle to solve problems.

The unit circle In right-angled triangles, we can use trigonometric ratios to describe the relationship between a given angle and two sides. The trigonometric ratios are:

Opposite

Hypotenuse

θ A

Adjacent

Exploration Two special triangles are each drawn in a circle with a radius of 1. 1

θ = 45°

θ = 30° −1

1 0

−1

1 45°

60° 1

−1

Calculate the lengths of the legs of each triangle, keeping your answer in fraction form.

Write the coordinates of the point where each hypotenuse meets the circle.

Write the coordinates of the point where each hypotenuse meets the circle in terms of sine and cosine.

8.02 Optional: The unit circle mathspace.co

397

If we place a right-angled triangle on the Cartesian plane at the origin and make the hypotenuse have a length of 1, we can write trigonometric ratios in terms of x and y. The trigonometric ratios become:

P (x, y) 1

θ −1

−1

If we look at any right-angled triangle with a hypotenuse of 1, the endpoint of the hypotenuse will lie on a circle with radius 1, known as the unit circle. The radius of the unit circle, which creates the angle with the positive x-axis, is called the terminal arm. unit circle

terminal arm −1

P (cos θ ,sin θ )

θ is the angle that the terminal arm makes with the positive x-axis when rotating anticlockwise.

Since cos(θ ) = x and sin(θ ) = y, the coordinates of the endpoint can be represented by

θ x

P(cos θ, sin θ )

The ratio , which is the gradient of the hypotenuse, can be represented by −1

This will be true in all quadrants.

Example 1 The graph shows an angle a with its terminal side intersecting the circle at

. 1 a −1 −1

398

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

a Find the value of sin a.

Create a strategy The value of sin a is the y-coordinate of point P.

Apply the idea

Reﬂect and check If we draw a vertical line from point P to the x-axis, we get the triangle shown.

b Find the value of cos a.

Create a strategy The value of cos a is the x-coordinate of point P.

Apply the idea

Reﬂect and check We can check this answer by using the diagram from the previous part.

8.02 Optional: The unit circle mathspace.co

399

c Find the value of tan a.

Create a strategy , the gradient of the line from the origin to the point.

Apply the idea Write the gradient formula

Substitute values of x and y

Evaluate the division

Reﬂect and check Checking the answer with the diagram once more:

Example 2 Find the coordinates of P(x, y), rounded to two decimal places.

y 1

P 72°

−1 −1

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

Create a strategy When the angle in the first quadrant is not a known angle (0°, 30°, 45°, 60°, 90°), use a calculator to evaluate. Recall cos θ = x and sin θ = y.

Apply the idea

Reflect and check

Using a calculator, we find:

When using a scientific or graphing calculator, be sure it is set to degree mode. The other mode (radian mode) will give you different values since it is used for different units.

cos 72° = 0.31 sin 72° = 0.95 The coordinates of the point are (0.31, 0.95).

Idea summary The unit circle is a circle with a radius of 1. A point on the unit circle, after having rotated by an angle of measure θ in the anticlockwise direction, can be represented by P(cos θ, sin θ ). 1

P (x, y)

• •

•

1 −1

θ x

sin θ = y, or the height of the triangle. cos θ = x, or the length of the base of the triangle. , or the gradient of the hypotenuse.

−1

Trigonometric ratios in all four quadrants The unit circle shows us that sin θ, cos θ, and tan θ exist for angles larger than what can be contained in a right-angled triangle. (cosθ , sinθ )

Previously, we defined:

y 1

θ −1

1 x

−1

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401

We can still use these definitions in any quadrants, but the signs of x and y will change based on the quadrant they are in.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.02 to answer these questions. 1.

In quadrant 1, what do you notice about the signs of the trigonometric ratios (positive/negative)?

In quadrant 2, what do you notice about the signs of the trigonometric ratios (positive/negative)?

In quadrant 3, what do you notice about the signs of the trigonometric ratios (positive/negative)?

In quadrant 4, what do you notice about the signs of the trigonometric ratios (positive/negative)?

We know that sin θ is equal to the y-coordinate (height), cos θ is equal to the x-coordinate (length), and tan θ is the gradient of the line from the origin to the point. Second quadrant: • y is positive: sin θ is positive. • x is negative: cos θ is negative. • Gradient is negative: tan θ is negative.

y (cosθ , sinθ )

cosθ tanθ

sinθ

−1

−1 y

θ −1

Third quadrant: • y is negative: sin θ is negative. • x is negative: cos θ is negative. • Gradient is positive: tan θ is positive.

1 x

sinθ tanθ cosθ

(cosθ , sinθ )

402

−1

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fourth quadrant: • y is negative: sin θ is negative. • x is positive: cos θ is positive. • Gradient is negative: tan θ is negative.

−1

tanθ cosθ −1 S sin positive

(cosθ , sinθ )

A all positive (cosθ , sinθ )

θ −1

T tan positive

sinθ

−1

C cos positive

A helpful mnemonic for positive trigonometric function values is “All Stations To Central” (ASTC). This can help you remember the signs of the trigonometric ratios in each quadrant. • In quadrant 1, all ratios are positive. • In quadrant 2, only sin θ is positive. • In quadrant 3, only tan θ is positive. • In quadrant 4, only cos θ is positive.

Example 3 Will tan 195° have a positive or a negative answer?

Create a strategy Think about which quadrant the angle is in and what the sign of tangent will be in that quadrant.

Apply the idea

Reflect and check

The angle 195° is between 180° and 270°, so it lies in quadrant 3.

In quadrant 3, both x and y are negative, so

1 195°

−1

tan 195° will have a positive answer.

1 −1

If we graph the angle and terminal arm, we will see that the terminal arm is increasing from left to right, so it has a positive gradient.

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Idea summary y

We can determine the sign of trigonometric functions by relating the point P (x, y) on the unit circle to the trigonometric functions P (cos θ, sin θ ).

(−, +)

(+, +)

In the quadrants where x is positive or negative, cos θ will be positive or negative. −x

(−, −) III S sin positive

(+, −)

T tan positive

tan θ is the gradient of the line,

−y

−1

A helpful mnemonic for positive trigonometric function values is “All Stations To Central” (ASTC).

θ −1

In the quadrants where y is positive or negative, sin θ is positive or negative.

• • • •

In quadrant 1, all ratios are positive. In quadrant 2, only sin θ is positive. In quadrant 3, only tan θ is positive. In quadrant 4, only cos θ is positive.

C cos positive

Angles beyond the ﬁrst quadrant Angles between 0° and 90° are in quadrant 1. Second

First

Angles between 90° and 180° are in quadrant 2. Angles between 180° and 270° are in quadrant 3. Angles between 270° and 360° are in quadrant 4. A boundary angle marks the edge of a quadrant, so 0°, 90°, 180°, 270°, and 360° are boundary angles.

Third

404

Fourth

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.02 to answer these questions. 1.

What is the y-coordinate when θ = 30°? What is the reference angle for θ = 30°?

What is the y-coordinate when θ = 150°? What is the reference angle for θ = 150°?

What is the y-coordinate when θ = 210°? What is the reference angle for θ = 210°?

What is the y-coordinate when θ = 330°? What is the reference angle for θ = 330°?

What patterns do you notice?

What do you think the reference angle is?

For any angle beyond quadrant 1, we can draw an angle in the ﬁrst quadrant that creates a congruent triangle to the one formed by the original angle. In the congruent triangles, the angles at the origin will be congruent. y

60°

1 3 , 2 2

60° angle

1 3 P − , 2 2 120°

60°

120° angle

210° 60°

300° 60°

−

210° angle

−

300° angle

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The relative acute angle, or reference angle, is the smallest angle from the x-axis to the terminal arm. It is shown in each quadrant in the diagram.

90° y 1 B

180° −1

180° − θ

180° + θ

360° − θ

0°

In each quadrant, the reference angle is equivalent to: • Quadrant 1: θ • Quadrant 2: 180 − θ • Quadrant 3: 180 + θ • Quadrant 4: 360 − θ

D −1 270°

We can use the relative acute angle to ﬁnd trigonometric ratios in any quadrant. We can use these steps to evaluate any trigometric ratio: 1. Sketch the angle on the unit circle. 2. Identify the sign of the ratio based on the quadrant the terminal arm is in. 3. Find the reference (relative acute) angle 4. Rewrite the ratio using the sign and reference angle. 5. Evaluate using special angles or a calculator.

Example 4 Point P on the unit circle shows a rotation of 330°. What acute angle in the ﬁrst quadrant can 330° be related to?

Create a strategy We want to draw an acute angle with the closest x-axis.

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Apply the idea 330° is between 270° and 360°, so it is in the fourth quadrant.

So the positive x-axis is closest. x

This diagram shows that an acute angle 360° − θ is related to θ in the fourth quadrant.

360° − 330° = 30° which is the reference angle. 330°

30° P

Example 5 For sin 300°: a Write an equivalent ratio using an acute reference angle.

Create a strategy Find the quadrant the angle lies in and determine the sign of the ratio. Then we can ﬁnd the closest x-axis to ﬁnd the reference angle.

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Apply the idea Starting at the positive x-axis and rotating 300° anticlockwise, we move into the fourth quadrant.

90° y sin positive

all positive

θ θ

180°

θ θ

This diagram shows that an angle of 360° − θ in the fourth quadrant is equivalent to an acute angle of θ in the ﬁrst quadrant for the trigonometric ratios.

0° 360°

Sine is negative in the fourth quadrant. tan positive

cos positive 270°

sin 300° = −sin(360° − 300°) = −sin 60°

Find the acute angle and use the sign Evaluate the angle

b Evaluate sin 300°, without a calculator.

Create a strategy We can now use a special triangle to evaluate.

Apply the idea Using our special triangle, we can see that: 30° 2

60° 1

So we have that: From above

Using special triangles

Reﬂect and check We can check our answer using a calculator or the GeoGebra applet in this section and looking at the y-coordinate.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary The relative acute angle, or reference angle, is the angle made with closest to the x-axis. It is shown, in our diagram, in each quadrant

90° y 1 B

180° −1

180° − θ

180° + θ

360° − θ

0°

D −1 270°

First quadrant Second quadrant Third quadrant Fourth quadrant

Reference angle θ 180° − θ 180° + θ 360° − θ

Practice questions Choose your question path Explorer 1–4, 6ab, 7, 8a-d, 9a-d, 10, 11a-f, 12, 13, 16a-g, 17, 18, 20, 21, 23, 25a

Adventurer 1–5, 6bc, 7, 8c-g, 9c-g, 10, 11c-h, 14, 15, 16e-l, 18–24, 25a

Trailblazer 1–5, 6cd, 8c-h, 9c-h, 10, 11cdh-l, 14, 15, 16e-l, 18–20, 22–26

Understanding 1

In which quadrant is the angle located? a

299°

5°

160°

229°

40°

310°

138°

344°

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409

Use the special triangles to complete the table. sin

cos

30°

tan

30° 45° 60°

45°

60° 1

45°

Evaluate on your calculator to three decimal places. a

tan 300°

sin 25°

cos 108°

tan 171°

cos 344°

sin 226°

sin 270°

cos 360°

Is each statement true or false? a

All points on the unit circle have positive coordinates.

We cannot evaluate trigonometric ratios for angles greater than 90°.

All trigonometric ratios are positive in the first quadrant.

The reference angle is always acute.

The reference angle is made with the closest x-axis.

A point P (x, y), on the unit circle makes an angle of θ with the positive x-axis. a

What is the radius of the unit circle?

What is cosθ?

What is sinθ?

What is tanθ?

Write the coordinates of P (x, y) in terms of θ.

y 1

θ −1

x 1

−1

Fluency Ex 1

For each diagram, find the value of: i

sin a

cos a

tan a

P −

3 4 P  ,  5 5 θ

−1

iii

1 a

x 1

−1

410

21 20 , 29 29

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−1

x 1

−1

1 a

−1

1 −1

Ex 2

−1

Find the coordinates of P (x, y), rounded to two decimal places. a

y 1

−1

40°

160°

−1

Ex 4

sin 143°

tan 284°

cos 143°

cos 284°

sin 284°

cos 269°

sin 170°

cos 320°

Write the relative acute angle on the unit circle that is equivalent to: a

310°

100°

160°

245°

192°

138°

256°

289°

For each statement, fill in the blank with = or ≠. a

−1

Will the ratio have a positive or a negative answer?

d Ex 5a

x 1

−1

Ex 3

x 1

sin 215° ⬚ sin 35°

sin 150° ⬚ sin 30°

cos 300° ⬚ cos 60°

−cos 110° ⬚ cos 70°

Write an equivalent ratio using an acute reference angle. a

sin 106°

cos 149°

tan 124°

c f

tan 275° ⬚ tan 85°

tan 145° ⬚ −tan 35° d

sin 240°

cos 315°

tan 140°

sin 174°

cos 210°

tan 295°

sin 320°

cos 111°

tan 264°

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411

Consider the point (x, y): i

Find the value of x.

Find the value of y. y

(x, y) 1

135°

225°

x 1 (x, y)

The diagram shows a unit circle with point

y 1

marked on the circle. Point P represents a rotation of 45° anticlockwise around the origin from the positive x-axis:

P 45°

−1

1 x

−1

Find the exact value of sin 45°.

The 45° reference angle has been marked in all quadrants. For each quadrant, find the relative angle.

Quadrant 2

Quadrant 3

iii

Quadrant 4

y 1 Q −1

Write sin 315° in terms of an equivalent ratio of 45°.

Find the exact value of sin 135°.

45° 45°

Find the exact coordinates of point S.

P 1 x S −1

The diagram shows a unit circle with point

y 1

marked on the circle. Point P represents a rotation of 60° anticlockwise around the origin from the positive x-axis:

P 1

−1

60°

1 x

−1

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Find the exact value of the ratio tan 60°.

The 60° reference angle has been marked in all quadrants. For each quadrant, find the relative angle. i

Quadrant 2

Quadrant 3

iii

Quadrant 4

−1

Find the exact coordinates of point R.

Write tan 240° in terms of an equivalent ratio of 60°.

Find the exact value of the ratio tan 240°.

60°

−1

The diagram shows a unit circle with point

y 1

marked on the circle. Point P represents a rotation of

30° anticlockwise around the origin from the positive x-axis: −1

30°

−1

Ex 5b

Find the exact value of cos 30°

The 30° reference angle has been marked in all quadrants. For each quadrant, find the relative angle: i

Quadrant 2

Quadrant 3

iii

Quadrant 4

y 1 Q −1

Find the exact coordinates of point Q.

Write cos 150° in terms of an equivalent ratio of 30°.

Find the exact value of cos 150°.

P 30° 30°

30° 30°

x S

−1

Evaluate without using a calculator. a

sin 120°

tan 135°

cos 240°

cos 300°

sin 135°

cos 225°

tan 315°

sin 240°

cos 120°

tan 300°

sin 210°

tan 330°

Reasoning 17

In which quadrant is angle θ located? Explain. a

sin θ > 0 and cos θ < 0.

tan θ < 0 and sin θ > 0.

tan θ < 0 and cos θ < 0.

tan θ > 0 and sin θ > 0.

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413

The points (0, 1), (0, 1), (−1, 0) and (0, −1) are signiﬁcant points on the unit circle. a

Explain how the points can be used to ﬁnd the value of sin 0°, sin 90°, sin 180°, and sin 270°.

Use the points to ﬁnd the value of cos 0°, cos 90°, cos 180°, and cos 270°.

Using the Pythagoras’ theorem and the unit circle, show that sin2 θ + cos2 θ = 1.

y 1

(cosθ , sinθ )

−1

1 −1

The sine, cosine and tangent ratios are all positive for angles in the ﬁrst quadrant.

y 1

This unit circle is labelled with the letters A, S, T, C in the quadrants. Explain how these letters help identify the sign of the ratios for angles in each quadrant.

1 −1

−1

Problem-solving 21

414

Find, rounded to two decimal places:

The coordinates of P (x, y)

The new coordinates of P if it was reﬂected across the y-axis.

sin 114°

66°

cos 114°

tan 114°

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Suppose that a

Find sin θ.

, where 270° < θ < 360°. b

Find tan θ. θ

3 5

Given: w = 65°, x = 10°, y = 44°, z = 380°, a = 460°, b = 550° Is the statement true or false?

tan a, cos(90° + x) and sin(180° + y) are negative.

sin x, cos z, and tan b are all positive.

Angles w, x, and y are in Quadrant 1.

sin x, cos y, and tan w are all positive.

Angles z, a, and b are in Quadrant 1.

Given: w = 253°, x = 265°, y = 193°, z = 462°, a = 538°, b = 407° Is the statement true or false?

Angles z, a, and b are in Quadrant 3.

sin z and sin b are both negative.

tan(z − 90°) and tan(z + 90°) are both positive.

The angles w, x, and y all have negative cosine values.

Angles w, x, and y are in Quadrant 3.

The diagram shows points P, Q, R and S, which represent rotations of 62°, 118°, 242° and 298° respectively around the unit circle. a

If point P has coordinates (a, b), write the coordinates of the following points in terms of a and b: i

iii

90°

Q 180°

tan 118° tan 298°

P (a, b) 0°

−1

62° R −1

Is the ratio equivalent to −tan 62° or tan 62°? iii

tan 242° 270°

Express in terms of tan x, where x < 90°: i

tan(180° − x)

tan(180° + x)

iii

tan(360° − x)

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For the trigonometric expressions: i

Find the value of the relative acute angle.

Find the exact value of the trigonometric expression.

sin 480°

tan 540°

cos 780°

sin 720°

8.03 Optional: Graphs of sine and cosine After this lesson, you will be able to… • graph the values of sine and cosine functions on a Cartesian plane. • identify the key features of sine and cosine functions.

Graphs of trigonometric functions You may remember that the unit circle provides us with a visual understanding that the trigonometric functions of sin θ, cos θ and tan θ exist for angles larger than what can be contained in a right-angled triangle. The sine of the angle is defined to be the y-coordinate of the point on the unit circle. The cosine of the angle is defined to be the x-coordinate of the point on the unit circle. The tangent of the angle can be defined algebraically as the ratio

. This also represents

the gradient of the line that forms the angle θ to the positive x-axis. It can also be geometrically deﬁned to be the y-coordinate of a point Q, where Q is the intersection of the extension of the line OP and the tangent of the circle at (1, 0). As we move through different values of θ the value of cos θ and sin θ move accordingly between −1 and 1.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.03 to answer these questions. 1.

What sort of pattern do sin θ and cos θ make?

What values of y do sin θ and cos θ start at?

What relationship can you find between the graphs of sin θ and cos θ ?

416

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The unit circle produces wave like curves for y = sin θ and y = cos θ, where y = sin θ starts at y = 0 and y = cos θ starts at y = 1. The y = cos θ graph is a translation of the y = sin θ graph, 90° to the left. If we plot the values of sin θ and cos θ according to different values of θ on the unit circle, we get the following graphs: y = sin θ : y 2 1 −540°

−360°

−180°

0 −1

180°

360°

540°

−2

y = cos θ : y 2 1 −540°

−360°

−180°

0 −1

θ 180°

360°

540°

−2

Notice that all of these graphs are constructed with degrees on the horizontal axis. The function values behave in the same way as in the unit circle — for example, in the graph above of y = cos θ, we can see that it has negative y-values for all of the angles in the domain of 90° < θ < 180°. These are the values associated with the second quadrant where cos θ will be negative.

Idea summary The unit circle produces wave like curves for y = sin θ and y = cos θ. which means the graph of y = tan θ will have asymptotes whenever cos θ = 0.

Properties of trigonometric graphs The graphs of y = cos θ and y = sin θ have certain common properties. Each graph demonstrates repetition. We call the graphs of y = cos θ and y = sin θ periodic or cyclic. We deﬁne the period as the length of one cycle. For both graphs, the period is 360°.

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Here is an example of a cycle. y 2 1 −540°

−360°

0 −1

−180°

180°

360°

540°

−2

Because of the oscillating behaviour, both graphs have regions where the curve is increasing and decreasing. Remember that we say the graph of a particular curve is increasing if the y-values increase as the x-values increase. Similarly, we say the graph is decreasing if the y-values decrease as the x-values increase. Here is an example of where y = sin θ is decreasing. y 2 1 −540°

−360°

0 −1

−180°

180°

360°

540°

−2

In addition, the height of each graph stays between y = −1 and y = 1 for all values of θ, since each coordinate of a point on the unit circle can be at most 1 unit from the origin. This means, the range of both the sin θ and cos θ functions is between −1 and 1.

Example 1 For the graph of y = sin (x): y 1 x −360 −270 −180 −90

180

270 360

−1

a What is the x-value of the x-intercept in the region −360° < x < 0°?

Create a strategy Use the fact that x-intercept is the point at which the curve intersects the x-axis.

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Apply the idea The x-intercept in this region −360° < x < 0° is plotted. y 1 x −360 −270 −180 −90

180

270 360

−1

The x-value at this point is x = −180°. b Determine whether this statement is true or false: As x approaches inﬁnity, the graph of y = sin x stays between y = −1 and y = 1. A

True

False

Create a strategy

Apply the idea

Use the fact that the graph of y = sin x is periodic.

The graph of y = sin x is periodic, meaning that it will periodically return to the same y-values and will continue to oscillate between a height of −1 and 1. So the answer is option A, true.

c In which two of these regions is the graph of y = sin x increasing? A

90° < x < 270°

−270° < x < −90°

270° < x < 450°

−450° < x < −270°

Create a strategy Use the graph and choose the regions where the y-values increase from left to right.

Apply the idea Each region of values is shown on the graph: y B

A x

−360 −270 −180 −90 D

180

−1

270 360 C

The graph is increasing for the regions in options C and D.

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Idea summary The graphs of y = cos θ and y = sin θ periodic or cyclic. For both graphs, the period is 360°. −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1 for all values of θ.

Practice questions Choose your question path Explorer 1–9, 12, 13, 15, 18aceg, 19

Adventurer 2–9, 12, 13, 15, 16, 18aceg, 19, 21

Trailblazer 2–4, 7, 8, 10, 11, 13, 14, 16, 17, 18bdfh, 19–22

Understanding 1

Describe the general shape of the graph for each trigonometric function: a

Sine

Cosine

The value of sin θ can be represented on the xy-plane. a

For the curve y = sin θ, complete these statements: i

iii

rotation of 380° has the same rotation as the A acute angle with measure ⬚. So sin 380° = sin ⬚.

rotation of 480° has the same rotation as the A obtuse angle with measure ⬚. So sin 480° = sin ⬚.

rotation of 580° has the same rotation as the A reﬂex angle with measure ⬚. So sin 580° = sin ⬚°.

Describe the nature of the graph of y = sin θ for 0 ≤ θ ≤ 720°.

The number of degrees it takes for the curve to complete a full cycle is called the period of the function.

−1

x 1

−1

θ 90 180 270 360 450 540 630 720 −1 −2

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(cosθ , sinθ )

Determine the period of y = sin θ.

420

For the curve y = sin x, are these statements true or false?

y 2

The graph of y = sin x is periodic.

As x approaches inﬁnity, the height of the graph approaches inﬁnity.

The graph of y = sin x is increasing between x = −90° and x = 0°.

The graph of y = sin x is symmetric about the line x = 0.

The graph of y = sin x is symmetric with respect to the origin.

The y-values of the graph repeat after a period of 360°.

1 x −360 −180

The graph of y = cos x is cyclic.

As x approaches inﬁnity, the height of the graph approaches inﬁnity.

The graph of y = cos x is increasing between x = −180° and x = −90°.

The graph of y = cos x has reﬂective symmetry across the line x = 0.

The graph of y = cos x has rotational symmetry with respect to the origin.

The y-values of the graph repeat after a period of 180°.

360

−1 −2

For the curve y = cos x, are these statements true or false? a

180

y 2 1 x −360 −180

180

360

−1 −2

Fluency 5

For the graph of y = sin x: a

Determine the coordinates of the y-intercept.

Find the maximum y-value.

Find the minimum y-value.

y 2 1 x −360 −180

180

360

−1 −2

8.03 Optional: Graphs of sine and cosine mathspace.co

421

For the graph y = cos x:

Determine the coordinates of the y-intercept.

Find the maximum y-value.

Find the minimum y-value.

2 1 x −360 −180

180

360

−1 −2

For the equation y = cos x: a

Complete the table of values: x

0°

90°

180°

270°

360°

270°

360°

cos x b 8

Sketch the graph of y = cos x.

For the equation y = sin x: a

Complete the table of values: x

0°

90°

180°

sin x b 9

Sketch the graph of y = sin x.

Estimate these values using the graph of y = sin x: y 1 0.5 x 30°

60°

90°

120°

150°

180°

210° 240° 270° 300° 330° 360°

−0.5 −1

sin 120°

sin 300°

sin 40°

sin 340°

Two values of θ where sin θ = 0.9

Two values of θ where sin θ = −0.6

422

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Estimate these values using the graph of y = cos x: y 1 0.5 x −180° −150° −120° −90° −60° −30°

30°

60°

90°

120° 150° 180°

−0.5 −1

cos 30°

cos(−100°)

cos 160°

cos(−20°)

Two values of θ where cos θ = −0.75

Two values of θ where cos θ = 0.3

For the graph of y = cos x:

Is cos 162° positive or negative?

In which quadrant does an angle with measure 162° lie in?

2 1 x −360 −180

180

360

−1 −2

For the graph of y = sin x:

Is sin 200° positive or negative?

In which quadrant does an angle with measure 200° lie in?

2 1 x −360 −180

180

360

−1 −2

8.03 Optional: Graphs of sine and cosine mathspace.co

423

Ex 1

For the graph of y = sin x: a

If one cycle of the graph of y = sin x starts at x = 0°, when does the next cycle start?

Find the x-value of the x-intercept in these regions:

0° < x < 360°

−360° < x < 0°

True or false: As x approaches inﬁnity, the graph of y = sin x stays between y = −1 and y = 1.

Do the graph of y = sin x increase or decreases in these regions? i

90° < x < 270°

−270° < x < −90°

iii

−450° < x < −270°

−90° < x < 90°

270° < x < 450°

x −360 −180

180 −1 −2

For the curve y = cos x:

If one cycle of the graph of y = cos x starts at x = −90°, when does the next cycle start?

Find the x-values of the x-intercepts in these regions:

0° < x < 360°

360

−360° < x < 0°

x −540−360−180

180 360 540

True or false: As x approaches inﬁnity, the graph of y = cos x stays between y = −1 and y = 1.

−1

Does the graph of y = cos x increases or decreases in these regions?

−2

0° < x < 180°

−180° < x < 0°

iii

180° < x < 360°

−360° < x < −180°

Reasoning 15

Describe the relationship between the graphs of the sine and cosine functions and how they are interconnected.

Are these statements true or false? Justify your answer. a

The sine and cosine functions have the same amplitude because they both oscillate between −1 and 1.

The sine function has a period of 180° because it starts at 0 and reaches its maximum value at 90°.

The graphs of the sine and cosine functions intersect at every multiple of 90°.

The sine and cosine functions have the same period because they are both based on the unit circle.

The functions y = sin q and y = cos q intersect at two points in the region 0° ≤ θ ≤ 360°. Use the unit circle to justify why these are the only two points of intersection in this region.

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Problem-solving 18

Using relevant graphs of trigonometric functions, evaluate: a

sin 90°

cos 90°

sin 180°

cos 180°

sin 270°

cos 270°

sin 360°

cos 360°

For the function y = sin θ, ﬁnd θ if: a

sin θ = sin 390°, where θ is in the 1st quadrant.

sin θ = sin 480°, where θ is in the 2nd quadrant.

sin θ = sin 570°, where θ is in the 3rd quadrant.

For the function y = cos x for −180° ≤ x ≤ 180°: a

Sketch the graph of the function.

Determine the x-values for which: i

cos x = ±1

iii

For the function y = sin x for −180° ≤ x ≤ 180°: a

Sketch the graph of the function.

Determine the x-values for which: i

cos x = 0

sin x = 0

sin x = ±1

We can deduce from the unit circle that a

Construct a table of values for tan θ.

Make an attempt at sketching a graph for tan θ from the table of values.

Research the shape of of the graph for tan θ and discuss where you were and were not successful.

8.04 Optional: Trigonometric equations After this lesson, you will be able to… • solve trigonometric equations.

Solve trigonometric equations The method for solving trigonometric equations is roughly the same: • After any algebraic manipulation, use the positive value on the right-hand side of the equation to ﬁnd the size of the related acute angle. • Then, draw a quadrant (ASTC) diagram to locate the angles you require to make the equation true.

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• Write down your solutions, ensuring that they are in the domain specified in the question. The sign ( positive or negative) and the trigonometric function will determine which quadrants the solutions are in. There are many variations on trigonometric equations you need to be comfortable with. In this set, we will consider three types of equations. Type 1: Equations involving exact values • If a question doesn’t specify a need for your final answer to be rounded, this indicates that it involves an exact value solution. Type 2: Equations involving boundary angles • Questions where the value specified is 0 or ±1 land on what we can refer to as boundary angles. It might be easier for you to solve these questions by considering the sketch of the trigonometric function in question, rather than trying to use your calculator or a quadrant diagram. The exception to this is when tan x is equal to ±1, where it’s more valuable to consider the exact value triangles to find a solution. Type 3: Equations with a different domain • It doesn’t matter how you arrive at your final solutions here, as long as you get there in the end. Many students prefer to solve an equation as if the domain was 0° ≤ x ≤ 360° and then “move” these answers into the correct domain by adding or subtracting 360° until they find the solutions required. You might feel comfortable looking at your quadrant diagram after finding the relative acute angle and going straight to writing the solutions in the required domain. • Having a variety of domains might mean you have more or less than two solutions, which is typically what we expect when solving for 0° ≤ x ≤ 360°.

Example 1 for x where 0° ≤ x ≤ 90°.

Solve

Create a strategy Use the exact value triangles.

Apply the idea Using this exact value triangle, we can see that the acute value of x = 60° since

30°

Since the domain is 0° ≤ x ≤ 90° we do not need to consider any of the other quadrants, so the only solution is:

60° 1

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x = 60°

Example 2 for x where 0° ≤ x ≤ 360°.

Solve

Create a strategy Use exact value triangles to solve the equation for the acute angle. Then consider the sign to determine which quadrants x could be in.

Apply the idea Using this exact value triangle, we can see that the acute value will be x = 30° since

30°

60° 1

We need tan x to be positive since

This quadrant diagram shows us that we need to ﬁnd equivalent angles in the ﬁrst and third quadrant.

θ x

The angle in the ﬁrst quadrant is 30°, while the angle in the third quadrant is 180° + 30° = 210°. So, the solutions are:

x = 30°, 210°

Idea summary To solve a trigonometric equation, after any algebraic manipulation use the positive value on the right-hand side of the equation to ﬁnd the size of the related acute angle. Then, draw a quadrant (ASTC) diagram to locate the angles you require to make the equation true.

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Practice questions Choose your question path Explorer 1–4, 5abc, 6abce, 7ac, 8, 9acfhjl, 10ab, 12ace, 13ade, 14ac, 15ac, 16–18, 20

Adventurer 1–4, 5abc, 6abce, 7bd, 8, 9acfhjl, 10ab, 12ace, 13ace, 14ac, 15bc, 16–18, 20, 21

Trailblazer 1–4, 5def, 6efgh, 7bd, 8ac, 9bdegjl, 10cd, 11, 13ace, 14bd, 16–22

Understanding 1

What is the domain of a function?

Is the trigonometric function positive or negative over each domain?

0° < x < 90°

90° < x < 180°

iii

180° < x < 270°

sin x

cos x

tan x

270° < x < 360°

Write the steps for solving trigonometric equations in the correct order. • Write down the solutions, ensuring they are within the specified domain. • Use the positive value on the right-hand side of the equation to find the size of the related acute angle. • Perform any algebraic manipulation to isolate the trigonometric function. • Draw a graph or a quadrant diagram to locate the angles that make the equation true.

Match the solution to the correct equation over the domain 0° ≤ x ≤ 360°. i

sin x = 1

cos x = 0

iii

tan x = 0

cos x = -1

x = 0°, 180°, 360°

x = 90°

x = 90°, 270°

x = 180°

Fluency 5

Ex 1

428

Find the size of the acute angle θ : a

Solve for 0° ≤ x ≤ 90°: a

tan x = 1

2 cos x = 1

cos x − 1 = 0

sin x − 1 = 0

Solve for 0° ≤ θ ≤ 90°:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 2

Determine the number of solutions for θ : a

for 0° < θ < 180°

for 0° < θ < 90°

for 0° < θ < 180°

cos x = -1

Solve for 0° ≤ x ≤ 360°: b

cos x = 0

sin x = 0

cos x =

sin x =

sin x = 1

tan x =

tan x = 0

tan x =

Solve for 0° ≤ θ ≤ 360°: a

4 tan θ + 2 = −2

8 cos θ − 4 = 0

2 cos θ + 4 = 3

8 sin θ −

Find the value of y: for 0° < y < 90°

a 12

tan θ = −1 for 0° < θ < 90°

for 0° < y < 180°

Solve for 0° ≤ θ ≤ 90°, to the nearest degree: a

cos θ = 0.7986

sin θ = 0.6428

sin θ = 0.3584

tan θ = 2.2460

tan θ = 0.7265

sin θ = −0.6428

Solve for 0° ≤ θ ≤ 360°, to the nearest degree: a

cos θ = 0.9063

cos θ = −0.7986

tan θ = 0.7002

tan θ = −0.7265

sin θ = 0.9336

Estimate the solutions to these equations for 0° ≤ x ≤ 360° using the graph of y = sin x: y 1 0.5 x 30°

60°

90°

120°

150°

sin x = −0.2

180°

210° 240° 270° 300° 330° 360°

−0.5 −1

sin x = 0.6

sin x = −0.95

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Estimate the solutions to these equations for 0° ≤ x ≤ 360° using the graph of y = cos x: y 1 0.5 x 30°

60°

90°

120°

150°

cos x = −0.2

180°

210° 240° 270° 300° 330° 360°

−0.5 −1

cos x = 0.1

2 cos x = −1.2

Reasoning 16

Explain whether these equations have a solution: a

cos θ − 4 = 0

2 tan θ + 4 = 0

Are these statements true or false? Justify your answer. a

The equation tan x = −1 has solutions in the second and fourth quadrants.

The equation

has only one solution within the domain 0° ≤ x ≤ 360°.

The equation

has solutions in the ﬁrst and third quadrants.

The equation sin x = 1 has solutions in both the ﬁrst and second quadrants.

The equation cos x = 0 has solutions in all four quadrants.

Problem-solving 18

Dylan is solving the equation 2 sin2 θ + 7 sin θ + 5 = 0. After factorising it, he obtains the pair of equations: a

What are the two equations obtained after factorising the given equation?

Which of the two equations has a solution?

A car is parked on a hill with a gradient of 10°. The angle between the car’s line of sight to the top of the hill and the bottom of the hill is 170°. In which quadrant of the ASTC diagram does this angle lie? What does this tell us about the sine, cosine, and tangent values of this angle?

Given that

, where 90° < θ < 180°, find the value of cos θ.

Given that

, where 270° < θ < 360°, find the value of tan θ.

Given that tan θ = -1, where 90° < θ < 180°, find the value of sin θ.

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Chapter 8 review 1

(Optional) Copy and complete: a

45° 1

b c

tan 45° = ⬚

45° 1

(Optional) Find the exact value of the pronumeral: a

b 20

w 60°

30° 2x

(Optional) Evaluate the expressions, leaving your answer in exact and rationalised form:

a c 4

sin 30° cos 30° + tan 45°

sin 45° tan 60°

sin2 60° + cos2 60°

(Optional) A right-angled triangle has one angle of 60° and an adjacent side of length a

Find the exact lengths of the other two sides.

Find the area of the triangle.

(Optional) Find the exact value of the expression:

(Optional) Use the diagram to ﬁnd the value of: a

sin b

cos b

tan b

1 b −1

x 1

−1

7 24 ,−  25 25 

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(Optional) Will the ratio have a positive or a negative answer? a

cos 309°

tan 225°

cos 92°

sin 318°

(Optional) Write an equivalent ratio using an acute reference angle: a

sin 200°

cos 230°

sin 115°

tan 171°

(Optional) For the point (x, y): a

Find the value of x.

Find the value of y.

y (x, y) 1 120° x

(Optional) Evaluate without using a calculator: a

sin 330°

cos 150°

tan 240°

cos 315°

(Optional) For the equation y = sin x: a

Complete the table of values. Round to two decimal places where necessary. x

0°

30° 90°

135°

180°

240° 270° 300° 360°

sin x

Write the coordinates of the y-intercept.

Write the coordinates of the x-intercepts from 0° ≤ x ≤ 360°.

Sketch the graph of y = sin x.

(Optional) Refer to the graph of y = cos x: a

Is the sign of cos 237° positive or negative?

In which quadrant of the unit circle is the angle 237°?

Find the x-value of the x-intercepts in these regions: i

0° < x < 360°

(−360) ° < x < 0°

y 2 1 x −360 −180

180 −1 −2

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360

(Optional) Regarding the curve y = sin x: a

If one cycle of the graph of y = sin x starts at x = −360°, when does the next cycle start?

True or false: As x gets larger and larger, the y-values of the graph of y = sin x will approach positive ∞.

120° < x < 200°

(−270) ° < x < −120°

1 x

iii

300° < x < 450°

−360 −180

180

360

−1 −2

(Optional) Determine the values of θ where 0° ≤ θ ≤ 360°: b

4 cos θ + 2 = 0

3 tan θ + 1 = −2

(Optional) Determine the value of y for the following equations: a

Is the graph of y = sin x increasing or decreasing over these intervals:

a 15

(Optional) Estimate the solutions to these equations for −180° ≤ x ≤ 180° using the graph of y = sin x: y 1 0.5 x −150°−120° −90° −60° −30° −0.5

30° 60° 90° 120° 150°

−1

a 17

sin x = 0.6

(Optional) Given that

sin x = −0.4 , where 180° < θ < 270°, ﬁnd the value of cos θ.

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9 Exponentials Big ideas • Exponentials demonstrate consistent growth or decay through multiplication by a constant factor. • Mathematical models provide powerful tools for analysing and predicting phenomena characterised by continuous change and accumulated effects over time.

Curriculum Outcomes VC2M10A11 Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital tools as appropriate VC2M10A14 Solve simple exponential equations VC2M10A15 Use mathematical modelling to solve applied problems involving inverse proportion, growth and decay, including in financial contexts to establish the compound interest formula as repeated applications of simple interest; formulate problems, choosing to apply linear, quadratic or exponential models; interpret solutions in terms of the situation; evaluate and modify models as necessary and report assumptions, methods and findings VC2M10A16 Solve equations graphically or using systematic numerical guess-check-and-refine with digital tools, with consideration of whether all solutions have been found VC2M10N01 Recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations VC2M10AA05 Describe, interpret, and sketch parabolas, hyperbolas, circles and exponential functions and their transformations

Chapter outline 9.01 9.02 9.03 9.04 9.05 9.06

Exponential relationships Characteristics of exponential graphs Optional: Transformations of exponentials Exponential equations Compound interest Exponential models Chapter 9 review

436 446 460 484 493 507 522

Carbon-14 dating uses exponentials to help archaeologists date ancient artefacts that are up to 50 000 years old!

9.01 Exponential relationships After this lesson, you will be able to… • differentiate between linear and exponential relationships. • identify the characteristics of exponential growth and decay. • recognise the growth or decay factor in exponential relationships. • interpret tables, equations, and graphs that represent exponential relationships. • formulate equations for exponential relationships based on provided data or patterns. • describe real-world scenarios that can be modelled using exponential relationships.

Exponential relationships We have seen that for a linear relationship, growth or decay are constant. x

f (x)

−5

The table shows an example of linear growth, an increase of one unit in x has an increase of three units in y.

−5

The table shows an example of linear decay, an increase of one unit in x has an decrease of ﬁve units in y.

−5

Exponential growth is not linear. Each time x increases by one, we multiply by the same amount each time. The multiplication factor will be a number greater than 1. This is sometimes called the constant factor, common ratio, or growth factor. x

f (x)

×3 3 =3 1

436

The table shows an example of exponential growth, an increase of one unit in x means that we triple the y-value.

×3 ×3 9 27 =3 =3 3 9

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

If we plot the table of values on a graph, we get a curve that is increasing at an increasing rate.

Every one unit moved in the x-direction causes the y-coordinate to triple.

21 18

y = 3x

We are multiplying by 3.

12 9 6 3

x 1

We may also have exponential decay. Each time x increases by one, we multiply by the same amount each time, but this time we multiply by a number between 0 and 1. We sometimes call this the decay factor. x

100

0.1

The table shows an example of exponential decay, an increase of one unit in x means that the y-value is one-tenth of the previous value, so the constant factor is 0.1

×0.1 ×0.1 ×0.1 10 1 0.1 = 0.1 = 0.1 = 0.1 100 10 1 100

If we plot the table of values on a graph, we get a curve that is decreasing at an decreasing rate.

Every one unit moved in the x-direction means that the y-coordinate is divided by 10.

80 70 60 50

We multiply by 0.1 each time.

y = 100(0.1x)

40 30 20 10

x 1

We need to be able to recognise exponential relationships from equations, tables of values, and graphs.

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Example 1 For y = 2x: a Complete this table of values. −3

x y

−2

−1

Create a strategy Substitute in each x-value and then simplify.

Apply the idea For each x-value, we can evaluate 2x. Let’s show our work for the ﬁrst one. y = 2x =2

Write the equation

−3

Substitute the ﬁrst value of x from the table

= 0.125

Express as a decimal

−3

−2

−1

−3

−2

−1

23 8

2 y

2 0.125

2 0.25

2 0.5

2 1

2 2

2 4

b Is y = 2x a linear or an exponential relationship?

Create a strategy We will check if the y-values in the table are increasing by a constant amount or have a factor they are being multiplied by.

Apply the idea x y=2

−3 0.125

−2 0.25

−1 0.5

0 1

1 2

2 4

3 8

We can ﬁnd the ratio between each pair of numbers, if they are all the same, then it is exponential. We’ll show our work for the ﬁrst three pairs.

Since these are all the same this is an exponential relationship. The multiplying factor is greater than 1, so this is exponential growth to be more speciﬁc.

Reﬂect and check Without the use of a table, we can tell this is an exponential relationship as there is a variable in the exponent.

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Example 2 Which graphs show exponential relationships? A y

2 −4 −3 −2 −1 −2

x 1

−4

−6

−8

C y

D 8

−4 −3 −2 −1 −2

x 1

−4 −3 −2 −1 −2

2 −4 −3 −2 −1 −2

−4

−6

−8

Create a strategy Look at the shape of the graph. An exponential relationship will be increasing at an increasing rate or decreasing at a decreasing rate.

Apply the idea Graph B is a straight line, indicating a linear graph. Graph C has a U-shape, representing a quadratic graph. Graphs A and D are exponential curves, with A showing exponential decay and D showing exponential growth. So, the correct answers are options A and D.

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Example 3 For each table of values, decide whether the relationship is exponential or not. If it is exponential, ﬁnd what we multiply by each time. a

x y

−3 −2

−2 1

−1 4

0 7

1 10

2 13

3 16

Create a strategy Look at the pattern in the y-values.

Apply the idea As x increases by 1, y is increasing by 3. This is a linear relationship, so is not exponential.

x y

−3 0.015 625

−2 0.0625

−1 0.25

0 1

1 4

2 16

3 64

Create a strategy Look at the pattern in the y-values.

Apply the idea We can see from the right-hand side of the table the y-values: 1, 4, 16, 64 This suggests an exponential where we multiply by 4. To check: 0.015 625 × 4 = 0.0625 0.0625 × 4 = 0.25 0.25 × 4 = 1 This is an exponential relationship with a growth factor of 4.

x y

−3 8

−2 4

−1 2

0 1

1 0.5

2 0.25

3 0.125

Create a strategy Look at the pattern in the y-values.

Apply the idea If we look at the y-values, we can see that each value is half of the previous value. This is an exponential relationship with a decay factor of 0.5

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x y

−3 1

−2 2

−1 6

0 24

1 120

2 720

3 5040

Create a strategy

Apply the idea

Look at the pattern in the y-values.

Going from 1 to 2 means multiplying by 2, but going from 2 to 6 means multiplying by 3 The ratio is not the same, so this is not an exponential relationship.

Example 4 This pattern shows an exponential relationship.

Step 1

Step 2

Step 3

Step 4

a Find the growth or decay factor.

Create a strategy We can write the number of squares in each step in a table of values.

Apply the idea Step Number of squares

1 2

2 4

3 8

4 16

Looking at the number of squares, to get from one step to the next, we multiply by 2. The growth factor is 2.

Reﬂect and check We may do this without a table of values by observation.

b Find the number of squares for Step 5.

Create a strategy

Apply the idea

We need to use the growth factor to go from Step 4 to Step 5.

To go from one step to the next, we multiply by the constant factor. Step 5 = 2 × Step 4 Step 5 = 2 × 16 Step 5 = 32

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Idea summary Exponential relationships are relationships in which y-values change by multiplying by a ﬁxed amount when the x-value increases by 1. Exponential growth relationships look like curves increasing at an increasing rate.

18 y 16 14 12 10 8 6 4 2

Exponential relationships have an equation which has a variable in the exponent like y = 2x. x 1

Exponential decay relationships look like curves decreasing at a decreasing rate and have an equation which has a variable in the exponent like y = 0.5x.

18 y 16 14 12 10 8 6 4 2

x 1

Practice questions Choose your question path Explorer 1ab, 2–5, 6bc, 7–10, 13, 15

Adventurer 1ac, 2–5, 6ac, 7–10, 12, 13, 15

Trailblazer 1cd, 2–5, 6ad, 7, 10–15

Understanding 1

442

For each table of values: i

Describe the pattern in words.

x 0 1 2 3 y 5 10 20 40

x y

0 5

x 0 1 2 3 y 2.5 7.5 22.5 67.5

x y

0 40 000

Find the next two terms.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

1 10

2 15

3 20

1 2 3 4000 400 40

Describe the pattern in words.

Find the number of circles for Step 4.

Step 1

Describe the pattern in words.

Find the number of hexagons for Step 4.

Step 2

Step 1 Step 2

Step 3

Fluency 4 Ex 1

For y = 5x: a

Complete the table of values. x

0 50 1

5 y b Ex 2

1 51 5

5 55 3125

Is y = 5x a linear or an exponential relationship?

Are these functions linear, exponential or quadratic? a

y = 4x 2

y = 4x + 7 y = 4 × 7x

y = 4x + 7

−5 −4 −3 −2 −1 −2

x 1

2 3 4 5

50 40

−4

−6

−8

−10

x 1 2 3 f (x) 0.8 1.6 2.4

−4 −3 −2 −1

4 3.2

x f (x)

1 0.125

x 1

2 0.5

3 2.0

4 8.0

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6 Ex 3

For these exponential functions: i

Find the growth or decay factor.

Find f (5).

x 1 2 3 4 f (x) 0.123 0.246 0.492 0.984

x 1 2 3 f (x) 2 30 450

Ex 4

4 6750

f (x) x f (x)

1 10

2 35

1 4096

2 1024

3 4 122.5 428.75 3 256

4 64

Consider the pattern: a

Find the growth or decay factor.

Find the number of triangles for Step 4. Step 1 Step 2

The total volume of water that has dripped from a tap is measured each minute: a

Describe the relationship between time and volume.

Complete the values in the table.

Find the volume after 8 minutes.

Time (minutes) 1 2 3 4 5 6

Step 3

Volume (mL) 3 9 27 81

Reasoning 9

The cumulative rainfall over a series of 4 days is represented on the graph. a

Describe the relationship between time and volume.

Determine if the points represent a linear or exponential function. Explain your reasoning

Assuming the pattern continues over the next few days, ﬁnd the cumulative volume on day seven.

Volume (mL)

32 28 24 20 16 12 8 4

Time (days) 1

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Consider the graphs:

Find the rule for g (x).

Find the rule for f (x).

Determine which is an exponential function and which is a linear function. Explain your reasoning.

f (x)

16 14

g (x)

12 10 8 6 4 2

x 1

Consider the table of values for x ≥ 1 where f (x) = (1.05)x and g(x) = 5x. Is it correct to say that g(x) is always greater than f (x) for all values of x ≥ 1?

x f (x) g(x)

1 1.05 5

2 1.10 10

3 1.16 15

4 1.22 20

5 1.28 25

Explain your reasoning.

Problem-solving 12

Consider the 7 plotted points:

Find three points that form a linear relationship.

For each 1 unit increase in x, determine how much the linear function increases by.

Find the three points that form an exponential relationship.

For each 1 unit increase in x, determine the constant ratio for the exponential function.

F E

20 C

15 10

5 A

B 1

x 2

A population of bacteria doubles every hour. Initially (at time = 0), there are 10 bacteria. Create a table showing the population of bacteria for the next 5 hours and determine if the relationship between time and population is linear or exponential.

A radioactive substance decays by 20% every hour. Initially there are 100 grams of the substance. Create a table showing the mass of the substance for the next 6 hours and determine if the relationship between time and mass is linear, quadratic or exponential.

A pile of bricks is decreased by 500 bricks every pallet. Initially there are 20 000 bricks in the pile. Create a table showing the number of bricks remaining after the ﬁrst 5 pallets are loaded and determine if the relationship between number of bricks in the pile and number of pallets is linear, quadratic or exponential.

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9.02 Characteristics of exponential graphs After this lesson, you will be able to… • recognise the standard form of exponential relationships as y = bx. • identify the y-intercept and horizontal asymptote of exponential functions. • determine the behaviour of exponential functions based on the value of b. • use tables of values, technology, or equations to sketch graphs of exponential functions. • differentiate between exponential growth and exponential decay functions based on the value of b.

Characteristics of exponential graphs Exponential relationships are where we multiply to get from one y-value to the next for each increase in 1 in the x-values. An exponential relationship can be modelled by y = bx where b is the growth or decay factor. 8

All functions of the form y = bx have these features in common: • The growth or decay factor is b. • The y-intercept is at (0, 1). • There is a horizontal asymptote at y = 0.

7 6 5 0<b<1

The value of b determines whether the function increases or decreases. • If 0 < b < 1, the function decreases. • If b > 1, the function increases.

b>1

4 3 2 1

−4 −3 −2 −1

x 1

Notice the dashed line at y = 0. This line is called an asymptote. An exponential function can get inﬁnitely close to an asymptote, but it can never cross it. This means that an exponential function of this form will not have an x-intercept.

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.02 to answer these questions. 1.

Is the graph increasing faster when b = 2 or when b = 5?

In general, what happens to the graph as b increases from 1.1 to 5?

Is the graph decreasing faster when b = 0.8 or when b = 0.3?

In general, what happens to the graph as b decreases from 0.9 to 0.1?

As x increases in exponential growth functions, the graph increases faster for higher values of b. As x increases in exponential decay functions, the graph decreases faster for smaller values of b. To draw the graph of an exponential function, we can use a variety of strategies, including: • Completing a table of values for the function and drawing the curve through the points • Using technology, such as a graphing calculator • Identifying key features from the equation

Example 1 Consider the function y = 3x. a Complete the table of values. x

−5

−4

−3

−2

−1

Create a strategy Substitute x = −2, −1, 0, 10 into y = 3x.

Apply the idea Substitute x = −2

Apply the negative exponent law

Evaluate the exponent

Substitute x = −1

Apply the negative exponent law

Substitute x = 0

Apply the zero exponent law

9.02 Characteristics of exponential graphs mathspace.co

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, to save time for x = 1, 2, 3, 4, 5, take the reciprocals of the corresponding y-values

Since

for x = −1, −2, −3, −4, −5. y = 310

Substitute x = 10

= 59 049 x

−5

−4

Evaluate the exponent −3

−2

−1

243 59 049

b Graph the exponential function.

Create a strategy Plot the points from the table of values found in part (a).

Apply the idea Because the graph increases so quickly, we need to choose an appropriate scale. To see most of the points, we can plot the plots from x = −3 to x = 3. y 27 24 21 18 15 12 9 6 3 −4 −3 −2 −1

x 1 2 3 4

c Is y = 3x an increasing function or a decreasing function?

Create a strategy Consider the y-values in the table from part (a) from left to right.

Apply the idea The y-values in the table are in increasing order, so y = 3x is an increasing function.

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Example 2 Consider the function

a Determine whether the function is increasing or decreasing.

Create a strategy

Apply the idea x

For functions in the form y = b the function is increasing if b > 1 and the function is decreasing Since if 0 < b < 1.

, the function is decreasing.

b Identify the equation of the horizontal asymptote.

Create a strategy For any function in the form y = bx, the horizontal asymptote is on the x-axis.

Apply the idea The equation of the horizontal asymptote is y = 0.

Reﬂect and check Because an asymptote is a line, it must be represented by an equation. Since exponential functions have horizontal asymptotes, the equation of the line will be in the form y = c where c is any real number.

c State the y-intercept.

Create a strategy Substitue x = 0 to ﬁnd the y-value of the y-intercept.

Apply the idea Write the function

Substitute x = 0

Apply zero exponent law

The y-intercept is at (0, 1).

Reﬂect and check For any exponential function of the form y = bx, the y-intercept will be at (0, 1).

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d State one other point on the curve.

Create a strategy Choose any value of x and substitute it into the equation to ﬁnd y.

Apply the idea Let x = −1. Write the function

Substitute x = −1

Apply negative exponent law

Another point on the curve is (−1, 4). e Sketch the graph.

Create a strategy

Apply the idea

Plot the key points and key features found in parts (a) through (d).

7 6 5 4 3 2 1 −4 −3 −2 −1

x 1

Example 3 Consider the function y = 8x. a Can the value of y ever be negative or zero?

Create a strategy Recall the property of the asymptote of the exponential function y = bx.

Apply the idea Since y = 8x is an exponential function of the form y = bx, there is an asymptote at y = 0. This means its value of y can never be equal to or lower than 0. So it cannot be negative or zero.

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b As the value of x increases towards ∞, what can we say about the value of y? A

y tends towards 8.

y tends towards 0.

y tends towards ∞.

Create a strategy Substitute some positive values of x into y = 8x.

Apply the idea We can use x = 5, 10. y = 85

Substitute x = 5

= 32 768 y=8

Evaluate

Substitute x = 10

= 1 073 741 824

Evaluate

We can see the values are increasing into inﬁnite values or ∞. The answer is option B. c As the value of x decreases towards −∞, what value does y approach? A

∞

Create a strategy

Apply the idea

Recall the property that as x decreases in exponential growth functions, y approaches the horizontal asymptote.

The answer is option A.

d Determine the y-value of the y-intercept of the curve.

Create a strategy Substitute x = 0 into y = 8x.

Apply the idea y = 80 =1

Substitute x = 0 Apply the zero exponent law

So, the y-intercept is at (0, 1). e How many x-intercepts does the curve have?

Create a strategy

Apply the idea

Use the answer from part (a).

We knew from part (a) that y = 8x can never have y = 0 because of the horizontal asymptote. However, x-intercept can only exist when y = 0. So the curve has 0 x-intercepts.

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Which of these could be the graph of y = 8x? A

2 −4 −3 −2 −1 −2

x 1

−4

−6

−8

−4 −3 −2 −1 −2

x 1

−4 −3 −2 −1 −2

2 −4 −3 −2 −1 −2

−4

−6

−8

Create a strategy Use the features found from previous parts.

Apply the idea The key features of y = 8x are: • The y-values can never be negative. • As x approaches ∞, y approaches ∞. As x approaches −∞, y approaches 0. • The previous feature tells that the curve is an increasing function. • The y-intercept is at (0, 1). All of these features can be found in option D.

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Example 4 Consider the function

a Sketch a graph of y = 2x on the Cartesian plane.

Create a strategy Plot the y-intercept and substitute x = 1, 2 into y = 2x for more points.

Apply the idea Recall that the y-intercept for functions of the form y = bx is at (0, 1). For more points, substitute x = 1, 2 into y = 2x: y = 21 =2 y=2 =4

Substitute x = 1 Evaluate the exponent

Substitute x = 2 Evaluate the exponent

So we can plot the points (0, 1), (1, 2), and (2, 4). y 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

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b Sketch

on the same Cartesian plane.

Create a strategy Plot the y-intercept and substitute x = −1, 1 into

for more points.

Apply the idea Recall that the y-intercept for functions of the form y = bx is at (0, 1). For more points, substitute x = 1, 2 into

Substitute x = −1

Apply the negative exponent law

Substitute x = 1

Evaluate the exponent

So we can plot the points (−1, 2), (0, 1), and

. y 4 3 2 1

−4 −3 −2 −1

−1

x 1

−2 −3 −4

c Compare the graphs of y = 2x and

Create a strategy Use the graph from part (b) to compare the key features of each function.

Apply the idea The two graphs are reﬂections across the y-axis. The graph of y = 2x is increasing while the graph of

is decreasing.

Both graphs have the same horizontal asymptote and y-intercept.

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Idea summary An exponential relationship can be modelled by y = bx

All functions of the form y = bx have these features in common:

• •

The y-intercept is at (0, 1) There is a horizontal asymptote at y = 0

5 0<b<1

b>1

4 3 2 1

−4 −3 −2 −1

x 1

The value of b determines whether the function increases or decreases. • •

If 0 < b < 1, the function decreases If b > 1, the function increases

As x increases in exponential growth functions, the graph increases faster for higher values of b. As x increases in exponential decay functions, the graph decreases faster for smaller values of b.

Practice questions Choose your question path Explorer 1–4, 5acef, 6acd, 7abf, 8, 10, 11, 14–16, 18, 19

Adventurer 1–4, 5aceg, 6bce, 7bef, 8, 9, 11, 12, 14, 15, 17–19

Trailblazer 1–4, 5efgh, 6bef, 7beh, 8, 11–15, 16cd, 17cd, 19–21

Understanding 1

What part of the exponential function f (x) = bx determines whether it increases or decreases?

What is the name of the line that the graph of an exponential function approaches but never crosses?

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Is each statement true or false for y = bx? If it is false, correct the statement. a

If b > 1, then it is exponential growth.

If b < 0, then it is exponential decay.

If b is a fraction, then it is exponential decay.

If b > 0, then it is exponential growth.

The y-intercept is always (0, 1).

Describe what happens to y as the value of x increases. a

y 5

5 y = 4x

1x 4

−5 −4 −3 −2 −1 −1

2 3 4 5

−5 −4 −3 −2 −1 −1

2 3 4 5

−2

Determine whether these are increasing or decreasing exponential functions: a e

y = 3x

b f

y = 0.97x

y = 0.5x

y = 1.5−x

y = 1.05x

Fluency 6 Ex 1

456

For each exponential function: i

Complete the table of values.

Graph the exponential function.

iii

Determine whether the function is increasing or decreasing.

x −1 0 1 y = 3x

x −1 0 1 y = 10x

−1

x −1 0

−1

y = 5x

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

7 Ex 2

For each function: i

Determine whether the function is increasing or decreasing.

Identify the equation of the horizontal asymptote.

iii

Find the y-intercept.

Find one other point on the curve.

Sketch the graph.

f (x) = 2x

e Ex 3

f (x) = 4x

f (x) = 7x

f (x) = 8x

For the function y = 6x: a

Can the value of y ever be zero or negative?

As the values of x get larger and larger, what can we say about the value of y?

As the values of x get smaller and smaller, what value does y approach?

Find the y-value of the y-intercept of the curve.

How many x-intercepts does the curve have?

Sketch the graph of y = 6x.

Consider the expression 2x. a

Evaluate the expression when x = −4.

Evaluate the expression when x = 0.

Evaluate the expression when x = 4.

For what values of x will 2x have a negative value?

Describe what happens to the value of 2x as x gets larger.

Describe what happens to the value of 2x as x gets smaller.

The graph of y = 4x is shown.

Is each y-value of the function positive or negative?

What is the equation and name of the horizontal line, which y = 4x gets closer and closer to but never intersects?

5 4 3 2 1 x −3

−2

−1

11 Ex 4

Consider the functions f (x) = 5x and a

Sketch the graph of f (x) = 5x.

Sketch the graph of

Compare the graphs of f (x) = 5x and

. .

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Reasoning 12

For the functions a

and

Is each statement true or false? i

All of the curves have a maximum value.

All of the curves pass through the point (1, 2).

iii

All of the curves have the same y-intercept.

None of the curves cross the x-axis.

What is the y-intercept of each curve?

Describe what happens to the values of y as x gets increasingly larger.

For

, which is decreasing more rapidly for x > 0? Justify your

and

answer. 14

The graphs of two exponential functions, R and S, are shown.

One of the graphs is of y = 3x and the other graph is of y = 5x. Which is the graph of y = 5x?

For negative values of x, which function will have smaller values of y? Explain your reasoning.

R S

Do either of the functions f (x) = 9x or

have x-intercepts? Explain your reasoning.

Problem-solving 16

Find the missing coordinate in each ordered pair so that the pair is a solution to y = 5x: a

(3, ⬚)

(0, ⬚)

(⬚, 5)

(−1, ⬚)

Find the missing coordinate in each ordered pair so that the pair is a solution of a

(⬚, 9)

If the point (3, m) lies on the curve y = 2x, solve for m.

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The graph of y = 2x is shown: a

Find the value of y when x = 7.

Find the value of x when y = 256.

6 5 4 3 2 1

−5 −4 −3 −2 −1

The graph of f (x) = 3x is shown. Find the length of the interval PQ.

2 3 4 5

y P

Q −5 −4 −3 −2 −1

For the function, y = 3−x: a

Rewrite 3−x with a positive exponent using the exponent laws.

Sketch the graph of y = 3−x.

Compare the graphs of y = 3x and y = 3−x.

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9.03 Optional: Transformations of exponentials After this lesson, you will be able to… • describe the effect of vertical translations on exponential graphs. • analyse vertical dilations of exponential graphs and distinguish between vertical stretches, compressions and reflections. • determine the equation of an exponential function from a given graph by identifying the y-intercept value and the growth or decay factor. • apply combined transformations to exponential graphs, understanding the sequence of dilation, reflection and translation.

Vertical translation of an exponential graph The graph and key features of y = bx depend on the value of b. The key features of y = bx for b > 1 are: • Increasing at an increasing rate • y-intercept is at (0, 1) • The x-axis, y = 0, is the horizontal asymptote • A point on the graph is (1, b)

y 12 10 8 6 4 2

−4 −3 −2 −1 −2

−4

The key features of y = bx for 0 < b < 1 are: • Decreasing at an decreasing rate • y-intercept is at (0, 1) • The x-axis, y = 0, is the horizontal asymptote • A point on the graph is (−1, b)

y 12 10 8 6 4 2 −4 −3 −2 −1 −2

x 1

−4

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.03 to answer these questions. 1.

When c is positive, how does the graph compare to the original?

When c is negative, how does the graph compare to the original?

As the value of c changes, how does the equation change?

As the value of c changes, what happens to the horizontal asymptote?

The general rule for a vertical translation is: y = bx + k y

4 2

−4 −3 −2 −1

−4 −3 −2 −1 −2

Vertical translation of 3 units upwards: f (x) = 2x + 3

Vertical translation of 3 units downwards: f (x) = 2x − 3

The y-intercept and the horizontal asymptote will move k units vertically with the curve, for example: y

The graph shown is

Notice the horizontal asymptote has moved downwards from y = 0 to y = −4.

10 8

The y-intercept has moved downwards from (0, 1) to (0, −3).

6 4 2 −4 −3 −2 −1

−2

x 1

The curve has the same shape as

−4

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Example 1 This is a graph of y = 3x. a How do we shift the graph of y = 3 to get the graph of y = 3x − 4?

Move the graph 4 units to the right.

Move the graph downwards by 4 units.

Move the graph 4 units to the left.

Move the graph upwards by 4 units.

−4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy Use the fact that when a graph of y = f (x) is moved downwards by k units, its new equation is y = f (x) − k.

Apply the idea Since the new equation is y = 3x − 4 and k = −4 < 0, then it means that we need move the graph downwards by 4 units to get the graph of y = 3x − 4. So, the correct answer is option B.

b Plot y = 3x − 4 on the same graph as y = 3x.

Create a strategy Plot the graph of y = 3x, then shift each point down 4 units to get the graph of y = 3x − 4.

Apply the idea 5

4 3 2 1 −5 −4 −3 −2 −1 −1

Substitute x = 0 and x = 1 to ﬁnd two points on the curve y = 3x.

y = 3x

Then, shift each point down 4 units. The horizontal asymptote for y = 3x − 4 is y = −4. x 1

2 3 4 5

−2 −3 −4 −5

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Connect the points to draw the curve y = 3x − 4.

4 3 2 y = 3x

−5 −4 −3 −2 −1 −1

2 3 4 5 y = 3x − 4

−2 −3 −4 −5

Example 2 The graph of

is shown. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

that would obtain

a Describe a transformation of the graph of

Create a strategy Use the fact that when a graph of y = f (x) is moved upwards by k units, its new equation is y = f (x) + k.

Apply the idea Since the new equation is

and k = 3 > 0, then it means that the transformation of the

graph of g (x) is a translation up 3 units from

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b Sketch the graph of

on the same set of axes as

Create a strategy Plot the graph of

. Use the transformation found from part (a) to plot the graph of

Apply the idea From the graph of

, we move this graph upwards by 3 units. 5

Substitute x = −1 and x = 0 to ﬁnd two points on the

curve

4 3 2 1

1 f ( x ) =    2

−5 −4 −3 −2 −1 −1

Then, shift each point up 3 units.

The horizontal asymptote shifts up 3 units.

2 3 4 5

−2 −3 −4 −5

8 7

Connect the points to draw the curve

6 x 5  1 +3 ( ) g x =   4  2

x 2  1 1 f ( x ) =  2 

−5 −4 −3 −2 −1 −1

2 3 4 5

−2

c Write the equation of the horizontal asymptote of

Create a strategy

Apply the idea x

For any function in the form y = b ± k, the horizontal asymptote is at the y = ±k.

464

The equation of the horizontal asymptote is y = 3.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary If the function bx is translated vertically, the equation will be of the form y = bx + k If k > 0, it is translated upwards. If k < 0, it is translated downwards. The key features become: • A y-intercept at (0, 1 + k) • Horizontal asymptote at y = k

Vertical dilation of an exponential graph We will now look at exponential functions of the form y = a (bx).

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.03 to answer these questions. 1.

When ∣a∣ > 1, how does the graph compare to the original?

When 0 < ∣a∣ < 1, how does the graph compare to the original?

When a is negative, how does the graph compare to the original?

As the value of a changes, how does the equation change?

As the value of a changes, what happens to the horizontal asymptote?

The graph of y = a (bx) is a vertical dilation of y = bx. We can compare y = 3 (2x) to y = 2x. This will be a vertical stretch. Every point on the graph of y = 3(2x) is three times as far away from the x-axis as the point on y = 2x at the corresponding x-coordinate.

y 12 10

For example, we can see (2, 4) → (2, 12).

8 6 4 x

y = 3 (2 )

−6 −5 −4 −3 −2 −1 −2

The y-intercept for y = 2x is at (0, 1), and the y-intercept for y = 3 (2x) is at (0, 3).

y = 2x x

Both curves have y = 0 as the horizontal asymptote.

1 2 3 4 5 6

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465

We can compare y = 0.5 (2x) to y = 2x. This will be a vertical compression. Every point on the graph of y = 0.5 (2x) is half as far away from the x-axis as the point on y = 2x at the corresponding x-coordinate.

y 12 10

For example, we can see (3, 8) → (3, 4).

The y-intercept for y = 2x is at (0, 1), and the y-intercept for y = 0.5 (2x) is at (0, 0.5).

6 y = 2x

4 2

y = 0.5(2 x )

−6 −5 −4 −3 −2 −1 −2

Both curves have y = 0 as the horizontal asymptote.

1 2 3 4 5 6

We can compare y = −3 (2x) to y = 2x. This will be a vertical stretch and a reﬂection across the x-axis. There is still a dilation by a scale factor of 3, but the negative also produces a reﬂection in the x-axis.

y 8

y = 2x

For example, we can see (2, 4) → (2, −12). The y-intercept for y = −3 (2x) is at (0, −3).

4 x −6 −5 −4 −3 −2 −1 −4

1 2 3 4 5 6 y = −3(2x )

Both curves have y = 0 as the horizontal asymptote.

−8 −12

Example 3 Of the two functions y = 2x and y = 3 × 2x, which is increasing more rapidly for x > 0? A y = 2x

y = 3 × 2x

Create a strategy Create a table of values to compare the values of y between the two functions y = 2x and y = 3 × 2x.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea For x = 1: y = 2x = 21

Substitute 1 to x

Use the power of a power rule

=2 y = 3 × 2

Evaluate x

= 3 × 21

Substitute 1 to x

=3×2

Use the power to a power rule

Evaluate

Similarly, by substituting the remaining x-values into y = 2x and y = 3 × 2x, we get: x

y = 2x

2 6

4 12

8 24

16 48

32 96

y = 3 × 2x

Based on the table, the function y = 2x is increasing more rapidly for x > 0. So, the correct answer is option B.

Example 4 Consider the function

a Find the coordinates of the y-intercept.

Create a strategy Substitue x = 0 to ﬁnd the y-value of the y-intercept.

Apply the idea Write the function

Substitute x = 0

Use zero exponent law

Evaluate

The y-intercept is at (0, 2).

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b What is the equation of the horizontal asymptote?

Create a strategy For any function in the form y = a (bx), the horizontal asymptote is at the y = 0.

Apply the idea The equation of the horizontal asymptote is y = 0.

c Is the function increasing or decreasing?

Create a strategy For functions in the form y = bx the function is increasing if b > 1 and the function is decreasing if 0 < b < 1.

Apply the idea The constant factor of this function is . Since

, the function is decreasing.

d Graph the function.

Create a strategy Plot the key points and key features found in parts (a) through (c). Use the fact that a function of the form y = a (bx) is dilated by a factor of a.

Apply the idea , we multiply the y-values by 2.

From the graph of 11 10 9 8 7 6 5 4 3 2 1 −4 −3 −2 −1

468

Substitute x = −1 and x = 0 to ﬁnd two points on the

graph of

Then, multiply the y-values of the points by 2.

1 f ( x ) =   5

x 3

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

11 10 9 8 7 6 5 4 3 2 1 −4 −3 −2 −1

Connect the points to draw the curve of

The horizontal asymptote stays at y = 0.

x 1

Idea summary If the function bx is dilated or reﬂected vertically, the equation will be of the form y = a (bx) If ∣a∣ > 1 then it is stretched and if 0 < ∣a∣ < 1 then it is compressed. Negative values of a result in a reﬂection acress the x-axis. The key features become: • •

A y-intercept at (0, a) Horizontal asymptote at y = 0

Find the equation In the previous section, we saw that functions of the form y = a (bx) always have a y-intercept at (0, a). Recall that the value of b is the ratio between successive terms in an exponential function. These two facts can help us identify the equation of any exponential function that has been dilated or reﬂected.

Example 5 The equation of the graph is of the form y = a (b)x.

3 2 1 −4 −3 −2 −1

x 1

2 3 (1, −1)

−1 −2 −3 −4 −5 (0, −5) −6 −7 −8 −9

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469

a What is the y-intercept? What does this tell us about the equation?

Create a strategy

Apply the idea

Check the point where the curve crosses the y-axis.

Based on the graph, the y-intercept is at (0, −5). This tells us that a = −5.

b What is the decay factor? What does this tell us about the equation?

Create a strategy Use the coordinates of two successive points on the graph of the exponential function, and ﬁnd the ratio of their y-values.

Apply the idea We are given the points (0, −5) and (1, −1), which are successive points since x = 0 and x = 1. Dividing the y-value of the point on the right by the y-value of the previous point gives us

This tells us

Reﬂect and check A different method would be to use the fact that a = −5 to rewrite the equation as y = −5 (bx). Then, substitute one of the points and solve for b. Substituting one of the points, let’s say (1, −1), to y = a(b)x where a = −5, we will get the value of b. Substitute the values

Evaluate the power

Divide both sides by −5

Evaluate

Since

then this means that the decay factor is .

c What is the equation?

Create a strategy Use the y-intercept and the decay factor found in parts (a) and (b).

Apply the idea Since a = −5 and

470

, the equation is given by:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary For any function of the form y = a (bx), we can ﬁnd the equation from a graph by identifying the y-value of the y-intercept and the growth or decay factor. The y-value of the y-intercept is a in the equation, and the growth or decay factor is b.

y = a (bx) a b

y-value of the y-intercept growth or decay factor

Apply both transformations We will now explore exponential functions of the form y = a (bx) + k.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.03 to answer these questions. 1.

Which variable(s) change the horizontal asymptote?

Which variable(s) change the shape?

Which variable(s) change the y-intercept?

Consider graphing y = 3 (2x) − 4. We can start by graphing y = 2x with: • y-intercept at (0, 1) • Horizontal asymptote at y = 0 • Another point at (2, 4)

y 12 10 8

y = 2x

6 4 2 −4 −3 −2 −1 −2

x 1

−4

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We can can then apply the dilation to get y = 3 (2x) with: • y-intercept at (0, 3) • Horizontal asymptote at y = 0 • Another point at (2, 12)

y 12

y = 3(2x)

10 8 6 4

y = 2x

−4 −3 −2 −1 −2

−4

We can then apply the translation to get y = 3 (2x) − 4 with: • y-intercept at (0, −1) • Horizontal asymptote at y = −4 • Another point at (2, 8)

y 12 10 8 6 4 x

y = 3(2 )

−4 −3 −2 −1 1 2 3 −2 y = 3(2x) − 4

x 4

−4

In general, for y = a (bx) + k • The asymptote would be y = k. • The y-intercept will be at (0, k + a), so it is above the asymptote if a is positive, and below the asymptote if a is negative. If we are doing multiple transformations, we should go in order of dilation, reflection and translation.

Example 6 Sketch the graph of y = −0.5 (2x) + 1.

Create a strategy We can consider the transformations to get an idea of what the graph will look like and then identify the key features from the equation.

Apply the idea Starting the function y = 2x, ﬁrst there was a compression with a scale factor of 0.5. Next, there was a reﬂection across the x-axis. Finally, there was a vertical translation upwards by 1 unit.

472

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Let’s graph using the key features. To ﬁnd the y-intercept we can set x = 0 and evaluate. y = −0.5 (2x) + 1 y = −0.5 (20) + 1 y = −0.5 (1) + 1 y = −0.5 + 1 y = 0.5 The horizontal asymptote is y = 1. To ﬁnd another point we can pick an x-value and evaluate. y = −0.5 (2x) + 1 y = −0.5 (21) + 1 y = −0.5 (2) + 1 y = −1 + 1 y=0 We now have two points and the asymptote so can draw the graph. 4

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Reﬂect and check We can see each transformation applied one at a time as well. 4

Original function of y = 2x.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

9.03 Optional: Transformations of exponentials mathspace.co

473

Compression with the scale factor of 0.5.

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

Reﬂection across the x-axis.

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

Translation upwards 1 unit.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

474

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary A function of the form y = a (bx) + k has both: • •

Vertical translation, moving the graph up or down. Vertical dilation, stretching the curve in the y-direction.

We can ﬁnd the values of a, b, and k for a given curve. • •

y = k will be the asymptote. The y-intercept is at (0, a + k).

We can ﬁnd b using any other point on the curve.

Practice questions Choose your question path Explorer 1–3, 5–9, 12, 13, 14deg, 16, 17, 19ab, 21, 22, 24, 26, 27

Adventurer 1–3, 5–11, 13, 14abg, 16, 18, 19bc, 20, 22, 24, 26, 28

Trailblazer 1–11, 14ceh, 15, 16, 18, 19cd, 20, 23, 25, 27, 29–31

Understanding 1

For y = 2x and y = 2x + k: a

Are these key features the same or different? i

Rate of change

Asymptote

iii

y-intercept

Shape

Increasing or decreasing

What happens if k is positive?

What happens if k is negative?

For y = 2x and y = a × 2x, with a > 0: a

Are these key features the same or different? i

Rate of change

Asymptote

iii

y-intercept

Shape

Increasing or decreasing

What happens if a > 1?

What happens if 0 < a < 1?

What would happen if a was negative?

9.03 Optional: Transformations of exponentials mathspace.co

475

Match each graph to one of the equations: i

y = 2x

y = −2x

iii b

2 1

−4

−6

−8

−4 −3 −2 −1 −2

−4

−6

−8

For f (x) = 3x, will the transformation change the equation of the horizontal asymptote? a

A vertical dilation by a factor of 5

A translation down 2 units

A refection across the x-axis

A vertical dilation by a factor of

Fluency Ex 1

Describe how the graph of y = 2x is transformed to get each graph: a

476

y = 2x + 4

y = 2x − 6

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

y = 5 (2)x

Ex 2

The graph of a

is shown.

Describe a transformation of the graph of that would obtain

on the same

Sketch the graph of .

set of axes as c

Write the equation of the horiztonal asymptote of .

Ex 3

The graph of f (x) = 5x is shown. a

Describe a transformation of the graph of f (x) = 5x that would obtain g (x) = −5x.

Sketch the graph of g (x) = −5x on the same set of axes as f (x) = 5x.

As x increases, what happens to the function values of g (x)?

Of the two functions y = 4x and

The graph of f (x) = 8x is shown. a b

y 5 4 3 x 2 1 f ( x) = 1 3 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

2 3 4 5

f ( x) = 5 x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

, which is increasing more rapidly for x > 0?

Describe transformations of the graph of f (x) = 8x that would obtain g (x) = 0.5 (8x). x

Sketch the graph of g (x) = 0.5 (8 ) on the same set of axes as f (x) = 8x.

6 4 2 −4 −3 −2 −1 −2

f ( x) = 8x

x 1

−4 −6 −8

9.03 Optional: Transformations of exponentials mathspace.co

477

The graphs of A, B, and

are shown. B

What transformation has been applied to to get the graph of A?

11 10 9

What transformation has been applied to to get the graph of B?

8 7 y=

1 6

5 4

3 2 1

−1

For each graph, describe the transformation applied to graph A to produce graph B. a

1 −4 −3 −2 −1

−1 B

x 1

−2

−3

−4 −3 −2 −1

478

−4 −3 −2 −1

y 11 10 9 8 7 6 5 4 3 2 1

−4

x 1

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−2

12 11 10 9 8 B 7 6 5 4 3 2 1 −1

x 2

x 1

Two exponential functions, R and S, are graphed.

One of the graphs is of y = 5x and the other graph is of y = 5x − 2. Which is the graph of y = 5x − 2?

x S R

For the function g (x) = 3 (4)x: a

What transformation has been applied to f (x) = 4x to obtain the graph of g (x) = 3 (4)x?

Complete the table of values: −2

x f (x) = 4x

−1

g (x) = 3 (4)x c Ex 4

For each exponential function: i

Find the coordinates of the y-intercept.

What is the equation of the horizontal asymptote?

iii

Is the function increasing or decreasing?

Graph the function.

y = 3x − 5

e 15

Sketch the graphs of f (x) = 4x and g (x) = 3 (4)x on the same Cartesian plane.

y = 2 (5)x

The equation of the graph is of the form y = a (bx). a

What is the y-intercept? What does this tell us about the equation?

What is the growth factor? What does this tell us about the equation?

y = 4x + 3

(1, 20)

18 16 14 12 10

What is the equation?

8 6 (0, 5) 4 2 −2

−1

x 1

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479

Ex 5

The equation of the graph is of the form y = a (b)x. What is the y-intercept? What does this tell us about the equation?

1 −2

What is the decay factor? What does this tell us about the equation?

What is the equation?

−1

y x 1

−1 −2 (1, −2) −3 −4 (0, −4) −5 −6 −7 −8 −9

This table of values is for an exponential function. x

Find the growth or decay factor.

Find the y-intercept.

What is the exponential equation for the table of values?

This table of values is for an exponential function. x

−4

−3

−2

−1

112

What is the exponential equation for the table of values? Ex 6

For each exponential function: i

Find the coordinates of the y-intercept.

Find the equation of the horizontal asymptote.

iii

Describe the transformations from y = 2x.

Graph the function.

y = 3 (2x) + 1

y = −(2x) + 4

y = 0.5 (2x) − 3

y = −3 (2x) − 1

Reasoning 20

When the only transformation applied to the function f (x) = bx is a dilation by a factor of a, what are the coordinates of the y-intercept? Explain your reasoning.

How could we use the graph of y = 7x to draw the graph of y = 7 − x?

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The graphs of two exponential functions, A and B, are shown.

One of the graphs is of y = 3x and the other graph is of A

Which function has its graph below that of the other when x < 0? Explain your reasoning.

Consider the equation y = −10x. a

Jeannine thinks she has found a set of solutions for the equation as shown in the table: −2

−1

−10

−100

−1000

he notices that all the y values are negative and concludes that for any value of x, y S must always be negative. Is she correct? Explain your reasoning.

Graph y = −10x.

Find the values of x for which y = 0.

The function y = 3 (2x) is shown. a

10 8

Describe a transformation of the graph of y = 3 (2x) that would obtain y = −3 (2x). Sketch the graph of y = −3 (2 ) on the same set of axes as y = 3 (2x).

The number of bacteria over time is to be modelled by an exponential function, with x representing time and y representing the number of bacteria. Which function should be used? Explain.

4 2

−1

y = 3x

y = 3x − 1

y = 2 (3x) − 1

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

Emir graphed y = 2 (3x) − 1 by transforming a few points. His work is shown. x

y = 3(2x)

2 4 6 8 10

4 3 2 1

−4 −3 −2 −1

−1

x 1

−2 −3

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481

Yalile graphed y = 2 (3x) − 1 by ﬁnding some key features. Her work is shown. y-intercept: (0, 1) y = 2 (30) − 1

y = 2 (1) − 1

y = 1

Horizontal aymptote: Translated downwards 1 unit to y = −1

2 1

One other point: (1, 5) a

Did Emir or Yalile draw the graph correctly?

Explain and correct the error.

−4 −3 −2 −1

x 1

−1

−2 −3

Problem-solving 26

The graph of f (x) = 2x is shown. a

Describe a transformation of the graph of f (x) = 2 that would obtain g (x) = 2− x. −x

Sketch the graph of g (x) = 2 axes as f (x) = 2x.

Is g (x) = 2−x an increasing or decreasing function?

on the same set of

2 1 −5 −4 −3 −2 −1 −1 −3 −4 −5

For the function f (x) = 4x: a

Sketch the graph of f (x) = 4x.

Complete the table of values for x

−3

−2

−1

g (x)

482

Sketch the graph of

Compare the graphs of f (x) = 4x and

How can

. .

be transformed to get the graph of

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

f (x) = 2x

−2

x 1

2 3 4 5

The equation of the original function is form a

. The equation of the new function is of the

Complete the table. Point on original graph

(−1, 2)

(0, 1)

Point on new graph

(−1, ⬚)

(0, 3)

(2, ⬚)

Find the equation of the new function.

Sketch the graphs of the original and new function on the same set of axes.

For positive x-values, is the graph of the new function above or below the graph of

For negative x-values, is the graph of the new function above or below the graph of

What is the equation of this exponential graph? y

−3 −2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

(2, ⬚)

The equation of the graph is of the form y = a (bx) + k. x

−1

y = a (bx) + k y = a (bx) y = bx a

⬚ ⬚

−1

⬚ ⬚

What is the asymptote? What does this tell us about the equation?

Complete the row of the table for y = a (bx).

What is value of a?

Complete the row of the table for y = bx.

What is the equation of the graph?

y (2, 8)

8 6 4 (1, 2)

x −2

−1 −2

1 (0, −1)

9.03 Optional: Transformations of exponentials mathspace.co

483

Find the equation of the graphs below in the form y = a (bx) + k. a

8 6 4 2

10 5 −3

−2

−1

(1, 0) 2

−5 −10

−2

−1

−2 −4 −6 (−1, −8) −8 −10 −12

x 1

9.04 Exponential equations After this lesson, you will be able to… • understand the definition of an exponential equation. • apply a systematic approach to solving an exponential equation. • use the power of a power law to rewrite exponential expressions with the same base. • solve exponential equations graphically.

Exponential equations An exponential expression is an expression of the form bx where the base, b, is a positive number and x is a variable. An exponential equation is an equation in which one, or both, sides are exponential functions, for example: 3x + 2 = 81. The key property we need is that: If bx = by, then x = y To solve an exponential equation, we can: 1. Write in the form bx = c. 2. Rewrite in the form bx = b⬚.

3. Set x = ⬚, and solve if necessary.

We may need the power of a power law, (am)n = amn, to rewrite with the same base.

484

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 Solve 4x = 48.

Create a strategy Set the exponents equal and solve for x.

Apply the idea Both sides of the equation are exponentials with the same base. So the powers must be equal. 4x = 48

Write the equation

x=8

Equate the exponents

Example 2 Solve

Create a strategy Use the negative exponent law to rewrite with the same base.

Apply the idea Write the equation

Use the negative exponent law

Equate the exponents

Example 3 Solve 25x + 1 = 1253x − 4.

Create a strategy Write both sides as exponents with the same base.

9.04 Exponential equations mathspace.co

485

Apply the idea 25 and 125 are both powers of 5 since 52 = 25 and 53 = 125. So we can write both sides with a base of 5. 25x + 1 = 1253x − 4

Write the equation

2 x+1

3 3x − 4

Write the bases as powers of 5

3(3x − 4)

Use the power of a power rule

9x − 12

Use distributive law

2x + 2 = 9x − 12

Equate the powers

(5 )

2(x + 1)

2x + 2

= (5 ) =5

2 = 7x − 12

Subtract 2x from both sides

14 = 7x

Add 12 to both sides

x=2

Divide both sides by 7

Reﬂect and check We can check our answer by substituting x = 2 into the original equation. 25x + 1 = 1253x − 4

Write the equation

2+1

3(2) − 4

= 125

Substitute x = 2

25 = 125

Simplify the exponents

15 625 = 15 625

Evaluate on a calculator

Idea summary An exponential equation is an equation in which one, or both, sides are exponential functions, for example: 3x + 2 = 81. The key property we need that: If bx = by, then x = y To solve an exponential equation, we can: 1. Write in the form bx = c. 2. Rewrite in the form bx = b⬚.

3. Set x = ⬚, and solve if necessary.

We may need the power of a power law, (am)n = amn, to rewrite with the same base.

486

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Solve exponential equations graphically When we cannot solve equations that involve exponentials by expressing both sides of the equation with the same base, one way that we can get an approximate solution is to use a graph. For example, suppose we were asked to solve 2x + 2 = 3−x We can plot the graph of y = 2x + 2 and the graph of y = 3−x on the same axes. The solution to the equation is the x-coordinate of the point of intersection of the two curves.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.04 to explore the how to use graphs in estimating the solution of exponential equations.

Example 4 For the equation 2x = 3x + 1: a Using technology, graph y = 2x and y = 3x + 1 on the same axes. Use a window of x from −5 to 5, and y from −5 to 20.

Create a strategy We need to input each equation separately and then set the zoom.

Apply the idea 18 16 14 12 10 8 6 4 g (x) = 3x + 4 2 −4 −3 −2 −1 −2

f (x) = 2x x 1

9.04 Exponential equations mathspace.co

487

b How many solutions are there to the equation 2x = 3x + 1?

Create a strategy We can see solutions on a graph by looking for points of intersection.

Apply the idea 18 16 14 12 10 8 6 4 g (x) = 3x + 4 2 −4 −3 −2 −1 −2

f (x) = 2x x 1

We can see two points of intersection on the graph, so there would be two solutions to the equation (as far as we can tell from this graph).

c Using technology, ﬁnd each solution rounded to one decimal place, where appropriate.

Create a strategy The solutions we can see from the original graph are close to −1, and close to 4. We need to zoom in on each solution.

Apply the idea 0.9 0.8 0.7

The curves cross between x = −1.2 and x = −1.1, closer to x = −1.2, so we could give the solution as x = −1.2 to one decimal place.

0.6 0.5 0.4 0.3 0.2 −1.4 −1.2

488

−1 −0.8 −0.6

0.1 x

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The solution appears to be exactly at x = 4.

We can show that this solution is exactly at 4. 24 = 16 and 3(4) + 4 = 16 16

x 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4

Idea summary We can get approximate solutions to equations that involve exponentials by graphing the function on each side of the equation using technology and ﬁnding the x-coordinate of any points of intersection.

Practice questions Choose your question path Explorer 3–8, 9adefh, 10bc, 11aceg, 12ad, 13, 14ab, 15, 17, 19, 22, 24, 25afg

Adventurer 1–4, 6–8, 9bcfgk, 10ace, 11bdfh, 12be, 13, 14bc, 15, 17–19, 23, 24, 25bdf

Trailblazer 1–4, 6, 9eijkl, 10def, 11fhij, 12eh, 14cd, 15, 16, 18, 20, 21, 23, 24, 25deh, 26

Understanding 1

Which statements are true? a

If bx = ay, then x = y

If ax = ay, then x = y

If ax = bx, then a = b

, then x = y

When solving an exponential equation with a variable in the exponent, why do we want the bases on both sides of the equation to be the same?

Rewrite each number using the given base. a

125 with a base of 5

32 with a base of 2

64 with a base of 4

81 with a base of 3

with a base of 6

with a base of 10

9.04 Exponential equations mathspace.co

489

Complete each statement: a

89 = 8⬚

73 = (⬚)3

4⬚ = 64

10⬚ = 1

(4)2 = (22)⬚

(32)2 = 9⬚

10x = 0.01

10x = 0.0001

6x = 6–3

3x = 27

2x = 64

5n = 625

Fluency 5

Solve: a

Ex 1

Ex 2

10x = 100

Solve: a

3x = 36

7x = 1

For each equation: i

Rewrite both sides of the equation with a base of 2.

Solve for x.

8x = 4

5x = 0.2

9y = 27 5x − 10

a x − 1 = a4

25y = 125 2 x+7

4x + 4 = 8

(2 )

= 3125

510x + 33 = 125

9x + 3 = 27x

36x − 1 = 2162x + 5

8x + 3 = 322x − 1

81x − 1 = 93x + 5

−3x − 1

Solve: a

Solve: e

5 × 10x = 500 40 × 3

x+1

= 3240

b f

10 × 2x = 80 2x − 1

18 × 5

= 450

30 × 2x − 6 = 15 x+2

24 × 2x − 6 = 12

−8=1

4x + 32 = 288

2x × 2x + 3 = 32

2x × 8x + 1 = 16

Solve: a e

490

Solve: a

Ex 3

9y = 81

2x × 2x + 2 = 16

b f

4−2x × 4x = 64

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Consider the equation 2x = 8. a

Sketch the graph of y = 2x.

Using the graph, ﬁnd the x-value of the function when y = 8.

Solve the equation 2x = 8.

For each equation: i

Identify two functions that could be graphed to solve the equation.

Using technology, graph both lines and estimate the solution to two decimal places.

2x = 5

4 × 3x = 20

Consider the equation

Ex 4

2x − 2 = 2x + 5

Using technology, ﬁnd the solution(s) to the equation, rounding the answers to one decimal place. 16

Consider the equation

−3(2x) = −3x − 4

Using technology, ﬁnd the solution(s) to the equation, rounding the answers to one decimal place.

Reasoning 17

Explain why 23x − 1 = 27x − 9 cannot be solved using exponent laws, while 23x − 1 = 8x − 9 can.

Explain the reasoning for each step of this solution. 1

Given equation

2 3 4 5 6 7 19

Lorena was trying to ﬁnd the solution to the equation Her working is shown.

Find and explain Lorena’s mistake.

Correct her solution.

3 4 5

9.04 Exponential equations mathspace.co

491

Explain how to solve the equation roots.

Julian wants to ﬁnd the solution to the equation 200 (1.02)x = 220.

using your knowledge of squares and square

To do this, he uses his graphing calculator to graph y = 200 (1.02)x, then uses his calculator to generate a table of values. His calculator shows this table of values for y = 200 (1.02)x. Explain how Julian can use the table to estimate the solution to the equation 200 (1.02)x = 220.

x 0 1 2 3 4 5 6 7

y 200 202 208.08 212.24 216.49 220.82 225.32 229.74

Problem-solving 22

A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by: N (t) = 2t Find the time when the number of cells will reach the following amounts: a

1024

Given the points (3, n), (k, 16) and value of: a

4096

all lie on the curve with equation y = 2x, ﬁnd the

The frequency, f (Hz), of the nth key of an 88-key piano is given by

Find the frequency of the forty-ninth key.

Find the frequency of the 40th key to the nearest whole number.

Use technology to ﬁnd the value of n that corresponds to the key with a frequency of 1760 Hz.

Solve these exponential equations: a

27 (2x) = 6x

Using the digits 1 to 9 each at most once, ﬁll in the boxes to write a valid exponential equation.

492

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

9.05 Compound interest After this lesson, you will be able to… • apply the formula for calculating simple interest. • apply the compound interest formula for varying compounding frequencies. • convert interest rates and time periods based on different compounding intervals. • solve real-world problems involving simple and compound interest.

Repeated applications of simple interest Simple interest is calculated as a ﬂat percentage of the amount that was borrowed or invested, called the principal. We can calculate the simple interest of a loan using the formula:

I = PRT I P R T

is the total interest earned is the principal (the initial amount borrowed/invested) is the interest rate, as a decimal is the number of time periods (the duration of the loan/investment)

So the total amount in the account will be:

A=P+I A P I

future value in the account principal amount interest accumulated

Exploration Taryn and Elijah each open an account with $1 000 that interest at 10% per annum. 1.

Taryn earns simple interest. Complete the table to show how the account balance grows over 5 years. Year 1 2 3 4 5

Opening balance 1000 1100 1200

Interest 100 100

Closing balance 1100 1200

9.05 Compound interest mathspace.co

493

Elijah’s interest payment changes each year as he earns interest on the closing balance in the account from the previous year instead of on the principal. Complete the table to show how Elijah’s account balance grows over 5 years now. Year 1 2 3 4 5

Opening balance 1000 1100 1210

Interest 0.1 × 1000 = 100 0.1 × 1100 = 110 0.1 × 1210 = 121

Closing balance 1000 + 100 = 1100 1100 + 110 = 1210

Plot the closing balances for the ﬁrst 5 years for Taryn and Elijah’s accounts on the same Cartesian plane. How does the growth of Taryn’s account compare to the growth of Elijah’s account?

Which investment option would earn more money in the long run?

When an account earns simple interest, it grows by the same amount each time. The growth in the balance of a simple interest account is linear. When the interest for the next period is calculated on the principal plus the previously earned interest, this is known as compound interest. The growth in the balance of a compound interest account is exponential.

Example 1 The graph shows both an investment with simple interest, and one with compound interest, labelled Investment A and Investment B respectively.

Investment ($1000) 10 8 6

Investment A

4 Investment B

Time (years) 2

a Which investment has a higher principal amount?

Create a strategy Choose the graph with the higher y-intercept.

Apply the idea The graph for Investment A has a higher y-intercept than the graph for Investment B. So, Investment A has a higher principal amount.

494

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

10 12

b Which investment has a higher ﬁnal amount after 10 years?

Create a strategy Find the point on each graph that lines up with x = 10.

Apply the idea From the Cartesian plane we can see that when x = 10 the value of Investment B is higher that the value of a Investment A.

Investment ($1000) 10 8

So, Investment B has a higher ﬁnal amount after 10 years.

Investment A

4 Investment B

Time (years) 2

10 12

c After how many years are the investments equal in value?

Create a strategy Find the intersection point of the graphs.

Apply the idea

From the the Cartesian plane we can see that the two graphs intersect at a point where x = 9.

So this means that after 9 years the investments are equal in value.

Investment ($1000)

Investment A

4 Investment B

Time (years) 2

10 12

9.05 Compound interest mathspace.co

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Example 2 $3200 is invested for three years at a rate of 6% p.a., compounding annually. a Complete the table to determine the ﬁnal value of the investment. Round to the nearest cent for each year. First year Second year Third year Fourth year

Create a strategy

Balance + Interest − $3200 + $192 $3392 + ⬚ ⬚

Total balance $3200 $3392 ⬚ ⬚

Interest earned $192 $203.52 ⬚ −

To get the value in the “Interest earned” column, calculate the simple interest on the amount in the “Total balance” column. The values of “Total balance” and “Interest earned” are copied down to the next row into the “Balance + Interest” column, and add them together to give the new “Total balance”.

Apply the idea We bring down the “Interest earned” of $203.52 from the row above then evaluate the sum $3392 + $203.52 = $3595.52 to ﬁll in the second box. First year Second year Third year Fourth year

Balance + Interest − $3200 + $192 $3392 + $203.52 ⬚

Total balance $3200 $3392 $3595.52 ⬚

We calculate interest on $3595.52 and put it in the interest earned column.

Interest earned $192 $203.52 ⬚ −

$3595.52 × 0.06 = $215.73

First year Second year Third year Fourth year

Balance + Interest − $3200 + $192 $3392 + $203.52 ⬚

Total balance $3200 $3392 $3595.52 ⬚

Interest earned $192 $203.52 $215.73 −

Total balance $3200 $3392 $3595.52 $3811.25

Interest earned $192 $203.52 $215.73 −

We bring down the “Total balance” of $3595.52 and the “Interest earned” of $215.73 from the row above then evaluate the sum $3595.52 + $215.73 = $3811.25 to ﬁll in the second box. First year Second year Third year Fourth year

496

Balance + Interest − $3200 + $192 $3392 + $203.52 $3595.52 + $215.73

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Calculate the total interest earned over the three years.

Create a strategy Add all the values on the “Interest earned” column from part (a).

Apply the idea Total interest earned = 192 + 203.52 + 215.73 Add the values = $611.25

Evaluate

Idea summary We can calculate simple interest with the formula:

I = PRT I

interest accumulated

principal amount

rate of interest, as a decimal

time in years

Simple interest pays the same amount of interest in each period because it is based on a percentage of the principal only.

The growth in the balance of a simple interest account is linear.

Compound interest pays increasing amounts of interest in each period because it is based on a percentage of the balance in the account, which grows as interest is added.

Investment value 8 Simple interest

Compound interest

2 2

10 12

Time 14

The growth in the balance of a compound interest account is exponential.

Compound interest formula It can be time consuming to use a table of values to calculate the balance for an account, especially as the time increases. We could use a spreadsheet to do repeated calculations, but a formula would be helpful. Let’s look investing $1000 at 10% p.a. using compound interest. Using the idea that A = P + I in a table, we get: Year 1 2 3

Opening balance 1000 1100 1210

Compound interest calculation 1000 × 0.1 = 100 1100 × 0.1 = 110 1210 × 0.1 = 121

Closing balance 1000 + 100 = 1100 1100 + 110 = 1210 1210 + 121 = 1331

Note that we can go straight to the closing balance from the opening balance by multiplying by 1.1 since we are doing 1000 + 0.1(1000) = 1.1(1000) to get the closing balance. 9.05 Compound interest mathspace.co

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Year 1 2 3

Opening balance 1000 1100 1210

Closing balance 1000 × 1.1 = 1100 1100 × 1.1 = 1210 1210 × 1.1 = 1331

This is a quicker way of calculating, but we can go one step further. Year 1 2 3

Opening balance 1000 1100 1210

Closing balance 1000 × 1.1 = 1100 1000 × 1.1 × 1.1 = 1210 1000 × 1.1 × 1.1 × 1.1 = 1331

Note from the table that to get to the closing balance in year 3, we have multiplied the principal by 1.1 three times. $1 331 = $1 000 × 1.1 × 1.1 × 1.1 = $1 000 × 1.13 Generalising this, we can write the compound interest formula as:

A = P (1 + r)n A P r n

closing balance or future value of the investment principal amount interest rate, as a decimal number of time periods

If we want to ﬁnd the amount of interest earned, we can subtract the principal amount from the future value of the investment. Interest = Future value − Principal amount

Example 3 Valerie borrows $1250 at a rate of 7.6% p.a., compounding annually. a Which of these expressions represents the amount Valerie must repay after 15 years, assuming that she hasn’t paid anything back? A

1250 × (1 + 0.076) × 15

1250 × (1 + 0.076)15

Create a strategy Use the compound interest formula: A = P (1 + r)n

Apply the idea • P = 1250 • • n = 15

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A = P (1 + r)n

Write the formula

= 1250 (1 + 0.076)15

Substitute the known values

The correct answer is option D. b How much interest was generated on the loan over the ﬁfteen years? Round your answer to two decimal places.

Create a strategy Subtract the principal from the future value.

Apply the idea In part (a), we have the future value, 1250 (1 + 0.076)15. So we have: Interest = 1250 (1 + 0.076)15 − 1250

Substitute the values

= 3750.54 − 1250 Subtract the principal = $2500.54

Evaluate

The interest generated on the loan over the fifteen years was $2500.54

Idea summary We can calculate compound interest with the formula:

A = P (1 + r)n A P r n

closing balance or future value of the investment principal amount interest rate, as a decimal number of time periods

If we only want to ﬁnd the amount of interest earned, we can simply subtract the principal amount from the future value of the investment. Interest = Future value − Principal amount

Different periods The examples we have looked at so far have all had interest calculated annually. However, interest can be calculated over various time periods. To convert the annual interest rate to a periodic interest rate, we can divide it by the number of compounding periods in a year. For example, for an investment that earns an interest rate of 6% p.a., compounding quarterly. Since there are 4 quarters in a year:

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It is also important to match the total number of periods to the duration of the investment. To do this, we multiply the number of years by the number of compounding periods per year. If interest is calculated quarterly for 5 years, there will have been: 4 quarters per year × 5 years = 20 time periods

Example 4 A $9450 investment earns interest at 2.6% p.a., compounded monthly over 14 years. What is the future value of the investment?

Create a strategy Use the compound interest formula: A = P (1 + r)n

Apply the idea We are given that: P = 9450

n = 14 × 12 months = 168 months

Substitute the values

Evaluate

The future value is $13 593.90

Example 5 Victoria borrows $35 000 at a rate of 4.8% p.a., compounded monthly. a After 4 months, Victoria repays the loan all at once. How much money does she pay back in total?

Create a strategy Use the compound interest formula: A = P (1 + r)n

Apply the idea We are given that: P = 35 000 r = 0.048 n = 4 months

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Substitute the values

Evaluate

The total amount that Victoria repaid after four months is $35 563.37

b How much interest was generated on the loan over the four months?

Create a strategy Use the fact that the amount of interest generated is the difference between the ﬁnal amount repaid and the initial amount borrowed.

Apply the idea Use the total amount that Victoria repaid after four months from part (a). Interest = 35 563.37 − 35 000 = 563.37

Subtract the two values Evaluate

Idea summary If interest is compounded daily, weekly, fortnightly, monthly, quarterly, or half-yearly, then we need to convert the interest rate and number of time periods to the same units. To convert the interest rate r, divide the annual rate by the number of compounding periods in a year. To convert the number of time periods n, multiply the number of years by the number of compounding periods in a year. We usually use the following values when converting the rate and periods: • • • • • •

There are 365 days in a year. There are 52 weeks in a year. There are 26 fortnights in a year. There are 12 months in a year. There are 4 quarters in a year. There are 2 half-years in a year.

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Practice questions Choose your question path Explorer 1–3, 5–12, 14–17, 20–23

Adventurer 1–3, 5–7, 9–11, 13–16, 18, 19, 21–24

Trailblazer 1, 2, 4–6, 8, 10, 11, 14–16, 18, 19, 21–25

Understanding 1

For compound interest, are these statements always true, sometimes true, or never true? a

The account balance grows exponentially.

The amount of interest accrued for the ﬁrst and second periods will be the same.

Assuming the account remains untouched, the amount of interest accrued per period increases.

The interest rate is applied to the initial principal.

The interest rate is applied to the current balance.

The ﬁnal balance of an investment is more than an investment with simple interest.

The table shows the bank balance of two accounts. Xuan invested in a simple interest yielding account for 3 years at 7%, and Florence invested in a compound interest account, compounding annually for 3 years also at 7%. Account A Account B

Principal $5000 $5000

Year 1 $5350.00 $5350.00

Year 2 $5724.50 $5700.00

Year 3 $6125.22 $6050.00

According to the data, identify which account is Xuan’s. 3

$1000 is invested for 10 years with an interest rate of 10% p.a., compounded annually. Which graph represents the growth of the investment? Graph 1 Graph 2 Investment ($)

Investment ($)

3000

2500

2000

1500

1000

500

500 T (years) 1

502

2 3 4 5 6 7 8 9 10

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T (years) 1

2 3 4 5 6 7 8 9 10

Which graph could represent a savings account that earns compound interest over 50 years? a

Balance ($)

Time (years)

Balance ($)

Time (years)

Frasier is calculating the compound interest accumulated on his loan. He types this into his calculator: 9000

(

0.07

)

(

)

Identify how much Frasier borrowed in dollars.

Identify the annual interest rate as a percentage.

Identify if the interest is being compounded weekly, monthly, quarterly or annually.

Identify how many years he is calculating the interest for.

Fluency Ex 1

Investment ($1000) The given graph shows two investments, one with simple interest (Investment A) and another with compound 10 interest (Investment B):

Which investment has a higher principal amount?

Which investment has a higher ﬁnal amount after 10 years?

After how many years will the investments be equal in value?

2 Time (years) 2

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$3200 is invested for three years at a rate of 5% p.a., compounded annually. a

Complete the table to determine the ﬁnal value of the investment. Round to the nearest cent for each year. First year Second year Third year Fourth year

b 8 Ex 2

−

$3500 is invested for three years at a rate of 3.8% p.a., compounded annually. a

Complete the table to determine the ﬁnal value of the investment. Round to the nearest cent for each year. Balance at beginning of year $3500 $3633

Interest earned $133 $138.05 −

Calculate the total interest earned over the three years.

A $2090 investment earns interest at 4.2% p.a. compounded annually over 5 years. a

Complete the table to determine the ﬁnal value of the investment if each period is rounded to the nearest cent. Year Start of investment, 0 First year, 1 Second year, 2 Third year, 3 Fourth year, 4 Fifth year, 5

Balance + interest − $2090 + $87.78

Value $2090 $2177.78

Interest earned $87.78 $91.47

−

Complete the table to determine the ﬁnal value of the investment if each period is rounded to the nearest dollar. Year Start of investment, 0 First year, 1 Second year, 2 Third year, 3 Fourth year, 4 Fifth year, 5

504

Interest earned $160 $168

Calculate the total interest earned over the three years.

First year Second year Third year Fourth year

Balance at beginning of year $3200 $3360

Balance + interest − $2090 + $88

What is the difference in the two values after 5 years?

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Value $2090 $2178

Interest earned $88 $91

−

10 Ex 3

11 Ex 4

14 Ex 5

Tauren borrows $500 000 at a rate of 1.2% p.a., compounding annually. a

Write an expression that represents the amount Tauren must repay after 24 years, assuming that he hasn’t paid anything back.

How much interest was generated on the loan over the twenty four years? Round your answer to two decimal places.

Calculate the future value, A, of each investment: a

A $7150 investment earns interest at 2.5% p.a., compounded weekly over 3 years.

A $5420 investment earns interest at 5% p.a., compounded monthly over 4 years.

Tiara borrows $5000 at a rate of 4.5% p.a., compounded annually. a

After 3 years, Tiara repays the loan all at once. How much money does she pay back in total?

How much interest was generated on the loan over the three years?

Ivan borrows $3000 at a rate of 6.9% p.a., compounded annually. a

After 6 years, Ivan repays the loan all at once. How much money does he pay back in total?

How much interest was generated on the loan over the six years?

Vera borrows $25 000 at a rate of 6% p.a., compounded monthly. a

After 5 months, Vera repays the loan all at once. How much money does she pay back in total?

How much interest was generated on the loan over the ﬁve months?

Calculate the interest that each investment accumulates: a

A $5970 investment earns interest at 2.1% p.a., compounded annually over 17 years.

A $5460 investment earns interest at 2.8% p.a., compounded monthly over 10 years.

A $6240 investment earns interest at 2.1% p.a., compounded weekly over 6 years.

Reasoning 16

Evangelina wants to invest $1400 at 5% p.a for 5 years. She has two investment options, compounding quarterly or compounding daily. a

Calculate the value of the investment if it is compounded quarterly.

Calculate the value of the investment if it is compounded daily.

Calculate how much extra the investment is worth if it is compounded daily rather than quarterly.

What effect does the frequency of compounding have on the total amount of interest earned?

Jadria applies for a loan of $9000 and she is offered the choice between two loan plans: • Plan 1: Simple interest at a rate of 2.9% • Plan 2: Compound interest at a rate of 2.75% compounded monthly a

Determine which plan Jadria should choose if she wants to pay back the loan over 3 years. Explain your reasoning.

Jadria decides that she needs longer to pay off the loan. Determine if the plan Jadria chose in part (a) will always be the better choice. Explain your reasoning.

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Hassan plans to borrow $25 000 to buy a new car. After shopping around, he ﬁnds three different offers: • 4.75% compounded annually • 4.7% compounded quarterly • 4.65% compounded monthly Which offer should Hassan choose? Explain your reasoning.

Dorian’s investment of $6000 earns interest at 2% p.a., compounded annually over 3 years. He uses this table to predict the value of his investment. Year Start of investment, 0 First year, 1 Second year, 2 Third year, 3

Balance + interest − $6000 + $120 $6120 + $122 $6242 + $125

Value $6000 $6120 $6242 $6367

Interest earned $120 $122 $125 −

Find the value of the investment after 3 years using the compound interest formula.

Explain why there is a difference between your answer to part (a) and the value of the investment after 3 years in the table.

Problem-solving 20

Rory is planning to make an investment which accumulates monthly compound interest at a rate of 4.8% p.a. If she aims to have her investment reach $90 000 after 11 years, ﬁnd the principal amount that she should invest.

Cade has a savings account which accumulates compound interest yearly at a rate of 2.9% p.a. If he aims to have $60 000 in his account after 4 years, ﬁnd the principal amount that he should invest.

Han opened a savings account at the beginning of February 2023, where the interest is compounded monthly. His statements for March, April and May show his account balance at the beginning of each month:

Find the monthly interest rate, r, for his investment.

How much did Han deposit into this savings account when he opened it?

Month March April May

Balance $3825 $3901.50 $3979.53

Meerah invests $600 into an account which accumulates interest at a rate of 5.25% p.a., compounding annually. a

Use technology to graph of the relationship between the number of years passed and the Meerah’s account balance.

How many years will it take for the account to reach a total value of $2000?

Dietrich wants to put a deposit on a house in 5 years. In order to ﬁnance the $80 000 deposit, he decides to put some money into a high interest savings account that pays 5.5% p.a. interest compounded monthly. If P is the amount of money that he must put into his account now to accumulate enough for the deposit, ﬁnd P.

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A high interest savings account is advertised as earning 5% interest compounded monthly. At the end of the year, the actual interest earned is higher than 5%. Find the actual interest rate after one year. Write your answer as a percentage, rounded to three decimal places.

9.06 Exponential models After this lesson, you will be able to… • differentiate between exponential growth and exponential decay. • identify and define key features of exponential functions. • express exponential functions algebraically, highlighting either the growth or decay factor or rate. • apply knowledge of exponential functions to solve real-world problems. • interpret the implications of the growth or decay rate in practical scenarios.

Exponential growth 8

An exponential function can be expressed as:

f (x) = abx • a is the y-intercept, so this curve would intercept the y-axis at (0, a). • b tells us how fast it is growing.

7 6 5

y = 3(1.6x)

4 3 2 1 −4 −3 −2 −1

x 1

When 0 < b < 1 we get exponential decay, when b > 1 we get exponential growth. When we considered compound interest, we derived a formula: A = P (1 + r)n and noted that compound interest is an example of exponential growth where r was the growth rate of the investment. Using b = 1 + r where b is the growth factor and r is the growth rate, we can express our exponential function f (x) = abx as f (x) = a (1 + r)x Notice that the base is (1 + r). This is a growth factor because it is greater than 1. We can use the key features of an exponential function to build its equation. First, we must find the y-intercept, then find the growth factor or growth rate. The specific form of the equation we use will depend on whether we want to highlight the growth factor or the growth rate.

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Exploration Here are three exponential functions represented in three different ways: Function 1: 8

Function 2: x y

0 1

1 1.15

2 1.3225

3 1.5209

4 1.7490

Function 3:

Reagan decided to get a part-time job to begin saving for a new car. He was hired by a company that pays $21 500 in the ﬁrst year, and he will receive a 5% raise each year after that.

5 4 3 2 1

x 1

Which of the functions would you model using y = abx and why?

Which of the functions would you model using y = a (1 + r)x and why?

In what type of situations would you prefer knowing the growth factor, b, over the growth rate, r?

In situations where growth is given as a percentage, it is easiest to model the equation in the form y = a (1 + r)x. When trying to determine the equation from a table or a graph, it is easiest to ﬁnd the growth factor and convert it to the growth rate if needed.

Example 1 Identify the y-intercept and the growth rate for each exponential functions: a f (x) = 3 (1.725)x

Create a strategy This function is in the form y = abx. We can identify the y-intercept and the growth factor, but then we need to use b = 1 + r to solve for the growth rate.

Apply the idea The growth factor, b, is 1.725 1.725 = 1 + r r = 0.725 The y-intercept is (0, 3), and the growth rate is 72.5%.

Reﬂect and check In the formula, the growth rate is represented as a decimal. When we are discussing the rate outside of the formula, it is best to state it as a percentage.

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Create a strategy This function is in the form y = abx. We can identify the y-intercept and the growth factor, but then we need to use b = 1 + r to solve for the growth rate.

Apply the idea For y = abx, we can use that a is the y-intercept and b is the growth factor.

To ﬁnd the growth rate: b=1+r

Equation for growth factor

2=1+r

Substitute b = 2

1 = r

Subtract 2 from both sides 2=1+r r=1

The y-intercept is

and the growth rate is 1 or 100%.

Reflect and check When something grows by 100%, it is being increased by its own value. In other words, we are doubling the output each time.

Example 2 Justin purchased a piece of sports memorabilia for $2900, and it is expected to increase in value by 9% per year. a Write a function, V, to represent the value of the piece of sports memorabilia after t years.

Create a strategy Since the memorabilia is predicted to increase in value by a percentage, we will use the growth rate form of the function, y = a (1 + r)x or with the given variables V = a (1 + r)t.

Apply the idea The initial value of the memorabilia is $2900. This represents the value when t = 0, so this is the y-intercept. The growth rate is 9% which is 0.09 as a decimal, so the function is V = 2900 (1 + 0.09)t.

Reﬂect and check Notice the similarities to the compound interest formula A = P (1 + r)n.

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b Evaluate the function for t = 8 and interpret the meaning in context.

Create a strategy

Apply the idea

In this context, t represents the time in years, V = 2900 (1 + 0.09)8 ≈ 5778.43 which tells us and the output, V, represents the value of that the memorabilia will be worth approximately Justin’s sports memorabilia. We will evaluate the $5778.43 after 8 years have passed. function and apply these units to interpret the meaning of the solution.

Idea summary An exponential growth function can be expressed as f (x) = a (1 + r)x a

is the y-intercept, sometimes called the initial value

is the growth rate

b=1+r

is the growth factor

Exponential decay 8

We have seen that exponential functions can be expressed in the form:

f (x) = abx

where: • a is the y-intercept, so this curve would intercept the y-axis at (0, a). • b tells us how fast it is decaying.

5 4 3 2 1 −4 −3 −2 −1

x 1

For exponential growth, we saw that for a growth rate r: f (x) = a (1 + r)x Exponential functions can be classiﬁed as exponential decay based on the value of the decay factor. We get exponential decay when 0<b<1 Using b = (1 − r) where b is the decay factor and r is the decay rate, we can express our exponential function f (x) = abx as f (x) = a (1 − r)x Notice that the base of the exponent is (1 − r). This is the decay factor because it is less than 1.

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Exploration Consider these exponential functions: Function 1: A ball is dropped from a ledge. The height of the ball rebounds to 80% of its previous height after each bounce. 9

x 2

Function 2

Function 3

Function 4: Function 5: x y

0 10

Which ones can be classiﬁed as decay functions?

Explain how you identiﬁed the decay functions.

1 9.6

2 9.216

3 8.847 36

An exponential decay function will have a decay factor than is less than 1, which will cause the y-values to decrease over time. When a percentage is used, sometimes it is the value of r, while other times it is the value of b. For example, these two scenarios are the same: • Dara has a bag of 400 cookies. Each day she leaves 80% for the next day. c(x) = 400 (0.80)x = 400 (1 − 0.20)x • Dara has a bag of 400 cookies. Each day Dara eats 20% of the cookies in her bag. c(x) = 400 (1 − 0.20)x = 400 (0.80)x We can use the key features of an exponential function to build its equation. 1. Find the y-intercept. 2. Find the decay factor or decay rate. 3. The speciﬁc form of the equation we use will depend on whether we want to highlight the decay factor or the decay rate.

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Example 3 Identify the y-intercept and the decay rate for each of these exponential functions: a

Create a strategy This function is in the form y = abx. We can identify the y-intercept and the decay factor, but then we need to use r = 1 − b to solve for the decay rate.

Apply the idea r=1–b

Equation for decay rate

r = 1 − 0.625

Substitute b = 0.625

r = 0.375

Evaluate the subtraction

As a percentage, the decay rate is r = 37.5%. Since

, the y-intercept is

Create a strategy This function is in the form y = abx. We can identify the y-intercept and the decay factor, but then we need to use r = 1 − b to solve for the decay rate.

Apply the idea We see a = 1 and

. Next, we need to solve for the decay rate. Equation for decay rate

Substitute

Evaluate the subtraction

The y-intercept is at (0, 1), and the decay rate as a percentage is 75%.

Reﬂect and check Notice that the original equation tells us that we are keeping losing 75%, as shown with the decay rate.

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. This means that we are

Example 4 A frog population, F, has been following the model F (n) = 480(0.952)n, where n is the number of years. a What is the initial population of frogs?

Create a strategy The initial value will be when time is 0.

Apply the idea

Reﬂect and check

When n = 0, F = 480, so the population was 480.

Since negative values for time do not make sense, the y-intercept will be the leftmost point on the graph. In this case, we refer to it as the initial value.

b What is the decay rate? Explain what it means in this context.

Create a strategy

Apply the idea

We can ﬁnd the decay rate using that for y = abx, b = 1 − r has decay rate of r.

Since b = 0.952: b=1−r 0.952 = 1 − r r = 1 − 0.952 r = 0.048 This means that the decay rate is 4.8%. The frog population is declining at 4.8% per year.

c Find the population of frogs after 3 years, rounding to the nearest integer.

Create a strategy We can substitute n = 3, and then evaluate using a calculator.

Apply the idea F (n) = 480(0.952)n F (3) = 480(0.952)3 F (3) ≈ 414 After 3 years, there would be 414 frogs remaining.

d Using technology, graph the frog population for 0 < n < 120.

Create a strategy We can input the equation and then adjust the view using the zoom so we can see from n = 0 to n = 120.

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Apply the idea A good window setting is 0 < n < 120 and 0 < F (n) < 480. 480 440 400 360 320 280 240 200 160 120 80 40

x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

e Use your graph to determine approximately how many years it will take for the population to reach half of the original population.

Create a strategy We want to ﬁnd half of 480 on the vertical axis and then slide over to ﬁnd where the graph reaches that value.

Apply the idea Half of 480 is 480 440 400 360 320 280 240 200 160 120 80 40

x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

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Using our original scale we can see this is between n = 10 and n = 15, so we can zoom in to get a better estimation. 275 y 270 265 260 255 250 245 240 235 230 225 220 215 210 205

x 11

From the zoomed in graph we can see that it is just after 14 years where the population reaches half of the original value.

Reﬂect and check We can solve for the exponent using a graph, table of values, or writing both sides of the equation with the same base if possible.

Idea summary An exponential decay function can be expressed as

f (x) = a (1 − r)x a is the y-intercept r is the decay rate b = 1 − r is the decay factor

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Practice questions Choose your question path Explorer 1–9, 11, 13–16, 19–21, 23, 25, 26

Adventurer 1–4, 6–9, 11, 13, 14–17, 19, 20, 22, 23, 25, 26

Trailblazer 1–4, 6–8, 10–13, 15, 17, 18, 19, 22–24, 26, 27

Understanding 1

Are these situations exponential growth, exponential decay, or neither? a

A store owner checked the sales report for the previous month and found that each week the sales were

Ex 1, 3

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found to triple each hour.

A sequence starts with 2, 3 and the next term is always the product of the previous two terms.

A piece of land was purchased for $80 000. The value of the land has slowly been decreasing by 2% annually.

Michael has $1300 in his checking account. He gets paid $600 every 2 weeks.

For each exponential functions, where x represents time: i

Is it exponential growth or exponential decay?

What is the initial amount?

iii

What is the growth or decay rate as a percentage?

f (x) = 0.723 (1 − 0.05)x

c 3

f (x) = 5.9 (1.025)

f (x) = 2500 (0.877)x

f (x) = 10 000 (2)x

The local rat population is changing according to the exponential function f (t) = 840 (1 − 0.04)t where t is the number of years that have passed. a

What is the initial size of the rat population?

At what rate is the population is decreasing?

This is a table of values for an exponential function: x f (x)

516

of the previous week’s sales.

1 5

2 7

3 9.8

4 13.72

5 19.208

What is the growth factor?

What is the constant percent rate of change?

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Fluency 5

The mass in kilograms of a baby orangutan at n months of age is given by the function M (x) = 1.8 (1.1n), for ages up to n = 6 months. a

Find the mass of a baby orangutan at 3 months of age to the nearest tenth of a kilogram.

Complete the table. Round your answers to four decimal places. Months Mass (kg)

c 6 Ex 2

Sketch a graph of M (x).

Marlow purchased a piece of sports memorabilia for $1300, and it is expected to increase in value by 8% per year. a

Write an equation in terms of y to represent the value of the piece of sports memorabilia after v years.

Find the value of the piece of sports memorabilia after 10 years.

The population of Juanita’s home town is 27 400 and has a growth rate of 6% each year. Estimate the population after 7 years.

The population of some native bees is declining at a rate of 8% per year. If there are 10 000 bees in a hive now, ﬁnd the number of bees in 5 years time.

A property was valued 9 years ago to be worth $755 000. Its value depreciated at 4.4% p.a. due to poor maintenance. Calculate its depreciated value. Round your answer correct to the nearest dollar.

The price of train tickets in 1996 was $3. If the value inﬂated at an average rate of 6.3% per annum, what would the price have been in 2005?

An item costs $14 000 today. Find its price, P, 6 years ago if the inﬂation rate was an average of 2.5% per annum. Round your answer to the nearest dollar.

Winston and Rasiah want to organise a tournament. They want a knockout tournament, where the winner of each round progresses to the next round until there are only two players left. The diagram shows the draw for the final, semi-final and quarter final rounds. a

Semi-Final

Quarter-Final

Final

Winner

Complete the table of values for the total number of players, p, that the competition can accommodate given a number of rounds, r. Number of rounds (r) Number of players ( p)

Winston and Rasiah want to make sure that each round of the tournament has every spot ﬁlled. Find the values of p for which a tournament can be formed.

Write an equation relating r and p.

Over two weeks, they can ﬁt in 8 rounds of play. How many players can they accept into the tournament? 9.06 Exponential models mathspace.co

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13 Ex 4

The number of fungal cells, N, in a colony after t minutes is given by N = 5000 (1 + 0.023)t. a

What is the initial population of fungal cells?

What is the constant percent rate of change? Explain what it means in context.

Find the population of fungal cells after 3 hours, rounding to the nearest integer.

Graph the fungal cell population for 0 < t < 120.

Use your graph to determine approximately how many minutes it will take for the population to reach 10 times the original population.

Djamila loves treating their dog. They realise that they are currently giving their dog 2 kg worth of treats every week, which is way too much. Their plan is to cut the amount of treats they give each week in half until they are giving less than 50 g of treats per week. a

Use a table of values to determine how many weeks it will take to get down to less than 50 g of treats per week.

Write an equation relating week number to the weight of treats eaten.

Reasoning 15

Explain the difference between a decay rate and a decay factor.

When a heated substance such as water starts to cool, its temperature T at time t minutes after being left to cool is given by an exponential function. a

The function can be written in the form T = a (1 − r)t. Explain what the variables a and r mean within context.

Substance P starts off at a temperature of 145 °C and is left to cool in a room whose temperature is a constant 0 °C. By using the table, ﬁnd the equation for the temperature of the substance. Minutes passed (t) Temperature (T )

0 145

1 87

2 52.2

3 31.32

4 18.792

Find the temperature after 10 minutes, correct to two decimal places.

Will the temperature of the substance will ever reach 0 °C? Explain your reasoning.

The temperature of another substance Q which also starts off at a temperature of 145 °C is modelled by T = 145 (1 − 0.1)t. Which substance will cool more rapidly?

An online business that sells phones online has been growing very quickly over the last year. They have asked you to model their growth over the next three quarters, and further into the future. a

Looking at the quarterly data, you start by assuming the number of customers changed in the same way each quarter over the ﬁrst year.

Complete the table under this assumption to predict the number of phones sold over the next three quarters. Quarter (N ) Online customers (C) b

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1 480

2 2880

3 17 280

4 103 680

Find an equation for the number of sales (C) in terms of the number of quarters that have passed (N ).

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Predict the number of sales they will have 2.75 years after they started the business.

Using your answer to part (c), explain whether or not this is a reasonable goal for the company.

A teaspoon of sugar, weighing 4 grams, is poured into a cup of hot water and immediately begins to dissolve. Each second the amount of undissolved sugar remaining is amount present in the previous second. a

of the

Complete the table of values: Seconds passed (t) Undissolved sugar in grams ( y)

Write an equation linking undissolved sugar, y, and time, t.

Describe the change in the amount of undissolved sugar over time.

Use the model to detemine if all the sugar will eventually dissolve. Explain your reasoning.

The water is heated more so that when the experiment is performed again, the sugar dissolves twice as quickly. Will more or less sugar dissolve between the second and third seconds? Explain your reasoning.

Problem-solving 19

In the diagram, the number of squares at each stage increases exponentially. a

Find the number of squares in the next stage of the sequence.

The number of squares can be modelled by y = 4x − 1 where x is the number of the stage. At what stage will there be 16 384 squares?

One person in a city is infected with a virus. During the ﬁrst day, they infect ﬁve more people with the virus. a

Copy and complete the table showing the number of newly infected people for that day. Day Number of people infected

0 1

If this trend continues, write an expression for the number of people infected on day n.

There are 345 800 people in the city in total. Given that the trend continues, after how many days will the entire city have been infected?

If everyone takes a daily vitamin C supplement, they can restrict the spread so that each new person only infects an average of two new people on their ﬁrst day. How many people would be infected on day 10 if this was followed from the beginning?

The value of a painting valued at $700 appreciates in value at a rate of 9% per year. a

Sketch the relationship between the number of years passed and the painting’s value on a Cartesian plane.

How many years will it take for the painting’s value to reach a total value of $2100?

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Futsal is a modiﬁed form of football, typically played indoors on a smaller pitch with fewer players. A futsal ball has less bounce than a full size soccer ball and is expected to bounce back off the ground to

of the height from which it falls after every subsequent bounce.

Write an equation to represent the height of the nth bounce if it was dropped from an initial height of 5 metres.

After how many bounces will the ball have a height of

metres?

The population of a certain city is doubling each year. In 2012, the population was 10 000 people. In what year will the population reach 2 560 000?

To test the effectiveness of a new antibiotic, a bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are 19 bacteria in the body. After four hours, the number is found to double. After how many hours will there be 1216 bacteria in the body?

Juanita and Pedro are taking part in a study, and have to eat the same food and in equal quantity. They were chosen such that Juanita incorporates lots of sugar into their diet while Pedro consumes limited amounts of sugar. After eating, their blood sugar level peaked, and was then continually measured, with the following functions modelling their blood sugar level at time t hours: • Juanita: f (t) = 179 (0.3)t • Pedro: g (t) = 132 (0.6)t a

Find the highest blood sugar level measured for: i

Juanita

Pedro?

Pedro

Compare the reactions to sugar between Juanita and Pedro.

A new car purchased for $38 200 depreciates at a rate r each year. The value of the car for the ﬁrst two years is shown in the table: Years passed (n) Value of car ($A)

520

Find the rate, as a percentage, that the blood sugar level decreased each hour for: i

Juanita?

0 38 200

1 37 818

2 37 439.82

Find the value of r.

Write the rule for A, the value of the car n years after it is purchased.

Find the value of the car after 6 years.

A new motorbike purchased for the same amount depreciates according to the model V = 38 200 (1 − 0.03)n. Which vehicle depreciates more rapidly?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

In a memory study, subjects are asked to memorise some content and then asked to recall what they can remember each day after. The given graphs represent the percentage of content remembered by two participants, Maximilian (P ) and Lucy (Q), after t days. Recall percentage 100% 80% 60% 40% Q

20% P 2

Days 4

Use the given model or create new models to determine the percentage of the content Maximilian and Lucy still recall, respectively.

According to the model, determine if either of them will completely forget the content and explain your reasoning. Is this model realistic?

Another study, similar to the one above, was conducted by teachers to see how much of the content being taught was retained by students. The given graphs represent the percentage of content remembered by two students, Raymond (R) and Sunyi (S), after t days. Recall percentage 100% 80% 60% 40% S

20%

Days 2

Do the given functions accurately model the percentage of the content Raymond and Sunyi recall, respectively: R (t) = 100 (1 − 0.15)t and S (t) = 100 (1 − 0.13)t? Explain your reasoning.

Compare the amount of information retained by Maximilian, Lucy, Raymond, and Sunyi. Identify and explain various factors that could account for any differences.

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Chapter 9 review 1

Are these functions linear, exponential or quadratic? a

y = 5x + 7

f (x) 1 2 3 x 0.8 1.6 2.4

4 3.2

y = 2x2 + 7

f (x) x

1 0.125

T F 40

−8

−10

A R

−4 −3 −2 −1

For these exponential functions: i

Find the growth or decay factor.

Find f (5).

x 1 2 3 f (x) 2000 500 125

4 31.25

Find the growth or decay factor.

Find the number of cubes for Step 4.

f (x)

1 8

2 20

Step 1

522

1 2 3 4 5

−6

Consider the pattern:

−4

4 8.0

3 2.0

−5 −4 −3 −2 −1 −2

2 0.5

3 50

4 125

Step 2

Step 3

For each exponential function: i

Complete the table of values.

Graph the exponential function.

iii

Is the function increasing or decreasing?

x −1 0 1 y = 6x

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−1

For the function f (x) = 5x: a

Identify the equation of the horizontal asymptote.

Find the coordinates of the y-intercept.

Find the coordinates of one other point on the curve.

Sketch the graph.

The graph of

is shown.

How many x-intercepts does the function have?

Is each y-value of the function positive or negative?

T F −3 −2

A R

The graph of y = 4x is shown. a

Find the value of y when x = 4.

Find the value of x when

D 8

x 1

−1

16 14 12 10 8 6 4 2

−5 −4 −3 −2 −1

x 1 2 3 4 5

(Optional) Describe how the graph of y = 10x is transformed to get each graph: a

−1

9 8 7 6 5 4 3 2 1

y = 10x + 2

y = −10x

y = 3 (10)x

(Optional) The graph of f (x) = 7x is shown. a

Describe a transformation of the graph of f (x) = 7x that would obtain g (x) = 7x − 4.

Sketch the graph of g (x) = 7x − 4 on the same set of axes as f (x) = 7x.

Write the equation of the horiztonal asymptote of g (x) = 7x − 4.

y 7 6 5 4 3 2 1 f (x) = 7x −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

1 2 3 4 5

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(Optional) Two exponential functions, X and Y, are graphed.

One of the graphs is of y = 3x and the other graph is of

Which is the graph of

? x

(Optional) For the exponential function a

Find the coordinates of the y-intercept.

What is the equation of the horizontal asymptote?

Is the function increasing or decreasing?

Graph the function.

8x = 88

6x−4 = 36

Solve:

D b

3x × 9x+1 = 27

5−4x−1 = 1

105x+18 = 100x−3

all lie on the curve with equation y = 3x, ﬁnd the c

$4500 is invested for three years at a rate of 4% p.a., compounded quarterly. a

Complete the table to determine the ﬁnal value of the investment. Round to the nearest cent for each year. First quarter Second quarter Third quarter Fourth quarter

524

3x = 81

Given the points (0, n), (k, 243) and value of: a

2x × 2x+4 = 8

T F

A R

Solve:

a 14

Balance at beginning of year $4500 $4680

Interest earned $180 $187.20

Calculate the total interest earned over the ﬁrst three quarters.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

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Lara borrows $30 000 at a rate of 6.2% p.a., compounding annually. a

Write an expression that represents the amount Lara must repay after 25 years, assuming that she hasn’t paid anything back.

How much interest was generated on the loan over the 25 years? Round your answer to two decimal places.

Calculate the future value, A, of each investment: a

A $1380 investment earns interest at 3.9% p.a., compounded weekly over 4 years.

A $9670 investment earns interest at

p.a., compounded monthly over 2 years.

Weylan opened a savings account at the beginning of August 2019, where the interest is compounded monthly. His statements for September, Month Balance October and November show his account balance at the September $6375 beginning of each month: October $6502.50 a Find the monthly interest rate, r, for his investment. November $6632.55 b How much did Weylan deposit into this savings

T F

account when he opened it? 19

Yexel has a savings account which accumulates compound interest yearly at a rate of 3.8% p.a. If he aims to have $40 000 in his account after 5 years, ﬁnd the principal amount, P, that he should invest.

A piece of artwork was purchased for $7700, and it is expected to increase in value by 5.3% per year.

A R

Write an equation in terms of y to represent the value of the piece of artwork after n years.

Find the value of the piece of artwork after 15 years.

A car was valued 7 years ago to be worth $46 000. Its value depreciated at 17.4% p.a. Calculate its depreciated value. Round your answer to the nearest dollar.

The population of a colony of bees is given by P = 10 300 (1 − 0.098)t, where t is the time in years. a

What is the initial population of bees?

Is the population increasing or decreasing?

What is the percent rate of change?

Find the population of bees in 4 years time, rounding to the nearest integer.

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10 Logarithms Big ideas Understanding logarithms and their relationship with exponential functions is essential for simplifying and comparing numbers on different scales, solving exponential equations and analysing growth patterns.

Curriculum Outcomes VC2M10M02 Interpret and use logarithmic scales in applied contexts involving small and large quantities and change VC2M10AA04 Explore the inverse relationship between exponential functions and logarithmic functions and the solution of related equations VC2M10AN03 Use the definition of a logarithm to establish and apply the laws of logarithms and investigate logarithmic scales in measurement

Chapter outline 10.01 10.02 10.03 10.04 10.05 10.06

528 537 546 550 555 562 576 582

Logarithms help measure earthquakes, turning massive vibrations into manageable numbers!

10.01 Logarithms After this lesson, you will be able to… • represent numbers as powers of ten and find their base ten logarithms. • explain what is base two logarithms and represent numbers as powers of two. • recognise the importance and application of logarithmic scales in graphing large ranges of values. • use the appropriate mathematical notation for representing logarithms. • evaluate logarithms using technology. • rewrite equations involving logarithms in exponential form.

Logarithms If a number is a power of ten such as 10, 1000, or 0.01, the number can be written as a base ten logarithm. The base 10 logarithm is the power of 10 that makes the number. For example: log10 (1000) = 3 because 1000 = 103. log10 (1) = 0 because 1 = 100. log10 (0.01) = −2 because 0.01 = 10−2. In general:

log10 (10a) = a a

Base 10 logarithm

Here is a table with the logarithms of numbers that can be written with base ten: Basic number 1 000 000 100 000 10 000 1000 100 10 1 0.1 0.01 0.001 0.0001 0.00 001 0.000 001

528

Power of 10 106 105 104 103 102 101 100 10−1 10−2 10−3 10−4 10−5 10−6

Logarithm 6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −6

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

If the number is not a power of ten, then the logarithm can be found using the log button on the calculator. For example, log10(250) = 2.3979 because 102.3979 = 250. log10(0.0023) = −2.6383 because 10−2.6383 = 0.0023.

Example 1 Consider the number 0.000 001. a Express the number as a power of 10.

Create a strategy Count the number of decimal places, and write as a power of 10 with a negative index.

Apply the idea Since the number is smaller than 1, the power is negative. There are six decimal places in the answer, so the power is −6. 0.000 001 = 10−6

Express as a power of 10

b Find the base ten logarithm of the number.

Create a strategy The base ten logarithm is the exponent when written as a power of 10.

Apply the idea

Reﬂect and check

The base ten logarithm is −6.

We can write this as an equation using proper notation: log10 (0.000 001) = log10 (10−6)= −6

Idea summary The base 10 logarithm of a number is the exponent when the number is written as a power of ten. If the number cannot be written as a power of ten, the logarithm can be found using a calculator. For example, log10 (250) = 2.3979 because 102.3979 = 250.

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Base two logarithms The base two logarithm of a number is equal to the power of 2 that makes the number. For example, log2 8 = 3 because 23 = 8. log2 32 = 5 because 25 = 32. log2 0.125 = −3 because

Some useful base two logarithms to know are shown in the table: Basic number 64 32 16 8 4 2 1 0.5 0.25

Power of 2 26 25 24 23 22 21 20 2−1 2−2

0.125

2−3

Logarithm base 2 6 5 4 3 2 1 0 −1 −2 −3

Example 2 Evaluate:

Create a strategy Find the power to which 2 must be raised to get .

Apply the idea We can write base 2 of

as 2−2. Therefore, the logarithm

is equal to −2.

Reﬂect and check The answer makes sense because 2 to the power of −2 equals . Therefore, the logarithm base 2 of

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

is indeed −2.

Example 3 Rewrite the equation log2 0.125 = -3 in exponential form.

Create a strategy Exponential form will be 2⬚ = ⬚, since 2 is the base of the logarithm.

Apply the idea

The exponential form will be 2⬚ = ⬚, so we can use the logarithm is equal to the index on the 2.

The logarithm is equal to −3, so we have 2−3 = ⬚.

We are taking the base 2 logarithm of 0.125, so we have 2−3 = 0.125.

Reﬂect and check The answer makes sense because

Idea summary The base two logarithm of a number is equal to the power of 2 that makes the number. 23 = 8 therefore log2 8 = 3 In general, the base of the logarithm and the base of the exponential form must be the same. What the logarithm is equal to will become the index of that base.

a x = y ←→ loga y = x base

a > 0, a ≠ 1, y ≠ 0

Logarithmic scales on graphs Logarithmic scales are often used when there is a large range of values involved with the variables under consideration. Consider the set of ﬁve ordered pairs: x y

1 20

2 200

3 3631

4 52 481

5 250 000

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As seen on the graph of the ﬁve points, the scale on the y-axis is so huge, that the ﬁrst 3 points look like they almost have the same y-value. y (5, 250 000)

250 000 200 000 150 000 100 000 (4, 52 481)

50 000

(1, 20)

(2, 200)

(3, 3631)

To avoid losing data because of the scale we can use the base 10 logarithm of y instead of the actual y-values as shown in the table: x

1 1.301

log10 y

2 2.301

3 3.560

4 4.720

5 5.398

Using the logarithm of the y-values gives us this graph: Even though the y-values are far more manageable in this form, we need to remember that the actual data point y-values, rounded to 3 decimal places, are given by 101.301, 102.301, 103.560, 104.720, and 105.398.

(5, 5.4)

(4, 4.72)

4 (3, 3.56) 3 (2, 2.3)

(1, 1.3)

x 1

Example 4

Number of countries

The histogram shows the area (in km2) for 12 countries, plotted using a log scale. 10 9 8 7 6 5 4 3 2 1 0

3 Log of area

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

How many countries have an area of between 10 000 km2 and 100 000 km2?

Create a strategy Count the number of 0s of each number and ﬁnd their corresponding value in the vertical axis.

Apply the idea 10 000 has 4 zeros while 100 000 has 5 zeros. We can see from the histogram that 4 − 5 is corresponding to 3 in the vertical axis. So there are 3 countries with an area between 10 000 km2 and 100 000 km2.

Idea summary We can use logs to transform either (or both) axes depending on what is needed to better understand the data.

Practice questions Choose your question path Explorer 1–6, 7ab, 8ab, 9–12, 14, 15, 17, 19

Adventurer 1–6, 7bc, 8bc, 9–12, 14, 17–19

Trailblazer 1–6, 7cd, 8cd, 9–16, 18–20

Understanding 1

Evaluate: a

102

10−1

104

103

100

101

10−2

105

The base ten logarithm of a number is the power of ten that makes the number. Copy and complete: a b c d e f

The base ten logarithm of 100 is ⬚ because 10⬚ = 100

The base ten logarithm of 1000 is ⬚ because 10⬚ = 1000 The base ten logarithm of 0.01 is ⬚ because 10⬚ = 0.01 The base ten logarithm of 1 is ⬚ because 10⬚ = 1

The base ten logarithm of 65 is 1.8129 because 10⬚ = 65

The base ten logarithm of ⬚ is 2.3502 because 102.3502 = ⬚

The base two logarithm of a number is the power of two that makes the number. Copy and complete: a b

The base two logarithm of 8 is ⬚ because 2⬚ = 8

The base two logarithm of 16 is ⬚ because 2⬚ = 16 10.01 Logarithms mathspace.co

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c d e

The base two logarithm of 0.5 is ⬚ because 2⬚ = 0.5 The base two logarithm of 1 is ⬚ because 2⬚ = 1

The base two logarithm of 0.125 is ⬚ because 2⬚ = 0.125

Fluency Ex 1

Ex 2

For each number: i

Express the number as a power of 10.

Find the base ten logarithm of the number.

1 000 000

Express the number as a power of 2.

Find the base two logarithm of the number.

0.001

0.125

256

log10 100 = 2

log10 1 = 0

log10 10 000 = 4

log10 0.01 = −2

log10 45

log10 7692

log10 5.16

log10 0.005 79

Use technology to ﬁnd the missing number, rounding your answer to three decimal places: log10 ⬚ = 1.57

log2 4 = 2

log10 ⬚ = −1.2

log2 1 = 0

log10 ⬚ = 0.8

log2 16 = 4

log10 ⬚ = 2.31

log2 8 = 3 can be written in exponential form as 23 = 8. Write these logarithms in exponential form:

For each point, which coordinates represent the true value, (x, y)? a

(1, 2)

(1, 100)

(1, 20)

(4, 3981)

(2, −1)

(12 − 10)

(7, −2.5)

log10 y L J

−1 −2 −3

(4, 36)

(4, 3.6)

(7, 0.00316)

(7, 316.228)

Point M: A

log2 0.5 = −1

Point L: A

Point K: A

4 3

Point J: A

534

Use technology to ﬁnd the base ten logarithm, rounding your answer to four decimal places:

a 10

log10 1000 = 3 can be written in exponential form as 103 = 1000. Write these logarithms in exponential form:

a Ex 3

a 8

0.000 1

For each number:

a 7

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−4

x 1

3 K

8 M

Ex 4

The table shows the area (in km2) for 12 countries, using a log10 scale: a

How many countries have an area between 10 000 km2 and 100 000 km2?

How many countries have an area between 10 000 km2 and 1 000 000 km2?

Log of area

Number of countries

3–4 4–5 5–6 6–7

1 3 7 1

The histogram has been plotted on a log10 scale: a

What is the total frequency of scores between 1 and 100?

What is the total frequency of scores larger

6 5

than c

Frequency 7

What is the total frequency of scores smaller than

2 1 Log value −3

−2

−1

The histogram has been plotted on a log2 scale: a

What is the total frequency of scores between 1 (20) and 4 (22)?

What is the total frequency of scores larger than 0.5 (2−1)?

What is the total frequency of scores less than 2 (21)?

Frequency 7 6 5 4 3 2 1 Log value −2

−1

Reasoning 14

The populations of 9 Australian cities are shown in the table: a

Suppose you want to create a frequency table or frequency histogram. Explain why using a regular scale would not be ideal for your model.

Convert the data to a log10 scale.

Create a frequency table or frequency histogram of the data.

City Melbourne Adelaide Wollongong Geelong Hobart Toowoomba Ballarat Bunbury Warragul

Population 4 929 200 1 396 783 310 479 278 708 231 707 146 955 105 924 80 106 44 625

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In the graph shown, the horizontal scale is linear and the vertical scale is logarithmic.

A B

Which of the three plots represents exponential growth?

Which of the three plots represents growth that is less than exponential?

Which of the three plots represents growth that is greater than exponential?

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found to follow a base 2 logarithmic scale. a

Complete the table of values. Number of hours passed Number of bacteria using a log2 scale Actual number of bacteria

0 0

1 1

2 2

3 3

4 4

At this rate, how many bacteria will be present in the petri dish after 7 hours?

Describe the rate of increase of the actual number of bacteria.

Problem-solving 17

A ﬁnancial planner has clients with a large range of annual incomes. To manage the data more easily, the income (in dollars) of each client is ﬁrst converted to a log10 scale. a

What income would result in 4.9 on a log10 scale?

What range of incomes corresponds to a range of 5 to 6 on the log10 scale?

The Richter scale is used to measure the intensity of earthquakes. The formula for the Richter scale rating of a quake is given by R = log10 x, where x is the relative strength of the quake. The table shows how quakes are categorised according to their Richter scale rating: Richter rating

Light 4–4.9

Moderate 5–5.9

Strong 6–6.9

A seismograph measures the intensity of a quake to be x = 5711. Determine the Richter scale rating R of this quake to one decimal place.

In which category does the quake fall?

The lifespans of various animals and insects are shown in the table: Animal/Insect Ant African elephant Bowhead whale Bumblebee

536

Minor 2–3.9

Lifespan (days) 20 25 500 71 400 21

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Lifespan in a log10 scale 4.41 1.32

Animal/Insect Chicken Dog Emperor penguin Fly Giraffe Hamster Ladybug Little penguin Monkey Shark

Lifespan (days) 2920 5110 8500 18 10 120 780 365 2190 7300 10 980

Lifespan in a log10 scale 3.47 3.93

2.89 3.34 4.04

Complete the table by converting the lifespans to a log10 scale, rounding each value to two decimal places.

Create a histogram to represent this data on a log10 scale.

The number of layers, y, resulting from a rectangular piece of paper being folded in half x times, is shown in the graph. a

Plot the data on a coordinate plane where the horizontal axis is the number of times the paper is folded and the vertical axis is log2 y.

What does the point (5, 5) on your graph represent in context?

If a rectangular piece of paper is folded so it has 1024 layers, how many times it was folded.

If a rectangular piece of paper of thickness 0.02 mm is folded to a thinkness of 40.96 mm, how many times it was folded?

y 10 9 8 7 6 5 4 3 2 1

x 1

10.02 Order of magnitude and logarithmic scales After this lesson, you will be able to… • determine the order of magnitude of a given number. • describe the purpose and application of logarithmic scales. • use logarithmic scales to compare and differentiate between numbers with a broad range. • understand the concept behind the Richter scale and its logarithmic nature. • interpret sound levels using the decibel scale and recognise the logarithmic basis behind it.

10.02 Order of magnitude and logarithmic scales mathspace.co

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Orders of magnitude The order of magnitude of a number gives an indication of the size of the number using powers of ten. It is a useful way to compare numbers. 5000 has an order of magnitude of 3 because it can be written as 5 × 1000 = 5 × 103. 0.007 has an order of magnitude of −3 because it can be written as 7 ÷ 1000 = 7 × 10−3. The table shows a comparison between the basic numbers, powers of 10, and order of magnitude. Basic number

Power of 10

Order of magnitude

1 000 000 100 000 10 000 1 000 100 10 1 0.1 0.01 0.001 0.000 1 0.000 01 0.000 001

106 105 104 103 102 101 100 10−1 10−2 10−3 10−4 10−5 10−6

6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −6

Increasing order of magnitude

Decreasing order of magnitude

The order of magnitude forms a logarithmic scale of base 10.

Example 1 What is the order of magnitude of the number 6 × 109?

Create a strategy

Apply the idea

If a number is written in scientiﬁc notation, the order of magnitude is the power.

The number 6 × 109 has an order of magnitude of 9.

Example 2 A bakery sells 450 cupcakes per week. What is the order of magnitude of the total number of cupcakes sold over 8 weeks?

Create a strategy Multiply 450 by 8 and write the product in scientiﬁc notation.

Apply the idea Total = 450 × 8

Multiply

= 3600

Evaluate 3

= 3.6 × 10

Write in scientiﬁc notation

The order of magnitude is 3.

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Idea summary The order of magnitude of a number gives an indication of the size of the number using powers of ten. If a number is written in scientiﬁc notation, the order of magnitude is the power of 10.

Compare using logarithms Orders of magnitude and other logarithmic scales can be useful when comparing numbers that have a very large range. For example, we can compare the weight of a mouse (19 grams) with that of a cat (3600 g). Steps Original information Write using scientific notation Orders of magnitude

Cat 3600 g 3.6 × 103 3

Mouse 19 g 1.9 × 101 1

The two animals differ by 2 orders of magnitude. That means the mouse is roughly 100 times smaller than the cat. • An increase by 1 order of magnitude means the number is 10 times larger. • A decrease by 1 order of magnitude means the number is 10 times smaller. Similarly, we can use logarithmic scales to compare numbers with a large range, like the distance of the planets from the sun. Distance from the sun, d (km) 57 910 000 108 200 000 149 600 000 227 900 000 778 500 000

Planet Mercury Venus Earth Mars Jupiter

log10 (d) 7.76 8.03 8.17 8.36 8.91

Consider the difference between Mercury and Jupiter’s distances from the sun. The log values differ by a little bit more than 1, so Jupiter’s distance from the sun is a little more than 10 times the distance of Mercury. To ﬁnd a more exact answer: 8.91 − 7.76 = 1.15 1.15

= 14.13

Find the difference between Jupiter and Mercury Find 10 to the power of 1.15 and evaluate

Therefore, Jupiter is just over 14 times the distance of Mercury from the sun. • On a log scale with base ten every 1 unit up means increasing by a factor of 10, every 1 unit down means decreasing by a factor of 10. • On a log scale with base two every 1 unit up means increasing by a factor of 2, every 1 unit down means decreasing by a factor of 2. The Richter scale for measuring earthquakes and the decibel scale for measuring sound are two common application of logarithmic scales.

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Measuring the size of earthquakes Since the 1930’s the Richter magnitude scale has been used to describe the intensity or ‘size’ of an earthquake. Due to the number of variables involved in measuring an earthquake its value on the Richter scale is calculated using the log scale, the size of the seismic wave recorded and the distance the recording was from the source of the earthquake. The value on the Richter scale indicates the intensity and can be compared with other earthquakes using powers of 10 to describe how much stronger it was. For example, an earthquake of size 6 on the Richter scale is 103 times more intense than an earthquake of size 3. Measuring level of sound Sound levels are a measure of the intensity of the sound waves on your eardrum. The numbers given by the decibel (dB) scale indicate an intensity 10 times stronger for every 10 dB. The table shows the difference in intensity of a sound of a particular size when compared to another sound. Decibels 50 60 70 80 90

50 0 −10 −100 −1000 −10 000

60 +10 0 −10 −100 −1000

70 +100 +10 0 −10 −100

80 +1000 +100 +10 0 −10

90 +10 000 +1000 +100 +10 0

An sound with a reading of 70 on the decibel scale is 102 times more intense than a reading of size 50. An sound with a reading of 90 on the decibel scale is 103 times more intense than a reading of size 60.

Example 3 The Richter Scale is a logarithmic scale used to measure the magnitude of an earthquake or tremor. An earthquake that registers 4.0 on the Richter scale is 10 times stronger than an earthquake that measures 3.0. How many times stronger is a quake of 7.6 than one of 5.2? Round your answer to the nearest whole number.

Create a strategy Convert the values to linearly comparable measures: 10x and 10y.

Apply the idea

Convert to base 10

Use the exponential rule

Evaluate the exponent

Evaluate the expression

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Example 4 The decibel scale, which is used to record the loudness of sound, is a logarithmic scale. • In the decibel scale, the lowest audible sound, with intensity 10−12 watts/m2 is assigned the value of 0. • A sound that is 10 times louder is assigned a decibel value of 10. • A sound 100 (102) times louder is assigned a decibel value of 20. • A sound 1000 (103) times louder is assigned a decibel value of 30. If the sound of a normal speaking voice is 50 decibels (dB), and the sound in a bus terminal is 80 decibels (dB), then how many times louder is the bus terminal compared to the speaking voice?

Create a strategy Subtract the dB of the normal speaking voice from the dB of the bus terminal sound.

Apply the idea Decibel value = 80 – 50 = 30 dB

Subtract 50 dB from 80 dB Evaluate

On the decimal scale, a sound 1000 (103) times louder is assigned a decibel value of 30, so the bus terminal sound is 1000 louder than the normal speaking voice.

Idea summary Orders of magnitude and logarithmic scales are useful to compare very large and very small numbers. An increase by 1 order of magnitude means the number is 10 times larger. A decrease by 1 order of magnitude means the number is 10 times smaller. On a log scale with base b, every 1 unit up means increasing by a factor of b, every 1 unit down means decreasing by a factor of b.

Practice questions Choose your question path Explorer 1–3, 5–9, 11, 12, 14, 17, 18, 22

Adventurer 1abgh, 2abgh, 3–9, 11–15, 17, 18, 22

Trailblazer 1cdef, 2cdfg, 3, 4, 6, 7, 9–11, 14, 16, 18–23

Understanding 1

Write as powers of 10: a

100

1000

1 000 000

0.01

0.1

0.001

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Write using scientiﬁc notation: a

600

7000

4 800 000

230

0.0076

0.012

0.00009

0.1

0.5

Find the base ten logarithm of the number. a

1 000

0.000 01

Find the base two logarithm of the number. a

Fluency Ex 1

Ex 2

What is the order of magnitude in the number 6 × 10−7?

For each number: i

Write the number using scientiﬁc notation.

Identify the order of magnitude.

8 000

964 400

0.0008

0.2574

A coffee shop sells 580 cups of coffee in a week. What is the order of magnitude of the total number of coffees sold over 12 weeks?

One university has 6000 students enrolled, while another university has 7000 students enrolled. What is the order of magnitude of the total number of students enrolled at both universities?

The average distance of Mercury to the Sun is 5.791 × 107 km while the average distance of Uranus to the Sun is 2.871 × 109 km. By what order of magnitude does the distance between these planets and the Sun differ?

The scale shown is logarithmic with base two:

11 Ex 3

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Find the value of A.

Find the value of B.

Find the value of C.

How many times larger is the number represented by A than the number represented by B on the scale?

–5 –4 –3 –2 –1 0 1 2 3 4 5

The Richter Scale is a base 10 logarithmic scale used to measure the magnitude of an earthquake, given by R = log10 x, where x is the relative strength of the quake. Some past earthquakes and their Richter scales are shown in the table below: a

How many times stronger was the earthquake in Sumatra compared to the earthquake in China? Round your answer to the nearest whole number.

How many times stronger is a quake of 7.6 than one of 5.2? Round your answer to the nearest whole number.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Earthquake Sumatra 2004 China 2008 Haiti 2010 Italy 2009

Richter scale 9.1 7.9 7.0 6.3

Earthquakes are often preceded by lesser tremors, known as foreshocks, and followed by lesser tremors known as aftershocks. The graph shows some readings from an area before, during and after an earthquake. a b

The main earthquake occurred at 5:00 a.m. What was its magnitude? The last aftershock was measured at 9:30 a.m. How many times larger than this aftershock was the largest foreshock?

5.5

Magnitude

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

Time (hours) 1 2 3 4 5 6 7 8 9 10

14 Ex 4

The pH scale is a base 10 logarithmic scale used to measure the acidity (concentration of hydrogen ions) of a solution. The lower the pH, the higher the concentration of hydrogen ions. Jasper is a chemistry student who has 3 solutions. One is marked as having a pH of 10, one has a pH of 8 and one has a pH of 5.4. a

How many times greater is the concentration of hydrogen ions in the pH 8 solution compared to the pH 10 solution?

How many times greater is the concentration of hydrogen ions in the pH 5.4 solution compared to the pH 10 solution?

How many times greater is the concentration of hydrogen ions in the pH 5.4 solution compared to the pH 8 solution?

The decibel scale, which is used to record the loudness of sound, is a logarithmic scale with base 10. In the decibel scale, the lowest audible sound is assigned the value of 0. The table shows the decibel value for various types of noise. a

How many times louder is the sound of industrial noise than the sound of a wind turbine?

If the sound of a normal speaking voice is 50 decibels, and the sound in a bus terminal is 80 decibels, how many times louder is the bus terminal compared to a normal speaking voice?

Type of noise Jet plane Pneumatic drill Industrial Stereo music Inside a car Office Houehold Wind turbine Bedroom Falling leaves

Decibel value 150 120 110 100 90 70 60 50 30 20

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The decibel scale for sound is a base 10 logarithmic scale. Sound Intensity (watts/m2)

Relative intensity (decibels)

102 101 100 10−1 10−2

jack hammer rock concert ambulance siren school dance

140 130 120 110 100

10−3

electric drill

10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12

lawn mower hair dryer normal conversation refrigerator library

80 70 60 50 40 30 20 10 0

whisper at 2 m normal breathing barely audible

How many times more intense is the sound of a refrigerator compared to normal breathing?

How many times more intense is the sound of a rock concert compared to an electric drill?

How many times more intense is the sound of a lawn mower compared to a whisper at 2 m?

How many times more intense is the sound of a jack hammer compared to a normal conversation?

The decibel (dB) scale for sound is based on a base 10 logarithm. Three sounds are measured for their intensity. • Sound A was measured to be 125 dB. • Sound B was measured to be 97 dB. • Sound C was measured to be 61 dB. a

How many times more intense was sound A than sound B?

How many times more intense was sound B than sound C?

How many times more intense was sound A than sound C?

Reasoning 17

The diameter of a human strand of DNA is about 2.5 × 10−9 m while the diameter of a hydrogen nucleus is about 1.8 × 10−15 m. By how many orders of magnitude does the diameter of these objects differ? Explain your answer.

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The Richter scale is a base 10 logarithmic scale. Richter magnitude 1–2 3 4 5 6 7

Possible effects Usually not felt except by instruments May be felt but rarely causes damage Like vibrations from heavy traffic Strong enough to wake people sleeping Ground cracks, houses collapse Few buildings survive, landslides

How much bigger is an earthquake that is strong enough to crack the ground compared to one that is like heavy traﬃc?

How much bigger is an earthquake of magnitude 6.3 than one of magnitude 4.8?

Would an earthquake of magnitude 6.2 be suﬃcient to crack walls in buildings? Justify your reasoning.

The time taken (t years) for A grams of a radioactive substance to decompose down to y grams, where k is a constant related to the properties of a particular substance is given by:

Use these substances to answer the questions below: • R1 has a constant of k = 0.000 43 • R2 has a constant of k = 0.000 47 a

Use technology to ﬁnd how long it takes 140 units of R1 to decompose down to 105 units. Round your answer to the nearest year.

Use technology to ﬁnd how long it takes 200 units of R2 to decompose down to 150 units. Round your answer to the nearest year.

Use technology to ﬁnd the half-life of each substance. That is, determine how long it takes a quantity of a substance to decompose down to half the original quantity. Round your answers to the nearest year.

Will it take twice the half-life for each substance to decompose completely? Explain your answer.

Problem-solving 20

The table of values shows a log scale relating x and y. Form an equation relating x and y. Express the equation in logarithmic form. log scale measure ( y) 0 1 2 3 4

= = = = =

linear measure (x) 5 50 500 5000 50 000

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The table shows the time it takes, in years, for a $1 investment to grow to I at different annual interest rate, using a log2 scale: a

log2 I 0 1 2 3 4 5

Find the number of years it will take for a $1 investment to double in value at the given annual interest rates: i

iii

7% 0 10.24 20.49 30.73 40.98 51.22

8% 0 9.01 18.01 27.02 36.03 45.03

9% 0 8.04 16.09 24.13 32.17 40.22

How many years will it take for a $1 investment to quadruple in value at 9% annual interest rate?

The Richter Scale is a base 10 logarithmic scale used to measure the magnitude of an earthquake, given by R = log10 x, where x is the relative strength of the quake. This means an earthquake that measures 4.0 on the Richter Scale will be 10 times stronger than one that measures 3.0. The aftershock of an earthquake measured 7.9 on the Richter Scale, and the main quake was 6 times stronger than the aftershock. Find the magnitude of the main quake on the Richter Scale, to one decimal place.

The population of shrimp is tracked over time and recorded as shown: Days Shrimp

1 4

2 16

3 64

4 256

5 1024

Plot the data on a coordinate plane where x is the number of days and y is the population of shrimp.

Convert the data to a base 4 logarithmic scale.

Plot the data on a coordinate plane where x is the number of days and y is the number of shrimp using a base 4 logarithmic scale.

10.03 Optional: Other bases After this lesson, you will be able to… • extend the understanding of logarithms to bases other than ten. • rewrite expressions in exponential form and logarithmic form. • evaluate logarithms with different bases using both calculation and estimation methods. • apply understanding of logarithms to solve real-world problems.

Logarithm to bases other than ten The logarithm of a number to base 10 is the power of 10 that makes the number. 102 = 100 can be written as a logarithm as log10 100 = 2 or just log 100 = 2. 546

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If we wanted to find log 81, then we can press the “log” button on a calculator and then enter 81. This gives us 1.908 to three decimal places. If there is no base specified then the base is assumed to be 10. Logarithms can also be written with other bases. For example, log5 25 = 2, since 52 = 25. The base here is 5. log3 81 = 4, since 34 = 81. The base here is 3. The definition of a logarithm is that if y = Bx then logB y = x. y = Bx is called exponential form and logB y = x is called logarithmic form.

Example 1 Evaluate

Create a strategy Use the deﬁnition: ‘If y = Bx then logB y = x.’

Apply the idea Let x be the index or the subject of the equation. So we have: Write the equation

Write in exponential form

Write the denominator as a power of 8

Use the negative index rule

Equate the indices

Example 2 Consider the expression log 8892. a Within what range is the value of log 8892? A

between 0 and 1

between 4 and 5

between 2 and 3

between 3 and 4

between 1 and 2

Create a strategy Consider some powers of 10. Use the fact that 100 = 1, 101 = 10, 102 = 100, and so on.

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Apply the idea Since 8892 is greater than 103 = 1000 but less than 104 = 10 000, then the range of the value of log 8892 is between 3 and 4. So, the correct answer is Option D.

b Find the value of log 8892, rounded to four decimal places.

Create a strategy

Apply the idea

Use a calculator and enter the expression into your calculator by pressing the log button.

log 8892 = 3.9490

Idea summary The deﬁnition of a logarithm is that if y = Bx then logB y = x. In logB y, B is the base of the logarithm. By convention, if the base is not speciﬁed then B = 10. y = Bx is called exponential form and logB y = x is called logarithmic form.

Practice questions Choose your question path Explorer 1–3, 4acegik, 5abe-h, 6a-cfi, 8a, 9a, 10, 11, 15aef, 16abe-hj, 17abf

Adventurer 1–3, 4bdfhjl, 5a-dgh, 6acefik, 7abc, 8a, 9ab, 10, 11, 15abdg, 16abefij, 17ace, 18a

Trailblazer 1–3, 4bdfhjl, 5acdfgkl, 6b-dikl, 7acegik, 8, 9cd, 10, 12–14, 15b-dfh, 16h-l, 17egh, 18

Understanding 1

What number makes both equations true? a

2 3

7⬚ = 7

log2 16 = ⬚

4⬚ = 64

3⬚ = 81

3⬚ = 27

log3 81 = ⬚ log3 27 = ⬚ g

log7 7 = ⬚ log4 64 = ⬚

To evaluate the expression log2 8, we can ask ourselves: “⬚ to what power is ⬚?” Identify the base of the expression: a

548

2⬚ = 16

log5 125

log9 3

loga bc

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

log 100

Fluency 4

Rewrite in logarithmic form: a

52 = 25

4−2 = 0.0625

f j

i 5

Ex 1

20 = 1

9x = 81

4−1 = 0.25

30 = 1

(9.8)0 = 1

log4 16 = 2

log5 5 = 1

log2 0.125 = −3

log6 36 = 2

log2 2 = 1

log8 1 = 0

log5.8 33.64 = 2

log8.4 70.56 = 2

Evaluate: a

log2 8

log4 16

log5 1

log5 125

log2 16

log3 3

log5 0.2

log2 0.25

Evaluate: a

log36 6

log25 5

log0.2 25

log0.9 0.81

log8 2

log27 3

Find the value of x: a

Ex 2

61 = 6

i 8

42 = 16

Rewrite in exponential form:

i 7

3x = 81

8 log10 (x) = 16

4 log3 (x − 6) = 16

Without evaluating, ﬁnd the two integers that the value of each logarithm lies between. a

log2 (13)

log10 (539)

log7 (23)

log10 (−1)

iii

log10 (0)

Reasoning 10

Evaluate: i

log10 (−100)

Explain your results from part (a).

If logx y = 2, ﬁnd the value of logy x. Justify your answer.

If log7 x = y, ﬁnd the value of log7 x2. Justify your answer.

Consider the expression 3log3x.

log10 (−10)

Let 3log3x = a. Rewrite this as an equation in logarithmic form.

Use your answer from part (a) to simplify the expression 3log3x. Explain your reasoning.

Describe the possible values of y in the equation log4 y = x, for all values of x. 10.03 Optional: Other bases mathspace.co

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Problem-solving 15

Evaluate: a

(log5 625) (log5 125)

loga (a9)

For each logarithmic equation: i

Rewrite in exponential form.

Find the value of x.

log4 (64) = x

log3 (243) = x

log125 (5) = x

logx 64 = 2

logx 125 = 3

log2 x = 5

log3 x = 4

log25 125 = x

log4 32 = x

logx 64 = 1.5

logx 243 = 2.5

log4 (x + 6) = 3

Find the value of x.

log5 (x2 + 4) = 3

log2 (1 − 3x) = −4

log2 (4x − 16) = 5

log9 (7x + 652) = 3

log3 (log3 x) = 1

log7 (10x + 2311) = 4

Solve for x: a

9 log8 (4x) − 18 = 0

INVESTIGATION

Logarithms Investigate logarithms and understand the heating and cooling formula.

Objectives To investigate logarithms To understand the heating and cooling formula To practise recognising logarithmic data To investigate the relationship between exponents and logarithms

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Materials •

Paper

•

Scissors

•

Pen

•

Printed out times for the size group you have

•

Calculator

•

Cooking thermometer

•

Opaque bag

•

Stopwatch

Murder Mystery For this investigation we are going to be solving a murder mystery! You and your class have rented out a mansion for an end-of-year celebration. Everyone is having a lot of fun hanging out, playing games, and eating food.

Who did it? Suddenly you all hear a loud scream coming from the upstairs of the mansion. Everyone rushes upstairs to find one of your classmates, who was on his way to the bathroom, has discovered a dead body in the hallway, outside the library. The man who has died is the owner of the mansion and no one seems to know who did it.

Among us The only thing everyone agrees on is that it must be someone in the class because there was no one else in the mansion. At the time the body was found the entire class is present.

Time of death One of your friends comes up with the suggestion that if you knew the time of death, you may be able to determine who was with the mansion owner at the time, and most likely the murderer.

Measurements Another one of your friends has the idea that knowing the temperature of the body would help to figure out the time of his death, so immediately a thermometer is found and body temperature of the corpse is taken.

Temperature Based on the thermometer reading, the body was 33.89 °C at 6 p.m. when the body was found. Two hours later the body’s temperature is taken again and is now found to be 30.33 °C.

Pre-Investigation Work together in small groups to answer the following questions. We want to come up with a method to determine what time the man was killed.

Investigation: Logarithms mathspace.co

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Let’s look at a smaller example first to get the idea. Do you think that a corpse will cool off in a similar way to things like coffee or tea? How would it be similar? How would it be different? What is a good estimate for the starting temperature of coffee or tea? After what amount of time do you think a cup of coffee or tea would get cold? At what temperature would you consider a cup of coffee or tea to be cold? Why? Next, Would you consider the same temperature as being cold for a body? What factors do you think might affect how fast something cools off? On your piece of paper graph your estimated change in temperature over time, plotting temperature on the y-axis and time on the x-axis. Be sure to label your axes and title your graph. Do you think a linear representation would be useful in modeling the change in temperature over time? If not, what type of graph would make the most sense for this situation? Explain. Optional: If you have a cooking thermometer available complete the following activity to test your findings. Other thermometers may break during this experiment so make sure it is a cooking thermometer. Take note of the thermostat in the room you are in. If there is no thermostat be sure to get a reading of the room’s temperature. Make a cup of coffee or a cup of tea. Measure and record the temperature of the drink immediately after it is made, then continue measuring the temperature: 2 minutes after it has been made 4 minutes after it has been made 5 minutes after it has been made 10 minutes after it has been made 15 minutes after it has been made 20 minutes after it has been made 30 minutes after it has been made 60 minutes after it has been made

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Investigate Using the data that you have collected or been given, answer these questions: 1.

Was your guess of the initial temperature close to the real initial temperature?

Was your guess of how long it would take the drink to reach the temperature you considered to be cool close?

Make a graph to represent the data you are using. Be sure to label the axes and title the graph. Like your first graph, time should be on the x-axis and temperature should be on the y-axis.

Did the graph look as you expected it to? Why or why not?

Make some observations about the graph in the context of the situation.

What temperature do you think the drink would be after 2 hours? What about after 3 hours? How do you know?

Investigation Now let’s apply this knowledge about heating and cooling to the murder mystery in the mansion. Use the facts below about the crime scene to help you in your investigation. Work in groups to solve the murder mystery before the killer strikes again. The facts The mansion owner was found in the hallway near the library. There is only one pathway in front of the library and it leads to the bathroom. The temperature of the body was 33.89 °C at 6 p.m. The temperature of the body was 30.33 °C at 8 p.m. The thermostat was found to be set at 21.11 °C. After much discussion, everyone was able to remember the times that they were up in the hall passing by the library to walk to the bathroom. Cut out the time cards provided based on the number of students in your group. If you have more participants than cards, some people should pair up. In this case, they will be assumed to have been going to the bathroom together. Place the cards with times on them into an opaque bag. Each member (or pair) of the group should pick a piece of paper from the bag. This will indicate what time you were passing through the hall to go to the bathroom.

Did you know? An exponential equation such as ex = 3 can be rewritten as the equivalent logarithmic equation as x = loge3. We often use the special notation in for the natural logarithm, instead of loge.

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Newton’s formula for cooling The mansion owner was found in the hallway near the library. There is only one pathway in front of the library and it leads to the bathroom. The temperature of the body was 33.89 °C at 6 p.m. The temperature of the body was 30.33 °C at 8 p.m. The thermostat was found to be set at 21.11 °C. Newton’s formula for cooling:

T (t) is the temperature of the body at time t, Tenv is the temperature of the surrounding environment, T0 is the initial temperature of the body (look up average body temperature for this), t is the time after death in hours, and k is a constant that can be calculated using the formula:

Let t represent the time (in hours) after death that the body was found. Using this fact, at time t when the body was found, the temperature of the body was 33.89 °C. Two hours later, at time t + 2, the temperature of the body was 30.33 °C. Substitute the values you know and solve for t to find the time that elapsed before the body was found.

Investigate Answer these questions. 1.

What time was the mansion owner killed?

Based on everything you have found, who was the murderer?

Answer these questions after you have completed the activity.

Discussion

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Summarise what you have learnt about the heating and cooling formula.

Do all things heat and cool at the same rate?

What considerations need to be made when using the formula?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

10.04 Optional: Logarithm laws After this lesson, you will be able to… • understand the special cases of logarithms. • describe and prove the product law of logarithms. • describe and prove the quotient law of logarithms. • describe and prove the power law of logarithms. • apply the laws of logarithms to simplify expressions and solve problems without using technology.

Special cases First, consider the deﬁnition that if y = Bx then x = logB y. It follows that, logB Bx = x Substituting x = 0 and x = 1 gives the following special cases: logB 1 = 0 logB B = 1

Example 1 Evaluate log3 3.

Create a strategy Use the logarithm law: logB B = 1.

Apply the idea log3 3 = 1

Idea summary From the deﬁnition that if y = Bx then x = logB y we can show, logB Bx = x logB 1 = 0 logB B = 1

Product, quotient and power laws The product law of logarithms states that if you have the logarithm of a product you can write it as the sum of the logarithms of the factors.

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logB ( pq) = logB p + logB q m

To prove this, let p = B and q = B so that logB p = m and logB q = n. Bm × Bn = Bm + n m

logB (B × B ) = logB B logB (B × B ) = m + n

Start with the index multiplication law m+n

Take the logarithm of both sides Apply logB Bx = x

logB pq = logB p + logB q Substitute p = Bm and q = Bn The quotient law of logarithms states that if you have the logarithm of a quotient you can write it as the difference of the logarithms of the numerator and denominator.

To prove this, let p = Bm and q = Bn so that logB p = m and logB q = n. Start with the index division law

Take the logarithm of both sides

Apply logB Bx = x

Substitute p = Bm and q = Bn

The power law of logarithms states that if you have a logarithm to a power you can write it as the product of the logarithm and the power. logB pn = n logB p (Bm)n = Bmn mn

logB (B ) = logB B

Start with the index power of a power law mn

logB (B ) = mn logB pn = n logB p

Take the logarithm of both sides Apply logB Bx = x Substitute p = Bm

Example 2 Simplify each of the following expressions without using a calculator. Leave answers in exact form. a log10 11 + log10 2 + log10 9

Create a strategy Use the logarithm law: logB p + logB q = logB ( p × q).

Apply the idea log10 11 + log10 2 + log10 9 = log10 (11 × 2 × 9) = log10 198

556

Use the law: logB p + logB q = logB ( p × q) Simplify

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b log10 12 − (log10 2 + log10 3)

Create a strategy and logB p + logB q = logB ( p × q)

Use the logarithm laws:

Apply the idea Use logB p + logB q = logB ( p × q)

Simplify

Use

Simplify

Example 3 Simplify log 2x + log 50y to an expression with a single logarithmic term.

Create a strategy Use the logarithm law: logB pq = logB p + logB q

Apply the idea log 2x + log 50y = log (2x × 50y)

Use the law: logB pq = logB p + logB q

= log (100xy)

Simplify

= log 100 + log xy

Use the law: logB pq = logB p + logB q

= log 10 + log xy

Write the ﬁrst log as a power of 10

= 2 + log xy

Use the law: logB Bx = x

Example 4 Simplify

Create a strategy Use the logarithm law: logB pn = n logB p

Apply the idea Write the log as a power of 7

Use the law: logB pn = n logB p

Remove the common factor: log4 7

10.04 Optional: Logarithm laws mathspace.co

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Idea summary The deﬁnition of logarithms and the index laws give us the following results: Product law of logarithms

Quotient law of logarithms

Power law of logarithms

Practice questions Choose your question path Explorer 1–3, 4aceg, 5acegik, 6acegik, 7aceg, 8acfg, 9acegh, 10a-d, 11a-d, 12abe, 13ac, 14, 20, 22aceg, 23ac, 24abeg

Adventurer 1–3, 4aceg, 5acegik, 6acegik, 7aceg, 8acfg, 9aegim, 10bcfg, 11c-f, 12abef, 13aceg, 14, 15, 20, 22aceg, 23ab, 24abej

Trailblazer 1–3, 4bdfh, 5bdfhjl, 6bdfhjl, 7bdfh, 8cfh, 9dhlp, 10bdehj, 11h-l, 12cdeh, 13bdfh, 15–17, 20, 21, 22bdfh, 23bd, 24h-l, 25

Understanding 1

Complete each statement with the correct operation symbol: a c

logb (xy) = logb (x) ⬚ logb ( y)

logb (xy ) = y⬚ logb (x)

Identify what law was used to get the expression on the right-hand side of each equation. a

log10 25 = 5 log10 2

log10 (50) = log10 10 + log10 5

log2 7 + log2 11 = log2 77

log5 100 − log5 4 = log5 25

Fluency 3

Identify if each expression is equivalent to log 35: a

558

log 30 + log 5

log 70 − log 2

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

log 7 + log 5

Identify if each expression is equivalent to 1: a

log 2 − log 1

4 log 10 − 3

2 log 10 − log 10

log2 2a

log5 53

log9 9

2 loga a

log20 20

log7 1

logk 1

log78 1

−5 log7 7

7 log4 4

Rewrite the logarithm as the sum or difference of two or more logarithms. a

log7 ( fg)

log2 (3s)

log8 (xy)

logb (5 × 4)

log2 (39)

logb (20 ÷ 7)

log2 (11cd)

Rewrite the logarithm without any powers. log2 ( y3)

log2 (r6)

log4 ( y7)

loga (B–5)

log (2x + 3)−1

log (x4)

Rewrite the logarithm without any powers or surds. a

log2 (x + 7)6

log (2x + 5)6

c g

log (4x + 3)−8

Rewrite the logarithm as the sum or difference of logarithms without any powers. a e

log5 1

log7 7x

log12 3 + log12 4

i 7

Simplify:

i 6

log7 7

e Ex 1

log (u3 v5)

log (7x)6

log (7x4)

log (5x−1)

log (3x)−7

m log (7x−6)

log (2x)−1

Write the expression as a single logarithmic term. a

log 7 + log 12

log 11 + log 5 + log 3

log10 42 − log10 7

log10 8 − log10 32

log10 12 − (log10 2 + log10 3)

log10 2 + log10 3 − log10 7

5 (log10 3 + log10 6)

3 (log10 6 − log10 3)

3 log10 22 − 3 log10 11

7 log10 5 − 21 log10 25

10.04 Optional: Logarithm laws mathspace.co

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Ex 2

Evaluate: b

loga (a9)

log3 (108) − log3 (4)

log10 (2) + log10 (5)

log (10) + log (10)

log6 (4) + log6 (9)

log2 (15) − log2 (30)

k Ex 3

Write the expression as a single logarithmic term. 3 log (x5) − 2 log (x4)

Ex 4

logb (b6 + b4) − logb (b5 + b3)

6 log x + 5 log y

log (x4) + log (x2)

log (2x) + log (50y)

log (x) + 1

log (x) − 1

Evaluate: a

Reasoning 14

Pei has attempted to fully simplify an expression and showed their work: log3 (72) − log3 (8) + log3 (3) = log3 (72) − log3 (24)

Use logB ( pq) = logB p + logB q = logB p − logB q

= log3 (3)

Use logB

Use logarithm law logB B = 1

Identify where Pei has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Cinzia and Uri use different methods to simplify the logarithmic expression: log10 (x7) − log10 (x4) • Cinzia uses the logarithm law logB = logB p − logB q ﬁrst, then the logarithm law n logB p = n logB p. • Uri uses the logarithm law logB pn = n logB p ﬁrst, then combines the like terms. Determine whether either of Cinzia or Uri’s methods are invalid. If a method is valid, show the work for that method.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

= logB p − logB q is shown. Justify each step in the

The proof for logarithm law logB proof.

Let p = Bm and q = Bn. Then, logB p = m and logB q = n. Then:

Therefore, logB 17

= logB p − logB q.

The product law of logarithms states that: logB ( pq) = logB p + logb q Show this product law is true.

Using the product property of logarithms that states logB ( p) + logB (q) = logB ( pq), prove that logB pn = n logB p.

Using the power property of logarithms that states logB pn = n logB p, explain why loga (1) = 0.

Explain how 210 = 1024 can be used to make the estimation 10 log10 (2) ≈ 3.

Using the proof shown: Let logx y = b. Then, xb = y by the deﬁnition of a logarithm. By taking the log of both sides, we maintain equality. So, xb = y is equivalent to log xb = log y. By using the power law that states: loga pn = n loga p we have b log x = log y Solving for b, we ﬁnd a b c

Complete the proof by ﬁlling in the blank: Therefore, logx y = ⬚

Use the proof to rewrite the expression log2 9.

Use a calculator to evaluate the expression from part (b) to two decimal places.

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Problem-solving 22

When rounded to three decimal places, we have: • log (2) = 0.301 • log (5) = 0.699 • log (3) = 0.477 • log (7) = 0.845 Estimate the logarithm, rounded to two decimal places:

log (6)

log (25)

log (42)

log (500)

log5 (7)

log3 (28)

log10 (21)

log8 (81)

Suppose that logk (a) = 1.64. Evaluate the expression, rounded to 2 decimal places where necessary.

logk (ka4)

loga (k5) − logk (a)

Solve for x: a

logk (k2 a3) + logk (a2)

3 log (x) = log (125)

log5 (4x2) − log5 (x) = 3

log7 (2x) + log7 (3) = 3

log3 (8x) − log3 (40) = 2

log10 (x) + log10 (9) = log10 (45)

log10 (48) − log10 (3) = log10 (x)

log3 (x) + log3 (7) = log3 (56)

log5 (84) − log5 (x) = log5 (7)

x−2

loga (a

) = 10

log4 (27) − log4 (x) = 3

log6 (x) + log6 (x − 5) = 2

log2 (x) − log4 (x) = log64 (x)

For equation 2 logb (x) = 1 + logb (6x − 8b) a

Write the equation without logarithms.

Find x in terms of b where b is a positive constant and x is positive.

10.05 Optional: Connect exponentials and logarithms After this lesson, you will be able to… • understand the relationship between exponential and logarithmic functions. • identify the general forms of exponential and logarithmic functions and their properties. • find inverse functions • use function notation for inverse functions. • interpret tables of values to understand the relationship between a function and its inverse. • graph inverse functions and identify their line of reflection. • write the equation of an inverse function.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Exponential and logarithmic functions An exponential function with base b, has the form: y = bx A logarithmic function with base b has the form: y = logb x We use logarithmic functions when we are interested in ﬁnding out the power needed ( y) to raise base b to equal the value x. Consider the table of values for the function y = 3x x

−4

−3

−2

−1

The graph of the function looks like this: 80 y 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 −3

−4

−2

−1

y = 3x

x 1

We call y = 3x an increasing function because the y-values increase as the x-values increase. If the base is less than 1 the function will be decreasing. There is a horizontal asymptote at y = 0 because it is not possible for a base raised to a power to equal a number less than zero. Now consider the table of values for the logarithmic function with the same base, y = log3 x. x y

−4

−3

−2

−1

The graph of the function looks like this: 4

y = log3x

2 1 −1

x 5

20 25 30 35 40 45 50 55 60 65 70 75 80

−2 −3 −4

10.05 Optional: Connect exponentials and logarithms mathspace.co

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The graph of y = log3 x is also an increasing function. If the base is less than 1 then it will be a decreasing function. The graph has a vertical asymptote at x = 0 because we can only take the logarithm of a positive number with a positive base that’s not equal to one. We say the set of possible x-values for the function y = logb x is: all real values where x > 0 and b > 0, b ≠ 1

Example 1 Consider the function f(x) = log10(3x + 9) Evaluate f (4) to two decimal places.

Create a strategy Substitute x = 4 into f (x) = log10 (3x + 9).

Apply the idea f (4) = log10 (3 × 4 + 9)

Substitute 4 for x

= log10 21

Evaluate what’s inside the brackets

≈ 1.32

Evaluate the logarithm

Example 2 Consider the function y = log3 x. a Complete the table of values: x y

⬚

Express all values as simplified fractions.

Create a strategy Substitute the values of x into y = log3 x.

Apply the idea For

, Substitute

to x

Express

Use the law logB pn = n logB p

Evaluate

564

as powers of 3

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Substituting the other values of x into y = log3 x, the complete table of values is given by: x −5

−1

b Consider the behaviour of log3 x as x changes. Which of the following statements is correct? A

log3 x is a decreasing function.

log3 x is increasing in some intervals of x, and decreasing in other intervals.

log3 x is an increasing function.

Create a strategy Consider the table of values from part (a).

Apply the idea Based on the table of values from part (a), as x increases the value of y also increases. x −5

−1

The correct answer is option C.

c What value does log3 x approach as x approaches 0? A

−∞

−3

∞

Create a strategy

Apply the idea

Consider the graph of log3 x. Consider what happens to the value of y as you substitute in smaller and smaller values of x based on its graph.

Based on the graph of log3 x, as x approaches 0 the value of y approaches −∞. y 4 3 2 1 −1

−1

x 1

−2 −3 −4

The correct answer is option A.

10.05 Optional: Connect exponentials and logarithms mathspace.co

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d What happens when x = 0? A

y = −3

y = −∞

y is undeﬁned

Create a strategy

Apply the idea

Check for intervals of x where the values of y are deﬁned based on the graph of y = log3 x.

In the expression log3 x, x is the result when 3 is raised to some power and x must be positive. y 4 3 2 1 −1

−1

x 1

−2 −3 −4

This means that y is only deﬁned for positive values of x, so the correct answer is option C.

Idea summary The general form of an exponential function is:

y = bx b is the base x

is the index

is the result of raising the power

The general form of a logarithmic function is:

y = logb x

b is the base

is the result of raising the power

is the index

An exponential function is the result of raising the base to a power while a logarithmic function is equal to the index (or power). A number raised to a power is always positive, therefore y = bx has a horizontal asymptote of the x-axis. You cannot ﬁnd the logarithm of numbers less than or equal to zero so y = logb x has a vertical asymptote of the y-axis. Both functions are increasing functions for b > 1 and decreasing functions for b < 1.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Inverse functions Notice that the x- and y-values are swapped in the table of values for the two functions y = 3x and y = log3 x. The functions are inverse functions which are like the opposites of each other, so the coordinates are in opposite order. When we have an equation in two variables, such as y = 3x, we say y is a function of x. We sometimes write functions using function notation. For example, f (x) = 3x means f is a function of x. Function notation gives us greater ﬂexibility when communicating mathematically. The inverse of function f (x) is written using the notation f −1 (x). The inverse function of f (x) = 3x is f −1 (x) = log3 x. f (4) = 34 = 81 so f (x) contains the ordered pair (4, 81). f −1 (81) = log3 81 = 4 so f −1 (x) contains the ordered pair (81, 4).

Example 3 Find the inverse function of f (x) = 2x

Create a strategy Switch the x and y terms in y = 2x and ﬁnd the equation of y.

Apply the idea y = 2x x = 2y

Switch the x and y y

log2 x = log2 2

Get the logarithm of base 2 on both sides

log2 x = y log2 2

Use the law logB pn = n logB p

y = log2 x

Evaluate x

The inverse function of y = 2 is y = log2 x.

Example 4 Consider the function y = log4 x shown in the graph.

y 4 3 2 1 −2 −1 −1

x 1 2 3 4 5 6 7 8 9

−2 −3 −4

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a Complete the table of values. x

⬚

Create a strategy

Substitute the values of x into y = log4 x.

Apply the idea For x = 1, y = log4 1 = log4 4

Substitute 1 to x 0

Express 1 as powers of 4

= 0 × log4 4

Use the law logB pn = n logB p

Evaluate

Substituting the other values of x into y = log4 x, the complete table of values is given by: x

b The given curve is the inverse of log4 x. Notice that the points on the curve are the inverses of the points found in part (a).

y 4 3 2 1 x −4 −3 −2 −1 −1

1 2 3 4 5 6

−2

Find the equation of the inverse function of y = log4 x.

Create a strategy Switch the x and y terms in y = log4 x and ﬁnd the equation of y.

Apply the idea y = log4 x x = log4 y

Switch the x and y

We know that if logB p = A, then BA = p. This means that the inverse function of y = log4 x is given by: y = 4x

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary Inverse functions use the notation f −1 (x), and have opposite pairs of x- and y-values. y = bx and y = logb x are inverse functions.

Graphs of inverse functions 4

When inverse functions are graphed on the same Cartesian plane it can be seen that they are reﬂections of each other about the line y = x

y = 4x y=x

3 2 1

−4 −3 −2 −1 −1

2 3 4 y = log4x

−2 −3 −4

Example 5 Consider the graph of 3x.

y 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

a Sketch the inverse function on the same coordinate plane.

Create a strategy Use the fact that the points on the curve of y = 3x are the inverse of the points on the curve of the inverse function of y = 3x.

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Apply the idea Based on the graph of y = 3x, the points (0, 1) and (1, 3) lie on the curve of y = 3x.

y 4 3

(1, 3)

2 1 (0, 1) −4 −3 −2 −1 −1

x 2

−2 −3 −4

The inverse points corresponding to (0, 1) and (1, 3) are (1, 0) and (3, 1), respectively. So, these points can be used to sketch the graph of the inverse function of y = 3x.

y 4 3

(1, 3)

2 1 (0, 1) −4 −3 −2 −1 −1

(1, 0) 1 2 3

(3, 1)

−2 −3 −4

b Write the equation of the inverse function.

Create a strategy Switch the x and y terms in y = 3x and ﬁnd the equation of y.

Apply the idea y = 3x x = 3y

Switch the x and y

log3 x = log3 3y

Find the logarithm of base 3 on both sides

log3 x = y log3 3

Use the law logB pn = n logB p

y = log3 x

Evaluate x

The inverse function of y = 3 is y = log3 x.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary When inverse functions are graphed they are reﬂections of each other about the line y = x.

Practice questions Choose your question path Explorer 1–3, 4ab, 5ab, 6ab, 7, 8ac, 9, 11a, 12a, 13, 16, 19, 21, 23a, 24

Adventurer 1–3, 4ac, 5ac, 6ac, 7, 8ac, 9, 11, 12b, 13, 14, 16, 17, 19, 21, 23ab, 24

Trailblazer 1–3, 4bd, 5bd, 6cd, 7, 8bd, 10, 12cd, 13–18, 20, 22, 23bc, 24, 25

Understanding 1

Fill in the blank to complete the sentence:

The inverse of an exponential function is a ⬚ function of the same base.

For the function f (x) = loga x: a

Does the value of a make f (x) = loga x an increasing function? i

iii

a=2

Does the value of a make f (x) = loga x a decreasing function? i

a=0

iii

a=2

When a = 1, what does the graph of f (x) = loga x look like?

Determine whether the function could represent a logarithmic function. a

1 −4 −3 −2 −1 −1

x 1

1 −4 −3 −2 −1 −1

−2

−3

−4

x 1

10.05 Optional: Connect exponentials and logarithms mathspace.co

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1 −4 −3 −2 −1 −1

g 4

−2

−3

−4

−4 −3 −2 −1 −1

8 3

f (x) = log8 (x)

f (x)

25 = 52

64 = 43

log4 (16) = 2

Find the value of y when x = 5. a

7 Ex 2

y = log3 (5x + 2)

⬚

Sketch the graph of the function on a coordinate plane.

Is the function increasing or decreasing?

What value does log3 x approach as x approaches 0?

What happens when x = 0?

⬚ 5

Write the inverse function. a

572

y = log6 (x − 4)

Complete the table of values:

y = log2 (x2 − 9)

Using y = log3 x. x

Ex 3

Fluency 6

Rewrite in exponential form. a

Ex 1

Rewrite in logarithmic form. a

x 1 2 4 f (x)

f (x) = 5x

f (x) = log7(x)

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

log7 (1) = 0

Ex 4

Using f (x) = log2 x. a

Write the inverse, g (x), of the given function.

Complete the table of values for the inverse. x

−2

−1

g (x)

Ex 5

Use the inverse to create a table of values for f (x) = log2 x.

Draw the graph of f (x) = log2 x.

Find the x-intercept for f (x) = log2 (x).

Write the equation of the vertical asymptote for f (x) = log2 (x).

Is f (x) = log2 x an increasing or decreasing function?

Using

Find the inverse of the given function.

Graph the inverse.

Use the inverse to create a table of values for

Draw the graph of

Find the function value when x = 0.

on the same coordinate plane.

an increasing or decreasing function?

For each graph: i

Sketch the inverse function on the same coordinate plane.

Write the equation of the inverse function.

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

x 1

−2

−3

−4

For each logarithmic equation: i

Write the inverse in exponential form.

Graph the function and its inverse on the same coordinate plane.

f (x) = log10 x

f (x) = log4 (x)

f (x) = log5 (x)

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Using

Find the x-value of the x-intercept.

Complete the table of values: x

State the equation of the vertical asymptote.

Sketch the graph of the function on a coordinate plane.

an increasing or decreasing function?

The graph of f (x) = logk x is shown. a

Determine the value of the base, k.

Write the equation of f (x).

5 4 3 2 1 −1 −1 −2 −3 −4

x 1 2 3 4 5 6 7 8 9

−5

The graph of line y = x are shown.

and its reﬂection across the

Are the two curves inverse functions of each other?

Write the equation of the reﬂection of across y = x.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Reasoning 16

Explain why f (x) = 2x and g (x) = log2 x are inverse functions.

For the function y = loga x, where a is a value greater than 1:

574

For what values of x will loga x be negative?

For what values of x will loga x be positive?

Is there a value that loga x will always be greater than? Use a graph to explain your answer.

Is there a value that loga x will always be less than? Use a graph to explain your answer.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For the functions f (x) = log3 x and g (x) = log5 x: a

Which function has larger function values for values of x that are greater than 1? Explain your reasoning.

Justify your answer to part (a).

Problem-solving 19

Find the value of h, given the point

Given the points (8, n), (k, 4) and (m, −2) all lie on the curve with equation y = log2 x, ﬁnd the value of: a

lies on the curve

For the equation 2x = 8: a

Sketch the graph of y = 2x.

Sketch the inverse of y = 2x.

Using the graph from part (b), ﬁnd the value of the function at x = 8.

Solve the equation 2x = 8.

Given the equation log9

Sketch the graph of y = log9 x.

Sketch the inverse of y = log9 x.

Using the graph from part (b), ﬁnd the value of the function at

Solve the equation log9

For each logarithmic equation: i

Use technology to graph of each side of the equation on the same coordinate plane.

Estimate the solution to the equation.

log2 (x) = 4.2

log3 (x + 1) = −1

−log2 (x + 3) = 2.4 d

log4 (x − 3) = 0.3

For y = 2 log10 x a

Complete the table of values for y = 2 log10 x x

0.01

0.1

100

y b

Draw the graph of y = 2 log10 x

Draw the graph of y = log10 x on the same coordinate plane.

Explain how the y-values of y = 2 log10 x compare to the y-values of y = log10 x for the following values of x: i

0<x<1

x=1

iii

x>1

10.05 Optional: Connect exponentials and logarithms mathspace.co

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For these functions: • f (x) = log2 x • g (x) = log2 x + 5 • h (x) = 5 log2 x a

Determine the y-values for each function at x = 4, x = 128 and x = 21000 f (x)

i b

g (x)

iii

h (x)

Describe the relationship between the function values of f (x), g (x) and h (x) as x increases.

10.06 Optional: Solve exponential equations with logarithms After this lesson, you will be able to… • apply the method of taking logarithms to solve exponential equations. • solve real-world problems involving exponential equations.

Solve exponential equations We previously saw that one method to solve an exponential equation is to write both sides of the equation with the same base. 2x = 128 x

Both sides can be written with same base

2 =2

Write both sides as base 2

x=7

Equate the powers

It’s not always possible to write both sides of the equation with the same base. In this situation we use the idea that logarithms and exponentials are the inverse of each other and we can solve the exponential equation by taking the logarithm of both sides of the equation. Both sides cannot be written with same base

Take log of both sides

Use the log power law

Isolate the variable

Evaluate with calculator

Note that the logarithm of any base can be used to solve the equation, but it makes sense to use base ten logarithms as the calculator log button can be used, evaluating the answer to a number of decimal places.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 Consider the equation 2x = 11. a Rearrange the equation into the form

Create a strategy We make x the subject of the equation by using the laws of logarithm.

Apply the idea Start with the equation

Take the logarithm of both sides of the equation

Use the law logB pn = n logB p

Divide both sides by log 2

Simplify the expression

b Evaluate x to three decimal places.

Create a strategy

Apply the idea

Evaluate the expression found from part (a) using a calculator.

Start with the equation Evaluate and round to three decimal places

Example 2 Solve 3 (10x) = 6 for x. Round your answer to two decimal places.

Create a strategy Take the logarithm of base 10 on both sides of the equation and use the law logB pn = n logB p.

Apply the idea Start with the equation

Divide both sides by 3

Evaluate

Take the logarithm of both sides

Use the law logB pn = n logB p

Use the law logB B = 1

Evaluate and round to two decimal places

10.06 Optional: Solve exponential equations with logarithms mathspace.co

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Reﬂect and check We used the law log An = n log A which is a consequence of inverse functions. The reason we chose base 10 was to make it easier to calculate the solution using a calculator. In other cases it might be better to use a logarithm with the same base as the exponential so that the solution is a logarithmic expression.

Example 3 Solve 5y − 4 = 6y + 3 for y. Round your answer to two decimal places.

Create a strategy Take the logarithm of base 10 on both sides of the equation and use the law logB pn = n logB p.

Apply the idea Start with the equation

Take the logarithm of both sides

Use the law logB pn = n logB p

Use the distributive property

Add 4 log 5 to both sides

Subtract y log 6 from both sides Factor out the common term y in the left-hand side

Divide both sides by log 5 − log 6

Evaluate and round to two decimal places

Idea summary Exponentials and logarithms of the same base are inverse functions of one another. If we cannot solve an exponential equation by writing both sides with the same base, then the method of taking the logarithm of both sides can be used to solve the equation.

578

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1, 2, 3abdg, 4acegj, 5aceg, 6ace, 8ac, 9ab, 10ab, 11ab, 13, 14, 16a, 18–20

Adventurer 1, 2, 3aceg, 4abehk, 5bdfh, 6bdf, 8ac, 9ab, 10ac, 11ac, 13, 14, 16a, 18–21

Trailblazer 1, 2, 3adfh, 4behil, 5i-l, 6dgh, 8bd, 9cd, 10cd, 11cd, 12, 14, 15, 16b, 17, 18, 21–23

Understanding 1

Determine whether logarithms are required to solve the equation without a calculator: a

4x = 13

8x = 4

2 (3)x = 18

5x − 9 = 25

Using a calculator, determine x rounded to 3 decimal places. a

Write the ﬁrst step required to solve for x. a

7x = 2

5 (4)x = 100

10 (6)x − 12 = 8

d 9x = 27x − 1

72x + 3 = 1

3x = 2

4x = 6.4

49x = 169

(0.4)x = 5

Fluency 4

Write the equation in the form

Ex 1

Evaluate x rounded to three decimal places.

13x = 5

5x = 13

i 5 Ex 2

−x

= 12

29x = 5 x

5 + 4 = 3129

6 (2 ) = 42

27x + 4 = 1924

Solve for x, rounding to two decimal places. a

10x = 3.68

10x = 39.671

10x + 2 = 345.5

10x − 3 = 338

104x = 357

104x − 4 = 629

4 × 10x = 794.4

5 × 10x − 4 = 111

4 × 104x − 5 = 95

103x − 1 + 9 = 40

105x + 4 − 7 = 21

0.8x = 3

50 (0.8)x = 325

Solve for x, rounding your answer to two decimal places: a e

12x = 23 x

5 × 2 = 300

b f

5x = 9

c x

200 × 3 = 520

3x = 100 x

120 × 7 = 3

10.06 Optional: Solve exponential equations with logarithms mathspace.co

579

Solve for the exact value of x. a

12x − 40 = 120

e 8

Ex 3

3 (54x + 5) = 3

Make x the subject of the equation.

Evaluate x to three decimal places.

83x = 580

4x − 3 = 25

12x + 5 = 6.5

20 (3x) = 4

3x + 4 = 200

42x − 8 = 70

Solve for x, rounding your answer to two decimal places: 35x = 6

4x + 1 = 7

74x + 3 = 6

56a − 5 = 12

3y + 5 = 62y

25y + 2 = 34 − 3y

92y − 1 = 75 − y

Solve for the exact value of y. a

7x − 3 = 8.5

a 10

3y = 2y + 1

6y + 5 = 5y + 2

Solve for y, rounding your answer to two decimal places. a

6y − 3 = 11y

4y + 2 = 5y − 7

5y + 4 = 83y − 4

Akiko’s solution to the equation 311 − 4w − 7 = 6 is shown. Determine if her solution is correct by solving the problem a different way.

1 2 3 4 5

6 7

Reasoning 13

Explain why 23x − 1 = 27 requires logarithms to solve, while 23x − 1 = 8x − 9 does not.

Explain why 3x = −2 has no solution.

Use the equation Bx = A to show

For each of these incorrect sets of working:

580

for A, B ≥ 0.

Which step was incorrect? Explain your answer.

Rearrange the original equation into the form

iii

Evaluate a to three decimal places.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Hussam uses logarithms of base 3 to solve the equation 3x − 2 = 164x. His work is shown. Hussam has found the exact solution, but he wants to estimate the solution rounded to three decimal places. He does not know how to do this because his calculator only allows him to use logarithms of base 10. Explain how Hussam can rewrite the solution using base 10 logarithms.

1 2 3 4 5 6

Problem-solving 18

The population of mice, t months after initial observation, is modelled by: P (t) = 500 (1.2t). a

Find the initial population.

By what percentage is the population increasing by each month?

Find when the population reaches 1500, for the first time.

A population of wallabies, t years after initial observation, is modelled by P (t) = 800 (0.85t). a

Find the initial population.

By what percentage is the population decreasing each year?

Find when the population reaches 200, for the first time.

The growth of a population of mites modelled by P = 20 (3x), where P is the population after x weeks. After how many weeks will the population of mites have grown to 5000?

A microbe culture initially has a population of 900 000 and the population increases by 40% every hour. Let t be the number of hours passed. Find when the population reaches 7 200 000, for the first time.

A piece of art that costs $6000, appreciates at approximately 6.2% p.a. After how many full years, n, will the value of the piece of art be over $15 000?

Samir makes an investment of $1300, compounding monthly at a rate of 5.9% p.a. How many years will it take for this investment to reach $9000?

10.06 Optional: Solve exponential equations with logarithms mathspace.co

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Chapter 10 review

For each number: i

Express the number as a power of 10.

Find the base ten logarithm of the number.

0.01

10 000

For each number: i

Express the number as a power of 2.

Find the base two logarithm of the number.

0.5

T F

The histogram shows the population for 12 cities in Australia, using a log10 scale: How many cities have a population between 100 000 and 1 000 000?

A R

10 9 8 7 6 5 4 3 2 1 0

Number of cities

1 2 3 4 5 6 Log of population

The weights of various animals are shown in the table: Animal Ferret Cat Whale Rhinoceros Arabian wolf Hamster Shetland pony

Weight (in kg) 0.8 3.2 32 840

Weight in log10 scale

3.44 1.40 −0.82

127

Complete the table, rounding each value to two decimal places. 5

582

A maths tutor has clients with a large range of ages. To manage the data more easily, the age of each client is ﬁrst converted to a log2 scale. a

What range of ages corresponds to a range of 4 to 5 on the log2 scale?

What age would result in 3.2 on a log2 scale? Round to the nearest whole number.

For the number 0.000 04: a

Write the number using scientiﬁc notation.

Identify the order of magnitude.

A cafe sells 450 glasses of milk tea in a week. What is the order of magnitude of the total number of milk teas sold over 12 weeks?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The decibel scale, which is used to record the loudness of sound, is a logarithmic scale. In the decibel scale, the lowest audible sound is assigned the value of 0. The table shows the decibel value for various types of noise.

How many times louder is the sound of stereo music than the sound of a hair dryer?

If a sound is measured to be 136 dB and another sound is measured to be 85 dB, how many times more intense is the ﬁrst sound than the second sound? Write your answer as a power of 10.

(Optional) Evaluate: a

A R

D b

log12 1

log2 0.25

log2 (13)

log10 (0.073)

Solve for x.

Rewrite in exponential form.

log3 (27) = x

logx (16) = 4

log4 (x + 4) = 2

log7 (3 − 2x) = 1

(Optional) Rewrite the logarithm as a sum or difference of two or more logarithms. log7 (6s)

log4 (77)

(Optional) Rewrite the logarithm without any powers or surds. a

log3 27

(Optional) For each logarithmic equation:

a 14

T F

(Optional) Without evaluating, ﬁnd the two integers that the value of each logarithm lies between. a

Decibel value 150 120 110 100 90 70 50 30 20

The Richter Scale is a base 10 logarithmic scale used to measure the magnitude of an earthquake, given by R = log10 x, where x is the Earthquake Richter scale relative strength of the quake. Some past Mexico 2017 8.1 earthquakes and their Richter scales are shown Turkey 2023 7.8 in the table: Japan 2021 7.0 How many times stronger was the earthquake in 6.5 Turkey compared to the earthquake in Botswana? Botswana 2017 Round your answer to the nearest whole number.

Type of noise Jet plane Rock concert Ambulance siren Stereo music Electric drill Hair dryer Refrigerator Bedroom Falling leaves

log (a12)

log6 (x − 10)8

(Optional) Write the expression as a single logarithmic term. a

log2 4 + log2 a − log2 13

− log3 m − log3 n

log 5 + log x − 3 log y

Chapter 10 review mathspace.co

583

(Optional) Evaluate: a

log4 (32) − log4 (8) d

f (x) = log6 (x)

(Optional) Write the inverse function, g (x). a

logb (b7)

f (x) = 4x

(Optional) Using f (x) = log5 x. a

Write the inverse, g (x), of the given function.

Complete the table of values for the inverse: x

−2

−1

g (x)

Use the inverse to create a table of values for f (x) = log5 x.

State the equation of the vertical asymptote.

Sketch the graph of f (x) = log5 x on a coordinate plane.

Is y = log5 x an increasing or decreasing function?

A R

(Optional) The graph of f (x) = logk x is shown. a

Determine the value of the base, k.

Write the equation of f (x).

D 20

T F

5 4 3 2 1

−1 −1 −2 −3 −4 −5

1 2 3 4 5 6 7 8 9

(Optional) For each graph: i

Sketch the inverse function on the same coordinate plane.

Write the equation of the inverse function.

1 −4 −3 −2 −1

−1

−2

584

x −4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

(Optional) i

Write the equation in the form

Evaluate x, rounded to three decimal places.

17x = 3

9x = 529

(0.8)x = 5

5x + 2 = 600

104x+3 − 9 = −2

11x+1 = 4x

(Optional) Solve for x, rounded to two decimal places. a

10x = 2.65

5 × 102x−5 = 11

65x = 485

8x+11 = 57

32x−7 = 102

(Optional) A plant species initially has a population of 7000 and decreases by 5% each year. How many years will it take for the population to reach 600 plants? Round your answer to the nearest whole number.

T F

A R

Chapter 10 review mathspace.co

585

11 Circles Big ideas The equations representing circles provide a systematic way to understand their geometry and position in a Cartesian plane.

Curriculum Outcomes VC2M10A11 Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital tools as appropriate VC2M10AA05 Describe, interpret, and sketch parabolas, hyperbolas, circles and exponential functions and their transformations VC2M10AA10 Experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

Chapter outline 11.01 11.02 11.03

Circles at origin Translations of circles Optional: Transform circle equations Chapter 11 review

588 595 605 615

Circular dome structures, like Rome’s Pantheon, have incredible strength and stability!

11.01 Circles at origin After this lesson, you will be able to… • identify the equation of a circle centred at the origin. • identify the radius of the circle given points or equation. • graph circles centred at the origin. • analyse the position of a point relative to a circle (inside, on, or outside) using the circle’s equation.

Circles centred at the origin All points on a circle are the same distance from the centre. The radius tells us the distance from the centre to any point on the circle. A circle centred at the origin is represented by the equation, x2 + y2 = r2 where r is the radius. For example, the equation that represents the circle is x2 + y2 = 4.

y 2

Its centre is at (0, 0) and its radius is 2 units. 1 x −2

−1

−1 −2

588

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Exploration Match each equation with the correct graph. • x2 + y2 = 1

• x2 + y2 = 25 5 4 3 2 1

• x2 + y2 = 64

10 8 6 4 2

−5 −4 −3 −2 −1 −1 −2 −3 −4 − −5

Circle 1

x 2 4 6 8 10

Circle 2

10 8 6 4 2

−1

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

1 2 3 4 5

−1

• x2 + y2 = 100

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

Circle 3

x 2 4 6 8 10

Circle 4

What strategies did you use to match the equation to the graph?

If the radius of a circle centred at the origin is 3 units, what is the equation of the circle?

What is the radius of the circle with the equation x2 + y2 = 49?

To check whether a point is inside, on or outside a circle, we can compare the distance between that point and the centre of the circle to the radius. • If x2 + y2 < r2, then the point is inside the circle. • If x2 + y2 = r2, then the point is on the circle. • If x2 + y2 > r2, then the point is outside the circle. These conditions are the same as substituting the point into the equation of the circle and comparing the values on each side.

11.01 Circles at origin mathspace.co

589

Example 1 State the equation of the circle.

5 x −5

5 −5

Create a strategy Use the equation x2 + y2 = r2, and identify the radius.

Apply the idea Looking at the x-intercept and y-intercept, we can see that the radius of the circle is 7. x2 + y2 = r2

Write the equation

x2 + y2 = 72

Substitute r = 7

Evaluate 72

x + y = 49

Example 2 Consider the circle with centre (0, 0) that passes through the point (3, −4). a Find the radius of the circle.

Create a strategy Use the Pythagoras’ theorem: a2 + b2 = c2.

Apply the idea We can form a right-angled triangle with the distance between the origin and (3, −4) as the hypotenuse. Let r be the radius. a2 + b2 = c2 2

Write the formula

Substitute a = 3, b = −4, c = r

9 + 16 = r

Evaluate the squares

25 = r

Add the left-hand side

3 + (−4) = r

r=5

Take the square root of both sides

The radius of the circle is 5 units.

590

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Determine the equation of the circle.

Create a strategy 2

Apply the idea 2

Use the equation x + y = r .

From part (a), we know that r = 5. x2 + y2 = r2

Write the equation

x +y =5

Substitute the radius

Evaluate 52

x + y = 25

Example 3 Determine whether the point (−8, 5) lies inside, outside, or on the circle represented by the equation x2 + y2 = 117.

Create a strategy Substitute in the coordinates to determine whether the left-hand side is less than, greater than or equal to the right-hand side.

Apply the idea x2 + y2 = (−8)2 + 52

Substitute x = −8, y = 5

= 64 + 25

Evaluate the squares

= 89

Perform the addition

< 117

Compare to the circle equation

Recall that: • If x2 + y2 < r2, then the point is inside the circle. • If x2 + y2 = r2, then the point is on the circle. • If x2 + y2 > r2, then the point is outside the circle. Since 89 < 117, the point (−8, 5) lies inside the circle x2 + y2 = 117.

Idea summary All points on a circle are the same distance from the centre. The radius tells us the distance from the centre to any point on the circle. A circle centred at the origin is represented by the equation, x2 + y2 = r2 where r is the radius. To check whether a point is inside, on or outside a circle, we can compare the distance between that point and the centre of the circle the radius. • • •

If x2 + y2 < r2, then the point is inside the circle. If x2 + y2 = r2, then the point is on the circle. If x2 + y2 > r2, then the point is outside the circle

11.01 Circles at origin mathspace.co

591

Practice questions Choose your question path Explorer 1–6, 7abcd, 8, 9a, 10, 11, 14, 15, 18, 19, 21, 23

Adventurer 1–6, 7cdef, 8, 9b, 10–12, 15, 16, 18, 19, 21–23

Trailblazer 2, 3, 5, 6, 7cdef, 8, 9c, 10, 11, 13, 15–17, 19, 20, 22–24

Understanding 1

State the radius of each circle: a

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

10 8 6 4 2

d 16

8 4

x 3

9 12

−16 −12 −8 −4 −4

−6

2 4 6 8 10

−12 −9 −6 −3 −3

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

1 2 3 4 5

x 4

8 12 16

−8

−9

−12

−16

Use the graph of the circle to complete the statement: Every point on the circle is exactly ⬚ units away from the point (⬚, ⬚).

8 6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

592

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 2

If a point (x, y) lies on the circle with equation x2 + y2 = r2, what is the distance between the point and the centre of the circle?

Fluency 4

What is the equation of the circle with centre at the origin and a radius of 5 units?

What is the radius of the circle with equation x2 + y2 = 81?

A circle with centre (0, 0) has a y-intercept at (0, −8). Write the equation of the circle.

For the equations of circles given: i

Identify the radius of the circle.

Sketch the graph of the circle.

x2 + y2 = 36

x2 + y2 = 9

e Ex 1

x2 + y2 = 16

x2 + y2 = 0.64

Write the equation of each circle shown: a

y 8

4 3

6 4

2 1

−5 −4 −3 −2 −1 −1

−6 −8

y 12 10 8 6 4 2

−12−10−8 −6 −4 −2 −2 −4 −6 −8 −10 −12

−4

−4 −5

−8 −6 −4 −2 −2

2 3 4 5

−2 −3

Ex 2

x2 + y2 = 169

8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6

2 4 6 8 10 12

x 2 4 6 8 10

−8 −10

For circle with centre (0, 0) that passes through the points given: i

Find the radius of the circle.

(−3, 4)

(8, −6)

Find the equation of the circle.

(−5, −10) 11.01 Circles at origin mathspace.co

593

Ex 3

If a circle has its centre at the origin and passes through the point (6, 8), what is the equation of the circle?

Determine whether the following points are inside, outside, or on the circle with equation x2 + y2 = 25. a

(−3, 2)

(4, 3)

(1, 6)

(−5, 0)

Is the point (−4, 2) inside, outside, or on the circle with equation x2 + y2 = 21?

Is the point (7, 1) inside, outside, or on the circle with equation x2 + y2 = 50?

Reasoning 14

How can you determine if a point (a, b) lies inside, outside, or on the circle with equation x2 + y2 = r2?

A circle with its centre at (0, 0) passes through the points (7, 24) and (20, 15). Prove that the points are on the same circle.

Explain why the equation of a circle with its centre at the origin and a radius of r is given by x2 + y2 = r2.

A circle has its centre at the origin and passes through the point (5, 6). Determine the equation of the circle and justify your answer.

Problem-solving 18

A circle with centre (0, 0) has an x-intercept at (6, 0). a

Find the equation of the circle.

Find the exact area enclosed by the circle.

A circle with its centre at the origin has an area of 64π. Find the equation of the circle.

A circle with its centre at the origin has a circumference of 12π. What is the equation of the circle?

For circle with centre (0, 0) that passes through the point (−4, 4), ﬁnd the equation of the circle.

A circle with its centre at the origin has an equation of x2 + y2 = 9. Find the area of the part of the circle that is in Quadrant 1.

A circle with its centre at the origin has an equation of x2 + y2 = 81. Use technology to estimate the coordinates of the points where the circle intersects the line y = 3x.

594

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A circle with its centre at the origin has an equation of x2 + y2 = 64. Find the length of the horizontal line that is within the circle, parallel to the x-axis and has a y-intercept of 4.

y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

11.02 Translations of circles After this lesson, you will be able to… • identify the changes in the equation of a circle due to its translation. • understand the general equation of a circle and how it represents the centre of the circle. • write the equation of a circle that has been translated.

Translations of circles Similar to other shapes and functions, a circle can be transformed with horizontal and vertical translations.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 11.02 to explore how different parts of the circle’s equation relate to its graph.

11.02 Translations of circles mathspace.co

595

A circle can be vertically translated by increasing or decreasing the y-values by a constant number, k. When k is added to y, the circle is translated down by k. When k is subtracted from y, the circle is translated up by k. y

y x2 + ( y − 2)2 = 1

3 2

2 x2 + y2 = 1

x2 + y2 = 1

1 x

−2

−1

x −2

−1

−2

−3

The circle with equation x2 + y2 = 1 is translated up by 2.

x2 + ( y + 2)2 = 1

The circle with equation x2 + y2 = 1 is translated down by 2.

A circle can be horizontally translated by increasing or decreasing the x-values by a constant number h. When h is added to y, the circle is translated left by h. When h is subtracted from y, the circle is translated right by h. y

y 3

2 x2 + y2 = 1 1

2 x2 + y2 = 1 1 x

−3 −2

−1

1 −1 −2

x −3 −2

(x − 2)2 + y2 = 1

−1

(x + 2)2 + y2 = 1

−3

−1 −2 −3

The circle with equation x2 + y2 = 1 is translated right by 2.

The circle with equation x2 + y2 = 1 is translated left by 2.

In general, the equation of a circle is given by:

(x − h)2 + ( y − k)2 = r2 (h, k) coordinates of the centre of the circle (x, y) coordinates of any point on the circle r radius of the circle

596

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 1 The equation of a circle is given by (x − 4)2 + ( y + 1)2 = 36. a Find the coordinates of the centre of this circle.

Create a strategy Equations of the form (x − h)2 + ( y − k)2 = r2 represent circles with centre at (h, k).

Apply the idea The equation (x − 4)2 + ( y + 1)2 = 36 can be written as (x − 4)2 + ( y − (−1))2 = 62 and it is now of the form (x − h)2 + ( y − k)2 = r2 where h = 4 and k = −1. The centre of the circle is (4, −1). b Find the radius of the circle.

Create a strategy

Apply the idea 2

Equations of the form (x − h) + ( y − k) = r represent circles with radius of r.

In part (a) we rewrote the equation (x − 4)2 + ( y + 1)2 = 36 in the form: (x − 4)2 + ( y − (−1))2 = 62 So the radius of the circle is 6 units.

c Plot the graph of the circle.

Create a strategy Plot the centre of the circle then count the units equal to the radius in each direction (left, right, up, down), and draw a circular curve connecting them.

Apply the idea 5 4 3 2 1 −2 −1 −1

−2 −3 −4 −5 −6 −7

We can plot the point that represents the centre of the circle at (4, −1).

x 1 2 3 4 5 6 7 8 9 10

Now we count 6 units in all four directions to get the points (4, 5), (4, −7), (10, −1) and (−2, −1). Connect the points using a round curve passing through the four points to graph the circle.

11.02 Translations of circles mathspace.co

597

Example 2 A circle has its centre at (3, 3) and a radius of 6 units. a Plot the graph of the circle.

Create a strategy Plot the centre of the circle then count the units equal to the radius in each direction (left, right, up, down), and draw a circular curve connecting them.

Apply the idea 9 8 7 6 5 4 3 2 1 −3−2 −1 −1 −2 −3

We can plot the point that represents the centre of the circle at (3, 3).

Now we count 6 units in all four directions to get the points (3, 9), (3, −3), (9, 3) and (−3, 3). Connect the points using a round curve passing through the four points to graph the circle. x 1 2 3 4 5 6 7 8 9

b Write an equation of the circle, in the form (x − h)2 + ( y − k)2 = r2.

Create a strategy Remember the equation (x − h)2 + ( y − k)2 = r2 represents a circle, where (h, k) are the coordinates of its centre and r is its radius.

Apply the idea We have the centre at (3, 3) and the radius is 6 units. (x − h)2 + ( y − k)2 = r2 2

(x − 3) + ( y − 3) = 6

Rewrite the equation Substitute h = 3, k = 3, r = 6

(x − 3) + ( y − 3) = 36 Evaluate 62 The equation of the circle is (x − 3)2 + ( y − 3)2 = 36.

598

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 The graph of x2 + y2 = 4 is translated 3 units left and 6 units down. Find the new equation.

Create a strategy By using the equation of circle (x − h)2 + ( y − k)2 = r2, substitute the translated coordinates into the equation.

Apply the idea A translation of 3 units left means h = −3, and 6 units down means k = −6. So by substituting these values, we have the new equation: (x + 3)2 + ( y + 6)2 = 4

Reﬂect and check 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

Here is the graph of x2 + y2 = 4 before and after the translations:

y x 1 2 3

Idea summary The general equation of a circle is given by:

(x − h)2 + ( y − k)2 = r2 (h, k) (x, y) r

Coordinates of the centre of the circle Coordinates of any point on the circle Radius of the circle

Circles with equation x2 + y2 = r2 can be translated vertically or horizontally. • • • •

When k is subtracted from y, the circle is translated up by k units. When k is added to y, the circle is translated down by k units. When h is subtracted from x, the circle is translated right by h units. When h is added to x, the circle is translated left by h units.

11.02 Translations of circles mathspace.co

599

Practice questions Choose your question path Explorer 1–3, 4ab, 5abcd, 6ab, 7, 8, 9abc, 10ab, 11, 14ab, 15, 16

Adventurer 1–3, 4bc, 5bcde, 6bc, 7, 8, 9bcd, 10bc, 11, 13, 14bc, 15–17

Trailblazer 1–3, 4cd, 5cdef, 6cd, 7, 8, 9, 10cd, 12, 13, 14cd, 15, 17–19

Understanding 1

Determine if each circle has been translated away from the origin: a

x2 + y2 = 25

x2 + ( y − 2)2 = 4

(x − 5)2 + ( y + 1)2 = 1

(x + 1)2 + y2 = 16

Find the centre and radius of each circle: a

1 −6 −5 −4 −3 −2 −1 −1

3 2

−2

−3

−4

−4 −3 −2 −1 −1

−5

−2

−6

−3

2 1 x −3 −2

−1

−1 −2 −3 −4

600

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

x 1

x 1 2 3 4 5

Using the graph of the circle: a

What are the coordinates of the centre?

What is the length of the diameter?

6 5 4 3 2 1 −6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6

Fluency 4

Ex 1

For each equation, identify the: i

Centre of the circle

Radius of the circle

(x − 5)2 + ( y + 6)2 = 17

(x − 5)2 + ( y − 5)2 = 162

(x − 0.4)2 + ( y + 3.8)2 = 2

For each equation: i

State the centre of the circle.

State the radius of the circle.

iii

Sketch the graph of the circle.

(x − 2)2 + ( y + 2)2 = 36

(x − 2)2 + ( y + 2)2 = 16

(x + 1)2 + ( y − 3)2 = 16

(x + 3)2 + ( y + 4)2 = 25

(x − 6)2 + y2 = 49

e 6

Ex 2

Write the equation of the circle given its centre and radius: a

Centre (7, −1); r = 3

Centre (−2, −3); r = 5

Centre (−4.3, 0); r = 6

Centre (−10, 2); r = 12

Write the equation.

For each circle: i

Sketch the graph.

A circle whose centre is at (1, 2) and has a radius of 4 units.

A circle whose centre is at (3, −2) and has a radius of 4 units.

11.02 Translations of circles mathspace.co

601

Find the equation of each circle: a

−3 −2 −1 −1 −2 −3 −4 −5

9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7

−5 −4 −3 −2 −1 −−1

3 2 1

−3 −2 −1 −1 −2 −3 −4 −5 −6 −7

5 4 3 2 1

11 10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7

x 1 2 3 4 5

−1 −1

x 1 2 3 4 5 6 7 8 9 10 11

Write the new equation after the circle with equation x2 + y2 = 25 is translated: a

10 units upwards.

10 units downwards.

10 units to the right.

10 units to the left and 4 units upwards.

Describe how each of the circles has been translated from the graph of x2 + y2 = 1: a

12 10

2 x

6 −4

4 2 −12 −10 −8 −6 −4 −2 −2

x 2

−4

602

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−2

2 −2 −4 −6

1 −3 −2 −1

−1

10 9 8 7 6 5 4 3 2 1

x 1

−2 −3 −4 −5 −6

−7 −6 −5 −4 −3 −2 −1 −1

−7

x 1 2 3 4

Reasoning 11

Consider the circle with centre (−8, −6) and radius 10. a

Write the equation of the circle.

Does the circle pass through (0, 0)? Justify your answer.

A circle has the equation:

y 6 4

Find the radius of the circle.

The graph of the circle has the same centre as the circle in part(a) but not the same radius.

2 −2 −2

Would the circle with the equation lie inside or outside the

x 2

10 12

−4 −6 −8

circle? Explain your reasoning. 13

A circle has the equation (x + 8)2 + ( y + 1)2 = 1. a

How many x-intercepts does the circle have?

Use translations to explain why there are no y-intercepts.

Problem-solving Ex 3

Find the new equation after the graph of the circle with equation x2 + y2 = 1: a

Scaled by a factor of 3, then translated right 5 units and down 2 units.

Scaled by a factor of 6, then translated left 1 unit and up 4 units.

Scaled by a factor of 4, then translated down 3 units.

Scaled by a factor of 5, then translated right half a unit and up

units.

11.02 Translations of circles mathspace.co

603

When expanded, the equation of a circle is x2 − 6x + y2 + 8y = 9. a

Complete the square for the expression x2 − 6x then fill in the blanks: (x − ⬚)2 − ⬚ + y2 + 8y = 9

Complete the square for the expression y2 + 8y then fill in the blanks: x2 − 6x + ( y + ⬚)2 − ⬚ = 9

Use your answers from parts (a) and (b) to rewrite the equation in the form (x − h)2 + ( y − k)2 = r2.

Find the centre and radius of the circle.

The circle with equation (x − 3)2 + ( y + 1)2 = 9 is inscribed (ﬁtted exactly) inside a square. What are the coordinates of the vertices of the square?

A wind turbine has 20 m long blades that are attached to a tower 30 m high. The distance from the origin to the base of the wind turbine is 25 m.

20 m

Find the equation of the circle represented by the path of the blades of the wind turbine.

30 m x 25 m

A soccer match is being televised. One of the cameras is mounted on a drone which is programmed to zoom in on the ball only when it is inside the centre circle. The drone uses a coordinate system to track the position of the ball, where the origin is at the bottom left corner of the ﬁeld and each unit corresponds to 1 m. a

604

The centre of the “centre circle” is in the exact centre of the ﬁeld. What are the coordinates of the circle’s centre?

Centre Circle 9.15 m

What is the radius of the centre circle?

State the equation for the centre circle.

The drone can be programmed 99 m to only check a rectangular area of the ﬁeld for whether the ball is within the centre circle or not. What is the smallest area that the drone could check for this task?

75 m

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Aarushi needs to use a laser cutter to make a cork component for her design project. The intersection of two different-sized circles inscribed on a piece of cork gives the shape that she wants. Using technology to draw a diagram, she creates two circles with the equations (x − 12)2 + ( y − 8)2 = 49 (x − 20)2 + ( y − 8)2 = 36 where each unit is 1 cm and (0, 0) is the bottom left corner of the rectangle. In the diagram, the rectangle represents the piece of cork, and the shaded area represents the desired component. a

What is the width of the component that Aarushi wants to make?

Find the least area of material that could be used to make Aarushi’s shape, considering these constraints: • The laser cutter will only accept rectangular pieces of material. • The height of the component is 10.17 cm. • There must be a gap of at least 1 cm between any cut and the edge of the piece.

If the bottom left corner of the rectangular piece in part (b) is now (0, 0), state the new equations of the two circles.

11.03 Optional: Transform circle equations After this lesson, you will be able to… • apply the distance formula to find the radius of a circle when given two points. • rewrite a given equation of a circle in its standard form using the process of completing the square.

The circle equation and distance formula Recall that all points on a circle are the same distance from the centre. The radius tells us the distance from the centre to any point on the circle.

11.03 Optional: Transform circle equations mathspace.co

605

Exploration Consider the circle with a radius of 13 units: y

Find the distance from the the point (4, 8) to the centre of the circle.

Find the distance from any point, (x, y), on the circle to the centre of the circle.

Rewrite your equation to represent any circle with a centre at (h, k).

How does your equation relate to the equation of a circle?

10 5 x −10 −5 −5

15 20

(9, −4)

−10 −15 −20

Recall the distance formula is

If the ﬁrst point (h, k) is the centre of a circle, the second point (x, y) is a point on the circle, then the distance between the two points is the radius, r.

Squaring both sides relates the distance between the points back to the equation of a circle. (x − h)2 + ( y − k)2 = r2

Example 1 The centre of a circle is (−1, −27) and the point (13, 21) is on the circle. a What is the radius of the circle?

Create a strategy Use the distance formula: distance

Apply the idea Substitute the coordinates

Evaluate the subtraction

Evaluate the indices

Evaluate the square root

606

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Write the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

Create a strategy Use the centre of a circle (−1, −27) and the value of radius found from part (a).

Apply the idea (x − (−1))2 + ( y − (−27))2 = 502 2

(x + 1) + ( y + 27) = 50

Substitute the coordinates and the value of radius Evaluate the subtraction

Example 2 The endpoints of the diameter of a circle are (−5, −3) and (−15, −15). a What are the coordinates of the centre of the circle?

Create a strategy Use the midpoint formula:

Apply the idea We have x1 = −5, y1 = −3, x2 = −15 and y2 = −15. Substitute x1, y1, x2, y2

Evaluate the numerators

Simplify

The centre of the circle is (−10, −9).

b What is the exact value of the radius of the circle?

Create a strategy Use the distance formula: distance

Apply the idea We can use a point on the circle (−5, −3) and the centre (−10, −9) in part (a). Substitute x2, x1, y2, y1

Evaluate the subtraction

Evaluate the indices

Evaluate the addition

11.03 Optional: Transform circle equations mathspace.co

607

Reﬂect and check We can also ﬁnd the radius by calculating half of the distance between the two endpoints of the diameter so we have x1 = −5, y1 = −3, x2 = −15 and y2 = −15. By using the distance formula, we have: Substitute x2, x1, y2, y1

Evaluate the numerator

Simplify the numerator

Evaluate

c Find the equation of the circle which has (−5, −3) and (−15, −15) as endpoints of its diameter.

Create a strategy Remember the equation (x − h)2 + ( y − k)2 = r2 represents a circle, where (h, k) are the coordinates of the circle’s centre and r is its radius.

Apply the idea In part (a), we found the centre to be at (−10, −9). In part (b), we determined the radius to be

Write the equation

Substitute h = −10, k = −9, r =

Evaluate

Idea summary The distance formula is

If the ﬁrst point (h, k) is the centre of a circle, the second point (x, y) is a point on the circle, then the distance between the two points is the radius r.

Squaring both sides relates the distance between the points back to the equation of a circle. (x − h)2 + ( y − k)2 = r2

608

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Transform circle equations Exploration The equation x2 + y2 − 12x − 16y + 19 = 0 represents a circle.

Write the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

Expand the equation you wrote to show that it is equivalent to the given equation.

−16−14−12−10 −8 −6 −4 −2 −2 −4 −6

Transform the given equation into the form (x − h)2 + ( y − k)2 = r2 by completing the square.

−10

x2 − 12x + ⬚ − ⬚ + y2 − 16y + ⬚ − ⬚ + 19 = 0

−18

y x 2 4

−8 −12 −14 −16

(x − ⬚)2 − ⬚ + ( y − ⬚)2 − ⬚ + 19 = 0

(x − ⬚)2 + ( y − ⬚)2 = ⬚

The equation of a circle can be expressed in the form (x − h)2 + ( y − k)2 = r2 An equation of a circle that has been expanded will be in the form x2 + y2 + ax + by + c = 0 To convert this back into the form (x − h)2 + ( y − k)2 = r2, we can complete the square. Before completing the square, it helps to rewrite the equation so that the x-terms are together, and the y-terms are together. The constant can be moved to the right-hand side of the equation as well: x2 + ax + y2 + by = −c After it is in this form, we can complete the square for both the x-terms and the y-terms.

Example 3 Consider the equation of a circle given by x2 + 4x + y2 + 6y − 3 = 0. a Rewrite the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

Create a strategy Complete the square on both the x-terms and the y-terms.

Apply the idea x2 + 4x + y2 + 6y − 3 = 0 2

Start with the equation 2

x + 4x + 2 + y + 6y + 3 − 3 = 2 + 3

(x + 2)2 + ( y + 3)2 − 3 = 22 + 32 2

(x + 2) + ( y + 3) − 3 + 3 = 2 + 3 + 3 2

(x + 2) + ( y + 3) = 16

Add 22 and 32 to both sides Write in factorised form Add 3 to both sides Evaluate

11.03 Optional: Transform circle equations mathspace.co

609

b What are the coordinates of the centre of this circle?

Create a strategy

Apply the idea 2

Equations of the form (x − h) + ( y − k) = r are circles with centre at (h, k).

In part (a), we have the equation (x + 2)2 + ( y + 3)2 = 16 which can be written as (x − (−2))2 + ( y − (−3))2 = 42 and it is now of the form (x − h)2 + ( y − k)2 = r2. So we have h = −2 and k = −3. The centre of the circle is (−2, −3).

c What is the radius of this circle?

Create a strategy

Apply the idea 2

Equations of the form (x − h) + ( y − k) = r are circles with radius of r.

In part (b) we rewrote the equation as (x + 2)2 + ( y +3)2 = 42. So we have r = 4, and therefore the radius of the circle is 4 units.

Example 4 A circle has its centre at (3, −2) and a radius of 4 units. a Plot the graph of the circle.

Create a strategy Plot the centre of the circle. Count 4 units in each direction from the centre: left, right, up, and down and draw a round curve passing through each these points.

Apply the idea 4 3 2 1 −2 −1 −1 −2 −3 −4 −5 −6

610

The centre of the circle is at the point (3, −2). So we first plot this point (3, −2).

x 1 2 3 4 5 6 7 8

Now we count 4 units in all four directions to get the points (3, 2), (7, −2), (3, −6) and (−1, −2). Connect the points using a round curve passing through the four points to draw the circle.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Write the equation of the circle in the form x2 + y2 + ax + bx + c = 0.

Create a strategy Use the form (x − h)2 + ( y − k)2 = r2 and expand each perfect square.

Apply the idea We are given that the centre of the circle is (3, −2) and the radius is 4. So h = 3, k = −2 and r = 4. (x − h)2 + ( y − k)2 = r2 2

Write the formula

(x − 3) + ( y − (−2)) = 4 2

Substitute h, k, and r

(x − 3) + ( y + 2) = 16 2

Simplify

Expand the perfect squares

Combine like terms

x − 6x + 9 + y + 4y + 4 = 16 2

x − 6x + y + 4y + 13 = 16 2

x − 6x + y + 4y + 13 − 16 = 16 – 16 2

Subtract 16 from both sides

x − 6x + y + 4y − 3 = 0

Evaluate

Write in general form

x + y − 6x + 4y – 3 = 0

Idea summary The equation of a circle can be expressed in the form (x − h)2 + ( y − k)2 = r2 An equation of a circle that has been expanded will be in the form x2 + y2 + ax + by + c = 0 To convert an expanded circle equation into the form (x − h) + ( y − k)2 = r2, we must complete the square for the x-terms and the y-terms.

Practice questions Choose your question path Explorer 1, 2ac, 3ac, 4a, 5ab, 6ab, 7abc, 8a, 9ab, 10, 11ac, 12a, 14ac, 16, 19ab, 21a

Adventurer 1, 2ab, 3bc, 4a, 5bc, 6bc, 7bcd, 8a, 9bc, 10, 11cd, 12ab, 14bc, 15–17, 19cd, 20b, 21b

Trailblazer 1, 2cd, 3bd, 4b, 5cd, 6cd, 7def, 8b, 9cd, 10, 11cef, 12cd, 13, 14def, 16–18, 19d, 20d, 21d, 22, 23

11.03 Optional: Transform circle equations mathspace.co

611

Understanding 1

The points A (2, 1) and B (6, 5) are plotted in the Cartesian plane. a

If you were to draw a grid on the plane, and each square in the grid had a side length of 1 unit, estimate the length of the line segment AB by counting the number of squares it passes through.

Find the length of the line segment AB using Pythagoras’ theorem.

(x − 6)2 + ( y − 4)2 = 9 2

(x + 3) + ( y − 2) = 9

(x − 3)2 + ( y + 2)2 = 9

(x − 3)2 + ( y − 4)2 = 16

x2 + 18x

x2 − 8x

x2 + 10x + 31

(x − 4)2 + ( y + 2)2 = 25

(x + 1.5)2 + ( y − 6)2 = 12.25

A (−1, 9) and B (−4, 1)

Fluency

Ex 1

612

1 2 3 4 5 6 7

x2 + 14x + 47

Find the distance between Point A and Point B rounded to two decimal places. a A (1, 4) and B (7, 12) b A (4, 2) and B (−8, −7) c

Find the centre and radius of each circle: a

Rewrite each quadratic expression in the form (x + b)2 + c by completing the square: a

−4 −3 −2 −1 −1 −2 −3 −4

For the circle with equation (x − 3)2 + ( y − 4)2 = 9, describe the transformation that has taken place to get the graph of each equation. a

7 6 5 4 3 2 1

Using the centre and another point for each circle: i

Find the radius of the circle.

Find the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

Centre at (2, 4), passing through the point (3, 3)

Centre at (−2, 4), passing through the point (3, 5)

Centre at (−2, −5), passing through the point (−1, 0)

Centre at (−2, 5), passing through the point (3, −7)

Using the centre and another point for each circle: i

Find the radius of the circle.

Find the equation of the circle in the form x2 + y2 + ax + by + c = 0.

Centre at (1, 2), passing through the point (4, 2)

Centre at (−4, −3), passing through the point (−4, 1)

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 2

Ex 3

Ex 4

Centre at (5, 7), passing through the point (1, 10)

Centre at (1, 2), passing through the point (0, 1)

Centre at (3, −1), passing through the point (2, −5)

Centre at (−2, 3), passing through the point (−4, 4)

For each circle: i

Find the coordinates of the centre.

Find the exact value of the radius.

iii

Write the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

A circle with endpoints of the diameter at (10, 6) and (−12, −14).

A circle with the smallest radius that contains the points (1, 2) and (−5, 4) within or on its boundary.

For each equation: i

Rewrite the equation of the circle in the form (x − h)2 + ( y − k)2 = r2.

Find the coordinates of the centre of the circle.

iii

Find the radius of the circle.

x2 + 4x + y2 + 10y − 7 = 0

x2 + y2 − 14x − 10y + 73 = 0

A circle has its centre at (4, −6) and a radius of 3 units. a

Plot the graph of the circle.

Write the equation of the circle in the form x2 + y2 + ax + by + c = 0.

For each equation: i

Rewrite the equation in the form (x − h)2 + ( y − k)2 = r2 by completing the square.

Sketch the graph of this circle.

x2 + y2 − 12x + 2y + 1 = 0

x2 + y2 − 6x + 2y + 1 = 0

x + y + 4x − 14y = 11

x2 + y2 − 12x − 18y = −81

y2 + 2y + 3 = 10x − x2 + 58

y2 + 6y + 6 = −4x − x2 + 42

For each equation: i

Rewrite the equation in the form (x − h)2 + ( y − k)2 = r2 by completing the square.

Sketch the graph of this circle.

2x2 + 2y2 + 8x − 12y + 20 = 0

c 13

x2 + y2 + 18x − 16y + 141 = 0

4x + 4y + 12x − 36y + 41 = 0

2x2 + 2y2 − 8x + 4y + 5 = 0

6x2 + 6y2 − 36x + 24y + 54 = 0

For each equation: i

Rewrite the equation in the form (x − h)2 + ( y − k)2 = r2 by completing the square.

Sketch the graph of this circle.

11.03 Optional: Transform circle equations mathspace.co

613

Describe how each circle has been transformed from the graph of x2 + y2 = 1: a

x2 + y2 − 8y = 0

x2 + y2 − 10x − 75 = 0

x2 + y2 − 8x + 14y + 29 = 0

x2 + y2 + 1.5x − 0.4y − 0.3975 = 0

x + y + 4x + 8y + 11 = 0

Reasoning 15

Use the distance formula to explain why the equation (x − a)2 + ( y − b)2 = r2 represents a circle with its centre at (a, b) and radius r. A circle has a diameter with endpoints A (−1, 1) and B (9, −7). a

Determine the equation of the circle.

Show that the point P (8, 3) is outside of the circle.

A circle with centre (−7, 4) passes through the point (1, 4). Explain how to ﬁnd the radius without using the distance formula.

Is it possible to ﬁnd the centre and radius of a circle in the form x2 + y2 + ax + by + c = 0 without completing the square? Explain your answer.

Problem-solving 19

For each circle: i

Find the radius.

Find the circumference.

iii

Find the area.

x2 + y2 + 10x + 8y + 4 = 0

x2 + y2 − 10x + 8y + 40 = 0

x2 + y2 + 4x − 10y − 20 = 0

3x2 + 3y2 + 12x − 6y − 9 = 0

5x2 + 5y2 + 20x − 30y − 20 = 0

x + y − 8x + 6y − 4 = 0

For each circle: i

Find the radius.

Find the circumference.

iii

Find the area.

2x2 + 2y2 − 8x + 4y − 4 = 0

c 21

4x + 4y − 16x + 24y + 4 = 0

Does the point lie inside, outside, or on the given circle? a

Point: (0, 0) 2

Point: (−1, −4)

Equation: x + y − 2x + 4y + 3 = 0 Equation: x2 + y2 − 2x + 6y + 5 = 0 c

Point: Equation:

614

Point:

Equation:

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Darnell shines a torch at a wall which lights up a circular region with a diameter of 4 metres. The centre of the light is positioned 3 metres above the ground, and 5 metres horizontally from the left side of the wall. a

Let the bottom left corner of the wall be the origin. Determine the equation of the circle which describes the edge of the lighted area in the form x2 + y2 + ax + by + c = 0.

Yvonne has a height of 1.66 metres and is standing against the wall, 5 metres from the left side. Will any part of Yvonne be in the lighted area?

Kayoko is standing against the wall, 6 metres from the left side of the wall. Determine the greatest height that Kayoko can be without being in the lighted area. Round your answer to the nearest centimeter.

A circle passes through the points (0,0), (4,0) and (0,7). Find the equation of the circle and state the radius, and centre.

Chapter 11 review 1

For the equation x2 + y2 = 49: a

Identify the radius of the circle.

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

Sketch the graph of the circle.

Write the equation of each circle shown: a

y 8

T F 6 4

x 2 4 6 8 10

−8 −6 −4 −2 −2

A R

x 6

−4

−6 −8

For circle with centre (0, 0) that passes through the point given: i

Find the radius of the circle.

(6, 0)

(5, −12)

Find the equation of the circle.

(4, −2)

Determine whether the following points are inside, outside, or on the circle with equation x2 + y2 = 100. a

(7, 2)

(−8, 6)

(9, −5)

(−5, −5)

A circle with its centre at the origin has an equation of x2 + y2 = 64. Find the area of the part of the circle that is above the x-axis.

Write the equation of the circle given its centre and radius: a

Centre: (4, 9) ; r = 1

Centre: Chapter 11 review mathspace.co

615

For each equation: i

State the the centre of the circle.

iii

Sketch the graph of the circle.

(x − 3)2 + (y + 1)2 = 25

616

State the radius of the circle.

x2 + (y − 4)2 = 49

For each graph: i

Describe how the graph has been translated from x2 + y2 = 16.

Write the equation of the circle.

9 8 7 6 5 4 3 2 1

2 1 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

y x 1 2 3 4 5 6 7 8

T F

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1

A R

Find the equation after the graph of x2 + y2 = 1 is: a

Scaled by a factor of 3, then translated left 1 unit and down 7 units.

Scaled by a factor of 9, then translated right 5 units. 2

D 2

For the circle (x − 8) + (y + 6) = 100: a

What are the coordinates of the x-intercepts?

What are the coordinates of the y-intercepts?

(Optional) Using the given centre and another point for each circle: i

Find the exact radius of the circle.

Find the equation of the circle in the form x2 + y2 + ax + by + c = 0.

Centre at (−4, 4), passing through the point (1, 9)

Centre at (8, 10), passing through the point (6, 7)

(Optional) The endpoints of the diameter of a circle are (−9, 9) and (−1, 3). a

Find the coordinates of the centre of the circle.

Find the radius of the circle.

Write the equation of the circle in the form (x − h)2 + (y − k)2 = r2.

(Optional) For each equation: i

Rewrite the equation in the form (x − h)2 + (y − k)2 = r2.

Find the coordinates of the centre of the circle.

iii

Find the radius of the circle.

x2 + y2 + 14x + 24 = 0

3x2 + 3y2 − 12x + 6y − 33 = 0

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) Describe how each of the circles has been transformed from the graph of x2 + y2 = 1. a

x2 + y2 + 16x − 6y + 24 = 0

x2 + y2 − 0.5x − 1.5y − 0.375 = 0

(Optional) For each circle: i

Rewrite the equation in the form (x − h)2 + (y − k)2 = r2 by completing the square.

Sketch the graph of this circle.

x2 + y2 − 4x + 14y + 17 = 0

(Optional) Does the point lie inside, outside, or on the given circle? a

Point: (−5, 2)

Equation: x2 + y2 + 5x − 14y + 25 = 0

Point: (3, −2)

Equation: (x − 1)2 + (y + 3)2 = 25

Point: (1, 1.5)

Equation: (x − 2.5)2 + (y − 0.4)2 = 1.6

Point:

Equation:

T F

(Optional) A circle passes through the points (0, 0), (a, 0) and (0, b). Find the equation of the circle and state the radius, and centre.

A R

Chapter 11 review mathspace.co

617

12 Hyperbolas Big ideas Understanding and analysing the properties and transformations of hyperbolas provide insight into inverse relationships, allowing for the exploration and prediction of real-world scenarios involving direct and indirect proportions.

Curriculum Outcomes VC2M10A11 Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital tools as appropriate VC2M10AA05 Describe, interpret, and sketch parabolas, hyperbolas, circles and exponential functions and their transformations

Chapter outline 12.01 12.02 12.03

Direct and indirect proportion Hyperbolas Optional: Transformations of hyperbolas Chapter 12 review

620 631 643 656

Hyperbolas are used in the design of reflector telescopes for sharp, focused images of space.

12.01 Direct and indirect proportion After this lesson, you will be able to… • identify the relationship as a direct or indirect proportion. • calculate the constant of proportionality in a direct proportion. • express an inverse relationship mathematically and interpret its graph. • apply the knowledge of direct and indirect proportions to solve related problems.

Direct proportion Two quantities are said to be proportional (or have direct variation) if they increase or decrease together at the same rate. For example, if you earn $18 per hour, your income is directly proportional to the number of hours worked: income = $18 × hours worked, or y = 18x where y is the income and x is the number of hours worked. The constant of proportionality is the value that relates the two amounts. In that example, the constant of proportionality is 18. Notice that this constant is the gradient of the line. The y-intercept (c value) of a proportional relationship is zero. The general equation for variables that are directly proportional is:

y = kx k

is the constant of proportionality

We can write “y is directly proportional to x” as y ∝ x. The constant of proportionality is also called the unit rate. While it can be any non-zero number, positive or negative, we generally only consider relationships with a positive value of k.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 12.01 to answer these questions. 1.

What do you notice about position of the points in relation to the line?

Identify another Euro value and its corresponding British pound value.

What is the constant of proportionality? How did you find it?

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Example 1 Consider the values in each table. Which of them could represent a directly proportional relationship between x and y? Table A: x y

1 2

Table B: 2 8

3 18

x y

4 32

1 2

2 4

3 6

4 8

Create a strategy Determine which table shows that the y-values are a constant multiple of the x-values.

Apply the idea In table A, the ﬁrst y-value is equal to the corresponding x-value multiplied by 2: 1 × 2 = 2. The second y-value is equal to the corresponding x-value multiplied by 4: 2 × 4 = 8. The multiplier has changed from 2 to 4. In table B, the ﬁrst y-value is equal to the corresponding x-value multiplied by 2: 1 × 2 = 2. The second y-value is equal to the corresponding x-value multiplied by 2: 2 × 2 = 4. The multiplier has stayed the same. Table B could represent a directly proportional relationship between x and y.

Example 2 Consider the equation P = 90t. a State the constant of proportionality.

Create a strategy

Apply the idea

For equations in the form y = kx, k is the constant of proportionality.

The constant of proportionality in P = 90t is 90.

b Find the value of P when t = 2.

Create a strategy Substitute the given value into the equation.

Apply the idea P = 90 × 2 = 180

Substitute t = 2 Evaluate

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Example 3 Oliver is making cups of fruit smoothie. The amount of bananas and strawberries he uses is shown in the proportion table.

Strawberries Bananas

8 5.5

16 11

24 16.5

32 22

40 27.5

a Sketch the graph of this proportional relationship.

Create a strategy Use two pair of values from the table as coordinates of points in the graph.

Apply the idea 11 10 9 8 7 6 5 4 3 2 1

The points (8, 5.5) and (16, 11) have been plotted, and a line drawn through them.

Bananas

Strawberries 10 12 14

Reﬂect and check Conﬁrm that the other points (24, 16.5), (32, 22), and (40, 27.5) lie on the same line you have created. b What is the unit rate of this relationship?

Create a strategy The unit rate is the gradient or constant of proportionality.

Apply the idea 11 10 9 8 7 6 5 4 3 2 1

5.5 units

8 units

622

We can see in the graph that rise = 5.5 and run = 8.

Bananas

Strawberries 10 12 14

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

= 0.6875 bananas per strawberry

Reﬂect and check We can also substitute the second pair of values to confirm the unit rate.

= 0.6875 bananas per strawberry

c Select ALL the statements that describe the proportional relationship. A

For every 5.5 bananas Oliver uses, he adds 8 strawberries.

The unit rate of bananas with respect to strawberries is

For every 8 bananas, William uses 5.5 strawberries.

The unit rate of bananas with respect to strawberries is

. .

Create a strategy Use the table given in the question, and the unit rate from part (b).

Apply the idea From the table, when William uses 5.5 bananas, he adds 8 strawberries. So option A is correct. The unit rate is

bananas per strawberry. So option B is correct.

Reﬂect and check 11 10 9 8 7 6 5 4 3 2 1

We can also see in the graph that for every 5.5 bananas Oliver uses, he increases or adds 8 strawberries.

Bananas

5.5 units

This also shows why option A is correct.

8 units

Strawberries 10 12 14

Idea summary A directly proportional relationship is a linear relationship with a y-intercept of zero:

y = kx k

is the constant of proportionality

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Indirect proportion Some quantities have what we call an inverse relationship. As one of the quantities gets closer to zero, the other one gets further from zero, and vice versa. The relationship between speed and time is inversely proportional. For example, if a truck driver needs to drive 1000 km to deliver a load, the faster the driver travels, the less time it will take to cover this distance. Speed (km/h) Time (hours)

60 16.7

70 14.3

80 12.5

90 11.1

100 10

110 9.1

120 8.3

Plotting these values, the graph of the curve is given by: Each point on the curve corresponds to a different combination of speed and time, and all the points on the curve satisfy the equation

Time (hours) 35 30 T=

1000 S

, or TS = 1000, where T is the time

taken and S is the speed travelled.

(60, 16.7)

(100, 10)

10 5

Speed (km/h) 20 40 60 80 100 120 140 160

We can see from both equations that as average speed increases, the time taken will decrease. This is what we call inverse proportion, also known as inverse variation. Mathematically, we write this as y ∝

to say “y is inversely proportional to x”.

Two quantities x and y that are inversely proportional have an equation of the form: y = or xy = k or x = where k ≠ 0. When we graph this relationship, it will be non-linear.

Example 4 The equation y =

represents an inverse relationship between x and y.

a Complete the table of values: x y

−5

−4

−3

−2

−1

Create a strategy Substitute the values of x into y =

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Apply the idea

Reﬂect and check

For x = −1, we get y = 6.

We could check our values by sketching a graph of the relationship using technology and see

For x = −2, we get y = 3. Similarly, substituting the other values of x, the complete table of values is given by: −5

−4

−3

−2

−1

that y =

looks like this: 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

b Is y = A

x 1

2 3 4

increasing or decreasing when x < 0?

Increasing

Decreasing

Create a strategy

Apply the idea

Consider the table of values from part (a). Look what happens to y as the values of x increase.

Based on the table of values, y values increase as the values of x increase. The correct answer is Option A.

c Describe the rate of change of the function when x < 0. A

As x increases, y increases at a faster and faster rate.

As x increases, y increases at a slower and slower rate.

As x increases, y increases at a constant rate.

Create a strategy

Apply the idea

Look at how much the value of y changes by each time.

Let’s consider some values from the table: x y

−4

−3

−2

Based on the table, y increases by from x = −4 to x = −3. Then y increases by 1 from x = −3 to x = −2. This means that as x increases, y increases at a faster and faster rate, so the correct answer is Option A.

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Idea summary Two quantities, x and y, that are inversely proportional have an equation of the form: y=

or xy = k or x = ,

where k ≠ 0.

Practice questions Choose your question path Explorer 1−3, 4a-f, 5−7, 8ac, 9, 11, 13b, 15, 16ab, 17, 18ab, 19, 21, 26, 27, 29, 30

Adventurer 1−4, 5, 6, 8, 10−12, 13ab, 14, 15, 16cd, 17, 18, 21, 22, 24, 25, 27, 28

Trailblazer 1−4, 5, 6, 8bd, 10−15, 16cd, 17, 18, 20, 22, 23, 24, 25, 28, 31

Understanding 1

Is each statement true or false? a

In a directly proportional relationship, the constant of proportionality represents the gradient of the line.

The graph of any directly proportional relationship will go through the point (0, 0).

The graph of a directly proportional relationship can be curved.

A directly proportional relationship can be increasing or decreasing.

The equation y =

For an inversely proportional relationship, as x gets closer to zero, y gets further from zero.

represents a direct proportionality between two quantities, y and x.

Which graph indicate that y is directly proportional to x? A

B 4

4 3

1 −4 −3 −2 −1

−1

−2

626

x 1

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

x 1

−1

−2

−3

−4

Determine whether each of the following is an example of a direct variation or an inverse variation: a

The variation relating the distance between two locations on a map and the actual distance between the two locations.

The variation relating the number of workers hired to build a house and the time required to build the house.

Does each relationship between x and y show direct proportion, inverse proportion, or neither?: a

y = 6x

y = −x

x2 = y

y = 5x

y = 2x2 − 7x − 4

xy = −7

x = −2y

Fluency 5

Ex 1

Given that y = 0.4x, ﬁnd the missing values in the table.

x y

10 8

Does each table represent a direct proportional relationship between x and y? a

x 1 3 5 7 y 20 16 12 8

x y

1 16

5 12

6 8

20 4

x 1 2 3 4 y 5 20 45 80

x y

1 5

2 10

3 15

4 20

q is directly proportional to r, and q = 10 when r = 20. a

Graph the relationship between q and r.

Find the gradient of the line.

Express q in terms of r.

Find the value of q when r = 30.

Find the value of r when q = 22. 12.01 Direct and indirect proportion mathspace.co

627

Find the equation relating the directly proportional relationship using variables in each table. a

Ex 2

628

x 0 1 2 3 y 0 −2 −4 −6

15 1

Given A = 70w: a

What is the constant of proportionality?

Find the value of A when w = 4.

Given M =

What is the constant of proportionality?

Find the value of M when n = 50.

William is making rice. The amount of rice and water he uses is in the table. Rice (g) Water (mL)

Ex 3

a 0 1 2 3 b 0 9 18 27

100 210

110 231

120 252

130 273

140 294

Sketch the graph of the proportional relationship.

Find the unit rate of the relationship between rice and water.

Write two statements that describe the proportional relationship.

The table shows the proportional relationship between hours spent ﬁshing and the number of ﬁsh caught: a

Sketch the graph of the proportional relationship.

Find the unit rate of the relationship.

By considering parts (a) and (b), make a conclusion about the number of ﬁsh caught.

Hours Fish caught

4 9

8 18

12 27

For each table: i

Sketch the graph.

x 5 10 15 y 9.5 19 28.5

x 15 25 35 45 y 36 60 84 108

ii 20 38

If s varies directly with t, and s =

25 47.5 55 132

Find the unit rate of the relationship.

x y

10 7.5

20 15

x y

1 1.25

3 3.75

when t = 6:

Find the constant of proportionality, k.

Write the equation for s in terms of t.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

30 22.5 5 6.25

40 30

50 37.5

7 8.75

9 11.25

If y is inversely proportional to x, and y = 20 when x = 10: a

Find the constant of variation, k.

Express y in terms of x.

Find the value of y when x = 5.

The equation xy = 7 can be described as a function where y varies inversely with x and has a constant of proportionality, k = 7. Find the constant of proportionality, k, for the following:

Ex 4

y = 4 when x = 3

when x = −2

y = −2 when x = 3

when x = −3

Does each table represent an inversely proportional relationship between x and y? a

x 1 2 3 4 y 3 1.5 1 0.75

x y

1 36

2 18

3 12

4 9

x 1 5 6 10 y 3 75 108 300

x y

1 4

2 5

3 6

4 7

represents an inverse relationship between x and y.

The equation a

Complete the table of values:

Describe the rate of change of the function when x < 0.

x y

increasing or decreasing when x < 0?

−5

−4

−3

−2

−1

Reasoning 19

Two quantities are directly proportional. If one of the quantities increases by a factor of 7, what happens to the other quantity?

, where f is the centripetal force of an object of mass m moving along a circle of

radius r at velocity v, describe the relationship between the centripetal force and the mass. 21

Jimmy and Ben are both saving money for a vacation. Jimmy can represent his savings over , where x represents the number of days and y represents the

time using the equation

savings. Ben saves $7 every 4 days. a

Complete the table showing Ben’s savings over time. Days Amount saved

8 14

12 21

Graph the amount each of them have saved over time.

Explain how to determine who is saving more per day.

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A real estate agent advertised a house for sale online and in the newspaper. The agent received 4 online enquiries each day and 2 enquiries each day through the newspaper advertisement. Let y represent the number of online enquiries received x days after initially advertising the house. Write the equation for y based on x.

The line representing the number of online enquiries has been graphed. On the same set of axes, graph the line representing the number of newspaper enquiries after x days.

Online

20 15 10 5

Why do both lines pass through the origin?

35 30

Why is the line representing the number of online enquiries steeper than the line representing the number of newspaper enquiries?

y 40

x 2

Boyle’s law describes the relationship between pressure and volume of a gas of ﬁxed mass under constant temperature. The pressure for a particular gas can be found using

where P has units kg/cm2 and V has units cm3. a

What is the pressure if the volume is 1 cm3?

What happens to the pressure as the volume increases? Explain.

Can the pressure ever get to 0 kg/cm2? Explain.

Problem-solving 24

Write an equation involving constant k for the revenue of a company, r, that varies directly with the number of sales, n.

The values in the table represent the relationship between m and p, where m is inversely proportional to p. Find the values of a and b. p m

4 63

6 b

7 36

a 28

Harry and Carl love reading. Harry reads 16 books every 12 weeks. Carl has kept a table of his reading habits: Number of weeks Number of books read a

12 20

630

36 60

48 80

Complete this table for Harry: Number of weeks Number of books read

24 40 12 16

48 32

Who reads the fastest?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

60 80

Derek and Yuri are building robot kits. Derek can make 2 robots every 3 hours. Yuri can make 3 robots every 6 hours. a Sketch the graphs of both proportional relationships on the same Cartesian plane. b Who is the faster builder?

The time, t, taken by a typist to type up a document is inversely proportional to his typing speed, s. That is, the quicker the typing speed, the less time it will take. It takes a typist 20 minutes to type a particular document, typing at a speed of 61 words per minute. a Find the constant of variation, k. b How long, in minutes, will it take a typist with a typing speed of 30.5 words per minute to type up the document?

The height, h, of a regular tetrahedron varies directly with its side length, s. A particular tetrahedron with a side length of 3 cm has a height of 2.45 cm. a Find the constant of proportionality, k, rounded to two decimal places. b Express h in terms of s. c Find the height of a different tetrahedron with a side length of 2 cm.

The amount A (in grams) of a substance dissolved in water is directly proportional to the capacity C (in litres) of water used. When 5 litres of water is used, 15 grams of the substance is dissolved. a Find an equation relating the amount and capacity. b How much of the substance will be dissolved in 7 litres of water?

A particular shelter has cylindrical pillars that can hold a maximum load, L, that varies directly as the fourth power of the diameter, d, and inversely as the square of the height, h. If 6 m tall pillars with diameters of 1 m can support a maximum load of 18 metric tons, ﬁnd the maximum load that can be supported by a pillar 2 m tall and

m in diameter.

12.02 Hyperbolas After this lesson, you will be able to… • define what a hyperbola is and how it represents an inverse relationship. • identify key features of hyperbolas and use them to graph hyperbolas. • determine the domain and range of hyperbolas, and understand the restrictions on these. • apply the knowledge of hyperbolas to real-world situations.

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Hyperbolas When the inverse relationship call a hyperbola.

is graphed on the Cartesian plane it produces a graph that we

Exploration Here are some hyperbolas with equations of the form 1.

What key features do the graphs have in common?

What key features are different?

What values of k result in a graph that appears in quadrants 1 and 3?

What values of k result in a graph that appears in quadrants 2 and 4?

6 y −3 y= x

−1 x

y= −6

Ignoring the sign, what happens when the size of the constant of variation increases?

5 x

2 x x

−4

−2

Horizontal asymptote

2 −2 −4

Vertical asymptote

−6

This table shows the relationship x y

−4 −0.25

−2 −0.5

−1 −1

−0.5 −2

: −0.25 −4

0 undefined

0.25 4

0.5 2

Notice that for

1 1

2 0.5

4 0.25

This shows the points plotted on a Cartesian plane. 6 5 4 3 2 1 −6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6

If x is positive, y is positive. y=

1 x

1 2 3 4 5 6

If x is negative, y is negative. x

As x gets further from zero (in either direction), y gets closer to zero. As the value of one variable increases, the value of the other will decrease. The graph is split in two pieces and not deﬁned at x=0

Formally we deﬁne that an inversely proportional relationship is modelled by:

where x and y are any variable, and k is the constant of variation.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

has two asymptotes, a vertical asymptote at x = 0 and a horizontal asymptote

The function at y = 0.

A vertical asymptote is a vertical line that the graph of a function approaches, but does not cross, as the function values head towards positive or negative inﬁnity.

Vertical asymptote

Notice that , is undeﬁned if x = 0. This means its domain is all real values of x, x ≠ 0, and that there is vertical asymptote at x = 0. A horizontal asymptote is a horizontal line that the graph of a function approaches, but does not cross, as the domain values head towards positive or negative inﬁnity.

Horizontal asymptote

The end behaviour is y → 0 as x → −∞ and x → ∞. This means that there is a horizontal asymptote at y = 0. For

: 6 y −3 x

−1 x

y= y= −6

−4

5 x

2 x x

−2

Horizontal asymptote

2 −2 −4

Vertical asymptote

Key features: • Each hyperbola has two parts and they lie in opposite quadrants. • Vertical asymptote at x = 0 • Horizontal asymptote at y = 0 • The larger the size of the constant of variation, the further the graph is from the axes • The domain is all real values of x, x ≠ 0 • The range is all real values of y, y ≠ 0

−6

Graphs of

are symmetrical about the lines y = x and y = −x, and they have rotational

symmetry about the origin.

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Example 1 Consider the equation

a What is the value of the constant of variation?

Create a strategy

Apply the idea

We want to write this in the form the value of k.

and ﬁnd

We can write this as:

, so

b Complete the following table of values: x y

−3

−2

−1

Create a strategy Substitute the values from the table into the equation.

Apply the idea For x = −3: Write the equation Substitute x = −3

Evaluate

Similarly, by substituting the remaining x-values into x

−3

−2

−1

, we get:

c Sketch the graph.

Create a strategy Plot the points in the table of values and draw the curve passing through each plotted point.

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Apply the idea 4

The ordered pairs of points to be plotted on the coordinate plane are

, and

2 1 −4 −3 −2 −1

−1

x 1

, which are plotted on the graph. must pass

The curve of the hyperbola

through each of the plotted points and approaches the horizontal asymptote y = 0 and the horizontal asymptote x = 0.

−2 −3 −4

Reﬂect and check Notice that the graph is quite close to the axes since the constant of variation is quite small in size. Also, the graph is in quadrants 2 and 4 since the constant of variation is negative. d In which quadrants does the graph lie?

Apply the idea From the graph in part (b), we see that the hyperbola lies in quadrant 2 and quadrant 4.

Example 2 Consider the graph of the function f (x) = .

3 2 1 −4 −3 −2 −1

a For positive values of x, as x increases y approaches what value? A

−∞

∞

−1

x 1

−2 −3 −4

Create a strategy Substitute a large value of x into the equation.

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Apply the idea Let’s say we use x = 1000 so we have: Write the function

Subsitute x = 1000

Evaluate

If we keep increasing the value of x, the denominator of the function causing the value of y to approach 0. So, the answer is option A.

becomes smaller,

b As x takes small positive values approaching 0, what value does f (x) approach? A

∞

−∞

Create a strategy Substitute a small value of x into the equation.

Apply the idea Let’s say we use x = 0.001 so we have: Write the function

Subsitute x = 0.001

Evaluate

As x approaches 0 from the positive side, the denominator of the function large, causing the value of f (x) to approach ∞. So, the answer is option A.

becomes very

c What are the values that x and y cannot take?

Create a strategy Examine the function and its graph to identify any restrictions on the values of x and y.

Apply the idea The equation of the function is

. Since a division by zero is undeﬁned, x cannot take the value 0.

Additionally, from the graph, we can observe that y never touches the x-axis, that is, y can never be equal to 0. Therefore, the values of x and y cannot be equal to 0. d The graph is symmetrical across two lines of symmetry. State the equations of these two lines.

Create a strategy Observe the graph and identify the lines along which the graph’s symmetry occurs.

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Apply the idea is symmetric with respect to the lines y = x and y = −x. These

The graph of the function

lines represent the equations of the lines of symmetry.

Idea summary A reciprocal function that models inverse variation is the hyperbola,

Key features: • • • • • •

Each hyperbola has two parts and they lie in opposite quadrants. Vertical asymptote at x = 0 Horizontal asymptote at y = 0 The larger the size of the constant of variation, the further the graph is from the axes The domain is all real values of x, x ≠ 0 The range is all real values of y, y ≠ 0

Practice questions Choose your question path Explorer 1−4, 7, 8, 10ac, 11, 12ab, 14, 15, 17, 20, 21

Adventurer 1−5, 8, 9, 10bc, 11, 12bd, 14, 15, 17-20, 22, 23

Trailblazer 1−6, 8, 10cd, 11, 12b-d, 13, 14, 16, 18-20, 22, 23

Understanding 1

Which is a graph of the hyperbola in the form A

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

? 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

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−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

For the given hyperbola, complete the statement:

5 4 3 2 1

“Every point (x, y) on the hyperbola is such that xy = ⬚.”

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

For functions of the form

1 2 3 4 5

, what is the equation of the vertical asymptote?

Fluency 4

Consider the graph of a

638

. 4

What are the equations of the horizontal and vertical asymptotes?

What are the domain and range?

In which quadrants does the graph lies?

Identify the intervals where the function is increasing or decreasing.

Describe the end behaviour as x → ∞ and as x → −∞.

For the function

3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

For what value of x is the function undeﬁned?

As x approaches 0 from the positive side, what does y approach?

What y-value does the function get inﬁnitely close to, but never reach?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

Which equations are equivalent to x = 8y

Ex 1

represents an inverse relationship between x and y.

The equation

xy = 8

iii

y = 8x

Can x or y be equal to 0?

When x = 2, what is the value of y?

If x is a positive value, must the corresponding y value be positive or negative?

If x is a negative value, must the corresponding y value be positive or negative?

In which quadrants does the graph of

lie?

For each function: i

Complete the table of values. x

−4

−2

−1

Ex 2

Plot the points from the table of values on a Cartesian plane and sketch the graph of the function.

iii

In which quadrants does the graph lie?

Without graphing, determine the behaviour of each of the following hyperbolas: i

As x tends to 0 from the right (x → 0+)

As x tends to 0 from the left (x → 0−)

iii

As x tends to positive inﬁnity (x → ∞)

As x tends to negative inﬁnity (x → −∞)

Consider the functions

and

b .

For

, when x = 1, what is the y-value?

For

, when x = 1, what is the y-value?

Which graph lies further away from the axes?

For hyperbolas of the form

, what happens to the graph as the value of k increases?

Sketch the graph, label the asymptotes and two points on the curve. a

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The graphs of f (x) = , g(x) = , and h(x) =

are shown. 4

Match each graph with its equation.

3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

Find the equation of each hyperbola. a

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−2

−3

−4

1 −4 −3 −2 −1

−1

−2

640

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1

−1

−2

Delano is sharing a jar of jelly beans evenly between his Jelly beans friends. The graph shows the relationship between the 50 45 number of people receiving jelly beans and the number 40 of jelly beans they receive. Determine whether or not there is an inverse variation between the number of people and the number of jelly beans they receive. If so, ﬁnd the constant of variation.

35 30 25 20 15 10 5

People 5 10 15 20 25 30 35 40 45 50

Reasoning 14

Would a function of the form xy = k be a reasonable model for the following relationships? Explain why or why not. a

The relationship between the number of people working on a job and how long it will take to complete the job.

The relationship between the number of sales and the amount of revenue.

The relationship between the minutes of cycling and the amount of calories burned.

Ursula wants to sketch the graph of y = , but knows that it will look similar to many other hyperbolas. What can she do to the graph to show that it is the hyperbola y = , rather than any other hyperbola of the form

Consider the hyperbola. a

Every point (x, y) on the hyperbola is such that xy = k. What is the value of k?

When x increases, does y increase or decrease?

If the graph was reﬂected over the x-axis, what would the new equation be? Explain.

10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

For f (x) = . a

Sketch a graph of the function on a number plane.

Find an expression for f (−x).

How could you rotate or reﬂect the graph and have it look the same?

12.02 Hyperbolas mathspace.co

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For the graph of y = : a

Show that the line y = x is an axis of symmetry.

The graph has another axis of symmetry. What is the equation of the other one?

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Problem-solving 19

Ella is driving down the highway to her home. To drive safely, the speed limit is no more than 80 km/h and no less than 60 km/h. Ella is 30 kilometres from home and has already been driving for 0.4 hours. a

The relationship between speed, distance, and time is:

Write an equation relating y, the total time of Ella’s drive in hours, and x, Ella’s driving speed on the highway.

State the domain of the function using interval notation.

State the range of the function using interval notation.

Determine the number of minutes Ella’s drive home will be if she drives at 68 km/h on the highway.

The relationship between the current, C, (in amperes) and resistance, R, (in ohms) in an electrical circuit is given by:

where the voltage provided to the circuit is 240 V.

Graph this function for 0 ≤ R ≤ 50.

What happens to the current as the resistance increases?

The rent, electricity, telephone bill and other expenses for a ﬂat costs a total of $490 per week. These expenses are shared equally between the tenants of the ﬂat. a

How much will each occupant pay in dollars if the ﬂat is shared by two people?

Let the number of occupants be x, and the cost paid by each occupant be y.

Write a formula that relates the two pronumerals. c

Use this equation to ﬁll in the following table of values. x y

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Sketch the curve from your equation.

What type of relationship exists between x and y?

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Consider the hyperbola that has been graphed. Points A (4, 2), B(4, −2), C (−4, k) and D(−4, 2) form the vertices of a rectangle. a

Find the value of k.

Find the area of the rectangle with vertices ABCD.

y 6 4 D

x −8 −6 −4 −2 −2

−4 −6

The time it takes a commuter to travel 100 km depends on how fast they are going. We can where S is the speed in km/h and t is the time taken in

write this using the equation hours. a

Graph the relationship

What is the time taken if the speed travelled is 10 km/h?

What is the time taken if the speed travelled is 50 km/h?

If we want the travel time to decrease, what must happen to the speed of travel?

12.03 Optional: Transformations of hyperbolas After this lesson, you will be able to… • explore the effects of parameters on the graph of the hyperbola

• identify the transformations that result from changes in the parameters. • determine the vertical and horizontal asymptotes of transformed hyperbolas. • apply the understanding of transformations to solve problems involving hyperbolas.

Transformations of hyperbolas We have seen that the graph of

is a hyperbola which has a vertical asymptote at x = 0 and a

horizontal asymptote at y = 0. The product of the coordinates of any point on the hyperbola is the constant of variation, that is xy = k.

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 12.03 to answer these questions. 1.

How does k change the graph?

How does b change the graph?

How does c change the graph?

How can we find the vertical asymptote of

How can we find the horizontal asymptote of

The hyperbola

has been vertically stretched by a factor of k. 4

y 4 y= x

3 2

1 −4 −3 −2 −1

−1

−2

1 x

This graph shows

stretched vertically by

a factor of 4 to get

. To get

hyperbola x

y=−

2 x

−3

of 2 and reﬂected across the x-axis. :

• If k > 1, it is stretched vertically by a factor of k. • If k < −1, it is stretched vertically by a factor of k and reﬂected across the x-axis.

has been horizontally translated b units from

The hyperbola 1 x+2

1 −4 −3 −2 −1

−1

−2

This graph shows by 2 to get

3 2

1 x

In general, for x

1 x−3

translated horizontally left , and right by 3 to get :

• If b > 0, it is translated b units to the right. • If b < 0, it is translated b units to the left.

−3 −4

The hyperbola

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, the basic

is stretched vertically by a factor

In general, for

−4

has been vertically translated up by c units from

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

This graph shows 1 y= +2 x

3 2

1 −4 −3 −2 −1

−1

−2

, and down by 3 to get

2 to get In general, for

1 x

translated vertically up by

x 3

• If c > 0, it is translated up c units. • If c < 0, it is translated down c units.

1 y= −3 x

−3 −4

A hyperbola in the form

• If k > 1, is stretched vertically (away from the asymptotes). • If k < −1, is stretched vertically and reﬂected across the x-axis. • Has a vertical asymptote at x = b • Has a horizontal asymptote at y = c

Example 1 This is a graph of

3 2 1 −4 −3 −2 −1

−1

1 x

x 3

−2 −3 −4

a How do we shift the graph of

to get the graph of

Move the graph 3 units to the left

Move the graph upwards by 3 units

Move the graph downwards by 3 units

Move the graph 3 units to the right

Create a strategy

Apply the idea

A vertical translation occurs when a constant value is added or subtracted at the end of the equation.

In this case, represents a graph that has been translated vertically upwards by 3. So

A horizontal translation occurs when a value is added or subtracted with the x variable in the denominator.

the answer is option B.

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b Draw

on the same graph as

Create a strategy

Apply the idea

Based on our answer to part (a), the graph

Each point on the graph of up by 3.

translated vertically up by 3.

should show

must be shifted

1 +3 x

1 x

3 2 1 −4 −3 −2 −1 −1

x 3

−2 −3

Example 2 Consider the equation x y

−1

0.5

1.5

a Complete the table of values.

Create a strategy Substitute each value of x into the equation to ﬁnd the value of y.

Apply the idea at each value of x we get:

Evaluating the expression x y

−1

0.5

1.5

−1

−2

Reﬂect and check Notice that if we tried to substitute x = 1, y would be undeﬁned (since we can’t divide by zero).

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b Sketch a graph of the equation.

Create a strategy We can plot the points from the table of values and use this to help draw the curve.

Apply the idea 4

Based on the equation, we already know there will be a vertical asymptote at x = 1, where the equation is undeﬁned.

3 2 1 −4 −3 −2 −1

−1

−2

x 1

The points from the table of values conﬁrm that the vertical asymptote is between x = 0.5 and x = 1.5. The equation can also be written as

+ 0,

so the horizontal asymptote is at y = 0.

−3 −4

c Describe the transformation from the original function

Create a strategy Compare the graph of

with the graph of

Apply the idea Looking at the graph, and in particular the location of the vertical asymptote, we can see there was a horizontal translation 1 unit to the right.

Reﬂect and check If we consider the form

, the function

has k = 1, b = 1, and c = 0. This also

indicates a horizontal translation of 1 unit to the right (and no vertical stretch or vertical translation).

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Example 3 Describe the transformation that has occurred from

to the graph of

Create a strategy Compare the function

to the general form

Apply the idea There are no horizontal or vertical translations (b = 0 and c = 0). Since k = −3, the graph of across the x-axis.

has been stretched vertically by a factor of three and reﬂected

Reﬂect and check We can check our answer by sketching the graph of x

−3

−2

−1

undefined

−3

6 5 4 3 2 1 −6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6

using a table of values.

3 −1 When the graph of

graph of f (x) = x, we can see that for each x-value, the point is 3 times as far away from the x-axis, but on the other side of the x-axis.

1 x

1 2 3 4 5 6

y=−

is compared to the

3 x

This is most obvious if we compare (1, 1) on with (1, −3) on

This conﬁrms that the graph has been stretched vertically by a factor of three and reﬂected across the x-axis.

Idea summary A hyperbola in the form • • • •

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, compared to

If k > 1, has been stretched vertically (away from the asymptotes). If k < −1, has been stretched vertically and reﬂected across the x-axis. Has been horizontally translated by b units, so has a vertical asymptote at x = b. Has been vertically translated by c units, so has a horizontal asymptote at y = c.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1−3, 4ab, 5, 7ac, 9, 10, 12ab, 13, 15a, 17, 19, 20, 22, 23

Adventurer 1−3, 4bc, 6ab, 7bc, 8, 10, 12ac, 14, 15bc, 16a, 17, 19, 20, 22−24

Trailblazer 1−3, 4cd, 6cd, 7bd, 8, 11, 12cd, 15, 16b, 17−21, 23−25

Understanding 1

What is the equation for a hyperbola with a vertical translation of c units up?

What is the equation for a hyperbola with a horizontal translation of b units to the right?

In each case, where would the point (1, 3) be translated to? a

Vertical translation up 4 units

Horizontal translation 2 units to the left

Vertical translation down 10 units

Vertical translation up 3 units, then 5 units to the right

Is each statement true or false for a hyperbola in the form a

The vertical asymptote is y = b.

The horizontal asymptote is y = c.

If k < −1, the graph has been reﬂected across the x-axis.

The asymptotes cross at (c, b).

Fluency Ex 1

This is a graph of

. For each equation: 4

Describe the transformation from

Sketch the graph on the same Cartesian plane as

3 2

−4 −3 −2 −1

−1

1 x x

−2 −3 −4

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Ex 2

For the function a

Complete the table of values. x

Sketch the graph.

Describe the transformation from

For each equation: Describe the transformation from

Sketch the graph on the same Cartesian plane as b

Describe the transformation from

Sketch the graph on the same Cartesian plane as b

For the function

How can the graph of f (x) be obtained from the graph of

What is the equation of the vertical asymptote?

What is the equation of the horizontal asymptote?

Sketch the graph of y = f (x).

For the function

What is the equation of the vertical asymptote?

What is the equation of the horizontal asymptote?

Sketch the graph of the function.

For the hyperbola

How would the hyperbola relate to its asymptotes? A

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For each equation:

a 8

a 7

What are the equations of the vertical and horizontal asymptotes of

Sketch the graph of

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For the hyperbola a

How would the hyperbola relate to its asymptotes? A

What are the equations of the vertical and horizontal asymptotes of

Sketch the graph of

For the following functions: i

Write the equation of the vertical asymptote.

Write the equation of the horizontal asymptote.

a 13

For the graph of the hyperbola

c :

What would be the new equation if the graph was shifted upwards by 4 units?

What would be the new equation if the graph was shifted to the right by 7 units?

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

For the graph of the hyperbola

What would be the new equation if the graph was shifted down by 5 units?

What would be the new equation if the graph was shifted to the left by 9 units?

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

x 1 2 3 4 5

12.03 Optional Transformations of hyperbolas mathspace.co

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For each graph: i

State the vertical and horizontal asymptotes.

Describe the transformation from

iii

Write the equation of the transformed function.

(shown in blue and dashed).

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−2

−3 −4

−4 −3 −2 −1

−2 −3

−4

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

−2

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−1

−3

−1

−2

−4 −3 −2 −1

−2

−3

−1

−4

−4 −3 −2 −1

x 1

Each table of values represents a transformation of i

Determine the transformation of the graph of table.

Write an equation for f (x).

−3

−2

f (x) b

−7

−6

f (x) 17

−1

undefined

−5

−4

−3

−1

undefined

. which would give the values in the

−2

−1

For the hyperbolas:

What value cannot be substituted for x in

In which quadrants does

What value cannot be substituted for x in

In which quadrants does

How can the graph of

lie? ?

lie? be transformed to create the graph of

Without graphing, determine the behaviour of each of the following hyperbolas as: i

As x tends to 0 from the right, x → 0+

As x tends to 0 from the left, x → 0−

iii

As x tends to positive inﬁnity, x → ∞

Reasoning 19

Arik is asked to draw the graph of graph shown. Identify and correct his error.

. He drew the 4

3 2 1 −3 −2 −1 −1

x 1

−2 −3 −4

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Determine whether each statement is true or false. Justify your answer. a

A hyperbola with a horizontal translation has the equation

If a hyperbola is translated vertically down by 4 units, the equation becomes

A horizontal translation of 2 units to the left affects the constant value in the numerator.

If a hyperbola has the equation

. .

, it has been vertically translated by 2 units.

For a hyperbola in the form

What are the equations of the asymptotes of f (x)?

What are the domain and range of f (x)?

Problem-solving 22

Consider the table of values for f (x) and the graph of g (x) = f (x + k). x f (x) a b

−2 −6

−1 −12

0 undefined

1 12

12 9

2 6

Determine the type of transformation applied to f (x) to obtain g (x). Determine the value of k.

g (x)

6 3

−12 −9 −6 −3 −3

9 12

−6 −9 −12

A graph of the hyperbola

is shown:

Given points C (−4, 0) and D (2, 0), ﬁnd the length of AB.

D B

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Guido is organizing a yearbook order this year for his soccer club. There is a ﬁxed fee of $400 plus a cost of $25 per yearbook. The ﬁxed fee is evenly divided across everyone who orders a yearbook. a

Write the equation of a function that gives the cost per yearbook ( y) in terms of the number of yearbooks purchased (x).

There is a minimum purchase of 5 yearbooks. Draw the graph which represents this situation.

If the yearbook committee wants to get the price per yearbook down to $30, determine how many people would need to purchase a yearbook.

Explain why the cost will always be higher than $25.

A yoga studio offers classes on a pay what you can model. A third of the students paid $20 each for the class, a third of the students paid $7 each for the class, and a third of the students made various donations totalling $60. Determine the lower limit of the average amount paid per student for very large classes. Justify your answer.

12.03 Optional Transformations of hyperbolas mathspace.co

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Chapter 12 review 1

Determine whether the following is an example of a direct variation or an inverse variation: a

The time taken to dig a hole and the number of people digging

The number of macarons sold and the revenue earned

Does each table show a direct or inverse relationship between x and y? a

What is the constant of proportionality?

x y

1 6

3 2

4 1.5

Find the value of a when b = 6.

50 95

60 114

70 133

80 152

T F

90 171

A R

Sketch the graph of the proportional relationship.

Find the unit rate of the relationship between quinoa and water.

Write two statements that describe the proportional relationship.

If y is inversely proportional to x, and y = 30 when x = 5:

Explain how we can ﬁnd the constant of proportionality, k.

Express y in terms of x.

Find the value of y when x = 10.

The equation

represents an inverse

relationship between x and y.

2 3

Rose is making quinoa. The amount of rice and water she uses is in the table. Quinoa (g) Water (mL)

Given a = 30b: a

x 1 2 3 4 y 6 12 18 24

Complete the table of values.

Describe the rate of change of the function when x < 0.

x y

−5

−4

−3

−2

−1

increasing or decreasing when x < 0?

At ProPainters, painters must paint at a minimum rate of 60 m2/hr. The equation represents the time, t, it takes p painters to paint the outside of a house with a surface area of 1200 m2.

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Find the constant of proportionality, k.

Find the time it would take 4 painters to ﬁnish painting 1200 m2.

For the function

For what value of x is the function undeﬁned?

As x approaches 0 from the positive side, what can we say about the value of y?

What y-value does the function get inﬁnitely close to, but never reach?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

For each function: i

Complete the table of values. x

−4

−2

−1

Plot the points from the table of values on a Cartesian plane and sketch the graph of the function.

iii

In which quadrants does the graph lie?

Without graphing, determine the behaviour of each hyperbola: i

In which quadrants does the graph lie?

As x tends to 0 from the right, x → 0+

iii

As x tends to 0 from the left, x → 0−

As x tends to positive inﬁnity, x → ∞

T F

Sketch each graph, labelling the asymptotes and two points. a

A R

Find the equation of each hyperbola. a

y 4 3 2 1 −4 −3 −2 −1 (−1, −2)

−1

−2 −3

D (1, 2)

3 2

−4 −3 −2 −1 (−1, −1.25)

In an art competition, the total prize money is distributed evenly among the winners. The graph shows the relationship between the number of winners and the prize money each winner receives. Determine whether or not there is an inverse variation between the number of winners and the prize money each receives. If so, ﬁnd the constant of variation.

−1

x 1

−2 −3 −4

−4

(1, 1.25)

55 50 45 40 35 30 25 20 15 10 5

Prize money

No. of winners 5 10 15 20 25 30 35 40 45 50 55

Chapter 12 review mathspace.co

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The total costs of a seminar, including venue hire, materials, and refreshments, amount to $560. These costs are shared equally among the participants. a

What is the cost per participant if the seminar is attended by two people?

Let the number of participants be x, and the cost per participant be y. Formulate an equation to represent this relationship.

Complete the table with values based on the equation, rounding to two decimal places when needed. x y

Sketch the graph based on the equation.

Describe the relationship between x and y.

(Optional) This is a graph of

T F

For each equation: Describe the transformation from

Sketch the graph on the same Cartesian plane as .

3 2

1 x

−4 −3 −2 −1

A R

c 16

−1

x 3

−2 −3 −4

(Optional) Without graphing, determine the behaviour of each hyperbola: i

As x tends to 0 from the right, x → 0+

iii

As x tends to positive inﬁnity, x → ∞

As x tends to 0 from the left, x → 0−

(Optional) Tetay is asked to draw the graph of She drew the graph shown. Identify and correct her error.

. 4

3 2 1 −1 −1 −2 −3 −4

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

x 1 2 3 4 5 6 7 8 9 10 11 12 13

(Optional) Emilio is managing a custom footy guernsey order for his basketball team. There is an upfront design fee of $500 plus a cost of $30 per guernsey. The design fee is split equally among all orders. a

Write the equation of a function that represents the cost per guernsey, y, as a function of the number of guernseys ordered, x.

Assume the team decides on a minimum order of 10 guernseys. Draw the graph which represents this situation.

To lower the cost per guernsey to $35, how many guernseys need to be ordered?

Explain why the cost cannot go below $30.

T F

A R

Chapter 12 review mathspace.co

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13 Trigonometry Big ideas • Right triangles have special properties that allow their side lengths to be determined using the Pythagoras’ theorem or ratios. • Trigonometry connects the angle measures and side lengths of a triangle and provides a set of functions that can be used to find unknown angle measures and side lengths.

Curriculum Outcomes VC2M10A05 Substitute values into formulas to determine an unknown and rearrange formulas to solve for a particular term VC2M10M03 Solve practical problems by applying Pythagoras’ theorem and trigonometry to right-angled triangles, including problems involving direction and angles of elevation and depression VC2M10ASP02 Establish the sine, cosine and area rules for any triangle and solve related problems VC2M10ASP05 Apply Pythagoras’ theorem and trigonometry to solving three-dimensional problems in right-angled triangles

Chapter outline 13.01 13.02 13.03 13.04 13.05

Pythagoras’ theorem Trigonometric ratios Angles of elevation and depression Bearings Optional: Sine and cosine rules Chapter 13 review

662 671 685 695 711 730

Sundials, such as the famous Intihuatana of Machu Picchu, demonstrate the use of trigonometry to position the hour lines and determine the time of day!

13.01 Pythagoras’ theorem After this lesson, you will be able to… • apply Pythagoras’ theorem to identify right-angled triangles. • use Pythagoras’ theorem to find the length of the hypotenuse and other sides in a right-angled triangle. • apply knowledge of Pythagoras’ theorem to solve real-world problems.

Pythagoras’ theorem Hypotenuse

In a right-angled triangle, the largest angle in the triangle is 90°. The side opposite the right-angle will be the longest side. We call this side the hypotenuse.

Right angle

All three sides of a right-angled triangle are related by the equation shown. The two smaller sides will be called a and b, and the hypotenuse (the longest side) will be c.

a2 + b2 = c2 a, b, c

Sides of a right-angled triangle

a, b

The lengths of the shorter sides

The length of the hypotenuse

This relationship between sides in a right-angled triangle is called Pythagoras’ theorem. We can use this theorem to: • Check whether or not a triangle is a right-angled triangle • Find the length of the hypotenuse • Find the length of one of the shorter sides in a right-angled triangle

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 13.01 to answer these questions. 1.

What do you notice about the sum of the area of the squares of the two shorter sides and the area of the square of the hypotenuse?

If you subtracted the area of the shorter side sides from the largest side, what do you get?

Can you find another Pythagorean triple, where all three sides have integer values?

Example 1 Use Pythagoras’ theorem to determine whether this is a right-angled triangle.

16 9

a Let a and b represent the two shorter side lengths. First ﬁnd the value of a2 + b2.

Create a strategy Substitute the lengths of the two shorter sides and add them.

Apply the idea a2 + b2 = (9)2 + (16)2

Substitute the value of a and b

= 81 + 256

Evaluate

= 337

Evaluate the sum

b Let c represent the length of the longest side. Find the value of c2.

Create a strategy Substitute the length of the longest side.

Apply the idea c2 = (18)2 = 324

Substitute the value of c Evaluate

c Is the triangle a right-angled triangle?

Create a strategy

Apply the idea

Compare the results from parts (a) and (b) to determine if a2 + b2 = c2.

337 ≠ 324 The triangle is not a right-angled triangle because a2 + b2 ≠ c2.

13.01 Pythagoras’ theorem mathspace.co

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Example 2 Which of the following three lengths represent the sides of a right-angled triangle? Select the two correct answers. A (6, 8, 13)

(5, 12, 13)

(2, 4, 6)

(3, 4, 5)

Create a strategy Add the squares of the two smaller numbers. If the result is the same as the square of the biggest number, then it’s a Pythagorean triple.

Apply the idea The triple (a, b, c) is a Pythagorean triple if c2 = a2 + b2. 132 = 62 + 82

Substitute the sides from option A

169 > 100

Evaluate both sides

132 = 52 + 122

Substitute the sides from option B

169 = 169

Evaluate both sides

62 = 22 + 42

Substitute the sides from option C

36 > 20

Evaluate both sides

52 = 32 + 42

Substitute the sides from option D

25 = 25

Evaluate both sides

The correct answers are options B and D.

Example 3 Find the length of the unknown side as an exact value: a

Create a strategy We can use Pythagoras’ theorem: c2 = a2 + b2.

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Apply the idea c2 = a2 + b2 2

Write the formula 2

c = 12 + 16

Substitute a and b

c = 144 + 256

Evaluate the squares

c = 400

Evaluate the sum

Take the square root of both sides

c = 20

Evaluate the square root

Create a strategy Use Pythagoras’ theorem: a2 + b2 = c2.

Apply the idea We can substitute the following a = x, b = 6, and c = 10 into the formula: a2 + b2 = c2 2

Write the formula 2

x + 6 = 10

Substitute the values

x + 36 = 100

Evaluate the squares

x + 36 − 36 = 100 – 36

Subtract 36 from both sides

x = 64

Evaluate the difference

Take the square root of both sides

x=8

Evaluate the square root

Reflect and check The most important thing to remember when ﬁnding a short side is that the two lengths need to go into different parts of the formula. If we substitute incorrectly, we will end up with the square root of a negative number (calculator error) or a side which is longer than the hypotenuse.

13.01 Pythagoras’ theorem mathspace.co

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Example 4 A movie director wants to shoot a scene where the hero of the film fires a grappling hook from the roof of one building to the roof of another. If the first building is 37 m tall, the other building is 54 m tall and the street between them is 10 m wide, what is the minimum length, l, of rope needed for the grappling hook? Round your answer to two decimal places.

l 54 m 37 m

10 m

Create a strategy Use the Pythagoras’ theorem: c2 = a2 + b2.

Apply the idea Notice that we have a right-angled triangle, and the length of the grappling hook’s rope will be the hypotenuse. The one short side is the width of the street, 10, and the other short side is the difference in heights between the two buildings, 54 − 37. We can substitute the following a = 10, b = (54 − 37), and c = l into the formula: c2 = a2 + b2

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Write the formula

l = 10 + (54 − 37)

Substitute the values

l = 10 + 17

Simplify the brackets

l = 100 + 289

Evaluate the indices

l = 389

Evaluate the addition

Take the square root of both sides

l ≈ 19.72

Evaluate the square root

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary Pythagoras’ theorem relates the three sides of a rightangled triangle, a and b are the two smaller sides, and the longest side, called the hypotenuse, is c.

We can also test to see if a triangle is right- angled by checking to see if its three sides satisfy a2 + b2 = c2. b

To ﬁnd the hypotenuse: c2 = a2 + b2 To ﬁnd a shorter side use: a2 = c2 − b2 or b2 = c2 − a2 We can take the square root of both sides to give us the following formulas:

Practice questions Choose your question path Explorer 1–3, 4ace, 5a-dg, 7ac, 9, 10, 12ab i, ii, 13–15, 17

Adventurer 2cd, 3, 4bdf, 5deh, 6, 7d, 8a, 9, 10, 12ab ii, iii, 13–17

Trailblazer 2d, 3, 4gh, 5dfi, 6, 8, 9, 11, 12, 13b, 15, 16, 18

Understanding 1

Write Pythagoras’ theorem in a form you could use to ﬁnd: a

The length of c.

The length of a. c

Calculate the value of x, rounded to two decimal places. All values of x are positive. a

x2 = 22 + 32

82 = 52 + x2

13.01 Pythagoras’ theorem mathspace.co

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Ex 1

Use Pythagoras’ theorem to determine whether this is a right-angled triangle. a

Let a and b represent the two shorter side lengths. First ﬁnd the value of a2 + b2.

Let c represent the length of the longest side. Find the value of c2.

Is the triangle a right-angled triangle?

Fluency 4 Ex 2

Ex 3

Are these Pythagorean triples? a

(12, 16, 26)

(9, 12, 15)

(4, 8, 12)

(12, 16, 20)

(8, 15, 17)

(7, 7, 10)

(8, 11, 14)

(7, 24, 25)

Find the length of the unknown side as an exact value: a

d c

f 5

668

A right-angled triangle has a one side of length 6 and another side of length 4. Find the length of the hypotenuse.

A right-angled triangle has a hypotenuse of length 9 and another side of length 3. Find the length of the third side.

A right-angled triangle has a hypotenuse of length 17 and another side of length 14. Find the length of the third side.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Two poles, one of length 8 m, and the other of length 14 m, stand 8 m apart on level ground. Find the length of wire required to join the tops of the poles. 14 m 8m

Find the length of a diagonal of each shape, rounded to one decimal place where appropriate. a

10 cm

10 mm 5 mm

10 cm

21 m

10 cm 4 cm

20 m 16 cm

Find the distance between the points P and Q, rounded to one decimal place. a

4 cm 4 cm 6 cm

20 mm

P Q

10 mm

4 cm P

9 Ex 4

18 mm Q

6 mm 5 mm

Two buildings, one of height 300 m and the other of height 450 m, stand a distance apart on level ground. If a cable joining the tops of the buildings is 200 m long, ﬁnd, to the 300 m nearest metre, the distance between the buildings.

200 m 450 m

Reasoning 10

Can a triangle with sides of length 3 cm, 5 cm and 7 cm be a right-angled triangle? Use Pythagoras’ theorem to justify your answer.

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If AB = 10 cm, AD = 8 cm and ∠ADB is a right angle, show that D is the midpoint of BC.

A triangle with sides 3, 4, and 5 units is a right triangle. a

If the triangle is enlarged by a factor of 3, does Pythagora’s theorem still hold? Justify your answer.

Is a triangle with these sides a right triangle? 30, 40, 50

35, 45, 55

iii

3k, 4k, 5k

21a, 28a, 36a

Problem-solving 13

Find the value of x, rounded to one decimal place. a

5 cm

Find the length of the diameter AB, of the circle.

Find the exact length x.

24 cm O

A x

670

7 cm B 9 cm

Casper produces thin plastic triangles. a

If the triangles need to be edged by a material that costs 3 cents per cm, how much, in dollars, will it cost to edge 100 triangles?

If the triangles need to be painted with a coating that costs 2 cents per cm2, how much, in dollars, will it cost to coat 100 triangles.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

20 cm

29 cm

The diagram shows four squares. The outer square has side lengths of 20 cm. The next square is formed by joining the midpoints of the outer square, the third square by joining the midpoints of the second square, and the fourth square by joining the midpoints of the third square.

Tamsin is using wire to construct the shape in this diagram. Find, to the nearest centimetre, the length of wire she requires.

Tricia’s house has the outer dimensions as shown in the diagram:

A drainpipe needs to run from the vertex of the roof, indicated by point A in the diagram, vertically downwards until it is 30 cm above the ground. Find the length of pipe needed in centimetres, rounded to two decimal places.

A 5.6 m

5.6 m hm 3.6 m

3.6 m

10.2 m

Angelia and Beth start their hikes from the same starting point. Angelia walks 900 m south and then 900 m east, before stopping for a lunch break. Beth stops for a break having walked 1200 m south and then 600 m east. If they both walked back to the start point in a straight line, who would walk furthest, and by how far to the nearest metre?

13.02 Trigonometric ratios After this lesson, you will be able to… • identify and define sine, cosine, and tangent, and describe their relationship between angles and sides in right-angled triangles. • use trigonometric ratios to find an unknown side or angle in right-angled triangles.

Trigonometric ratios In right-angled triangles, we can use trigonometric ratios to describe the relationship between a given angle and two sides.

Hypotenuse

The trigonometric ratios are:

Opposite

θ A

Adjacent

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Exploration We have three right-angled triangles with acute angles of 30° and 60°. 1.

Is △ABC ∼ △DEF ∼ △GHI? Explain why or why not.

Find the ratio for sin 30° for each of the triangles.

30°

What do you notice about each ratio (use a calculator if you need to)?

Find the ratio for cos 30° for each of the triangles.

3 30° 2 60°

What do you notice about each ratio (use a calculator if you need to)?

Find the ratio for tan 30° for each of the triangles.

What do you notice about each ratio (use a calculator if you need to)?

Draw another similar triangle which would have the same ratios.

1 2

30°

2 3

3 2

60°

60° 1 F

The trigonometric ratios will be the same for similar right-angled triangles. This is because the size of the angles in a triangle determines the ratios between the side lengths of a triangle.

Example 1 Evaluate cos (θ ) within △ABC.

A α

Create a strategy Identify which sides we need for the cosine ratio and then ﬁll in the side lengths.

672

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea The angle θ is also called ∠B.

We are using the cosine ratio

Opposite

Hypotenuse

so we need the lengths of the adjacent and hypotenuse sides.

θ 5

Adjacent

With respect to the angle θ, the adjacent side is 5, and the hypotenuse is the longest side of 13. Deﬁntion of cosine

Substitute the sides

Idea summary

Hypotenuse

The trigonometric ratios are:

Opposite

θ A

Adjacent

Find a side If we are given one side length and one angle, we can use the trigonometric ratios to ﬁnd one of the missing side lengths. We can follow these steps to ﬁnd a missing side in a right-angled triangle: 1. Highlight the reference angle. 2. Identify which sides are the hypotenuse, opposite, and adjacent — label if desired. 3. Choose the ratio which has one of the required sides given and one is the missing side: • Sine: opposite and hypotenuse • Cosine: adjacent and hypotenuse • Tangent: opposite and adjacent 4. Set up the chosen ratio:

5. Solve for the unknown.

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Remember the following: • For a given angle θ, the value of each trigonometric ratio stays the same no matter the size of the triangle. • Your calculator must be set to degree mode when working with degrees in trigonometric ratios.

Example 2 Find the value of the variable, rounded to two decimal places: a

f 47°

Create a strategy Identify the given sides and write an equation using the appropriate trigonometric ratio.

Apply the idea With respect to the angle 47°, the length of the adjacent side is f, and the length of the hypotenuse is 8, so we can use the cosine ratio. Use the cosine ratio

Substitute the values and f

Multiply both sides by 8

Evaluate and round

11 g

42°

Create a strategy Use the trigonometric ratio of sine and substitute the given values.

674

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea With respect to the angle 42°, the length of the opposite side is 11, and the length of the hypotenuse is g, so we can use the sine ratio. Use the sine ratio

Substitute the values and g

Multiply both sides by g

Divide both sides by sin 42°

Evaluate and round

Reflect and check When the unknown is in the denominator, we need to take care to rearrange properly.

Idea summary We can use the trigonometric ratios to find an unknown side length of a right-angled triangle. Once we set up our equation with a variable representing the unknown side length, we can use inverse operations to make the variable the subject of the equation. Then we can evaluate the expression in our calculators to find the side length.

Find an angle If we are given two sides of a right-angled triangle, we can use the trigonometric ratios to find a missing angle. We can follow these steps: 1. Using the unknown angle as the reference angle, label the given sides as opposite, adjacent, or hypotenuse. 2. Fill in the trigonometric ratio with the given information. 3. Use the inverse ratio to solve for the angle.

Example 3 Find the value of x to the nearest degree.

A 7

C 25

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Create a strategy Use the cosine ratio to set up the equation then use the inverse ratio to ﬁnd the angle.

Apply the idea With respect to the angle of x, the length of the adjacent side is 7, and the length of the hypotenuse side is 25, so we can use the cosine ratio. Use the cosine ratio

Substitute the values

Apply the inverse ratio

Evaluate and round

Reflect and check If we get an error on our calculator, it usually means we haven’t used the right ratio or substituted incorrectly.

Idea summary To ﬁnd an unknown angle of a right-angled triangle, we use a trigonometric ratio to set up an equation relating the angle and two given sides. Then we can use the inverse ratio to ﬁnd the value of the angle.

Multi-step problems In some cases, a single application of trigonometry, on a single right-angled triangle, will not be enough to solve to the question.

x cm 43°

In this example, to find the length of h, we must first use our knowledge of trigonometry, in right-angled triangles, to find the length of x. Once we find the length of x, we add 70 to x, to get the length of h.

h cm 70 m

30 m

676

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

13°

16° C 2000 m

Airport

In this example, to find the distance between the two planes, AB, we will need to apply our knowledge of trigonometric ratios to find the distance of the first plane to point C (AC) and then again to find the distance of the second plane to point C (BC). Once we find the distance of both planes to point C, we subtract the distance of the second plane (BC) from the distance of the first plane (AC) to find distance between the two planes (AB).

Example 4 A girl is ﬂying a kite that is attached to the end of a 23.4 m length of string. The angle between the string and the vertical is 21°. If the girl is holding the string 2.1 m above the ground, ﬁnd the vertical distance from the ground to the place where the string attaches to the kite. Round your answer to two decimal places.

23.4 m 21°

2.1 m

Create a strategy Let’s call the vertical distance from the ground to the kite h. If we could ﬁnd the length of the vertical leg of the triangle, we calculate h. Let’s call the length of the vertical leg of the triangle x. First, calculate an acute angle in the right triangle to use as a reference angle, then use a trigonometric ratio to ﬁnd the value of x. Then, we can calculate the vertical distance from the kite to the ground.

23.4 m 21°

xm hm

2.1 m

Apply the idea Since the angle between the string and the horizontal is complementary to 21°, the reference angle in the triangle must be 69°.

Substitution

Multiply both sides by 23.4

Evaluate and round

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The vertical distance, h, from the ground to the kite is the sum of the distance from the ground to where the string starts and the vertical distance, x, from the string to the kite. That means h = 2.1 + x, where x = 21.85. So, h = 2.1 + 21.85 = 23.95. The kite is 23.95 m from the ground.

Example 5 Find the value of the variables x and y, rounded to the nearest whole number.

y cm

x cm 64°

Create a strategy Use the tangent ratio to ﬁnd y, then use the cosine ratio to determine x.

Apply the idea With respect to the angle of 64, the length of the opposite side is 8, and the length of the adjacent side is y, so we can use the tangent ratio. Use the tangent ratio

Substitute the values

Multiply both sides by y

Divide both sides by tan 64°

Evaluate and round

With respect to the angle of 64, the length of the adjacent side is 8, and the length of the hypotenuse is x, so we can use the cosine ratio. Use the cosine ratio

Substitute the values

Multiply both sides by x

Divide both sides by cos 64°

Evaluate and round

678

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary In multi-step problems, a single application of trigonometry will not be enough to solve the problem. Extra addition, subtraction or another application of trigonometry may also be needed.

Practice questions Choose your question path Explorer 1, 2, 3ace, 4, 5a, 6–8, 10–12, 14, 16, 17abc, 18, 20, 22

Adventurer 3df, 4, 5b, 7, 8, 10, 11a, 12, 14, 16, 17bcd, 18, 20, 22, 23

Trailblazer 3d, 4, 5b, 7, 9, 10, 11b, 13, 15, 16, 17cde, 19, 21–24

Understanding 1

Name the hypotenuse. a

A B

Which side is opposite to angle θ?

Which side is adjacent to angle θ?

Which side is opposite angle α?

Which side is adjacent to angle α?

Which angle is opposite the hypotenuse?

A α

θ B

With reference to the angle θ, ﬁnd the ratio: i

iii

13.02 Trigonometric ratios mathspace.co

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A 34

θ B

A 13 B

θ 5

θ B

25 A

Write down the indicated ratios for each triangle using the side labels: a

tan θ

tan α

sin θ

A α

θ B

cos θ

cos α

A α

θ B 680

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

sin α

Fluency Ex 1

Write down the indicated ratios for each triangle: a

tan θ

sin α

iii

cos θ

tan α

sin θ

cos α

iii

cos θ

sin α

B 17

θ°

α°

Find the value of tan θ. θ

25p 24p

7 Ex 2

Find the value of the variable, rounded to two decimal places: a

b 29° fm

7 cm

27°

f cm

4 cm

12 cm

44°

25° g cm 16.68 cm a cm

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, ﬁnd the value of b.

θ° 12 m

If tan (55°) = , ﬁnd the value of d.

A 2313 m 55° d

Calculate the size of angle θ.

Ex 3

For each triangle, ﬁnd the value of: i

θ 6

θ 3

If d is the distance between the base of the wall and the base of the ladder, ﬁnd the value of d, rounded to two decimal places.

1.88 m

38° dm

682

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A ladder is leaning at an angle of 44° against a 1.36 m high wall. Find the length of the ladder, rounded to two decimal places. 1.36 m

44°

A sand pile has base angles of 40° and is 10.6 m wide. Find the height of the sand pile, h, rounded to one decimal place.

h cm 40° 10.6 m

In the diagram, a string of lights of length 23.9 m joins the top of the tree. If the angle that the string of lights makes with the ground is θ °, ﬁnd θ rounded to two decimal places.

23.9 m 4.1 m

θ°

16 Ex 4

The top of the NJ building, which is 300 m high, has a 39° angle of elevation to the top of the LS building. The diagonal distance between the tops of the two buildings is 200 m.

200 m 39°

300 m

Find x, rounded to two decimal places.

Find the height of the LS building, rounded to two decimal places. x

17 Ex 5

Find the value of the variable(s), rounded to the nearest whole number: a

54°

26 mm

16 cm 42° f cm

h mm

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x cm

32 cm 52°

x cm

58° h cm

11 cm

82 cm

45 cm

A x

Reasoning 18

An isosceles triangle has equal side lengths of 10 cm and a base of 8 cm as shown. Explain how to calculate the size of angle A, rounded to one decimal place.

10 cm

A°

A° 8 cm

If tan θ = 0.4, explain how to ﬁnd the value of b. θ°

b cm

21.32 cm 7.9 cm

Problem-solving 20

, ﬁnd the value of cos θ and tan θ, without calculating the size of angle θ.

, ﬁnd the value of cos θ and sin θ, without calculating the size of angle θ.

684

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Consider the octagon:

10 cm

Use trigonometry to calculate the size of each internal angle rounded to the nearest whole number.

Conﬁrm your answer to part (a) using the formula to ﬁnd an internal angle of a polygon with n sides:

12 cm

A sporting ground is in the shape of an irregular hexagon. a

Find the perimeter of the ground.

Find the size of each of the four corner angles of the ground.

35 m 15 m

41 m

15 m 35 m

Calculate the size of angle DEC.

6 x+7 A

12 − x C B

13.03 Angles of elevation and depression After this lesson, you will be able to… • define angles of elevation and depression and understand their real-world applications. • model the relative position of two objects using a right-angled triangle. • identify and apply the three distances (horizontal, vertical, and direct) between two objects when given the angle of elevation or depression. • solve problems involving angles of elevation and depression using trigonometric ratios.

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Angles of elevation and depression An angle of elevation is the angle from the lower object to the higher one. The angle of elevation from point A to B is the angle between the horizontal line at A and the line connecting the two points.

θ A

An angle of depression is the angle from the higher object to the lower one. B

The angle of depression from point B to A is the angle between the horizontal line at B and the line connecting the two points.

Both angles are measured with respect to the horizontal.

Exploration Rectangle ABCD has a diagonal drawn on it.

What is the value of x? What property did you use to ﬁnd this? Is there another way to conﬁrm this?

What is the angle of elevation from point C?

What is the angle of depression from point A?

What do you notice about the angle of elevation and depression?

x 50°

50°

Notice that the angle of elevation between two points will always be equal to the angle of depression between those two points. This is because they are alternate angles on parallel lines (since all horizontal planes are parallel to each other).

θ θ A

When given the angle of elevation or depression between two objects, we will always be able to model their relative position using a right-angled triangle. There are three distances between two objects: horizontal distance, vertical distance and direct distance. These will represent the adjacent, opposite and hypotenuse sides respectively. If any two of these are given, then we can ﬁnd the angle of elevation or depression. B

Dir

ect

dis

Horizontal

686

θ Dir ect

tan

Horizontal

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

dis

tan

Vertical

Example 1 Find the angle of depression from point B to point D. Use x as the angle of depression and round your answer to two decimal places.

9 A

Create a strategy Identify where the angle of depression is and apply the appropriate trigonometric ratio.

Apply the idea The angle of depression is the angle ∠CBD, with adjacent side BC = 9 and hypotenuse BD = 19. We can let x be the unknown angle. Use the cosine ratio

Substitute the values

Apply inverse cosine on both sides

Evaluate using a calculator and round

Example 2 A fighter jet, flying at an altitude of 4000 m is approaching a target. At a particular time the pilot measures the angle of depression to the target to be 13°. After a minute, the pilot measures the angle of depression again and finds it to be 16°.

13° B

16°

4000 m

Airport

a Find the distance AC, to the nearest metre.

Create a strategy Use the trigonometric ratio of tangent.

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Apply the idea With respect to the given angle 13°, the opposite side is 4000, and the adjacent side is AC, so we can use the tangent ratio. Use the tangent ratio

Substitute the values

Multiply both sides by AC

Divide both sides by tan 13°

Evaluate using a calculator and round

b Find the distance BC, to the nearest metre.

Create a strategy Use the same method that we used to ﬁnd AC in part (a).

Apply the idea With respect to the given angle 16°, the opposite side is 4000, and the adjacent side is BC, so we can use the tangent ratio. Use the tangent ratio

Substitute the values

Multiply both sides by BC

Divide both sides by tan 16°

Evaluate using a calculator and round

c Now ﬁnd the distance covered by the jet in one minute to the nearest metre.

Create a strategy Find the difference between the length of AC and BC.

Apply the idea The distance covered by the jet in one minute is given by the length of AB. AB = 17 326 − 13 950 = 3376 m

688

Subtract BC from AC Simplify the difference

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 A room is rectangular and measures 5 metres in length and 4 metres in width. The angle of elevation from the bottom left corner to the top right corner of the room is 57°. a Draw a diagram of the room.

Create a strategy

Apply the idea

To draw a diagram of the room, we need to keep in mind that the room is rectangular in shape and the given dimensions are in metres. Additionally, the angle of elevation from one corner of the room to the other is given, which means we are considering a 3D model.

h 57°

b Find d, the distance from one corner of the ﬂoor to the opposite corner of the ﬂoor. Leave your answer in surd form.

Create a strategy Apply the Pythagora’s theorem, c2 = a2 + b2.

Apply the idea d2 = 52 + 42 2

d = 41 d=

Substitute the values

Evaluate the right-hand side Take the positive square root of both sides

c Find h, the height of the room. Give your answer to two decimal places.

Create a strategy Remember the mnemonic SOH-CAH-TOA. Set up an equation involving the height, h, the diagonal on the ﬂoor, , and the angle of elevation, 57°.

h 57°

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Apply the idea h

Substitute the values

Multiply both sides by

Evaluate using a calculator and round

57°

Idea summary We can solve problems involving angles of elevation and depression using these steps: 1. Draw a diagram if not given one. 2. Label all known information on the digram, including the right angle.

Angle of elevation

3. Identify which angles or sides need to be found and in what order.

Horizontal

Angle of depression

4. Set up the correct trigonometric ratio and solve for the missing side or angle. 5. Give your answer as a sentence.

Practice questions Choose your question path Explorer 1–7, 10–13, 15, 17–21

Adventurer 2, 3, 5–13, 15, 18–21, 23

Trailblazer 3, 5, 8–11, 13, 14, 16, 20, 22–24

Understanding 1

Is angle θ an angle of elevation or depression? Explain how you know.

4 2

690

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A man standing at the top of the tower at point B, is looking at the ground at point C. a

Identify the angle of depression in the ﬁgure.

Identify the angle of elevation in the ﬁgure.

What can be said about the size of the two angles?

θ A

Using the given diagram:

Identify the angle of elevation.

How could you draw an angle of depression from A to C?

Which angle in the diagram is the same size as the angle of depression from A to C?

θ σ

Fluency 4

x is the angle of elevation to point A from point C. a

Find x, rounded to two decimal places.

Find y, rounded to two decimal places.

y x C

5 Ex 1

x is the angle of depression from point B to point C. a

Find y, rounded to two decimal places.

Find x, rounded to two decimal places.

25 B

D y

x 5

The person in the picture sights a pigeon above him. Find the angle of elevation, θ, rounded to two decimal places. 30 m

θ 23 m

A helicopter is 344 m away from its landing pad. If the angle of depression to the landing pad is 32°, ﬁnd x, the height of the helicopter above the ground, rounded to the nearest metre.

32° xm

344 m

Landing pad 13.03 Angles of elevation and depression mathspace.co

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The ﬁnal approach of an aeroplane when landing requires the pilot to adjust the angle of depression to 3°, as shown in the diagram. If the plane is 12 m above the runway and has d m until touchdown, ﬁnd d rounded to the nearest metre.

12 m 3° dm

A man stands at point A looking at the top of two poles. Pole 1 has a height 8 m and an angle of elevation of 34° from point A. Pole 2 has a height 25 m and an angle of elevation of 57° from point A. a

Find the distance from A to B, rounded to two decimal places.

Find the distance from A to C, rounded to two decimal places.

10 Ex 2

11 Ex 3

692

Pole 2

Pole 1

25 m

8m A

Find BC, the distance between the two poles in metres. Round your answer to one decimal place.

A fighter jet, flying at an altitude of 2000 m is approaching an airport. The pilot measures the angle of depression to the airport to be 12°. One minute later, the pilot measures the angle of depression again and finds it to be 19°. a

Find the distance AC, rounded to the nearest metre.

Find the distance BC, rounded to the nearest metre.

Find the distance covered by the jet in that one minute.

12° B

19°

C 2000 m

Airport

A room measures 6 m in length and 5 m in width. The angle of elevation from the bottom left corner to the top right corner of the room is 60°. Rounding all answers to two decimal places: a

Find the distance from one corner of the ﬂoor to the opposite corner of the ﬂoor.

Find the height of the room.

Find the angle of elevation from the bottom corner of the 6 m long wall to the opposite top corner of the wall.

Find the angle of depression from the top corner of the 5 m long wall to the opposite bottom corner of the wall.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reasoning 12

An airtraffic controller is communicating with a plane, which is flying at an angle of elevation of θ ° from the control tower. The plane is 10 369 m above the ground and is still 23 444 m from the runway. Viggo claims that the angle of depression that the pilot should fly is equal to 90° − θ °. Explain why Viggo is incorrect and correctly calculate the angle of depression that must be flown.

10 369 m

θ° 23 444 m

Fiona says that the angle θ is not an angle of elevation as it is not measured from the ground. Is Fiona correct? Explain your answer. 18 m

27 m

The angle of elevation of the sun causing a tree’s shadow to be 20 m is 43°. a

Early in the morning, and late in the afternoon when the angle of elevation of the sun is low, what happens to the length of the shadow?

What happens to the length of the shadow when the sun is directly overhead?

Find tan 90° on your calculator. How does this relate to your answer to part b?

h 43° 20 m

Problem-solving 15

Fred is on a ship and observes a lighthouse on a cliff in the distance. The base of the cliff is 651 m away from the ship, and the angle of elevation of the top of the lighthouse from Fred is 35°. a

If the top of the lighthouse is x m above sea level, ﬁnd x rounded to two decimal places.

If the lighthouse is 30 metres tall, how tall is the cliff that the lighthouse stands on?

Round your answer to two decimal places.

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Buzz is standing 250 m from a building and measures the angle of elevation of the top of the building to be 27°. a

If the difference in height between the top of the building and Buzz’s eye is h m, ﬁnd h rounded to two decimal places.

If Buzz’s eye is 172 cm from the ground, ﬁnd the height of the building, rounded to one decimal place.

From the top of a rocky ledge that is 274 m high, the angle of depression to a boat is 15°. If the boat is d m from the foot of the cliff, ﬁnd d rounded to two decimal places.

Amelia measures the angle of elevation to the top of a tree from a point, 29 m away from the base, to be 31°. Find the height of the tree, h, rounded to the nearest metre.

The angle of elevation from an observer to the top of a tree is 18°. If the distance between the tree and the observer is d m and the tree is known to be 3.53 m high, ﬁnd the value of d rounded to two decimal places.

At a certain time of the day a light post, 6 m tall, has a shadow of 5.8 m. If the angle of elevation of the sun at that time is θ °, ﬁnd θ rounded to two decimal places.

A ramp of length 311 cm needs to ascend at an angle between 10° and 20° for it to be safe to use.

311 cm

If the height of the ramp is 152 cm, ﬁnd the angle the ramp makes with the ground, rounded to two decimal places.

If the height of the ramp is 100 cm, ﬁnd the angle the ramp makes with the ground, rounded to two decimal places.

If the height of the ramp is 25 cm, ﬁnd the angle the ramp makes with the ground, rounded to two decimal places.

At which height(s) is the ramp safe?

height

Two hot air balloons are tied to the ground below, each at a different location. An observer at each location measures the angle of elevation to the opposite balloon. The observers are 1600 m apart as shown in the diagram: a

Calculate the difference in height between the two balloons. Round your answer to the nearest metre.

Find the direct distance between the two balloons. Round your answer to the nearest metre.

3.38°

1.22° 1600 m

Roald is standing at point P and observes two poles, AB and A CD, of different heights. P, B, and D are on horizontal ground:

From P, the angles of elevation to the top of the poles at A and C are 29° and 18° respectively. Roald is 16 m from the base of pole AB. The height of pole CD is 7 m.

694

Calculate the distance from Roald to the top of pole CD, B rounded to two decimal places.

Calculate the distance from Roald to the top of pole AB, rounded to two decimal places.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

29° 16 m

45° P

18°

A ski slope is inclined at 15° to the horizontal. At the bottom of the slope, there is a starting point labeled as point A. From point A, the angle of elevation to the top of a pine tree, labeled as point D, with its base at point B on the slope, is 28°. The pine tree grows vertically from point B. After skiing 80 m along the slope and passing point B, the skier arrives at point C. From point C, the angle of elevation to point D, the top of the pine tree, is 58°. a

Find the value of ∠ADC.

Compute the distance, DA, rounded to one decimal place.

Determine the height of the pine tree, DB, rounded to the nearest metre.

13.04 Bearings After this lesson, you will be able to… • use the starting direction and acute angle to describe the location of a point. • write bearings using mathematical notation. • apply bearings in 3D problems by drawing relevant triangles and using trigonometric ratios. • draw a diagram of a 3D situation, find distances and calculate angles related to bearings.

Compass bearings In surveying and navigation, bearings are used to help identify the location of an object. The four main directions of a compass are known as cardinal directions. They are north (N), east (E), south (S), and west (W).

N NW

NNW

NNE

WNW

NE ENE

E WSW

ESE SSW

SSE

In between each of these cardinal direction are intermediate directions of northeast (NE), southeast (SE), southwest (SW), and northwest (NW) that are 45° from the two cardinal directions closest. There are smaller divisions shown on the compass as well.

S A compass bearing describes the location of a point using: • The starting direction of either north or south; • The acute angle needed to rotate • The direction to rotate, east or west.

Starting direction

Direction of rotation

Angle rotated by

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To describe the position of point B from A we would say: “Starting at South, then rotate 53° towards East.”

North

We can write this mathematically as: S 53°E A

West

East 53°

South

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 13.04 to answer these questions. 1.

If A is between north and east, what do you notice about the way the the compass bearing of A from B is measured?

If A is between south and east, what do you notice about the way the the compass bearing of A from B is measured?

If A is between south and west, what do you notice about the way the the compass bearing of A from B is measured?

If A is between north and west, what do you notice about the way the the compass bearing of A from B is measured?

Example 1 N

Consider the point A.

35° O

a For the compass bearing of point A from O, which option shows the correct letters that should be written either side of the angle? A

N⬚E

Create a strategy

N⬚W

S⬚E

Consider whether point A is closer to the north or south directions.

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S⬚W

Apply the idea Since A is closer to north, we start at North, rotate clockwise towards the East direction. So the letters that should be written is in option A, N ⬚ E. b Complete the compass bearing of point A from O. N⬚E

Create a strategy

Apply the idea

Write the angle measure shown in the ﬁgure.

N 35°E

Idea summary Starting direction

Direction of rotation

A compass bearing describes the location of a point using: • • •

Angle rotated by

The starting direction of either north or south The acute angle needed to rotate The direction to rotate, east or west.

Three-figure (true) bearings A three-ﬁgure bearing are: • Measured from north (N) • Measured in a clockwise direction • Written using three ﬁgures T is often, but not always, used to describe a true bearing. If the angle measure is less than 100° it would be written as 040° or 040°T. To use true bearings to describe the location of a plane at point B from the airport at point A: • Place the centre of a compass on the starting point, in this case the airport. • Starting at North, rotate clockwise until we get to the line AB. • Write the angle as the true bearing of point B. The true bearing of B from A is 127° or 127°T.

North

West

127° East B

South

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The diagram describes the bearing of P from O. Rotating clockwise from North, we get an angle of 55°.

North

West

55°

P East

Since this measure is less than three digits, we put 0 in front so the true bearing of P is 055°. For the true bearing of O from P, the true bearing would be 180 + 55 = 235°.

South

Example 2 Determine the true bearing of point A from point O. a A

55° T

035° T

055° T

35° T

325° T

125° T

35° O

Create a strategy

Apply the idea

Measure the bearing clockwise from North, and write it using three digits.

The angle between the North direction and the line OA measured clockwise is 35°. So the true bearing is in option B, 035°T.

N 42° O

Create a strategy

Apply the idea

Subtract the given angle from 360°.

The angle between the North direction and the line OA measured clockwise is 360 − 42 = 318°. So the true bearing is 318°T.

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Example 3 Consider the following diagram: A

B 42°

52°

a Find the true bearing of point A from point C.

Create a strategy We can find the angle between AC and north. Then, we can subtract that from 360°.

Apply the idea

52°

To find the angle that AC makes with true north, we can extend the north line from point C to segment AB. This will create a right-angled triangle.

Recall the acute angles in a right-angled triangle sum to 90°, so the angle that AC makes with true north is 90 - 42 = 48°.

B 42° 48°

Starting from point C and rotating clockwise from North gives us an angle of 360 - 48 = 312°. This means the true bearing is 312°T.

b Find the compass bearing of point A from point C.

Create a strategy Determine if going from C to A is north or south and east or west.

Apply the idea From C to A it is north and west which means it is N 48°W.

Reflect and check Notice going from A to C would be the opposite direction and would be S 48°E.

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Example 4 The position of a ship S is given to be 20 kilometres from P, on a true bearing of 049° T. The position of the ship can also be given by its (x, y) coordinates.

N x y

049° 20 km

a If the ship’s x-coordinate is x, ﬁnd x rounded to one decimal place.

Create a strategy Use the sine ratio given by: sin θ =

Apply the idea Based from the given diagram, we are given θ = 49°, opposite = x, and hypotenuse = 20. Substitute the values

Multiply both sides by 20

Evaluate the multiplication

Evaluate and round

b If the ship’s y-coordinate is y, ﬁnd x rounded to one decimal place.

Create a strategy Use the cosine ratio given by: cos θ =

Apply the idea Based from the given diagram, we are given θ = 49°, adjacent = y, and hypotenuse = 20. Substitute the values

Multiply both sides by 20

Evaluate the multiplication

Evaluate and round

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Idea summary True bearings are: • •

Measured from north (N) Measured in a clockwise direction

• •

Written using three ﬁgures Usually written with a T at the end

Bearings in 3D A problem in three dimensions may use bearings to describe the situation. It is helpful to ﬁrst draw a diagram if one is not given to us and then solving as usual for problems in 3D by drawing out triangles in 2D.

Example 5 A miner is mining 500 m below the surface. If you walk 300 m north and then 400 m west from the oﬃce, you will be directly above the miner. a Draw a diagram of the situation in 3D and the related triangles in 2D.

Create a strategy We can start by drawing the surface using the bearings and then placing the miner below the surface. It is helpful to label the vertices.

Apply the idea The whole scenario on one diagram.

N 400 m W

N 300 m Office

500 m

We can break this into two right-angled triangles which have a common side of WO. W

400 m

300 m 500 m O M

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b Find the distance from the oﬃce to the point on the surface directly above the miner.

Create a strategy We want to ﬁnd the length OW. We have the other two lengths on a right-angled triangle, so can use Pythagoras’ theorem.

Apply the idea For triangle ONW, we know two sides and have one missing side, let’s use Pythagoras’ theorem to ﬁnd OW. Pythagoras’ theorem

Substitute known values

Evaluate the indices and addition

Take the square root of both sides

Simplify

The distance from the oﬃce to the point on the surface above the miner is 500 m.

Reflect and check We may have noticed this is the Pythagorean triple, 3, 4, 5 multiplied by 100.

c Find the direct distance from the miner to the Oﬃce, to the nearest metre.

Create a strategy We want to ﬁnd the length OM. We have the other two lengths on a right-angled triangle, so can use Pythagoras’ theorem.

Apply the idea For triangle OWM, we know two sides and have one missing side, let’s use Pythagoras’ theorem to ﬁnd OM. Pythagoras’ theorem

Substitute known values

Evaluate the indices and addition

Take the square root of both sides

Simplify the surd

Evaluate using a calculator and round

The direct distance from the oﬃce to the miner is 707 m.

Reflect and check If we use this rounded value instead of the exact value of introduced.

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, there will be some error

d At what angle of depression from the oﬃce would you dig to reach the miner.

Create a strategy We are now trying to ﬁnd ∠WOM. Since this is a right-angled triangle, we can use trigonometric ratios to ﬁnd this angle.

Apply the idea We know all of the side lengths in this triangle, so we can use any pair. However, one of them is rounded, so we will use the two exact values.

θ 500 m

The exact value sides are the opposite and adjacent.

500 m

707 m

Write the ratio

Substitute known values

Use inverse tan ratio

Evaluate using a calculator

Reflect and check Instead of using trigonometric ratios, we could have recognised that this is an isosceles rightangled triangle, so both acute angles would be 45°.

Idea summary Although bearings refer to directions on the surface, we may use them in 3D problems where there is point above or below ground which is involved. We can draw relevant triangles in 2D to solve for one problem at a time.

Practice questions Choose your question path Explorer 1–6, 8, 9, 11–13, 15, 18–20, 21abc, 24, 25, 27, 29, 30

Adventurer 1–6, 8, 9, 11–13, 16, 18–20, 21bcd, 22, 24–27, 30

Trailblazer 3–5, 7, 8, 10, 14, 16, 17, 20, 21cd, 22–24, 26–28, 31, 32

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Understanding 1

Write the compass bearing for point P from point O. a

Write the true bearing of P from point O. a

N P N 26° W

O P

S 64° W

N 64° W

P S

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S 26° W S

Which of these diagrams correctly displays WNW? A

N NW

B NE

WNW

N NW

WNW W

WNW

What is the equivalent true bearing for these compass bearings? a

Southwest

Northwest

West-Northwest

South-Southeast

Northeast

Southeast

Fluency Ex 1, 2

Find the true bearing of point A from O.

Find the compass bearing of point A from O.

30° O

41° A

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A 40° O

O 35° A

Determine the true bearing for: a

Point B from point A.

Point C from point A.

Point C from point B.

B 37°

48°

Determine the true bearing for: a

Point A from point B.

Point A from point C.

Point B from point C.

B 47°

38°

Ex 3

Find the true bearing of point A from point C.

Find the compass bearing of point A from point C.

B 49°

39°

Find the true bearing of point B from point C.

Find the compass bearing of point B from point C.

B 37°

45°

Find the true bearing of point C from point A.

Find the compass bearing of point C from point A.

B 42°

53°

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Find the true bearing of point C from point B.

Find the compass bearing of point C from point B.

B 38°

49°

If a person walks at a compass bearing of SW, and then turns around to go back in the direction they came, what direction are they heading now?

An aeroplane sets off on a bearing of 051°T. After some time, it needs to turn back and head to its original position. On what true bearing must it travel?

A boat sets off on a bearing of 125°T. After some time, it needs to turn back and head to its original position. On what true bearing must it travel?

If the bearing of the clearing from the town is a°, ﬁnd a rounded to the nearest degree.

N 5.9 km 3.2 km

Clearing

a°

O Town

Luke sailed for 116 km on a bearing of 231°.

Find w, the distance in kilometres he has sailed west from his starting point, rounded to one decimal place. w 231° 116 km

Shortly after take-off, a plane is 42 km south and 57 km west of the airport in Sydney that it left from. a

Find the size of the angle marked b, to one decimal place.

Find the compass bearing of the plane from the airport.

What is the true bearing of the plane from the airport?

O b°

Plane

Sydney

42 km

57 km

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18 Ex 4

20 Ex 5

The position of a ship S is given to be 20 km from P, on a true bearing of 049° T. The position of the ship can also be given by its (x, y) coordinates.

Find the ship’s x-coordinate rounded to one decimal place.

Find the ship’s y-coordinate rounded to one decimal place.

049°

N B 1000 m

C 062° d m

From a point 15 m due north of a tower, the angle of elevation of the tower is 32°. Find the height of the tower h. Round your answer to two decimal places.

Find θ, the angle of elevation of the tower at a point 20 m due east of the tower. Round your answer to the nearest degree.

20 km

On an orienteering course, Beth runs 1000 metres north from point A to point B, then turns east and runs to point C. If the true bearing of C from A is 062°T and the direct distance Beth must run to get back to point A is d metres, ﬁnd d to the nearest metre.

North

32° hm 15 m

20 m

θ East

Reasoning 21

708

Given that a, b, c and d are angles between 0° and 90°: a

If a compass bearing is N a°W, write an expression for the true bearing.

If a compass bearing is S b°W, write an expression for the true bearing.

If a compass bearing is S c°E, write an expression for the true bearing.

If a compass bearing is S (90 − a) °E, write an expression for the true bearing.

xplain why the overall horizontal distance from A E to C is equal to 12 cos 45° + 16 cos 30°.

Write an expression using sin which will calculate the overall vertical distance from A to C.

Use Pythagoras’s theorem to calculate the direct distance from A to C.

N 45° 12 km

Find the bearing of A from C.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

N 60° B

16 km

Find the direct distance from A to C.

Find the bearing of A from C.

N N

145°

B 72°

600 m

800 m

Problem-solving 24

During a group hike, a hiker walked 11 km NE of base camp. In the evening the hiker was separated from the group and after a night lost in the wilderness, the hiker was found 11 km NW of the base camp. What compass bearing must the hiker have followed during the night?

During a rescue search, a helicopter ﬂew west from point X to point Y, then changed course and ﬂew 10.7 km north to point Z. If point Z is on a bearing of 335° T from point X:

N Z

Find the size of ∠Y XZ.

Find b, the distance from point Y to point X, rounded to one decimal place.

Calculate d, the distance that the helicopter must ﬂy between point Z and point X, rounded to one decimal place.

In remote locations, photographers must keep track of their position from their base. One morning a photographer sets out from base, B, to the edge of an ice shelf, S, on a bearing of 055°. She then walked on a bearing of 145° to point P, which is 916 m due east of base.

335°

145° S B

916 m

N B

Find the size of the angles: i

A ﬁshing boat in search of large schools of ﬁsh sails from point A for 26 km on a bearing of 205° to point C. It then sails a further 24 km on a bearing of 226° to point E. Rounding all answers to one decimal place: a

X b km

Rounding all answers to one decimal place: a State the size of ∠BSP. b Find the distance BS. c Find the distance SP. d If the photographer walks directly back to her base from point P, determine the total distance travelled. 27

d km

10.7 km

∠BAC

∠DCE

Find the distance the boat is from its starting point as a straight line from A to E. Find θ °, the true bearing of the boat from its starting point, to the nearest degree.

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A drone travels due east for 2.2 km and then travels on a bearing of S 32°E for 5.9 km. a

Given that the compass bearing is a°, write the bearing needed to return to the starting point, in terms Start of a.

Find x, the distance between the end point to the start point of the drone’s ﬂight. Round your answer to two decimal places.

c 29

N 2.2 km

x km

32° 5.9 km

Find a rounded to the nearest degree.

A yacht sailed in a direction so that its ﬁnal position was 248 km west and 225 km south of its starting point. a

Draw a diagram to represent the situation.

Find the true bearing on which the yacht sailed, rounded to one decimal place.

Find the total distance the boat has sailed, rounded to one decimal place.

Three television presenters are practising their navigation skills before heading off on an expedition to a remote location. Belinda at point B is positioned 17.6 m south of Amelia at point A. Carl at point C is due east of Belinda and on a bearing of S 38°E from Amelia. a

Draw a diagram to represent the situation.

If Amelia and Carl are d m apart, ﬁnd d to one decimal place.

A yacht travelled a distance of 30 km from port A at a bearing of 110°, then another 40 km to port C at a bearing of 130°. What is the distance between ports A and C?

A and B are two positions on level ground. From an advertising balloon at a vertical height of 740 m, A is observed at a bearing of 043° and B at a bearing of 133°. The angles of depression of A and B, as viewed from the balloon, are 30° and 20° respectively. Rounding all answers to the nearest metre:

710

Find the distance from point A to the point directly below the advertising balloon.

Find the distance from point B to the point directly below the advertising balloon.

Find the distance between points A and B.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

13.05 Optional: Sine and cosine rules After this lesson, you will be able to… • utilise the sine rule to find unknown sides and angles. • distinguish between the two forms of the sine rule and when to use each. • identify the conditions that lead to multiple solutions when using the sine rule. • define and explain the cosine rule and its applications in solving problems related to non-right-angled triangles. • use the cosine rule to find unknown sides or angles.

Sine Rule The sine rule is used in non-right-angled triangles that involve two sides and two angles. We are able to use this rule if we want to ﬁnd an unknown side or angle.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 13.05 to answer these questions. 1.

Which side (shortest, middle, longest) is across from the largest angle? Which side is across from the smallest angle?

What do you notice when you change the value of the angles (using angle C)? What changes? What stay the same?

What happens to the value of sin (C) when it is C = 90°?

What happens when C is 90° or 180°?

The sine rule, sometimes called the law of sines or sine law, tells us that:

A, B, C

the measures of the angles in the triangle

a, b, c

the sides in the triangle

We can also take the reciprocal of each fraction to give the alternate form:

B a

c A

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This alternate form is used to find angles. Suppose we know the side lengths a and b and the angle B, and we want to ﬁnd the angle A. Using the form of the sine rule sin

, we can solve for A.

We can make sin A the subject by multiplying both sides by a:

Then, to ﬁnd B itself, we need to take the inverse on both sides:

Example 1 Find the side length a using the sine rule. Round your answer to two decimal places.

a 69° 18

33°

Create a strategy Use the sine rule:

Apply the idea Substitute b = 18, A = 33, B = 69

Multiply both sides by sin 33°

Evaluate

Example 2 Find the value of the acute angle A using the sine rule. Round your answer to two decimal places.

A° 81 50° 98

Create a strategy Use the sine rule:

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Apply the idea Substitute a = 98, b = 81, B = 50

Multiply both sides by 98

Evaluate

Take the inverse sine of both sides

Evaluate

Idea summary The sine rule can be written in two forms. The ﬁrst form is best when solving for an angle. The second form is best when solving for a side.

A is an angle of a triangle opposite side a B is an angle of a triangle opposite side b C is an angle of a triangle opposite side c We will generally only use two angle-side pairs when solving. We need one side-angle pair both known and one other piece of information.

Ambiguous case of the sine rule When we are solving for angles using the sine rule, there may be two possible solutions.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 13.05 to explore the number of possible solutions when the information given.

The ambiguous case occurs when it is possible to have multiple solutions with the information given. When using the sine rule, it is possible to have no solution, one solution or two solutions: • Case 1: No triangle exists This is the case when an error occurs in the calculator while solving for the unknown angle. This means that no such triangle with the given side lengths and angle exists.

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• Case 2: Exactly one triangle exists This is the case when the sum of the given angle and the supplement to the solution calculated is equal to or exceeds 180°. • Case 3: Two possible triangles exist This is the case when the sum of the given angle and the supplement to the solution are less than 180°. Then, the possible triangles are as follows: • A triangle that includes the calculated solution • A triangle that includes the supplement to the calculated solution C1 b A

C2 b

a B1

Possibility 1

a B2 = 180° − B1

Possibility 2

To ﬁnd the second possible solution, subtract the ﬁrst value of B that you found from 180°.

Example 3 Find the value of x in degrees, given that x is obtuse. Round your answer to two decimal places.

25° 18.8 x A 12.6 C

Create a strategy Use the sine rule:

Apply the idea Substitute A = x, a = 18.8, b = 12.6, B = 25

Multiply both sides by 18.8

Evaluate

Take the inverse sine of both sides

Evaluate and round

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To ﬁnd the obtuse angle x we subtract the acute angle from 180°: x = 180 − 39.09 = 140.91°

Subtract 39.09 from 180 Evaluate

Example 4 If possible, solve the triangle where a = 85, b = 93, and ∠A = 61°. Round your answers to two decimal places.

Create a strategy Draw and label a diagram of the triangle; it does not need to be drawn to scale. B

85 61° A

From this diagram we can see that we were given two sides and their non-included angle. This is an example of the ambiguous case and when we solve we may ﬁnd one, two, or no solutions to this problem. We are given a pair of opposite sides and angles with side a and angle A in addition to side b. We will set up the equation:

to ﬁnd ∠B.

Apply the idea Sine rule

Substitute given values

Multiply both sides by 93

Apply the inverse sine to both sides

Evaluate and round

Now that we’ve found an acute solution to the problem, we need to ﬁnd its supplement and check if this will produce a valid second solution.

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The supplement of ∠B1 = 73.12° is ∠B2 = 180° − 73.12° = 106.88°. If we add these two angles together we get 61° + 106.88° = 167.88° which does not exceed the maximum 180° possible in a triangle. That means this triangle has two solutions. Solution 1: ∠B1 = 73.12° So far we have a = 85, b = 93, ∠A = 61° and ∠B1 = 73.12°. Use the triangle angle sum to solve for the missing angle, ∠C1 = 180° − (61° + 73.12°) = 45.88° and use the sine rule to ﬁnd the missing side, c1: Sine rule

Substitute known values

Multiply both sides by sin 45.88°

Evaluate and round

The three side lengths are a = 85, b = 93, and c1 = 69.77 and the three angles are ∠A = 61°, ∠B1 = 73.12° and ∠C1 = 45.88°. Solution 2: ∠B2 = 106.88° For the second solution we have a = 85, b = 93, ∠A = 61° and ∠B2 = 106.88°. Use the triangle angle sum to solve for the missing angle, ∠C2 = 180° − (61° + 106.88°) = 12.12° and use the sine rule to ﬁnd the missing side, c2: Sine rule

Substitute known values

Multiply both sides by sin 12.12°

Evaluate and round

The three side lengths are a = 85, b = 93, and c2 = 20.40 and the three angles are ∠A = 61°, ∠B2 = 106.88° and ∠C2 = 12.12°. The diagram shows a visual of both solutions: B1 c1

85 c2

61° A

61°

85 93

Reflect and check There will be no solution, or a calculator error, if we attempt to take the inverse sine of a value greater than 1. There will be exactly one solution if the solved angle is acute, but its supplement is too large to create a second valid triangle.

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Idea summary Sine rule (ambiguous case) If you are trying to find an angle using the sine rule (with two known sides and a known angle), use the following rule to find one value of B:

If the side opposite the known angle is the shorter side, you are in the ambiguous case. Subtract the ﬁrst value of B that you found from 180° to ﬁnd another possible solution. Check that the second solution is a valid solution using the sum of angles in at triangle.

Cosine Rule The cosine rule rule is used in non-right-angled triangles that involve three sides and one angle. We are able to use this rule if we want to ﬁnd an unknown side or angle. Two side lengths and one angle, opposite the unknown side, are known. x 10

32° 4

Three side lengths are known, and one angle is unknown. x°

The cosine rule says: c2 = a2 + b2 − 2ab cos C

b2 = a2 + c2 − 2ac cos B

a2 = b2 + c2 − 2bc cos A b

To ﬁnd a missing angle using the cosine rule, we need to know all three side lengths. When looking to solve for an unknown angle, the equation can also be rearranged and written as

13.05 Optional: Sine and cosine rules mathspace.co

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Example 5 Find the length of a using the cosine rule. Round your answer to two decimal places.

C a

13 A

33° 30

Create a strategy Use the cosine rule a2 = b2 + c2 − 2bc cos A.

Apply the idea Write the formula

Substitute b = 13, c = 30, A = 33

Evaluate

Take the square root of both sides and round

Example 6 Find the value of angle w using the cosine rule. Round your answer to two decimal places.

32 25 w 19

Create a strategy Use the rearranged formula of the cosine rule given by to C.

, where w corresponds

Apply the idea Write the formula

Susbtitute the values

Evaluate

Take the inverse cosine of both sides

Evaluate and round

718

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary If we have a problem involving two sides and the included angle of a non-right angled triangle, we can use the cosine rule to ﬁnd the unknown side length:

c2 = a2 + b2 − 2ab cos C c

is the side opposite angle C

a is another side length b is another side length If we have a problem involving 3 side lengths of a non-right angled triangle, we can use the cosine rule to ﬁnd the unknown angle:

is the side opposite angle C

a is another side length b is another side length

Practice questions Choose your question path Explorer 1, 3, 4, 5acdg, 6, 7, 9, 10ab, 11abdg, 13a, 14ab, 15, 16ace, 18, 20abce, 21, 23, 24, 29, 30, 32, 39, 40

Adventurer 1, 3, 4, 5aceg, 7–9, 10ab, 11bdfj, 13b, 14cd, 15, 16bdf, 19, 20cdef, 21, 23–25, 30–33, 37–39

Trailblazer 2–4, 5bdf, 8, 9, 10d, 11ij, 12, 14d, 16ef, 17, 19, 20dfg, 22, 25–28, 33–37, 41–43

Understanding 1

Which strategy would you use to ﬁnd the unknown: Pythagoras’ theorem, trigonometric ratios, sine rule or cosine rule? a

69° 18

33° 11

13.05 Optional: Sine and cosine rules mathspace.co

719

32 25

27° 19

For each triangle, determine if there is enough information to ﬁnd all the remaining sides and angles in the triangle using the: i

Sine rule

Cosine rule

Three sides are known.

wo of the angles and the side included T between them are known.

B c

A c

Two of the angles and a side not

included between them are known.

Two of the sides and an angle included between them are known.

C b

A b

Write an equation relating the sides and angles using the sine rule.

Write an equation relating the sides and angles using the cosine rule.

R r P

720

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fluency 5 Ex 1

Find the unknown side, rounded to two decimal places. a

b 13

64° 21

63°

28°

35°

d 104°

4.5 m

a 39°

25°

310 cm

25° a

20 D 41°

47°

C x

64°

42°

39°

12 55°

a 35°

48° 15

Find the length, rounded to two decimal places. a

K 46°

37° 531.36 km

97° J

For triangle ABC, where BC = 37.8 m, ∠BAC = 57° and ∠ACB = 42°, calculate the length of side AB, rounded to two decimal places.

13.05 Optional: Sine and cosine rules mathspace.co

721

For triangle PQR, where QR = 4.79 mm, ∠QPR = 33° and ∠PRQ = 124°, calculate the length of side PQ, rounded to two decimal places.

For triangle ABC with ∠C = 72.53° and ∠B = 31.69°, and one side length a = 5.816 m, find: a

∠A, rounded to two decimal places.

The length of side b, rounded to three decimal places.

The length of side c, rounded to three decimal places.

A c b

B a

Solve for x, given that all equations relate to acute-angled triangles, rounded to two decimal places. a

11 Ex 2, 3

For each triangle, ﬁnd the value of the variable using the sine rule. Round your answers to one decimal place. a

B 9 56°

A C

62° 11

B x C

d 39°

48°

75 m

9 cm 11.2 cm

x° 72 m

6.58

34°

x 58

116°

41°

287

275 B

722

3.69

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

25° 17

14 C

x 110°

30°

5.52 cm

2.34 cm

18.1 x A 12.4 C

For triangle MNP, where NP = 100 m, MN = 204 m and ∠NPM = 58°, calculate the size of the acute angle ∠NMP, rounded to the nearest degree.

For each triangle, calculate ∠B rounded to the nearest degree.

Ex 4

△ABC where ∠A = 57°, side a = 156 cm and side b = 179 cm

△ABC where ∠A = 48°, side a = 2.7 cm and side b = 1.9 cm

Solve for θ, given that all equations relate to acute-angled triangles, rounded to two decimal places. b

a 15

Which angle would need to be given to find the length of AC using the cosine rule?

B 31.05 cm 38.44 cm

16 Ex 5

For each triangle, ﬁnd the value of the variable using the cosine rule. Round your answers to two decimal places. a

C 11

19.4 A

54° 261

25°

q B

23.3

131 A

33°

16 53° 24

13.05 Optional: Sine and cosine rules mathspace.co

723

C 24°

42°

In △ABC,

, ﬁnd the exact length of side AB in centimetres.

A 7 cm

B 9 cm

Triangle QUV has side lengths v = 7, u = 10 and an angle at Q such that cos value of q, where q is the length of the side opposite angle Q.

Triangle QUV has side lengths q = 63, u = 36 and ∠V = 116°. Find the value of v, where v is the length of the side opposite ∠V. Round your answer to two decimal places.

For each triangle, ﬁnd the value of the variable in degrees. Round your answers to two decimal places.

Ex 6

b 25

31 22

w°

B° 17

12 7

11.2

w° 18.1

A°

5.2

9.43

7.8

724

A°

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A°

. Find the

45 4

8 C°

B°

Triangle QUV has side lengths q = 5, u = 10 and v = 12. Find the size of ∠V, where V is the angle opposite the side of length v. Round your answer to the nearest degree.

Triangle QUV has side lengths v = 23.5, u = 15.6 and q = 10.0. Find the size of ∠Q, where Q is the angle opposite the side of length q. Round your answer to the nearest degree.

Reasoning 23

Blain and Odett are trying to ﬁnd side c in △ABC where a = 15 m, b = 12 m and ∠C = 34°. Review the work, explain the errors (if any) and determine the correct value for the side c. Blain’s work:

Odett’s work:

A rhombus of side length 10 cm has a longer diagonal of length 16 cm. a

Describe two ways you could solve for d, the length of the shorter diagonal of the rhombus.

Solve for d.

θ° 16 cm 10 cm d x°

A teacher is writing exam questions for her maths class. She draws a triangle, labels the vertices A, B and C and labels the opposite sides a = 5, b = 8 and c = 15 respectively. She wants to ask students to ﬁnd the size of ∠A. Explain what the problem is with her question.

Use the cosine rule to prove that any angle in a triangle with three equal sides is equal to 60°.

13.05 Optional: Sine and cosine rules mathspace.co

725

After two meteoroids collide at point A, one starts travelling in the direction of point B, while the other starts travelling in the direction of point C, with an angle of 53° between the two directions. The meteoroid projected in the direction of B is moving at a speed of 7860 km/h, while the other is moving at a speed of 10 170 km/h.

53°

Describe a plan to ﬁnd the distance between the two meteoroids 29 minutes after the collision.

Find the distance between the two meteoroids 29 minutes after the collision, rounded to two decimal places.

Write an equation to ﬁnd the distance between the two meteoroids m minutes after the collision. Let d be the unknown distance.

A pole is seen by two people, Jenny and Matt: a

Matt is x m from the foot of the pole. Find x rounded to the nearest metre.

Find the height of the pole h rounded to the nearest metre.

Pole

hm Matt

Jenny

34°

100 m

52°

58° xm

Problem-solving 29

Find the value of θ, rounded to two decimal places.

15 cm 80°

62° 42°

35 cm

726

A radio signal is sent from a transmitter at tower T, via a satellite S, to a town W, as shown in the diagram. The town is 526 kilometres from the transmitter tower. The signal is sent out from the transmitter tower at an angle of 18°, and the town receives the signal at an angle of 26°.

18°

26°

Find ∠WST.

Find the distance, SW, that the signal travels from the satellite to the town. Round your answer to the nearest kilometres.

If the satellite is h kilometres above the ground, ﬁnd h. Round your answer to two decimal places.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

526 km

Two hikers on a large flat national park walk 10 km from point D on a bearing of S 40°E to a point E. They then walk on a bearing of N 30°E to point F. If the distance DF is 15 km, find the bearing of D from F.

Two wires help support a tall pole. One wire forms an angle of 36° with the ground and the other wire forms an angle of 70° with the ground. The wires are 29 m apart. a

a h

70°

36°

Find a, the angle made between the two wires at the top of the pole.

29 m

Find d, the length of the longest wire in metres. Round your answer to two decimal places.

Find h, the height of the pole. Round your answer to two decimal places.

Mae observes a tower at an angle of elevation of 12°. The tower is perpendicular to the ground. Walking 67 m towards the tower, she ﬁnds that the angle of elevation increases to 35°. D

35°

12° A

Find ∠ADB.

Find the length of the side a. Round your answer to two decimal places.

Find the height, h, of the tower. Round your answer to one decimal place.

Prove the sine rule using these steps:

For △ACD, ﬁnd an expression for sin A.

For △BDC, ﬁnd an expression for sin B.

Solve for x in the equation from part (b).

Substitute your expression for x into the equation from part (a) to prove the law of sines.

How many triangles are possible given this information? Explain your reasoning. Assume a is the side length opposite ∠A, b is the side length opposite ∠B.

a = 29, b = 28, and ∠A = 37°

a = 38, b = 29, and ∠B = 45°

a = 54, b = 65, and ∠A = 60°

The sides of a triangle are in the ratio 4: 7: 8. Find the size of x, the smallest angle in the triangle. Round your answer to the nearest degree.

13.05 Optional: Sine and cosine rules mathspace.co

727

For triangle △QUV where the side lengths q, u and v appear opposite the angles Q, U and V. If q = 16, sin V = 0.5 and sin Q = 0.8. Solve for v.

Two trains leave a station at the same time. Train H travels at an average speed of 80 km/h on a bearing of 035°T, while Train I travels at an average speed of 90 km/h on a bearing of 120°T. Find the distance between the two trains after 4 hours. Round your answer to the nearest kilometre.

During football training, the coach marks out the perimeter of a triangular course that players need to run around. The diagram shows some measurements taken of the course, where side length a = 14 m: a

Find ∠A.

Find the length of side b, rounding your answer to two decimal places.

Find the length of side c, rounding your answer to two decimal places.

Each player must sprint one lap and then jog one lap around the triangle. This process is to be repeated 3 times by each player.

55°

87° a = 14 m

If Tara can run 280 m/min, and can jog at half the speed she runs, calculate the time this exercise will take her, rounding your answer to one decimal place. 40

In a game of pool, a player has one last ball to sink into a corner pocket. The player must use a cue (stick) to hit the white ball so that it knocks the purple ball into the corner pocket.

1.5 cm

The player judges that the white ball is about 1.5 cm away from the corner pocket, and that the distance between the two balls is about 0.9 cm, while the purple ball is also 0.9 cm from the corner pocket.

0.9 cm

θ 0.9 cm

The player wants to ﬁnd the angle θ which is needed to knock the white ball against the purple ball. Round your answer to two decimal places.

Two straight paths to the top of a cliff are inclined at angles of 23° and 20° to the horizontal: a

728

If path A is 108 m long, ﬁnd the height h of the cliff, rounded to the nearest metre.

Find the length x of path B, rounded to the nearest metre.

Let the paths meet at 49° at the base of the cliff. Find their distance apart, y, at the top of the cliff, rounded to the nearest metre.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

xm 108 m Path A 23°

Path B

49°

20°

Two buoys, A and B, are observed from a lookout at the top of a 130 m high cliff. The bearing of buoy A is 337° and its angle of depression is 3°. The bearing of buoy B is 308° and its angle of depression is 5°: a

5°

Find the distance, rounded the nearest metre, from buoy A to the base of the cliff.

3°

Find the distance from buoy B to the base of the cliff, rounded to the nearest metre.

Find the distance between the two buoys, rounded to the nearest metre.

308°

130 m 337°

B A

From a point A, due north of a tower, the angle of elevation to the top of the tower is 51°. From a point B, which is on a bearing of 120° from the tower and is 100 m away from point A, the angle of elevation is 25°. Find the height of the tower rounded to the nearest metre.

13.05 Optional: Sine and cosine rules mathspace.co

729

Chapter 13 review 1

Find the length of the unknown side as an exact value: a

A right-angled triangle has a hypotenuse of length 26 and another side of length 24.

A right-angled triangle has a one side of length 6 and another side of length 5.

T F

Could a triangle with side lengths of 6 mm, 8 mm, and 13 mm be a right-angled triangle? Explain.

Rachel hikes south of her starting point for 834 m, then turns and walks 691 m east, before stopping for a lunch break. She then travels south again for 427 m before arriving at her ﬁnal destination.

A R

What is the shortest distance between where Rachel started and where she stopped for lunch? Round your answer to two decimal places.

What is the shortest distance between where Rachel started and where she ﬁnished her journey? Round your answer to two decimal places.

(Optional) A rectangular prism has dimensions as labelled on the diagram. Find each length in exact form: a

4 D

C 15

730

F 6

(Optional) A triangular prism has dimensions as shown. Explain the process for ﬁnding: a

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) A can has a height of 14 cm and a radius of 5 cm. Find L, the length of the longest stick that can ﬁt into the can. Round your answer down to the nearest cm, to ensure it ﬁts inside the can.

(Optional) Consider the cone with slant height of 10 m and perpendicular height of 8 m: a

Find the length of the radius, r.

Find the diameter of the cone’s base.

29° 5m

T F

A R

Find the value of f, rounded to two decimal places: a

sin θ

27°

10 cm

For △DEF, ﬁnd the exact value of : a

10 m

f cm

θ 13 F

Jason is ﬂying a kite that is attached to the end of a 25.2 m length of string. The angle between the string and the vertical is 40°. If Jason is holding the string 1.2 m above the ground: a

Find the angle of elevation from Jason to the kite.

Find x, rounded to two decimal places.

Find the height h of the kite above the ground, rounded to two decimal places.

25.2 m

40° hm

1.2 m

Chapter 13 review mathspace.co

731

Find the value of the variable(s), rounded to the nearest whole number: a

20 cm

58° 23 mm

h mm

67°

x cm

10 cm

(Optional) Consider the rectangular prism: a

Find the length x.

Find the length of the prism’s diagonal y.

Find angle θ to the nearest degree.

y cm 8 cm

T F

x cm θ

12 cm

(Optional) A, C, and X are three points in a horizontal plane and B is a point vertically above X as shown:

A R

Find these lengths to the nearest metre: a

9 cm B

37°

45°

Find the length of the edge AE using two different methods.

If θ is the measure of the surface ∠BEA, ﬁnd θ rounded to the nearest degree.

86 m

(Optional) A pyramid has a square base with side length 8 m and a vertical height of 25 m. a

25 m D A C 8m B

The person in the picture sights a pigeon above him. Find the angle of elevation, θ, to two decimal places.

24 m

θ 17 m

732

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

An aeroplane is ﬂying at an altitude of 3000 m and approaching an airport. The pilot measures the angle of depression to the airport to be 14°. One minute later, the pilot measures the angle of depression again and ﬁnds it to be 16°. a

Find the distance AC, to the nearest metre.

Find the distance BC, to the nearest metre.

Find the distance covered by the jet in that one minute, to the nearest metre.

14° B

16°

3000 m

Airport

At a certain time of the day a light post, 5 m tall, has a shadow of 9.4 m. If the angle of elevation of the sun at that time is θ°, ﬁnd θ rounded to two decimal places.

Dave is standing 270 m from a building and measures the angle of elevation of the top of the building to be 19°.

If the difference in height between the top of the building and Buzz’s eye is h m, ﬁnd h rounded to two decimal places.

If Dave’s eye is 161 cm from the ground, ﬁnd the height of the building rounded to one decimal place.

Find the true bearing of point A from O.

Find the compass bearing of point A from O.

N A

35° O

T F

A R b

Find the true bearing of point A from point C.

Find the compass bearing of point A from point C.

O 38° A

B 38°

49°

If a person walks at a compass bearing of NE, and then turns around to go back in the direction they came, what direction are they heading now?

A boat sets off on a bearing of 121 °T. After some time, it needs to turn back and head to its original position. On what true bearing must it travel?

Chapter 13 review mathspace.co

733

Uri sailed for 155 km on a bearing of 215°. N

w 215° 155 km

Find w, the distance in kilometres he has sailed west from his starting point, rounded to one decimal place.

T F

During a group hike, a hiker walked 12 km NE of base camp. In the evening the hiker was separated from the group and after a night lost in the wilderness, the hiker was found 12 km NW of the base camp. What compass bearing must the hiker have followed during the night?

(Optional) Find the unknown, rounded to two decimal places. a

41°

47°

A R

63°

(Optional) For each triangle, calculate ∠B to the nearest degree. a

△ABC where ∠A = 48° side a = 2.7 cm and side b = 1.9 cm

△ABC where ∠A = 57° side a = 156 cm and side b = 179 cm

(Optional) To use the cosine rule to ﬁnd the length of AC, which angle would need to be given? B 31.05 cm 38.44 cm A

734

28°

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) Find a using the cosine rule. Round your answers to two decimal places. C 11

33° A

(Optional) Triangle ABC has side lengths a = 83, b = 45 and c = 110. Find ∠C. Round your answer to the nearest degree.

T F

A R

Chapter 13 review mathspace.co

735

14 Measurement Big ideas • Physical objects can be modelled with 3D geometric figures whose properties can be applied to solve real-world problems. • Estimation and accuracy are fundamental concepts for practical decision-making and interpreting measurements across diverse real-world contexts. • Scale factors and ratios provide a framework for comparing and relating measurements, dimensions and volumes across different scales and contexts.

Curriculum Outcomes VC2M10A05 Substitute values into formulas to determine an unknown and rearrange formulas to solve for a particular term VC2M10M01 Solve problems involving the surface area and volume of composite objects using appropriate units VC2M10M04 Use mathematical modelling to solve practical problems involving direct and inverse proportion and scaling of objects; formulate problems and interpret solutions in terms of the situation, including the impact of measurement errors on the accuracy of results; evaluate and modify models as necessary, and report assumptions, methods and findings VC2M10N01 Recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations VC2M10AM01 Solve problems involving surface area and volume of right pyramids, right cones, spheres and related composite solids

Chapter outline 14.01 14.02 14.03 14.04

Surface area of prisms and cylinders Volume of prisms and cylinders Surface area and volume of composite solids Estimation and accuracy Chapter 14 review

738 748 758 770 779

If you unrolled a tennis ball, how big do you think its surface area would be? Roughly the same as a playing card, a piece of toast, or this textbook?

14.01 Surface area of prisms and cylinders After this lesson, you will be able to… • find the surface area of right prisms. • find the surface area of cylinders. • solve problems involving the surface area of right prisms and cylinders.

Surface area of right prisms Surface area is the total area of the surface of the solid. To ﬁnd the surface area of a prism, identify the shapes of the different faces, and then add the area of all faces.

Example 1 Consider the rectangular prism with a width, length and height of 5 m, 7 m and 15 m respectively.

5m 7m

Find the surface area. 15 m

Create a strategy We need to add the areas of the 3 pairs of equal rectangular faces: front and back, left and right, and top bottom.

Apply the idea Front and back = 2 × (15 × 5) 2

= 150 m

Sides = 2 × (15 × 7) 2

= 210 m

Top and bottom = 2 × (7 × 5) 2

= 70 m

Evaluate Multiply the area by 2 Evaluate Multiply the area by 2 Evaluate

Surface area = 150 + 210 + 70 2

= 430 m

738

Multiply the area by 2

Add the areas Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 2 The dimensions of a trapezoidal prism are given in the diagram (not to scale):

10 cm 6 cm

6 cm

5 cm 8 cm

14 cm

Find the surface area of the trapezoidal prism.

Create a strategy A trapezoidal prism consists of two trapezium faces and four rectangular faces.

The total surface area of the prism is the sum of the areas of all the faces.

Apply the idea Surface area = 112 + 2 × 48 + 80 + 2 ×

rea of four rectangles then A two trapeziums

= 112 + 96 + 80 + 24 × 5 cm2

= 112 + 96 + 200 cm2

Evaluate the product

Evaluate the brackets and product

= 408 cm

Evaluate

Idea summary To ﬁnd the surface area of a prism identify the shapes of the different faces, then add the areas of the faces.

Surface area of cylinders A cylinder has a curved face so it’s useful to remember what the net of a cylinder looks like to work out the surface area. A cylinder has two ends the shape of a circle, and a rectangular piece that creates the curved surface. The total surface area (SA) of a cylinder can be expressed as: SA = 2π r2 + 2π rh

2π r

where r is the radius and h is the height of the cylinder. 14.01 Surface area of prisms and cylinders mathspace.co

739

Example 3 This cylinder has a height of 74 cm and base radius of 21 cm. Find the total surface area in square centimetres. Round your answer to two decimal places. 74 cm

21 cm

Create a strategy The total surface area of a cylinder is given by the formula 2π r2 + 2π rh, where r is the radius and h is the height of the cylinder. We can substitute the given radius and height into the formula to ﬁnd the total surface area.

Apply the idea Surface area = 2π r2 + 2π rh Surface area of cylinder = 2π × 212 cm2 + 2π × 21 × 74

Substitute the values

= 12 534.95 cm

Evaluate the expression

Idea summary

Surface area of a cylinder = 2π r2 + 2π rh r is the radius h is the height of the cylinder

Practice questions Choose your question path Explorer 1–3, 4, 6acdf, 7a, 8ac, 10, 11ab, 15a, 17–20, 22–24, 27, 28, 30, 33, 35

740

Adventurer 1–4, 6b,d-f, 7b, 8bd, 10, 11cd, 12a, 13, 15b, 17, 21–24, 27–29, 31, 33–35

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 2–5, 6c-f, 7b, 8bd, 9, 11d, 12b, 13, 14, 15b, 16, 21, 22, 25–27, 29, 31, 32, 34–36

Understanding 1

A cube has a side length of x cm. a

How many faces does the cube have?

What is the area of one face?

Write an expression for the surface area of the cube.

x cm

To ﬁnd the surface area of this pentagonal prism, the area of how many faces must be added together?

For each 3D shape, identify the number and type of 2D shapes that make up the net. a

Cube

Cylinder

Square-based pyramid

Hexagonal prism

Fluency 4

Find the surface area of a cube with side length 5 cm.

Find the side length of a cube that has a surface area of 384 cm2.

Find the surface area:

Ex 1

7m 9.9 m 15 m 8.7 m 7.5 m

5 cm

4 cm

23 cm 6 cm

4 cm 14.01 Surface area of prisms and cylinders mathspace.co

741

11 cm

10 cm 3 cm

5 cm

9 cm

5 cm

16 cm

4 cm

For each triangular prism: i

Find the value of x.

4 cm

Find the surface area.

10 cm

x cm

16 cm

18 cm 12 cm

3 cm

8 Ex 2

Find the surface area of the prisms. Round answers to two decimal places. a

10 cm 5 cm 12 m

3 cm 13 cm

11 cm

10 cm

4 cm

8 cm 5 cm

9 cm

742

16 cm

3 cm 17 cm

For this triangular prism: a

Find the exact length of the unknown side, k.

Find the exact surface area of the triangular prism.

If the value of k was rounded to the nearest metre before calculating the surface area, how much would the surface area be overestimated by? Round your answer to two decimal places.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

10 m

For this rectangular prism: 5

Determine whether the net matches the rectangular prism. i

5 5 5 5

5 8

b 11 Ex 3

iii

7 5 7 5

7 5 7 7

7 8 7

8 8 8

Find the surface area of the rectangular prism.

Find the surface area of each of these cylinders rounded to two decimal places. a

b 6 cm

93 cm 12 cm

30 cm

9m 8m 10 m

Find the area of the curved rectangular face of these open-ended cylinders, rounded to two decimal places. a

2m 3m

14.01 Surface area of prisms and cylinders mathspace.co

743

For this cylinder:

Find the circumference of the circular base rounded to four decimal places.

Find the area of the curved face of the cylinder rounded to two decimal places.

4m 2m 4m

Find the height, h, of this closed cylinder if its surface area is 25 547 m2 and the radius of each circular face is 38 m. Round your answer to the nearest whole number.

38 m

For each cylinder: i

Find the value of x, rounded to two decimal places.

Find the total surface area, rounded to two decimal places.

4 cm

xm 23 m

13.6 cm

744

25 m

The area of the circular face on a cylinder is 8281π m2. The total surface area of the cylinder is 25 662π m2. a

x cm

Find the radius of the cylinder.

Find the height of the cylinder.

A cylinder has a height of 20 cm and a diameter of 12 cm. a

Find the exact surface area of the cylinder.

If the value of π was rounded to 3.1 before calculating the surface area, how much would the surface area be overestimated by. Give your answer rounded to two decimal places.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A birthday gift is placed inside the box shown, which has the shape of a rectangular prism.

12 cm 14 cm

Find the minimum amount of wrapping paper needed to cover this box. 5 cm

The roof of a shed is the shape of a triangular prism with dimensions shown in the diagram:

Find the surface area of the roof of the shed (do not include the base of the prism). Give your answer rounded to the nearest square metre.

10 m 6m

Laura is building a storage chest in the shape of a rectangular prism. The chest will be 55 cm long, 41 cm deep, and 39 cm high. Find the surface area of the chest.

A swimming pool has the shape of a trapezoidal prism 14 m long and 6 m wide. The depth of the water ranges from 1.2 m to 2.5 m, as shown in the diagram. The walls and base of the pool are going to be tiled. Calculate the total area inside the pool that is to be tiled. Give your answer rounded to one decimal place.

14 m

1.2 m

2.5 m

Find the surface area of the brickwork for this cylindrical silo. Assume that there is a brick roof but no ﬂoor. Give your answer rounded to two decimal places.

2.24 m

6.68 m

Reasoning 23

Here are three formulas: • SA = 2π r2 + 2π rh • SA = π r2 + 2π rh • SA = 2π rh How do you know which formula to use to calculate the surface area of a cylinder?

Scott uses the formula SA = 2π r(r + h) to ﬁnd the surface area of a closed cylinder. Lachlan uses the formula SA = 2π r2 + 2π rh. Who is correct?

14.01 Surface area of prisms and cylinders mathspace.co

745

For the composite shape:

2 cm

List the faces and their dimensions that make up the total surface area.

3 cm

Calculate the total surface area of the composite shape.

3 cm

10 cm

Write a simpliﬁed formula to ﬁnd the surface area of a cylinder that has its radius equal to its height.

Quicker Oatmeal Company is choosing between two new packages for their chocolate chip oats. • Option 1 is a canister with a diameter of 6 cm and a height of 10 cm. • Option 2 is a box with dimensions 3 cm × 6.5 cm × 9.5 cm. a

Which package should Quicker Oatmeal use in order to minimise the amount of materials used? Explain.

No matter which package Quicker Oats chooses, they will sell it for $4.99 per package. Which package should Quicker Oats use to maximise their proﬁts? Explain.

Problem-solving 28

Find the surface area of the triangular prism, rounded to one decimal place.

16 cm

46 cm

The Super Cereal Company sells their Frosty O’s cereal in a rectangular box with dimensions 12 cm × 10 cm × 2.5 cm. In order to reduce their environmental impact, Super Cereal wants to reduce the surface area of their boxes while maintaining the proportions of the current packaging. a

List three different scale factors that Super Cereal could use to reduce the total surface area of Frosty O’s boxes by somewhere between 15 cm2 and 25 cm2.

Give the dimensions for a Frosty O’s box that satisﬁes the required reduction in surface area.

A paint roller is cylindrical in shape and has a diameter of 6.8 cm and a width of 31.2 cm. Find the area painted by the roller when it makes 300 revolutions. Give your answer in square metres rounded to two decimal places.

746

h cm

21 cm

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

31.2 cm 6.8 cm

This water trough in the shape of a half cylinder is to be painted on the outside with two coats of paint. a

Find the surface area of the outside rounded to two decimal places.

If paint costs $24.50 for a 1 L tin that covers 6.5 m2, ﬁnd the cost to paint the trough.

0.93 m

Both popcorn bags are designed to carry 1680 cm3 of popcorn. Assume there’s minimal wastage of space when the popcorn is packed into each bag. a

Find the height h of bag A in centimetres.

Find the height h of bag B in centimetres.

Find the surface area of bag A.

Find the surface area of bag B.

Which bag should be used to reduce the amount of paper?

2.88 m

Bag A

Bag B

7 cm

8 cm 10 cm

10 cm

Find the surface area of the trapezoidal prism, rounded to one decimal place.

y cm

3 cm

10 cm

8 cm 13 cm

The square prism has a volume of 138.72 cm3 and the side length of the square base is 3.4 cm. Calculate the surface area of the prism.

Jenny wants to make several cans like the one shown. She plans to cut them out of a sheet of material that has an area of 1683 cm2. Determine the number of complete cans can she make.

5 cm

7 cm

If two identical spherical balls with radii of 2.6 cm ﬁt exactly inside a cylinder, ﬁnd the surface area of the closed cylinder. Give your answer rounded to one decimal place.

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14.02 Volume of prisms and cylinders After this lesson, you will be able to… • apply the formula to find the volume of various prisms. • apply the formula to calculate the volume of cylinders. • solve problems involving the volume of prisms and cylinders.

Volume of prisms We can ﬁnd the volume of any prism by multiplying the base area by the height of the prism. The height of the prism is the distance between the two base faces. The formula used to ﬁnd the volume of a prism is given by V = Ah, where A is the base area and h is the height of the prism.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 14.02 to answer these questions. 1.

Find the volume of a triangular prism, with dimensions b = 5, h = 5 and height = 10. If the height of the triangular prism is halved, what do you notice about the volume?

Find 3 different prisms that have the same volume but different dimensions. Is there a pattern with the dimensions that give the same volume?

Example 1 Find the volume of this triangular prism.

5 cm 10 cm 5 cm

Create a strategy Use the volume formula V = Ah, where the area of the base is given by

748

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea We are given base length b = 5, height of the triangular base H = 5, and height of prism h = 10. Write the formula

Substitute the values

Evaluate

Example 2 A prism has a volume of 1080 cm3. If it has a base area of 120 cm2, ﬁnd the height of the prism.

Create a strategy Use the volume of a prism formula V = Ah

Apply the idea We are given V = 1080 and A = 120. Write the formula

Substitute the values

Divide both sides by 120

Evaluate

Idea summary The volume of a prism is given by:

V = Ah A h

is the base area is the height of the prism

Volume of cylinders We can ﬁnd the volume of a cylinder using the same method, by multiplying the area of the circular base by the height of the cylinder. The formula for the area of a circle is A = π r2, therefore the formula for the volume of a cylinder is V = π r2h, where r is the radius and h is the height of the cylinder.

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 14.02 to answer these questions. 1.

How does the volume formula of a right cylinder relate to the volume formula of a right prism?

Find the volume of a right cylinder, where the radius = 2 and height = 5. If the height is doubled, what do you notice about the volume?

Example 3 Calculate the volume of the quarter cylinder. Round your answer to one decimal place.

14 cm

5 cm

Create a strategy Multiply the volume of a cylinder formula by a quarter.

Apply the idea We are given r = 5 and h = 14. Multiply the formula by

Substitute the values

Evaluate

Idea summary The volume of a cylinder is given by:

V = π r2h r h

750

is the radius of the cylinder is the height of the cylinder

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Practice questions Choose your question path Explorer 1–4, 5ace, 7, 8a-e, 9ac, 11, 12, 14, 16, 18, 20, 23–26

Adventurer 1–3, 5ace, 7, 8aceh, 9ac, 11, 12–14, 16, 18, 19, 21, 22, 24, 26, 27

Trailblazer 1–3, 5bdf, 6, 7, 8bcgh, 10, 11, 13, 15, 17, 25–30

Understanding 1

Are these statements true or false? a The volume of a prism or cylinder is calculated by multiplying the area of its base by its height. b The volume of a triangular prism can be found by multiplying the base area by one third of the height. c The volume of a cylinder can be determined by multiplying the area of its circular base by its height. d The volume of a rectangular prism can be calculated by multiplying the length, width and height together. e The volume of a cylinder is always greater than the volume of a rectangular prism with the same base area and height.

Explain the similarities and differences in ﬁnding the volume of a rectangular prism and the volume of a cylinder.

Which of these prisms would have a volume that would be found by calculating 4 × 3 × 5? A

B 4 cm 3 cm 5 cm

4 cm

5 cm 5 cm 4 cm 3 cm

3 cm

4 cm

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Fluency 4

What is the shape of the prism’s base?

What is the area of the prism’s base?

iii

What is the volume of the prism?

b 15 cm

10 cm

6 cm 5 cm 18 cm 6 cm

3 cm

5 Ex 1

Find the volume of these prisms: a

A = 38 cm2

A = 52 cm2 10 cm

3 cm

7 cm

11 cm

3 cm

4 cm

3 cm

14 cm 5 cm

10 cm 12 cm

6 cm 752

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A prism has a volume of 880 cm3. If it has a height of 11 cm, ﬁnd the base area of the prism.

A prism has a volume of 1200 cm3. If it has a base area of 240 cm2, ﬁnd the height of the prism.

Ex 2

8 Ex 3

Find the volume of these solids, rounding your answers to one decimal place. a

5 cm

10 cm

5 cm

17 cm

6 cm 8m 12 cm

3 cm

10 cm 13 cm 3 cm

16 cm 7 cm 19 cm

8 cm

14.02 Volume of prisms and cylinders mathspace.co

753

Find the volume of each of the cylinders. Round your answers to one decimal place. a

A cylinder with a radius of 6 cm and height of 15 cm.

A cylinder with a radius of 7 cm and height of 15 cm.

A cylinder with a diameter of 2 cm and height of 19 cm.

Find the volume of the ﬁgure shown, rounded to two decimal places.

5 cm

15 cm

A cylindrical spa has a radius of 1.8 m and the water is 160 cm deep. Find the volume of water in the spa in litres. Round your answer to two decimal places.

A cylindrical candle has a radius of 46 mm and a height of 0.13 m. Find the volume of the candle in cubic centimetres. Round your answer to two decimal places.

Reasoning 13

A right prism has a rectangular base and three dimensions: length, width and height. How does the volume of a right prism change if we double the length and width while keeping the height the same?

Oliver is helping his father build a wooden toy box in the shape of a rectangular prism for his younger brother. The planned dimensions for the box are 1.8 m in length, 1.1 m in width, and 0.9 m in height. They need to calculate the volume of the box to know how much space it will take up.

754

Calculate the volume of the box using the exact dimensions.

Oliver’s father prefers to work with rounded dimensions to make the measurements easier. He decides to round each measurement to the nearest metre. Calculate the volume of the box using these rounded dimensions.

Compare the two volumes. Which is greater, and by how much?

Edrian’s science class is conducting an experiment that involves ﬁlling a cylindrical container with water. The container has a radius of 2.5 m and a height of 1.9 m. The students need to know the exact volume of the container to make sure they have enough water for the experiment. a

Calculate the volume of the container using the exact value of π and the given dimensions rounded to two decimal places.

One of the students argues that using 3.14 for π will make the calculations easier. Calculate the volume of the container using 3.14 as the approximation for π rounded to two decimal places.

Compare the volumes calculated with the exact value of pi and the approximation. Which is greater, and by how much?

Discuss the potential implications of using rounded or approximated values in these real world situations.

Determine the capacity of the container in millilitres.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Problem-solving 16

A rectangular swimming pool has a length of 27 m, width of 14 m and depth of 3 m. Find the capacity of the swimming pool in litres, if 1 m3 = 1000 litres.

Sanford Seminole Aquatic Center has a swimming pool that is 25 metres by 55 metres, and 2.5 metres deep. Morcom Aquatic Center has a swimming pool with a volume of 6187.50 cubic metres. a

Which pool has the greater volume?

How many times bigger is the volume of the bigger pool than the smaller pool? Round your answer to one decimal place.

Find the exact scale factor between the pool at Morcom and the pool at Sanford Seminole.

An ice cream store has two different types of cones. One type uses a pointy cone while the other type has a ﬂat base, made up of two cylinders. Both types have a hemispherical scoop on top and are ﬁlled with ice cream. Both cones have the same thickness. Types of ice cream cone Point base cone

Flat base cone 6 cm

3 cm

3 cm 1 cm

8 cm

6 cm

4 cm

Vidar wants to have as much ice cream as possible. Determine which type he should get.

Yolanda wants to have as much cone as possible. Determine which type she should get.

There are two types of cylindrical soup cans available for Ray to purchase at his local store. One has a diameter of 16 cm and a height of 18 cm while the other has a diameter of 18 cm and a height of 16 cm. Which type of can holds more soup?

Jack’s mother told him to drink 3 large bottles of water each day. She gave him a cylindrical bottle with height 17 cm and radius 5 cm. a

Find the volume of the bottle. Round your answer to two decimal places.

Assuming he drinks 3 full bottles as his mother suggested, calculate the volume of water Jack drinks each day. Round your answer to two decimal places.

If Jack follows this drinking routine for a week, how many litres of water would he drink altogether? Round your answer to the nearest litre.

5 cm

17 cm

14.02 Volume of prisms and cylinders mathspace.co

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Find the capacity of the ﬁsh tank in litres: 50 cm 30 cm

25 cm

A cylindrical swimming pool has a diameter of 5 m and a depth of 1.8 m. a

How many litres of water can the pool contain? Round your answer to the nearest litre.

Express this amount of water in kilolitres.

Find the area of the trapezium-shaped block of land.

During a heavy storm, 63 mm of rain fell over the block of land. Find the capacity of water that landed on the property in litres.

10 m 17 m

25 m

The roof of a cottage has the shape of a triangular prism. Find the volume of the roof.

10 m

A ﬁsh tank designed to sit in the corner of a room has the shape of a right-angled triangular prism. Determine the capacity of the ﬁsh tank in litres.

50 cm 30 cm 90 cm

756

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Find the volume of the tent in cubic metres:

150 cm 290 cm 200 cm

The roof of a house is ﬂat and measures 16 m by 11 m in the shape of a rectangle. a

Find the area of the roof.

Find the volume that could be collected from 20 mm of rain.

How many litres of water is equivalent to the volume collected in part (a)?

Once the water tank is full, a further 5 mm of rain falls from the roof. How many litres of rainfall are not collected as a result?

The outline of a rice ﬁeld is shown. a

Find the area of the rice ﬁeld.

Throughout the monsoon season, the rice ﬁeld receives 16 cm of rain. What volume of water, in cubic metres, has fallen on the rice ﬁeld?

35 m 36 m 52 m

45 m 16 m 80 m

The outline of a construction site for the foundations of a large oﬃce building is shown. a

Find the area of the construction site.

Prolonged rainstorms bring 0.21 m of rain to the region and the foundations of the building are ﬂooded. Determine the volume of water of the construction site, in cubic metres.

85 m 85 m

85 m

13 m 84 m

The ﬂoor plan for a house as shown, with measurements given in metres. The annual rainfall over the house is 700 mm. Calculate the potential amount of water that could fall on the house over one year, if 1 m3 holds 1000 litres. Give your answer in kilolitres.

3.00 m

4.00 m 2.60 m

3.20 m

14.02 Volume of prisms and cylinders mathspace.co

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14.03 Surface area and volume of composite solids After this lesson, you will be able to… • recognise and define composite solids. • calculate the volume of composite solids by adding or subtracting individual volumes. • identify and compute the surface area of composite solids by summing the areas of their faces. • solve problems related to the volume and surface area of composite solids.

Volume of composite solids We can ﬁnd the volume of composite solids by ﬁnding the volume of each of the solids and adding them or subtracting them. To find the volume of this composite solid we can subtract the volume of a simple solid. This composite solid is a rectangular prism with half a cylinder subtracted from it.

Where a composite solid is also a prism with a composite shape as its base, we can find its volume the same way we’d find the volume of a prism.

Example 1 This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. Find the volume of the solid. 13 cm 2 cm

6 cm

11 cm

15 cm

Create a strategy Subtract the volume of the smaller rectangular prism from the volume of the larger rectangular prism.

758

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea Large prism = 6 × 15 × 13 3

= 1170 cm

Small prism = 11 × 2 × 15 3

= 330 cm

V = 1170 − 330 3

= 840 cm

Multiply the dimensions Evaluate Multiply the dimensions Evaluate Subtract the volume of one prism from the other Evaluate

Example 2 A wedding cake with three tiers rests on a table. The layers have radii of 50 cm, 54 cm and 58 cm as shown in the ﬁgure. If each layer is 21 cm high, calculate the total volume of the cake in cubic metres. Round your answer to two decimal places.

21 cm 21 cm 21 cm

r1 = 50 cm r2 = 54 cm r3 = 58 cm

r1 r2 r3

Create a strategy Add the three volumes and convert the units to cubic metres.

Apply the idea Add the volumes Substitute the radii and heights

Evaluate

To convert to cubic metres we need to divide by 100 . Convert to m3

Evaluate and round

14.03 Surface area and volume of composite solids mathspace.co

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Example 3 Find the volume of the ﬁgure shown, rounded to two decimal places:

10 m 4m

Create a strategy Add the volume of the square prism to the volume of the half cylinder.

Apply the idea The half cylinder has a diameter of 4 m which means its radius is 2 m. Write the formula

Substitute r = 2, h = 10

Evaluate and round

Write the formula

Substitute s = 4, h = 10

Evaluate Add the volumes

Evaluate

Reflect and check Since the composite solid was made by adding simple solids together, its volume is the sum of the volumes of the simple solids.

Idea summary Some composite solids are made by joining simple solids together. We can find the volume of these composite solids by finding the volume of each of the pieces and adding them. In the case where the composite solid is made by subtracting one simple solid from another, we can subtract volumes to find the volume of the composite solid.

760

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Surface area of composite solids The surface area of a composite solid is the sum of the areas of its faces. If the composite solid is a prism, we can find its surface area in the same way that we’d find the surface area of a prism. If the composite solid is not a prism, we can find the surface area by considering the areas of each face of the solid and adding them up.

Example 4 Find the surface area of the house. 3m

6m 9m 5m

Create a strategy Add all the faces together. The diagram shows which faces are equal.

Use Pythagoras’ theorem, c2 = a2 + b2, to ﬁnd the width of the roof panels.

Apply the idea Add the areas and multiply by 2 Substitute the dimensions Evaluate Side walls = 9 × 6 × 2 Multiply the length and width and multiply by 2 = 108 m2 Base = 5 × 9 2

= 45 m

Evaluate Multiply the length and width Evaluate

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Using Pythagoras’ theorem, we can find the width of the roof panels. On the diagram, we can see two right-angled triangles where the shorter sides are a = 3 and

Use Pythagoras’ theorem

Evaluate Square root both sides Find the area of the 2 roof sides Evaluate Add all the areas Evaluate

Idea summary The surface area of a composite solid is the sum of the areas of its faces.

Practice questions Choose your question path Explorer 1, 2, 3abde, 4ad, 5, 6ad, 7bcegh, 8, 9abde, 11, 12, 14ab, 16, 17

Adventurer 1, 2, 3a,c-e, 4ab, 5, 6bd, 7a,c-e,h, 8, 9adfg, 11, 12, 14ab, 15–17

Understanding 1

What is a composite solid? Give an example.

Explain what you would do to:

762

Calculate the surface area of a composite shape.

Calculate the volume of a composite shape.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 1, 2, 3c-f, 4bc, 5bc, 6bc, 7acefh, 9bcegh, 10, 11, 13–15, 18–20

Austin is testing out his new 3D printer by creating some geometric objects. What basic shapes are used to make each of these composite solids? a

5 cm

7 cm

3 cm

8 cm 2 cm

7 cm

5 cm

4 cm

10 cm

2 cm

4 mm 4 mm

7 cm

3 mm

4 mm

5 cm

7 mm

2 cm

8 cm

7 cm

4 km

5 mm 4 mm 3 mm 5 mm

8 km

Identify the solids used to make up each of these composite solids. a

b 4 cm 7 cm

15 m 7m

8 cm

11 cm 8 cm

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9 mm 16 mm

11 cm

10 mm

6 cm 4 cm

8 cm

53 mm

36 mm

Fluency 5 Ex 1

Find the volume of these composite solids, rounding to two decimal places. a

8 cm

12 cm 3 cm

3 cm

4 cm

10 cm

7 cm 15 cm 3 cm

10 cm

12 cm 20 cm 7 cm

16 cm

8 cm

5 cm

6 Ex 2

Find the volume of these composite solids, rounding to two decimal places. a

4 cm 7 cm 8 cm

b 11 cm

8 cm 5 cm

764

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

15 cm

9 cm

4 cm 2 cm

2 cm

7 Ex 3

2 cm

9 cm

12 cm 2 cm

Find the volume of these composite solids, rounding to two decimal places. a

8m 15 m 7m

5m 12 m 6m

d 10 cm

9m 4m

10 m 8m

5 cm

12 m

14 cm 5 cm

15 cm

10 cm

19.5 m

9m 8m

15 17 m

24 20

h 6 cm

11 cm

3 cm

3 cm 12 cm

6 cm 8 cm

16 cm 4 cm

14.03 Surface area and volume of composite solids mathspace.co

765

A wedding cake has three tiers. The layers have radii of 51 cm, 55 cm and 59 cm. If each layer is 20 cm high, calculate the total volume of the cake in cubic metres.

51 cm

20 cm 20 cm

Round your answer to two decimal places.

55 cm

20 cm

59 cm

9 Ex 4

Find the surface area of these composite solids, rounding to two decimal places. a

1 cm

7 cm

16 cm

13 cm

4 cm 10 cm

12 cm

1 cm

d 1 cm 5 cm

7 cm

4 cm

1 cm

7 cm

9 mm 16 mm 10 mm

12 m 36 mm

2 cm

53 mm

h 3 cm 3 cm 5 cm

10 cm 766

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

15 cm

By considering the water bottle as a composite solid, estimate its surface area.

2 cm 1 cm

4 cm

20 cm

3 cm

Reasoning 11

Brenton wants to know the volume of the basketball hoop stand at school, including the base, pole, backboard, and hoop. He’s conﬁdent that he can measure the dimensions of the different parts of the stand, but isn’t sure how to break up the volume into simple solids.

Using the image of the basketball hoop stand, identify the simple solids that make up the stand and describe the dimensions that Brenton would need to measure to ﬁnd their volume. 12

A triangular tunnel is made through a rectangular prism. a

Find the volume of the solid formed using two different methods: i ii

ind the volume of the rectangular prism and subtract F the volume of the triangular prism. ind the area of the shaded face and use Volume of F Prism = A × h.

3 cm

Explain how to find, and then calculate, the volume of the remaining solid. Round your answer to two decimal places.

8 cm

7 cm

Explain the process of finding the surface area of the solid formed, including the inside of the tunnel.

The larger prism has had two identical holes carved out of it, each of which is a rectangular prism. All measurements are in metres. a

3 cm 6 cm

6.5 5

14 6.5

Explain how to find, and then calculate, the surface area of the remaining solid.

14.03 Surface area and volume of composite solids mathspace.co

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Problem-solving 14

A swimming pool is the shape of a trapezoidal prism. a

Find the volume of the pool in m3.

If the pool is three quarters full, what is the volume of the unfilled space of the pool?

If the distance of the water level to the top of the pool is h m when it is is three quarters full, ﬁnd h.

51 m

3.3 m 50.7 m

A hollow pipe needs painting on both inside and outside surfaces. a

Find the external surface area of the curved surface.

Find the total surface area of the two end pieces.

Find the internal surface area.

Calculate the total surface area of the pipe.

1.4 m

10 m

24 cm

11 cm 10 cm

The lid of this treasure chest is found to be exactly one half of a cylindrical barrel. Find the surface area of the chest. 30 cm

42 cm

768

The diagram shows the design for a marquee (tent). The material to make the marquee costs $44 per square metre.

Find the area of the front of the marquee.

Find the surface area of one of the side walls (not including the roof).

Find the surface area of the entire roof.

Find the total surface area. Do not include the ﬂoor.

What is the total cost of material to make the marquee?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

59 cm

10 m 8m

A company manufactures nuts shaped like regular hexagonal prisms, with cylindrical bolt holes cut out of the centre. a

The total surface area of a nut before the bolt hole is drilled is 14.7 cm2. Find the surface area of a single nut after the bolt hole is drilled out, including the inside area of the hole.

0.5 cm

Each nut that is manufactured requires a zinc coating to prevent corrosion. If 1 kg of zinc is enough to coat a surface area of 1 m2, how many complete nuts can be coated with 1 kg of zinc?

A swimming pool is composed of a trapezoidal prism joined to a half cylinder.

1.0 cm 1.2 cm 7m 3m

Find the volume of the pool in cubic metres.

How many complete litres of water can fit in the pool?

If the pool is ﬁlled to a height 10 cm below the top, how many complete litres of water are in the pool?

After construction works at a neighbouring property, a crack opens in the bottom of the pool and water begins to leak from the pool. It is observed that the height of the surface of the water in the pool is decreasing by 7 cm each week. How much water is leaking out each week?

Assuming water continues to leak at this rate, at the end of which week will the pool become empty?

0.8 m

A steel shed is to be constructed with a rectangular cut-out at the front for the entrance. a

Determine the surface area of the shed, assuming the floor is separate.

Construction of the shed requires an additional 0.1 m2 of sheet metal for each 1 m2 of surface area, due to overlaps and wastage. How many complete square metres of sheet metal are required to construct the shed?

2.5 m 10 m 3m 4m

If the steel sheets cost $18 per square metre, calculate the total cost of the steel required to build the shed.

14.03 Surface area and volume of composite solids mathspace.co

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14.04 Estimation and accuracy After this lesson, you will be able to… • explore rounding and how it influences the accuracy of calculations, especially in large datasets or multi-step computations. • differentiate between rounding and truncating numbers. • define the term ‘accuracy’ in the context of measurement and discuss factors that can influence it.

Effects of rounding on accuracy Rounding numbers affects the accuracy of any calculation we do with those numbers. The effect of rounding becomes especially important when dealing with large sets of data or calculations involving multiple steps. Each instance of rounding can introduce a “rounding effect” that can cumulatively impact the ﬁnal result’s accuracy. When banks calculate how much interest to pay customers, it is routine to truncate rather than round. For example, they may calculate that you are due to receive $465.239 in interest. They would not round to $465.24 but would truncate to $465.23. Maintaining high precision in numbers can lead to time-consuming calculations. The minimal decrease in accuracy due to rounding is sometimes an acceptable trade-off for the simplicity it provides.

Example 1 A customer invests $5000 at 6.43% p.a. compound interest. This table shows the ﬁrst seven years of the investment. Year 1 2 3 4 5 6 7

770

Start of year 5000 5321.50 5663.67 6027.84 6415.43 6827.94 7266.97

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Interest 321.50 342.17 364.17 387.59 412.51 439.03 467.26

End of year 5321.50 5663.67 6027.84 6415.43 6827.94 7266.97 7734.23

a Show how to complete the calculation for the second row of the table.

Create a strategy Multiply the start of year 2 balance by 6.43% then truncate for the end of year 2 balance.

Apply the idea To get the interest earned, we multiply $5321.50 by 6.43%

We then truncate 342.172 45 to 342.17. To get the ending balance of the investment at year 2, we have: 5321.50 + 342.17 = 5663.67

b Use the compound interest formula A = P(1 + r)n where P is the amount invested, r is the interest rate as a decimal and n is the number of years of the investment, to calculate the value of the investment to the nearest cent after 7 years.

Create a strategy Substitute the values on the compound interest formula A = P(1 + r)n and round the answer to nearest cent.

Apply the idea We are given P = 5000, r = 0.064, and n = 7. A = 5000(1 + 0.0643)7

Substitute the values

A = $7734.25

Evaluate

c How much interest does the bank keep over 7 years?

Create a strategy Subtract the ending balance of the investment at year 7 from the value found from part (b).

Apply the idea The interest that the bank keeps over 7 years is given by: $7734.25 − $7734.23 = $0.02

14.04 Estimation and accuracy mathspace.co

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Idea summary Use the most accurate ﬁgures you have for calculations and perform any rounding necessary at the end. Rounding before a calculation can make a signiﬁcant difference to the answers, especially if: • •

The rounding is all in the same direction; for example all rounded up. There are several quantities involved in the calculation.

The more rounded quantities that are used in calculations, the larger the potential variances that can arise. Truncation makes small differences to individual accounts, but over long periods of time, the amounts can become signiﬁcant.

Accuracy of measurement devices We use different devices for measurement, including a ruler for length, a weighing scale for weight, a thermometer for temperature and a clock for time. The accuracy of these devices is important in providing reliable results. Accuracy refers to how close a measured value is to the true or actual value. The quality and design of the measurement device can inﬂuence the accuracy but no measuring device is perfectly accurate, there will always be some degree of uncertainty in every measurement. This uncertainty can be due to factors such as the limitations of the device itself, environmental conditions or even human error in reading the measurements.

Example 2 Lila and Alex are working on a geometry project where they measure various geometric quantities using different tools. Their measurements sometimes lead to slight discrepancies. Calculate the percentage error of Lila’s measurements, to two decimal places, if Alex’s measurements are used as the true values for the given scenarios. a The perimeter of a rectangular garden where Lila measures the sides as 68.2 m and 42.9 m, while Alex measures them as 68.4 m and 43.0 m.

Create a strategy Calculate the perimeter of the rectangular garden based on Lila and Alex’s measurements using the formula P = 2 × (length + width), then calculate the percentage error between their measurements.

Apply the idea Perimeter according to Lila: PLila = 2 × (68.2 + 42.9) Substitute the values = 222.2 m

772

Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Perimeter according to Alex: PAlex = 2 × (68.4 + 43.0) Substitute the values = 222.8 m

Evaluate

The percentage error is given by:

b The area of the circular pool where Lila measures the radius as 25.8 m and uses 3.14 as an estimate of π. Alex measures the radius as 26.0 m and uses the button π on the calculator.

Create a strategy Calculate the area of the circular pool based on Lila and Alex’s measurements using the formula A = π r2 then calculate the percentage error between their measurements.

Apply the idea Area according to Lila: ALila = 3.14 × (25.8)2 = 2090.11 m

Substitute the values Evaluate

Area according to Alex: AAlex = π × (26.0)2 2

= 2123.72 m

Substitute the values Evaluate

The percentage error is given by:

c The surface area of the rectangular box on which Lila measures the dimensions as follows: length 48.6 cm, width 37.5 cm, and height 29.8 cm. Alex’s measurements are slightly different: length 48.8 cm, width 37.6 cm, and height 29.9 cm.

Create a strategy Calculate the surface area of the circular pool based on Lila and Alex’s measurements using the formula SA = 2(wl + hl + hw) then calculate the percentage error between their measurements.

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Apply the idea Surface area according to Lila: SALila = 2 × (37.5 × 48.6 + 29.8 × 48.6 + 29.8 × 37.5) Substitute the values = 8776.56 cm2 Evaluate Surface area according to Alex: SAAlex = 2 × (37.6 × 48.8 + 29.9 × 48.8 + 29.9 × 37.6) Substitute the values = 8836.48 cm2 Evaluate The percentage error is given by:

d The volume of a cubic container where Lila measures the length, width, and height as 71.4 cm each. Alex measures them as 71.6 cm each. Round your answer to two decimal places.

Create a strategy Calculate the volume of the cubic container based on Lila and Alex’s measurements using the formula V = side3, then calculate the percentage error between their measurements.

Apply the idea Volume according to Lila: VLila = (71.4)3

Substitute the values 3

= 363 994.344 cm

Evaluate

Volume according to Alex: VAlex = (71.6)3

Substitute the values 3

= 367 061.696 cm

Evaluate

The percentage error is given by:

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Idea summary Meausuring devices can include rulers, weighing scales, thermometers, clocks, etc. These devices are not always accurate due to: • • •

Limitations of the device Environmental conditions Human error

Practice questions Choose your question path Explorer 1, 2, 4, 6ab, 7, 8, 10, 11, 13

Adventurer 1, 2, 4–11, 13, 15

Trailblazer 1–9, 11–15

Understanding 1

Are these statements true or false? a

Truncation and rounding errors can signiﬁcantly affect the ﬁnal results in a series of calculations.

The impact of compounding errors is negligible in ﬁnancial calculations.

The greater the number of decimal places in a calculation, the more accurate the result.

Truncation and rounding are the same things when it comes to calculations involving real numbers.

List all numbers with two decimal places that round to 6.4.

What is the smallest number with three decimal places that would give an answer of 12.72 when rounded to two decimal places?

What is the largest number with two decimal places that would round to 5.4?

Three two-digit numbers with two decimal places are first rounded to the nearest tenth and then added together. Describe the size of the error that could result.

Fluency 6

Tony is calculating the surface area of the solid as shown, where the height and diameter of the cylinder are 5 cm. a b

Find the smallest value that each of the measurements could be.

5 cm

8 cm

Find the largest value that each of the measurements could be. 6 cm 14.04 Estimation and accuracy mathspace.co

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7 Ex 1

8 Ex 2

Find the smallest possible surface area for the solid, rounded to the nearest whole number.

Find the largest possible surface area for the solid, rounded to the nearest whole number.

A customer invests $10 000 at 5.48% p.a. compound interest. The table shows the ﬁrst seven years of the investment. Year

Start of Year

Interest

End of Year

$10 000

$548.00

$10 548.00

$578.03

$11 126.03

$609.71

$11 735.74

$643.12

$12 378.86

$678.36

$13 057.22

$715.54

$13 772.76

$754.75

$14 527.51

Show how to complete the calculation for the second row of the table.

Use the compound interest formula A = P(1 + r)n where P is the amount invested, r is the interest rate as a decimal and n is the number of years of the investment, to calculate the value of the investment to the nearest cent after 7 years.

How much interest does the bank keep over 7 years?

If 20 million people have such an account, how much interest does the bank keep over 7 years?

Jasper has a ruler that is marked in cm. When he measures something with it he looks carefully to see whether the object is less than half way between two marks, in which case he rounds down, or more than half way between two marks, in which case he rounds up. When measuring the rectangle shown, he would record the length as 4 cm. 1

Betty has a ruler that is marked in both cm and mm, so that she can get lengths in cm to one decimal place. Calculate the percentage error, to one decimal place, in Jasper’s measurement compared to Betty’s measurement for:

776

The perimeter of a rectangle which Betty measures as being 5.3 cm long and 4.4 cm wide.

The area of a circle where Betty measures the radius as 4.3 cm, given that Betty uses the button π on the calculator and Jasper uses 3.1 as an estimate of π.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The surface area of a rectangular prism which Betty measures as 7.8 cm long, 5.7 cm wide and 4.1 cm high.

The volume of the rectangular prism in part (c).

Reasoning 9

A bank truncates interest rates at the third decimal place for daily calculations on a savings account. Over a year, how could this affect the total interest received by a large number of customers?

A company uses a faulty weighing scale that adds 0.1 kg to every measurement. How might this error inﬂuence their cost calculations if they are dealing with large quantities of materials?

Caleb measures the length of the classroom to be 7 metres. Ida says the same length is 6.8 metres and Lavinia gives her answer as 6.85 metres.

Explain how all three people could have different answers for the same measurement.

Write down the level of accuracy being used by each person.

Are all their answers correct? Explain your reasoning.

Alex and Liam are physics students conducting an experiment to measure the kinetic energy of a ball rolling down a ramp. They know the formula for kinetic energy is is the mass and v is the velocity.

mv2, where m

Alex has an old scale which only measures mass to the nearest 10 g (0.01 kg). Liam has an accurate scale which gives the mass in kilograms to three decimal places. To measure the velocity, Alex has a simple timer that gives the time it takes for the ball to roll a certain distance to the nearest second, whereas Liam has a radar gun that gives the velocity in m/s rounded to two decimal places. In one trial, Liam measures the mass of the ball as 0.365 kg and the velocity as 2.24 m/s. a

Find the measurements that Alex will use in his calculations.

Calculate the kinetic energy of the ball for: i

Alex

Liam

Using the answers from (b), and assuming Liam’s answer to be accurate, calculate the percentage error in Alex’s answer.

The students are then told that they can use the same data to calculate the momentum of the ball using the formula p = mv. Calculate the momentum of the ball for: i

Alex

Liam

Using the answers from (d), and assuming Liam’s answer to be accurate, calculate the percentage error in Alex’s answer.

What do you notice about the answers to parts (c) and (e)? Explain why this is the case.

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Problem-solving 13

Keepsafe Family Bank pay a small amount of interest on their customer’s accounts when the accounts are in credit (the account holder has money in the bank). The interest is added to the account daily, based on the balance in the account at the end of the day. Before they add the interest to the account, they truncate the amount, removing anything less than one cent, so if they calculated the interest to be $2.5387, the customer would receive $2.53, and the bank keeps $0.0087 or 0.87 cents. If the bank has 5 million customers with accounts in credit and the average amount truncated is $0.005, how much money does the bank keep from these interest payments in a standard 365-day year?

778

Prior to 1999, the estimated age of the universe was 13.5 Byr ± 6.5 Byr (Byr means billion years). a

The absolute error compared to 13.5 Byr is 6.5 Byr. Find the percentage error, rounded to two decimal places.

Improvements in astronomical data mean the current estimate of the age of the universe as 13.5 Byr with an absolute error of just 200 million years. Find the percentage error in the current estimates, rounded to two decimal places.

The dating of artefacts by Argon-Argon suggests that those found in volcanic deposits at a site called Tower Hill dated from 36.8 ± 3.8 thousand years ago. a

Find the earliest date (the longest time ago) that the artefacts can be from, measured in thousands of years ago.

Find the latest date (the shortest time ago) that the artefacts can be from, measured in thousands of years ago.

Express 3.8 as a percentage of 36.8, rounded to one decimal place.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Chapter 14 review 1

Find the surface area: a

16 m 6 cm

T F

13 cm

7 cm

5 cm

26 cm 8 cm

A R

12 cm

10 cm

8 cm

Find the height, h, of a closed cylindrical tower given its surface area is 28, 000 m2 and the radius of its circular base is 40 m. Round your answer to the nearest whole number.

Fruit and Nut Co. is deciding between two new packaging designs for their raisins. • Design A is a cylinder with a diameter of 10.2 cm and a height of 11.5 cm. • Design B is a rectangular prism with dimensions 4.4 cm × 11 cm × 12.9 cm.

Which design should Fruit and Nut Co. select if they want to use the least amount of material for the packaging? Justify your answer 4

For the cylinder: a

Find the value of x, rounded to two decimal places.

Find the total surface area, rounded to two decimal places.

5 cm

15.7 cm

x cm

Find the volume of these prisms: a

b 5 cm

A = 35 cm2 10 cm

14 cm 4 cm

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17 cm

4 cm

10 cm 8 cm

15 cm 7 cm

A storage box has a volume of 990 cm3. If its height is 9 cm, determine the base area of the storage box.

Find the volume of these solids, rounding your answers to two decimal places: a

T F

11 cm

20 cm

A R

12 cm

8 cm

A city park has a rectangular playground that measures 19 m by 12 m. a

If the playground is to be covered with a layer of sand for safety reasons, how many cubic metres of sand would be needed for a 15 mm thick layer across the entire playground?

If the workers mistakenly spread an additional 3 mm thick layer of sand across the playground, how many extra cubic metres of sand were used?

Find the volume. Round your answers to two decimal places where necessary. a

9 cm

12 cm

11 cm

9 cm

15 cm 5 cm

7 cm

5 cm

For each ﬁgure, ﬁnd to two decimal places: i

The volume.

6 cm

The surface area.

9 cm 6 cm

6 cm 15 cm

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6 cm

18 cm

A wedding cake has three tiers. The layers have radii of 15 cm, 20 cm and 25 cm. Each layer is 15 cm high. Calculate the total volume of the cake, rounded to two decimal places.

15 cm

Find the amount of yellow icing needed to cover the top tier of the cake, rounded to two decimal places.

15 cm

Find the amount of white icing needed to cover the middle layer of the cake, rounded to two decimal places.

20 cm 15 cm 25 cm

Find the amount of pink icing needed to cover the bottom layer of the cake, rounded to two decimal places.

The diagram shows the design for a shed. a

Find the total surface area of the shed.

T F

Do not include the ﬂoor. b

15 cm

If paint costs $24 for a 1 L tin that covers 6.5 m2, ﬁnd the cost to paint the shed, rounded to the nearest whole dollar.

A R

11 m 9m

Kareem, Alea, and Trina are calculating the volume of the solid shown. Kareem uses 3.14 for pi in his calculations, Alea uses

for pi in her

calculations, and Trina uses the exact value of pi in her calculations.

Whose estimation of the volume will be most accurate?

Whose estimation with be least accurate?

Find the difference between Alea’s estimation and Trina’s estimation, rounding each answer to three decimal places.

10 cm 4 cm

An online streaming service tracks the amount of time a person spends watching a video, rounded down to the nearest minute. Over a month, how might this affect the total viewing time calculations for a large number of users?

A bakery uses a timer that is consistently 30 seconds slow. How might this error impact the quality of the baked items?

Before 2005, scientists believed the oldest fossilized plant was 500 Myr ± 25 Myr (Myr means million years). a

Find the oldest age of the plant fossil.

Find the youngest age of the plant fossil.

Given the absolute error in relation to 500 Myr is 25 Myr, determine the percentage error, rounding to two decimal places.

With the discovery of new fossils, the current estimation for the age of the oldest plant fossil is 500 Myr with an absolute error of just 10 million years. Determine the percentage error of this newer estimation, rounded to two decimal places.

Chapter 14 review mathspace.co

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(Optional) For each sphere, ﬁnd to two decimal places: i

The surface area.

The volume.

4.7 cm

12 cm

(Optional) For the hemisphere: a

Find the surface area rounded to two decimal places.

Find the volume rounded to three decimal places.

T F

A R

4 cm

(Optional) Find the surface area of the composite shape rounded to two decimal places.

10.5 mm

(Optional) For each cone, ﬁnd to two decimal places: i

The surface area.

33.1 mm

The volume.

b 8 cm

9 cm

5 cm

3 cm

(Optional) Find the radius of a cone with volume of 232 cm3 and a height of 12 cm, rounded to two decimal places.

(Optional) For the composite ﬁgure: a

Find the surface area rounded to two decimal places.

Find the volume rounded to two decimal places. 11 cm 5 cm

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) For the pyramid shown:

y cm

Find the slant height labelled x, rounded to two decimal places.

Find the slant height labelled y, rounded to two decimal places.

Find the surface area, rounded to one decimal place.

Calculate the volume.

x cm

8 cm 11 cm 15 cm

(Optional) Find the volume of the shaded region.

12 cm

7 cm

T F

18 cm

A R

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15 Probability Big ideas • Physical objects can be modelled with 3D geometric figures whose properties can be applied to solve real-world problems. • Estimation and accuracy are fundamental concepts for practical decision-making and interpreting measurements across diverse real-world contexts. • Scale factors and ratios provide a framework for comparing and relating measurements, dimensions and volumes across different scales and contexts.

Curriculum Outcomes VC2M10A05 Substitute values into formulas to determine an unknown and rearrange formulas to solve for a particular term VC2M10P01 Use the language of ‘if … then …’, ‘given’, ‘of’ and ‘knowing that’ to investigate conditional statements and identify common mistakes in interpreting such language, and describe and interpret situations involving conditional probability; design and conduct simulations using digital tools to model conditional probability and interpret results VC2M10P02 Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events; investigate the concept of independence VC2M10AA02 Devise and use algorithms and simulations to solve mathematical problems VC2M10AP01 Explore counting principles, and factorial notation as a representation that provides efficient counting in multiplicative contexts, including calculations of probabilities

Chapter outline 15.01 15.02 15.03

Independent and dependent events Conditional probability Mixed probability concepts Chapter 15 review

786 800 812 827

Scientists use probability methods to determine the likelihood of space debris hitting land.

15.01 Independent and dependent events After this lesson, you will be able to… • differentiate between independent and dependent events. • recognise the role of replacement in probability scenarios. • calculate probabilities of events occurring with and without replacement. • apply mathematical formulas to determine the probability of combined events. • analyse real-world examples to identify whether events are independent or dependent. • use tree diagrams to visualise sequences of events and calculate probabilities.

Independent and dependent events Two events are independent if the outcome of each event does not affect the outcome of the other event. For example, landing on heads and landing on tails when tossing a fair coin are independent events because the first toss has no effect on the second toss. Two events that are not independent are called dependent. For example, receiving a first place prize and receiving a second place prize in a race are dependent events. Once someone has won the first place prize, the probability of winning the second place prize will increase for those who are still in the race. If two events are independent, the probability of both events occurring is: P ( A ∩ B) = P ( A) × P (B) The converse statement is also true. This means that if P ( A ∩ B) = P ( A) × P (B), then A and B are independent.

Example 1 A standard six-sided die is rolled 691 times. a If it lands on a six 108 times, what is the probability that the next roll will land on a six?

Create a strategy We can use the formula:

Apply the idea The previous results do not affect the outcome of the next roll. There are 6 numbers on a die and 1 of them is a six. Substitute the number of outcomes

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b Is the outcome of the next roll independent of or dependent on the outcomes of previous rolls?

Apply the idea The outcome of the next roll is not affected by the outcomes of previous rolls. So the outcome of the next roll is independent.

Example 2 Two events A and B are such that P ( A ∩ B) = 0.3 and P ( A) = 0.5. Calculate P (B) if events A and B are independent.

Create a strategy Use the fact that if the events A and B are independent, then P ( A) × P (B) = P ( A ∩ B).

Apply the idea The formula P ( A) × P (B) = P ( A ∩ B) can be rearranged to create an equation that will let us solve for P (B). Divide both sides by P ( A) Using the formula of P (B), the value of P (B) is given by:

Write the equation

Substitute the values

Evaluate

Idea summary Two events are independent if the outcome of the each event does not affect the outcome of the other event. Two events that are not independent are called dependent. To check if two events are independent you can use: P ( A ∩ B) = P ( A) × P (B)

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Replacement

Probability

A probability tree diagram is useful in tracking two events that happen one after the other. At each point, it branches out to all the possible events that could occur from that point. The probabilities of each event is written on the branches.

Branch

For example, selecting two cards from a deck of cards, one after the other, are events that occur in sequence. There are two ways this can happen: • Draw a card, then put it back in the deck before selecting another card • Draw a card and keep it, then select another card from the deck

0.1 Outcome 0.9 T

The ﬁrst method is described as “with replacement” because the card is placed back into the deck. The second method is called “without replacement” because the card is not placed back into the deck.

Exploration The jar of marbles shown contains 7 red marbles, 3 blue marbles, and 10 green marbles. 1.

A red marble is drawn from the jar, then placed back into the jar. How many marbles are in the jar?

A green marble is removed from the jar. How many marbles are left in the jar?

Suppose one marble is drawn and placed back into the jar, then a second marble is drawn. Let event A represent drawing a red marble and event B represent drawing a green marble. Are these events independent or dependent? Explain.

Suppose one marble is drawn, then a second marble is drawn. The ﬁrst marble was not replaced. Are events A and B independent or dependent? Explain.

When selecting objects from a group “with replacement”, selections are independent. Each time you select a card, you have the same probabilities. 26 52

will be

26 52

For example, selecting a red card from a standard deck every time if you replace the card.

26 52

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When selecting objects from a group “without replacement”, selections are dependent. Each time you select a card, you change the probabilities for the next selection.

25 51 26 51

26 52

For example, the probability of drawing a red card from a standard deck is

If we draw a red card and do not replace it, there are only 25 red cards left and only 51 cards left in the deck. The probability of selecting a second red card is

. This can be seen in the top branches of the tree

diagram.

25 51

This shows the probability of the second event is dependent on what colour card was drawn ﬁrst.

Example 3 A number game uses a basket with 10 balls, all labelled with numbers from 1 to 10. One ball is drawn at random, then another ball is drawn at random. a What is the probability that the ball labelled 2 is picked once if the balls are drawn with replacement?

Create a strategy Find the probability for each way that this could happen, then add them together.

Apply the idea This could happen in two ways: 1. P (2) × P (2′): The ball labelled 2 is picked ﬁrst and a ball not labelled 2 is picked second. 2. P (2′) × P (2): A ball not labelled 2 is picked ﬁrst and the ball labelled 2 is picked second. Find the probability of picking the ball with 2

Find the probability of not picking the ball with 2 Add the probabilities of the two ways

Substitute the probabilities

Evaluate and simplify

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b What is the probability that the ball labelled 2 is picked once if the balls are drawn without replacement?

Create a strategy Find the probability for each way that this could happen, then add them together. For the second event, the total number of balls will need to decrease.

Apply the idea Similarly to part (a) this could happen in two ways. However, for the probability of the second ball, the total number of outcomes will be 10 − 1 = 9 since the ball was not replaced. Add the probabilities of the two ways

Substitute the probabilities

Evaluate and simplify

Example 4 Three marbles are randomly drawn without replacement from a bag containing 6 red, 6 yellow, 4 white, 6 black and 4 green marbles. Find: a the probability of drawing 3 white marbles.

Create a strategy Remember that the number of white marbles and the total number of marbles in the bag decreases by 1 every time a white marble is drawn.

Apply the idea There are 4 white marbles and a total of 26 marbles inside the bag. Multiply the probabilities

Evaluate

b the probability of drawing 3 black marbles.

Create a strategy Remember that the number of black marbles and the total number of marbles in the bag decreases by 1 every time a black marble is drawn.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea There are 6 black marbles and a total of 26 marbles inside the bag. Multiply the probabilities

Evaluate

c the probability of drawing zero green marbles.

Create a strategy Remember that the number of non-green marbles and the total number of marbles in the bag decrease by 1 every time a non-green marble is drawn.

Apply the idea There are 22 non-green marbles inside the bag. Multiply the probabilities

Evaluate

d the probability of drawing zero yellow marbles.

Create a strategy Remember that the number of non-yellow marbles and the total number of marbles in the bag decrease by 1 every time a non-yellow marble is drawn.

Apply the idea There are 20 non-yellow marbles inside the bag. Multiply the probabilities

Evaluate

e the probability of drawing at least one red marble.

Create a strategy Use the fact that P (at least one red marble) is the complement of P (no red marbles).

Apply the idea There is a total of 20 non-red marbles inside the bag. Multiply the probabilities

Evaluate

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We now calculate P (at least one red marble): Subtract P (at least one red marble) from 1

Evaluate

Example 5 Bart is purchasing a plane ticket to Joh. He notices there are only 4 seats remaining, 1 of them is a window seat (W) and the other 3 are aisle seats ( A). His friend, Joh, gets on the computer and purchases a ticket immediately after. The seats are randomly allocated at the time of purchase. a Fill in the probabilities matching the edges of the probability tree for the seat Bart receives and the seat his friend receives:

A E

Create a strategy Use the fact that the ﬁrst event is Bart receiving his seat allocation and the second event is Bart’s friend (Joh) receiving a seat allocation.

Apply the idea W

1 4

There are 3 aisle seats and 1 window seat. This means that when Bart picks a seat, the probability of sitting on a window seat is

3 4

and the probability of sitting on an aisle seat is .

A E

A A

1 4 3 4

There are 3 aisle seats left when Joh picks a seat given that Bart already picks a window seat, so the probability of sitting on an aisle seat is 1.

A E

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1 4

There are 2 aisle seats and 1 window seat left when Joh picks a seat given that Bart already picks an aisle seat, so the probablity of sitting on a window seat is

and the

probability of sitting on an aisle seat is . 3 4

1 3 2 3

So, the probabilities matching the edges of the probability tree for the seats are , and

b What is the probability that Bart’s friend, Joh, has an aisle seat?

Create a strategy Find the probability for each way that this could happen, then add them together.

Apply the idea W

1 3

1 4 3 4

2 3

The events that ﬁnish on a node labelled A correspond to the friend having an aisle seat. That is, WA and AA. The probability of that event occurring is found by multiplying previous probabilities along that branch.

This means that the probability that Joh has an aisle seat is given by:

Evaluate the multiplication

Evaluate the addition

c What is the probability of Joh receiving an aisle seat if Bart has a window seat?

Create a strategy

Apply the idea

Use the probabilities found in part (a).

If Bart receives a window seat, only aisle seats are left. So, the probability of Joh receiving an aisle seat is 1.

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Idea summary Events “with replacement” occur when the item drawn is placed back into the group before each selection. Each selection is independent of the others. Events “without replacement” occur when the item remains outside of the group after selection. Each selection is dependent of the others. The probabilities of each selection will change depending on previous selections.

Practice questions Choose your question path Explorer 1, 2, 4, 6, 7, 9, 11–13, 15–17, 19, 22, 24, 25, 28–30, 33

Adventurer 1–7, 9–13, 15–17, 19, 22, 24, 25, 29, 31, 33

Trailblazer 1, 3, 5, 6, 8–10, 12, 14–16, 18, 20, 21, 23, 24, 26–29, 31, 32, 34

Understanding 1

Identify whether these events are independent or dependent: a

A card is randomly selected from a normal deck of cards, and then returned to the deck. The deck is shuﬄed and another card is selected.

Each student is allowed to randomly pick an item from the teacher’s prize bag.

A coin is tossed, the result noted, and then it is tossed again.

Ursula takes a bus to the station and then immediately gets on a train to work. Is the probability of her missing the train independent or dependent on her missing her bus?

On the island of Guam, the probability that it rains during rainy season is P (R). This is shown in the tree diagram. a

What do the two sets of branches represent?

Does the weather on the first day affect the chance of rain on the second day?

Which two events have the same probability?

R R

0.4

0.6 0.4

0.6

R S

0.6 0.4

794

Dean has a box that contains ﬁve blue marbles and four red marbles. He takes a marble from the box, then takes a second marble from the box. Determine whether these events are dependent or independent if: a

He replaces the ﬁrst marble before taking the second.

He does not replace the marble before taking the second.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A bag contains 3 pink balls and 2 orange balls. You draw one ball, note its colour, then draw another ball without putting the first ball back. a

Find the missing probability for drawing a second pink ball on the tree diagram.

Which outcome can be found by multiplying

Which outcome(s) has a probability of

Are the events of drawing the balls independent or dependent?

by ?

O O P O

Fluency Ex 1

Ex 2

On a roulette table, a ball can land on one of 18 red or 18 black numbers. a

If it lands on a red number on the ﬁrst go, ﬁnd the probability that it will land on a red number on the second go.

Are the successive events of twice landing on a red number dependent or independent?

From a standard pack of cards, 1 card is randomly drawn and then put back into the pack. A second card is then drawn. a

Find the probability that neither of the cards are spades.

Find the probability that at least 1 of the cards is a spade.

Are the two draws dependent or independent?

Two standard dice are rolled. One is red and one is white. a

Are the results from the red die and the white die dependent or independent?

Calculate the probability that: i

The same number is rolled.

The sum of the two numbers exceeds 9.

iii

The red die is 4 and the white die 6.

The red die is even and the white die is odd.

The sum of the two numbers is less than 2.

Two events A and B are such that P ( A ∩ B) = 0.1 and P ( A) = 0.2 Calculate P (B) if events A and B are independent.

Two events A and B are such that P ( A ∩ B) = 0.04 and P ( A) = 0.4 If P (B) = 0.1, state whether the events A and B are independent.

A number game uses a basket with 8 balls, all labelled with numbers from 1 to 8. Three balls are drawn at random, with replacement. Find the probability that the ball labelled 4 is picked: a

Exactly once

Exactly twice

Exactly three times

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Ex 3

Christa randomly selects two cards from a normal deck of cards. Calculate the probability, in each of the following, in the case that the cards are selected:

With replacement

Without replacement

The ﬁrst card is a Queen of Diamonds and the second card is a 10 of Spades.

Both cards are 10s.

The ﬁrst card is a Queen and the second card is a King

The ﬁrst card is red and the second card is not a spade.

Mae deals two cards from a normal deck of cards, without replacement. a

Are the events of drawing Card 1 and drawing Card 2 dependent or independent?

Calculate the probability that she deals: i

Two 10s

Two red cards

iii

Two diamonds

A 10 of spades and an Ace of diamonds, in that order

A 10 of clubs and an Ace of spades, in any order

Valentina randomly selects three cards from a normal deck of cards. Calculate the probability if the cards are selected:

Ex 4

796

With replacement

The cards are ﬁve of clubs, King of clubs, and Jack of spades, in that order.

The cards are all red.

The ﬁrst card is the 2 of clubs, the second card is a spade and the third card is red.

The cards are all spades.

None of the cards is a 8.

Without replacement

Three marbles are randomly drawn without replacement from a bag containing 5 red, 4 yellow, 5 white, 4 black and 5 green marbles. Find the probability of drawing: a

Three white marbles

Three black marbles

Zero green marbles

Zero yellow marbles

At least one red marble

A hand contains a 10, a Jack, a Queen, a King and an Ace. Two cards are drawn from the hand at random, in succession and without replacement. Find the probability that: a

The Ace is drawn.

The King is not drawn.

The Queen is the second card drawn.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A number game uses a basket with 5 balls, all labelled with numbers from 1 to 5. Three balls are drawn at random, without replacement. a

Find the probability that the ball labelled 4 is picked.

Find the probability that the ball labelled 4 is picked and the ball labelled 1 is also picked.

From a set of 10 cards numbered 1 to 10, two cards are drawn at random, in succession and without replacement. Find the probability that: a

Both numbers are even.

One is even and the other is odd.

The sum of the numbers is 12.

Three fair coins are tossed: a

Find the probability of obtaining at least 1 head.

Find the probability of obtaining TTH in this sequence.

with the probability of . The tree diagram shows all the possible outcomes and probabilities: a

Find the probability that he will make all three free throws.

Find the probability that he will miss at least one free throw.

Hit

1 5 4 5 Miss

1 5 4 5

Find the probability that James selects a blue jacket for himself. Find the probability that both jackets James selects are green.

T H

Miss

Hit

Miss

4 7 3 7

1 5 4 5 1 5 4 5 1 5 4 5 1 5 4 5

Hit

Find the probability that he will make at least one free throw.

James owns four green jackets and three blue jackets. He selects one of the jackets at random for himself and then another jacket at random for his friend: a

A basketball player is practicing his free throws. He makes the shot and misses

Find the probability of obtaining THH in this sequence.

with a probability of

Hit Miss Hit Miss Hit Miss Hit Miss

1 2 1 2 2 3 1 3

15.01 Independent and dependent events mathspace.co

G B G B

797

Ex 5

Han owns four green ties and three blue ties. He selects one of the ties at random for himself and then another tie at random for his friend. a

Write the probabilities for the outcomes on the edges of the probability tree diagram.

Calculate the probability that:

G B

Han selects a blue tie for himself.

Han selects two green ties.

An ice cream shop offers one flavour of ice cream at a discounted price each day. There are 6 flavours for the owner to choose from, that could be discounted. 3 of the flavours are sorbet and the other 3 are gelato. The three sorbet flavours are raspberry (R), lemon (L) and chocolate (C). The three gelato flavours are vanilla (V ), mint (M ) and chocolate (C). Each decision has equal probability. a

Write the sample space for which ice cream type and ﬂavour may be chosen.

Find the probability that the lemon sorbet is selected.

Determine the probability that the ﬂavour will be chocolate.

R L C

V M C

Reasoning 24

In a lottery there are 37 balls. a

Find the probability of the 22 ball being drawn ﬁrst.

A ball is discarded after it has been drawn. If ball number 22 is drawn on the ﬁrst go, ﬁnd the probability of ball number 15 being drawn next.

Is the probability of each successive ball drawn the same as the probability of the ﬁrst ball drawn?

Are the draws dependent or independent events? Explain your reasoning.

Two cards are to be chosen from a pack of cards numbered 1 to 11. Explain how to ﬁnd the probability of drawing a 1, 2 or 3 ﬁrst, and after replacing the card, drawing a 9, 10 or 11.

Recall that for mutually exclusive events, P ( A ∩ B) = 0. For two events A and B, where P ( A) ≠ 0 and P (B) ≠ 0, explain why the events cannot be both independent and mutually exclusive.

Given the Venn diagram shown, explain why events A and B are independent. A

6 4

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Problem-solving 28

Consider the word WOLLONGONG. If three letters are randomly selected from it without replacement, ﬁnd the probability that: a

The letters are W, O, L, in that order.

The letters are O, N, G, in that order.

All three letters are O.

None of the three letters is an O.

Sarah has a box of Favourites chocolates. In this box there are 30 chocolates, 5 of which are Turkish Delights. Sarah takes and eats a chocolate without looking until she gets a Turkish Delight. a

Find the probability she only eats one chocolate.

Find the probability she eats only two chocolates to the nearest percent.

Find the probability she eats ﬁve chocolates to the nearest percent.

As she eats more and more chocolates, state whether the probability of the next chocolate being a Turkish Delight is getting higher or lower.

How many chocolates must Sarah have eaten to be certain that the next chocolate will be a Turkish Delight?

Tom has shuﬄed a standard pack of 52 playing cards. He draws one card from the pack at a time, and then puts it aside. He will stop when he draws the 9 of Diamonds. a

Find the probability he only has to draw exactly 1 card.

Find the probability he has to draw exactly 2 cards.

Find the probability he has to draw exactly 3 cards.

Find the probability Tom will stop after the 44th card.

As more cards are drawn that aren’t the desired card, does the probability that the next card selected is the 9 of Diamonds increase or decrease?

A bag contains 3 green marbles and 4 purple marbles. Two marbles are drawn at random from the bag. a

If the ﬁrst marble is replaced before the second marble is drawn complete the table with the probabilities matching the edges of the probability tree.

Drawing two green marbles

Drawing two purple marbles

iii

Drawing marbles of different colours

Edge

Determine the impact that replacing the marble before drawing a second has on the probability of these events: i

If the ﬁrst marble is not replaced before the second marble is drawn complete the table with the probabilities matching the edges of the probability tree:

Probability

Sureya has three races to swim at her school swimming carnival. The probability that she wins a particular race is dependent on whether she won the previous races, as summarised: • The chance she wins the first race is 0.7 • If she wins the first race, the chance of winning the second is 0.8 • If she loses the first race, then the chance of winning the second is 0.4

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• If she wins the first two, then the chance of winning the third race is 0.9 • If she lost the first two then her chance of losing the third race is 0.9 • If she won only one of the first two races, then the chance of winning the third is 0.6

Construct a probability tree to represent all outcomes in this situation.

Calculate the probability Sureya won all three races rounded to three decimal places.

Calculate the probability Sureya won the third race, rounded to three decimal places.

For breakfast, Moriah has something to eat and drinks a hot drink. She will either eat toast or cereal and then will drink tea or Milo. • The chance of Moriah making toast is 0.7 • The chance of Moriah drinking Milo is 0.4

Construct a tree diagram illustrating all possible combinations of food and drink Moriah can have for breakfast and their associated probabilities.

Find the probability Moriah drinks tea and eats toast.

Find the probability Moriah drinks tea or eats toast.

The proportion of scholarship recipients at a particular university is

. The number of

students at the university is so large that even if a student is removed, we can say that the proportion of scholarship recipients remains the same. If three students are selected at random: a

Construct a probability tree showing all the possible combinations of recipients and nonrecipient.

Find the probability that at least one of the students is a scholarship recipient.

Find the probability that at least one of the students is a non-recipient.

Find the probability there is at least one recipient and one non-recipient in the selection.

15.02 Conditional probability After this lesson, you will be able to… • calculate probabilities using the conditional probability formula. • interpret two-way tables and Venn diagrams in the context of conditional probability. • solve real-world problems involving conditional probability. • identify the mathematical properties of conditional probability in independent events.

Conditional probability We are often interested in probabilities under certain conditions. For example, if an even number is rolled on a die, what is the probability that it is a six? Giving conditions will reduce the sample space we are concerned with.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Exploration A group of people were asked whether they went on holiday last summer. The results are provided in the given table: Teacher Not a teacher Total

Holiday 22 32 54

No holiday 26 20 46

Total 48 52 100

When randomly selecting a person from all those surveyed, how many people are in the sample space?

When randomly selecting a person from those who are not teachers, how many people are in the sample space?

Explain why your answers to the previous questions are different.

What is the probability of randomly selecting a person that is not a teacher and went on holiday?

Of those who are not teachers, what is the probability of randomly selecting a person that went on holiday?

Show that the probability from question 5 is equal to the probability from question 4 divided by the probability of randomly selecting someone who is not a teacher from the people surveyed.

For two events A and B, the probability of event A occurring given that event B has occurred has the notation P ( A∣B), which is read as the “probability of A given B”. This is known as conditional probability and is calculated by:

Note that if we have a Venn diagram or a two-way table, then we may use the number of favourable and total outcomes instead of their probabilities:

If we know the conditional probability, we can calculate the probability of A and B using: P ( A ∩ B) = P ( A∣B) × P (B) or P ( A ∩ B) = P (B∣A) × P ( A) For independent events, the probability of the second event does not depend on the first event, so P ( A∣B) = P ( A).

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Example 1 Find P ( A ∩ B) given these probabilities for an experiment in which A and B are two possible events: P (B∣ A) = 0.8 and P (A) = 0.5

Create a strategy Use the formula P (B∣A) =

Apply the idea Write the formula

Substitute known values

Multiply both sides by 0.5

Evaluate

Example 2 Use the two-way table to ﬁnd the probabilities. A A′ Total

B′

11 21 32

10 8 18

Total 21 29 50

a Find P ( A∣B).

Create a strategy Use the formula P ( A∣B) =

Apply the idea To use the formula for conditional probability, we need to ﬁnd P ( A ∩ B) and P (B). From the table, we see P ( A ∩ B) =

and P (B) =

. Write the formula

Substitute known values

Evaluate

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reﬂect and check We can also ﬁnd P ( A∣B) without the formula. Since we are given B as the sample space, the total number of outcomes is 32. Since the total number of desired outcomes in that sample space is A = 11, then P ( A∣B) =

b Find P ( A∣B′).

Create a strategy The formula for this conditional probability is P ( A∣B′) = P ( A ∩ B′) and P (B′).

, so use the table to ﬁnd

Apply the idea From the table, we see P ( A ∩ B′) =

and P (B′) =

Write the formula

Substitute known values

Evaluate

Simplify

Reﬂect and check Checking our answer, the total number of outcomes is B′ = 18 and there are A = 10 desired outcomes in that sample space, so P ( A∣B′) =

c Find P ( A′∣B′).

Create a strategy From the previous part, we know the sample space or the total number of outcomes is B′ = 18. Next, we use the table to ﬁnd how many outcomes of A′ are in B′.

Apply the idea From the table, we see there are A′ = 8 desirable outcomes in the sample space. So, Substitute known values

Simplify

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Reﬂect and check Using the formula to check our answer: Write the formula

Substitute P ( A′ ∩ B′) =

Evaluate

Simplify

and P (B′) =

Example 3 A basketball team has a probability of 0.8 of winning its first season and 0.15 of winning its first season and its second season. What is the probability of winning the second season, given they won the first?

Create a strategy Use the formula P (B∣A) =

with:

• Event A: Winning the 1st season • Event B: Winning 2nd season

Apply the idea From the given information we know that: P (Win 1st ∩ Win 2nd) = 0.15 and P (Win 1st) = 0.8. We want to ﬁnd P (Win 2nd∣Win 1st): Start with the conditional probability formula

Substitute the probabilities

Simplify

Example 4 A flight departs from Melbourne to Sydney. The probability that the flight departs on time, given the weather is fine in Melbourne is 0.9, and the probability that the flight departs on time, given the weather is not fine in Melbourne is 0.7. The probability that the weather is fine on any particular day in July is 0.4.

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a By constructing a tree diagram or otherwise, ﬁnd the probability that the ﬂight from Melbourne to Sydney departs on time in a day in July.

Create a strategy Use a tree diagram.

Apply the idea We can construct the following tree diagram for this situation: Fine 0.4 0.6 Not Fine

0.9

On time

0.1

Late

0.7 0.3

On time Late

The ﬂight can be on time in two different ways depending on whether the weather is ﬁne: P (On time) = P (Fine) × P (On time∣Fine) + P (Not fine) × P (On time∣Not fine) P (On time) = 0.4 × 0.9 + 0.6 × 0.7 = 0.78

Substitute the probabilities Evaluate

Reflect and check A two-way table could have been used instead, but the values in the table would need to be calculated first. From the information given, we know: • P (Fine) = 0.4 • P (Not fine) = 0.6 • P (On time∣Fine) = 0.9 • P (On time∣Not fine) = 0.7 The intersection of Fine and On time is: P (Fine ∩ On time) = P (On time∣Fine) × P (Fine) = 0.36 The intersection of Fine and On time is: P (Not fine ∩ On time) = P (On time∣Not fine) × P (Not fine) = 0.42 Filling in the table and finding the total of the On time column gives us: Fine Not fine Total

On time 0.36 0.42 0.78

Late 0.04 0.18 0.22

Total 0.4 0.6 1

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b Find the probability that the weather is ﬁne in Melbourne given that the ﬂight departs on time on a day in July?

Create a strategy We can use the formula: P ( A∣B) =

Apply the idea Write the formula

Substitute the probabilities

Evaluate the numerator

Multiply by

Divide by common factor of 6

Idea summary For two events A and B, the probability of A occurring given that B has occurred is given by:

If we know the conditional probability, we can calculate the probability of A and B using either: P ( A ∩ B) = P ( A∣B) × P (B) or P ( A ∩ B) = P (B∣A) × P ( A).

Practice questions Choose your question path Explorer 1–3, 5–9, 11–13, 15, 18, 20, 22, 23, 27–29

Adventurer 1–3, 5–9, 11–14, 16, 18, 20, 22, 24, 26, 27, 29

Trailblazer 1–4, 6–8, 10, 13, 14, 16, 17, 19, 20, 21, 24–27, 29, 30

Understanding 1

Are these statements true or false? a

806

If P ( A) = 0.5, P (B) = 0.3 and P ( A ∩ B) = 0.15, then the events are dependent.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Using the two-way table shown, P (Maths∣Fail) =

Pass Fail Total

Maths 25 15 40

Physics 7 12 19

Total 32 27 59

In a Venn diagram, if events A and B do not overlap, this means P ( A∣B) = 0.

The phrase “The probability of B given A” means that event A has already occurred and we want to know the probability that event B will occur next.

If one event is “a student drives to school” and a second event is “the weather is rainy”, write the formula for ﬁnding the probability of selecting a student who drives to school given the weather is rainy. Fill in the blanks with “drive” or “rain”:

If P (B∣A) = 0.2, what additional piece of information is needed to ﬁnd P ( A ∩ B)? P ( A)

A 4

P (B)

P ( A ∪ B)

P ( A∣B)

There are two positions available at a company and the applicants have been shortlisted down to 6: Patricia, Nadia, Amy, Aaron, Lachlan, Jimmy. The two people to ﬁll the position will be picked randomly. • Event A: Patricia is chosen • Event B: Aaron is chosen a

Describe the probability P ( A∣ B).

Is P ( A∣ B) less than, greater than or equal to P ( A ∩ B)?

Fluency 5

Find P (B∣A) given these probabilities for an experiment in which A and B are two possible events: P ( A ∩ B) = 0.48 and P ( A) = 0.6

Ex 1

Find P ( A ∩ B) given these probabilities for an experiment in which A and B are two possible events: P ( A∣B) = 0.6 and P (B) = 0.3

Ex 2

Use the two-way table to ﬁnd the probabilities. a

P ( A)

P (B)

P ( A ∩ B)

P ( A ∪ B)

P ( A′ ∩ B′)

P ( A ∩ B′)

P ( A∣B)

P (B∣A)

P ( A′∣B)

P (B∣A′)

P ( A′∣B′)

P (B′∣A′)

A A′ Total

B′

5 7 12

6 2 8

Total 11 9 20

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Using the Venn Diagram, ﬁnd: a

P ( A)

P (B′)

P ( A∣B)

P (B∣A)

P ( A∣B′)

B 0.2

0.2

0.4 0.2

A group of people were asked what type of exercise they like to do. The results are shown in the Venn diagram.

walk

A person is chosen from the group at random.

Find the probability that the person walks.

Find the probability that the person runs, knowing that they also walk.

At a university there are 816 students studying ﬁrst year engineering, 497 of whom say football is their favourite sport (set F ). Of the 348 students studying Civil Engineering (set C), 237 of them say football is their favourite sport.

Ex 3

run 10

C w

Find the value of: i

iii

Of the students whose favourite sport is not football, find the probability that a randomly selected student does not study Civil Engineering.

A player is rolling 2 dice and looking at their difference, that is, the largest number minus the smaller number. They draw a table of all the possible dice rolls for 2 dice and what their difference is: a

State the sample space for the difference of two dice.

Find the probability the dice will have a difference of 2.

Find the probability the dice will have a difference of 2 given that one of the dice rolled is a 3.

1 2 3 4 5 6

1 0 1 2 3 4 5

2 1 0 1 2 3 4

3 2 1 0 1 2 3

4 3 2 1 0 1 2

5 4 3 2 1 0 1

6 5 4 3 2 1 0

A card is randomly drawn from a standard 52 pack of cards. Find the probability that it is a Jack, Queen, King or Ace if: a

No additional information is known.

We know it’s a 10, Jack or Queen.

We know it’s a 9 or a Queen.

We know it’s not 2, 3, 4 or 5.

A software development team has a probability of 0.8 of successfully completing their first project and 0.55 of successfully completing both their first and second projects. Find the probability of successfully completing the second project, given they completed the first one successfully.

In a population, 35% of people have brown eyes, 45% of people have blonde hair, and 10% of people have both brown eyes and blonde hair. A person is chosen randomly from the population. Find the probability that they have:

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Ex 4

Blonde hair, knowing that they have brown eyes

Brown eyes, knowing that they have blonde hair

Two dice are rolled, one after another. Find the probability, in simplest form, of rolling: a

A pair of fours, given the ﬁrst die is a four.

A pair with a sum of 7 or more, given that the ﬁrst die is a two.

Four identical balls labelled 1, 2, 3 and 4 are in a bag. Two balls are randomly drawn from the bag in succession and without replacement. Find the probability that: a

The ﬁrst ball is labelled 4 and the second ball is labelled 2.

The sum of the numbers on the two balls is 5.

The second ball drawn is 1 given that the sum of the numbers on the two balls is 5.

Two cards are randomly drawn in succession and without replacement from a deck of cards numbered from 1 to 20. Find the probability that the second card is: a

An even number given that the ﬁrst card is a 10.

Less than 5 given that the ﬁrst one is a 14.

A number divisible by 5 given that the ﬁrst card is a 15.

A bus service operates from Town A to Town B. The probabilities shown were found with regard to departure times and traffic in June: • The probability that the bus departs on time, given the traffic is light in Town A is 0.8 • The probability that the bus departs on time, given the traffic is heavy in Town A is 0.7 • The probability that the traffic is light on any particular day in June is 0.4 Find the probability that:

The bus service from Town A to Town B departs on time on a day in June.

The traﬃc is light in Town A given that the bus departs on time on a day in June. Give the answer as a fraction in simplest form.

A museum has new software to determine whether a painting is fake or authentic. The system has an accuracy of 75%. Art experts have previously determined that 5% of paintings that the museum examines are fakes. Find the probability that:

The software correctly identiﬁes a painting as fake.

The software incorrectly identiﬁes an authentic painting as fake.

The software will identify a painting as fake.

A painting that is identiﬁed as fake is actually fake, rounded to two decimal places.

In a survey, 531 people are asked whether they watch My Kitchen Rules (MKR) or Masterchef (MC). Of the total respondents, 177 people watch both and 65 watch neither. The number who watch MKR is twice the number who watch both. a

Find the number of people who only watch MKR.

Of the people who watch MC, ﬁnd the probability that a randomly selected person also watches MKR.

Of those who don’t watch MC, ﬁnd the probability that a randomly selected person watches neither.

15.02 Conditional probability mathspace.co

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The probability of two independent events, A and B are P ( A) = 0.8 and P (B) = 0.5 Determine the probability of: a

Both A and B occurring

Neither A nor B

A or B or both

B but not A

A given that B occurs.

Reasoning 22

For P ( A ∩ B): a

Write the formula of independent events.

Write the formula of dependent events.

Which formula from part (a) or (b) can be used for both independent and dependent events? Explain.

Given the conditional probability formula,

show that, P ( A∣B) × P (B) = P (B∣A) × P ( A) 24

Use the Venn diagram shown, where B is a subset of A, to ﬁnd: a

P ( A∣B)

P (B∣A)

P ( A′∣B′)

P (B′∣A′)

Explain why the probability of A occurring given B has already occurred is always certain.

A Venn diagram is shown in which the size of the regions are proportional to the number of elements in that region.

A B 5

Which conditional probability is greater, P ( A∣B) or P (B∣A)? Justify your answer.

Problem-solving 26

810

Edwin watches three episodes of TV each night. He begins with News or Current Affairs, then the next two shows are either Comedy, Horror or Animation. He never watches more than one Horror show, and if he watched the News, he will follow this immediately with a Comedy. a

Construct a tree diagram of all possible options.

If, at every stage, the possible outcomes of each choice are equally likely, ﬁnd the probability Edwin watches two Comedies, given that he watched the News.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

When Fred gets ready for work in summer, he ﬁrst decides whether it will be a hot day (H) or not (N ) and then decides whether to wear a tie (T ) or to just dress casually (C). The chance of Fred deciding it will be a hot day is 0.7. If he decides it will be a hot day, there is a 0.2 chance he will wear a tie. If he decides it will not be a hot day, there is a 0.85 chance he will wear a tie. a Construct a probability tree for this situation. b Calculate the probability Fred decided it was a hot day given that he wears a tie. Round your answer to two decimal places.

For breakfast each morning, Marge eats porridge, toast or cereal. With that, she will either drink orange juice, tea or hot chocolate. Find the probability that Marge will: a Eat toast and drink tea b Eat cereal or drink orange juice c Eat porridge given that she drinks hot chocolate

Two hundred people were questioned about whether they voted for Labor, Liberal or Greens last election and who they’ll vote for this election.

Labor (last election) Liberal (last election) Greens (last election) Total a b c d

Labor (next election) 26

Liberal (next election) 40

1 5 10

Total 70 90 200

Complete the table. Find the probability that a randomly selected person will vote Labor next election given that they voted Liberal last election. Find the probability that a randomly selected person did not vote Green last election given that they will vote Green next election. If a person voted Labor or Liberal last election, ﬁnd the probability they’ll vote Liberal next election.

A class of 13 students had their height, shoe size and 400 m race time collected. The results are shown in the table: a Let T be the event that a randomly chosen student from the class has a race time greater than 90. Find the value of P (T ). b Let H be the event that a randomly chosen student from the class has a height greater than 130. Find the value of P (H). c Find the value of P (T∣H). d Let S be the event that a randomly chosen student from the class has a shoe size greater than or equal to a size 10. Find the value of P (H∣S). e

140

Greens (next election)

Which of the events appear to be dependent, H and S or H and T?

Athlete Aaron Tobias Lisa Georgia Ben Patricia Vanessa Sandy Adam Amelia Rosey Neil Roald

400 m Height time (sec) (cm) 94 137 62 165 89 100 64 155 98 100 127 102 84 147 98 144 54 154 84 147 81 126 104 134 67 127

Shoe size 8 10 11 7 10 11 7 8 7 8 12 9 8

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15.03 Mixed probability concepts After this lesson, you will be able to… • use Venn diagrams, two-way tables, and tree diagrams for organising data. • perform probability calculations using appropriate formulas. • interpret results in the context of the problem.

Mixed probability concepts Many situations in probability can be modelled by a Venn diagram, an array or two-way table, and/or a tree diagram to organise the information, determine the size of different groups, and do calculations. Probability notation can also help us to organise the information given in a question, and these formulas can help us calculate different probabilities. If two events are independent: P ( A ∩ B) = P ( A) × P (B) When two events are dependent, the probability of each event will change, depending on the first event. P ( A ∩ B) = P ( A∣B) × P (B) For conditional probability:

For mutually exclusive events: P ( A ∪ B) = P ( A) + P (B) For non-mutually exclusive events: P ( A ∪ B) = P ( A) + P (B) − P ( A ∩ B)

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Example 1 A bag contains the letters A, B, and C. A letter is randomly drawn out of the bag without replacement and placed on the table. This is done three times.

B C A C A B

a Find the probability that A is drawn before B is drawn.

C B C A B A

Create a strategy Count the number of branches where A appeared before B.

Apply the idea

Reﬂect and check

The sequences where A is drawn before B are ABC, ACB, and CAB. There are 3 desired outcomes and 6 total outcomes possible.

Notice each sequence has a probability of

Therefore, the probability is .

Since there are 3 possible sequences, the probability of drawing A before B is:

b Find the probability that two A ’s are drawn.

Create a strategy

Apply the idea

Note that the letters are not replaced after being As there is only one A inside the bag, it is drawn. impossible to randomly draw two A ’s without replacement. So the probability is 0.

c Find the probability the word CAB is spelled out in that order.

Create a strategy Consider the number of outcomes that spell out CAB.

Apply the idea Only one of the six outcomes spells CAB. The probability is .

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Reflect and check We can check the answer by thinking about the probability of drawing C first, then A, then B. The probability of drawing C first is . For the second draw, there are 2 letters left. The probability of drawing A next is . There is only one letter left for the third draw, so the probability of drawing B third is 1. So the probability of drawing the letters CAB in that order is:

d Find the probability that B is drawn last, given that A is drawn ﬁrst.

Create a strategy The probability of selecting B last given A was drawn ﬁrst can be calculated using the formula P (B last∣A first) =

Apply the idea There is only one way to select B last and A ﬁrst, so P (B last ∩ A first) = . For the ﬁrst letter, A is one of the three letters, so P ( A first) = . Dividing these, we get:

. Start with the conditional probability formula

Substitute known values

Evaluate the division

Simplify

Reﬂect and check With conditional probability, the given condition determines the total number of outcomes. If we know A was drawn ﬁrst, there are only 2 possible outcomes: drawing B last or drawing C last. Since drawing B last is one of those outcomes, the probability is .

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Example 2 At a restaurant, 50 people are asked about whether they would like to add rice (R) or broccoli (B) to their meal. Some of the results are shown in the Venn diagram.

14 15

a Find the number of people in R ∩ B′.

Create a strategy This is the blank section of the Venn diagram. We can subtract the amounts in the diagram from the total number of people surveyed.

Apply the idea

Reﬂect and check

The number of people in R ∩ B′ is

We can now complete the Venn diagram.

50 − 9 − 14 − 15 = 12

n (R ∩ B′) = 12 12

14 15

b Find P (B∣R).

Create a strategy Use the formula P (B∣R) =

Apply the idea

Reﬂect and check

Using the Venn diagram from part (a), we know the number of people in B ∩ R is 9 and the number of people in R is 9 + 12 = 21.

Checking our answer, we can look at the Venn diagram to conﬁrm that of those who chose rice (21 people), 9 of them also chose broccoli.

Start with the conditional probability formula

Substitute known values

Evaluate the division

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c Find P (R∣B′).

Create a strategy Use the formula

Apply the idea Using the Venn diagram from part (a), we know the number of people in R ∩ B′ is 12 and the number of people in B′ is 12 + 15 = 27. Start with the conditional probability formula

Substitute known values

Evaluate the division

Simplify

Reﬂect and check Checking our answer, we can look at the Venn diagram to conﬁrm that of those who did not choose broccoli (27 people), 12 of them chose rice.

Example 3 600 people were questioned as to whether they read novels in paperback form or on an e-reader. • 120 said they do both. • 180 said they read paperbacks. • The probability that a person who didn’t read paperbacks also didn’t read on an e-reader was 30%. a Complete the table for these results: Paperback

Not paperback

E-reader Not e-reader

Create a strategy Add a total row and column to the table to make it easier to ﬁnd missing values. Paperback

Not paperback

E-reader Not e-reader Total

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Total

Apply the idea The first piece of information goes in the first cell of the table. The second piece of information helps us fill in the first column and last row.

E-reader Not e-reader Total

Paperback 120 60 180

Not paperback

Total

420

600

The third piece of information can help us ﬁll in the rest of the table. Of the people who didn’t read paperback books, 30% did not read on an e-reader. 30% × 420 = 126

E-reader Not e-reader Total

Paperback 120 60 180

Not paperback 294 126 420

Total 414 186 600

b Calculate the probability that a randomly selected person reads novels on an e-reader but not in paperback.

Create a strategy Use the table from part (a) to ﬁnd the number in the row for E-reader and the column for Not paperback, E ∩ P′.

Apply the idea In the table from part (a), we see 294 people read novels on an e-reader but not in paperback. Divide desirable outcomes by total outcomes

Simplify

c Of those that read on e-readers, what percentage read in both formats? Give your answer correct to the nearest percent.

Create a strategy Divide the number of people who read on both formats by the number of people who read on an e-reader.

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Apply the idea The number of people who read on both formats is 120. The number of people who read on an e-reader is 414. Use a calculator to estimate the probability as a percentage, and round to the nearest whole percentage:

Reﬂect and check We can check this answer by using the formula for conditional probability: P (P∣E) = Start with the conditional probability formula

Substitute known values

Evaluate the division

Convert to a percentage

Evaluate using calculator and round

Example 4 Vanessa has 12 songs in a playlist. Four of the songs are her favourite. She selects shuffle and the songs start playing in random order. Shuffle ensures that each song is played once only until all songs in the playlist have been played. Find the probability that: a The first song is one of her favourites.

Create a strategy The probability can be calculated using

Apply the idea Vanessa has 4 favourite songs and there are 12 songs in the playlist. The probability that the ﬁrst song is one of her favourites is

b Two of her favourite songs are the ﬁrst to be played.

Create a strategy The probability for the second song will be dependent on the ﬁrst song that is played.

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Apply the idea From part (a), we know the probability that the ﬁrst song is one of Vanessa’s favourites is . This leaves 3 favourite songs out of 11 songs that have yet to play. So, the probability that the second song is also her favourite is

Multiplying these probabilities together gives us:

c At least 1 of her favourite songs is played in the ﬁrst three.

Create a strategy The probability that at least 1 of the songs is her favourite is the complement of the probability that no songs are her favourite.

Apply the idea The probability that the ﬁrst song is not a favourite is

Since there are 7 non-favourite songs left and 11 songs left in the playlist, the probability that the second song is not a favourite is

For the third song, there are 6 non-favourites left and 10 songs left in the playlist. The probability the third song is not a favourite is

The probability that no songs are Vanessa’s favourite is:

The probability that at least one of the songs is Vanessa’s favourite is:

Idea summary Venn diagrams, arrays, and tree diagrams can be used to organise information and help solve problems in probability. We can also use the formulas for complementary events, independent events and conditional probability to solve problems. For independent events: P ( A ∩ B) = P ( A) × P (B) For dependent events: P ( A ∩ B) = P ( A∣B) × P (B) For conditional probability: P ( A∣B) = For mutually exclusive events: P ( A ∪ B) = P ( A) + P (B) For non-mutually exclusive events: P ( A ∪ B) = P ( A) + P (B) − P ( A ∩ B)

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Practice questions Choose your question path Explorer 1–3, 6–8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 26, 28

Adventurer 1–6, 8, 10–13, 15–18, 21, 22, 24, 26–28

Trailblazer 1–6, 9, 11, 12, 14, 16, 19, 20, 22, 23, 25, 27–29

Understanding 1

Match each of the Venn diagrams to the correct statement. i

B A

ii A B

0.32

0.08

0.12 0.48

iii

A B

iv A

0.32

0.08

0.48 0.12

Events A and B are mutually exclusive.

Events A and B are independent.

P ( A∣B) = 1

P ( A ∩ B) ≠ P ( A) × P (B)

Match each formula to the probability described. i

P ( A ∪ B) = P ( A) + P (B) − P ( A ∩ B)

P ( A ∩ B) = P ( A) × P (B)

iii

P ( A) + P (B) = 1

P ( A ∩ B) = P ( A∣B) × P (B)

Conditional probability

Complementary events

Non-mutually exclusive events

Independent events

Which statement is equivalent to this statement? Find the probability that a randomly selected person is female, given they are a team leader.

820

Of the females, ﬁnd the probability the person is a team leader.

If a person is a team leader, then ﬁnd the probability that they are female.

Find the probability that a randomly selected person is male knowing that they are a team leader.

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Fluency 4

Find the values of w, x, y, and z, given that P ( A ∩ B) = 0.3, P (B) = 0.6, and that the events A and B are independent.

Use the Venn Diagram to ﬁnd P ( A∣ B). Give the answer as a fraction in simplest form.

B w x

A A′

0.3

B′ y z

0.25

0.15

0.3

Ex 1

Three cards labelled 2, 3 and 4 are placed face down on a table. Two of the cards are selected randomly to form a two-digit number. The outcomes are displayed in the tree diagram. a

List the sample space of two digit numbers produced by this process.

Find the probability that:

4 2 4 2 3

2 appears as a digit in the number.

The sum of the two selected cards is even.

iii

The number formed is greater than 40.

2 is the second digit, given 3 is the ﬁrst digit.

Students were asked what they are allergic to. The table shows the results: Allergic to dairy Not allergic to dairy

Allergic to nuts 14 25

Not allergic to nuts 12 25

If a student is chosen at random, ﬁnd the probability that:

Ex 2

The student is allergic to dairy.

The student has an allergy.

The student does not have an allergy.

The student is also allergic to dairy if he or she is allergic to nuts.

In a survey, 100 oﬃce workers were asked about whether they buy a coffee (set C) or lunch (set L) during the work day. Some of the results are shown in the Venn diagram. Find: a

The number of workers in (L ∩ C′)

P ( C∣L)

P ( L∣C′)

L 43

47 7

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Of the 100 students starting kindergarten surveyed, 80% believe in Santa, 41% believe in the Tooth Fairy and 31% believe in both. a

Complete the table: Believe in Santa

Do not believe in Santa

Total

Believe in Tooth Fairy Do not believe in Tooth Fairy Total

Ex 3

100

Of those who believe in Santa, ﬁnd the probability that they also believe in the Tooth Fairy. Write your answer as a percentage rounded to two decimal places.

Of those who believe in either Santa or the Tooth Fairy, what is the probability they believe in both?

Of those who believe in the Tooth Fairy, what is the probability they do not believe in Santa?

Two sets of numbers, A and B, are such that set A contains the even numbers from 1 to 20, inclusive, and set B contains the factors of 48 from 1 to 20, inclusive. a

List the numbers in set A.

List the numbers in set B.

How many numbers are unique (only appear in one set)?

Find the probability that a randomly selected number is in set B, given that it is in set A.

In a survey, 166 children were asked whether they had read or watched at least one of the stories in a popular children’s series: • 62 hadn’t read any of the books. • 39 had read and watched at least one of the stories. • 97 had read the book. • 4 had never read or watched any of these. a

Complete the table for these results: Read the book Didn’t read the book Total

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Watched the movie 39

Didn’t watch the movie 4 62

Find the probability that a randomly selected student had: i

Read the book but hadn’t watched the movie.

Read the book, given that they had watched the movie.

iii

Watched the movie, given that they hadn’t read the book.

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Total 97 166

In a survey, 1000 people were asked about whether they bought (set B) or grew (set G) their own herbs. The results are listed: • 4% said they do neither. • 40% said they do both. • Of those who bought their herbs, the probability that they also grew herbs was . In this Venn diagram, where x, y and z represent numbers of people: a

Find the value of: i

iii

Find the number of people who only grew their own herbs.

Of those who grew herbs, what is the probability that they also bought herbs?

G z

y x

In a group of 69 people, 45 said they had been to Europe (E), 26 said they had been to Asia ( A) and 8 people said they had been to neither Europe nor Asia. Find: a

P ( A ∩ E′)

The number of people in (E ∩ A)

P ( E∣A)

P ( A∣E′)

At an Italian restaurant, Alessandra orders an entree, main and dessert. She has 2 entrees, 3 mains and 2 desserts to choose from:

• Entrees: calamari, soup • Mains: spaghetti, lasagna, risotto • Dessert: gelato, tiramisu

The possible combination of meals that Alessandra could order is shown in the tree diagram.

spaghetti

gelato

lasagna

tiramisu gelato

If each combination of dishes is equally likely, ﬁnd the probability Alessandra orders lasagna given that she ordered calamari.

calamari

risotto

Find the probability Alessandra ordered spaghetti and tiramisu, given that she ordered soup. Find the probability Alessandra ordered risotto or spaghetti, given that she ordered soup.

tiramisu gelato

spaghetti

tiramisu gelato

lasagna

tiramisu gelato

soup risotto

tiramisu gelato tiramisu

Two marbles are randomly drawn without replacement from a bag containing 1 blue, 2 red and 3 yellow marbles. a

Construct a tree diagram to show the sample space.

Find the probability of drawing these: i

A blue marble and a yellow marble, in that order

A red marble and a blue marble, in that order

iii

2 red marbles

No yellow marbles

2 blue marbles

A yellow marble and a red marble, in that order

vii A yellow and a red marble, in any order

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In tennis, if the ﬁrst serve is a fault (out or in the net), the player takes a second serve. A player serves with these probabilities: • First serve in: 0.55 • Second serve in: 0.81

Construct a probability tree showing the probability of the ﬁrst two serves either being in or a fault.

Find the probability that the ﬁrst serve is a fault.

Find the probability that the player makes a double fault (both serves are a fault).

Nine pilots from Star Jet and seven pilots from Air Tiger offer to take part in a rescue operation. If 2 pilots are selected at random: a

Construct a probability tree showing all possible combinations of airlines from which the pilots are selected.

Find the probability that the two pilots selected are from: i

Ex 4

Different airlines

A carton contains a dozen eggs, of which 4 contain a double yolk. If 3 eggs are chosen at random for a cake, find the probability that: a

The same airline

They all have a double yolk.

At least 1 has a double yolk.

From a standard pack of cards, 1 card is randomly drawn by Andrea and then put back into the pack. A second card is then drawn by Bridie. • Event A: Andrea selects a heart. • Event B: Bridie selects a heart.

Calculate the probability that neither of their cards are hearts

Calculate the probability that at least 1 of the cards is a heart.

The probability that Joanna, Nadya and Adam get permission to go on their school trip are 0.8, 0.1 and 0.2, respectively. Find the probability that at least 1 of them gets permission.

Reasoning 21

A sample of 20 students, A-T, from the same school were asked whether they had a haircut in the last month (H) and whether they had been sick in the last month (S). The results are shown in the given tables:

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Student Hair Sick

A ✓ −

B ✓ ✓

C − −

D ✓ ✓

E − −

F ✓ −

G ✓ ✓

H −

Student Hair Sick

K −

L −

M −

✓

O − −

P ✓ −

Q −

✓

N ✓ ✓

Find the probability that a student: i Had a haircut in the last month ii Had been sick in the last month iii Was sick and had a haircut in the last month

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✓

I − −

J ✓ −

R ✓ ✓

S − −

T − ✓

Find the value of P (H) × P (S).

Does the data suggest that getting haircuts and being sick in the last month are independent or dependent? Explain your reasoning.

In a memory game, 16 pairs of identical cards are randomly placed face down. When someone has a go, they turn one card over and then turn a second card over to try to ﬁnd an identical pair. If an identical pair are found, they are removed from the pack. a

At the beginning of the game, Kathleen turns a card over. Find the probability of her selecting a card that makes an identical pair.

The ﬁrst two cards Kathleen picks are identical. Is the probability of picking any more pairs of identical cards independent or dependent on the number of pairs picked before? Explain your reasoning.

Candidates for a professional examination are allowed a maximum of two attempts at the exam. Over a period of time, it has been shown that 60% of the candidates pass on the ﬁrst attempt and, if a second attempt is required, 70% of the candidates pass. Assume that these probabilities apply to a randomly chosen candidate.

Why does the second set of branches only have one fork?

0.6

Given that a candidate passes, calculate the probability that they passed on the ﬁrst attempt. Give the answer as a fraction in simplest form.

0.4

Explain why it is inappropriate to assume that the probabilities apply to a randomly chosen candidate.

0.7

0.3

Problem-solving 24

A school consisting of 410 primary students and 410 secondary students offers everyone an optional end of year project to improve their grade. There are various types of projects they can do. A summary of the number of student submissions are shown in the given table: Primary Secondary a b c d

Essay 18 76

Creative story 64 68

Art work 46 73

Other Project 24 45

Did not submit 258 148

Find the probability of a selecting a student that submitted a project. Find the probability of a selecting a student that submitted a project given that they are in a secondary year (S). Find the probability that a submitted project is an essay. Find the probability that a submitted project is an essay given that the project was submitted by a primary student (P).

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Find: a

P ( A ∩ B ∩ C′)

P (( A ∪ B) ∩ C′)

P ( B∣C)

P ( A′∣C)

P (( A ∪ B)∣C′)

B 0.17 0.18 0.1 0.07 0.11 0.12 0.05

0.2 C

Yvonne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror. a

How many of the movies are horror ﬁlms?

How many of the movies ﬁt into only one genre?

Find the probability of selecting a film that fits into only one genre given that it fits into the horror genre.

Comedy 10

Action 4

5 3

6 13 Horror

In a particular city, 55% of the population are adults. 8% of the children and 2% of the adults are left-handed. a

What percentage of the city’s population is left-handed?

If a person is chosen at random and is left-handed, what is the probability that the person is an adult? Give the answer rounded to two decimal places.

The probability of spotting a rare bird species in a national park on any given day is 0.3. If one is spotted one day, the probability of spotting another the next day increases to 0.6, due to ﬂock movement. However, if none are spotted, the probability of spotting one the next day decreases to 0.15. What is the probability of spotting exactly one rare bird over two consecutive days?

In a speciﬁc deck of 10 cards numbered from 1 to 10, the probabilities of drawing each number are in the ratio 1: 2: 3: 4: 5: 6: 7: 8: 9: 10. If two cards are drawn with replacement, what is the probability that they add up to 15?

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Chapter 15 review 1

Construct a two-way table based on the information in the Venn diagram.

A spinner is divided into 6 sections. 3 sections are red and 3 sections are yellow. a

If the spinner lands on yellow on the ﬁrst spin, ﬁnd the probability that the second spin will be yellow.

Are the successive events of spinning yellow twice dependent or independent?

a 4

A R

Exactly once

At least once

If three letters are randomly selected from the word CIRCUMFERENCE without replacement, ﬁnd the probability that: a

T F

A photo album contains 5 photos, all numbered from 1 to 5. Four photos are drawn at random, with replacement. Find the probability that the photo with the number 2 is drawn:

The letters are R, C, U in that order.

All three letters are E.

Find P (A ∩ B) given these probabilities for an experiment in which A and B are two possible events: P ( A ∣ B) = 0.4 and P (B) = 0.5

Use the two-way table to ﬁnd the probabilities. a

P (A)

P (B∣A′)

P (A ∩ B)

P (A′∣B′)

A A′ Total

Use the Venn Diagram to ﬁnd: a

P (B′)

P ( A ∪ B)

P ( A∣ B)

B′

14 6 20

8 12 20

0.35

Total 22 18 40

0.1

0.2

0.35

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Amy owns ﬁve necklaces and four earrings. She selects one of the items at random for herself and then another item at random for her sister: a

Find the probability that Amy selects a necklace for herself.

Find the probability that both items Amy selects are earrings.

5 9

1 2

4 9

5 8

Find the values of j, k, l, and m, given that P (A ∩ B) = 0.4, P (B) = 0.5, and that the events A and B are independent.

T F

B j l

A A′

B′ k m

In a survey, 640 participants were asked whether they prefer watching basketball or volleyball. Of the total respondents, 120 enjoy watching both sports and 70 enjoy neither. The number who prefer watching basketball is three times the number who enjoy watching both sports.

A R

Complete the Venn diagram.

Of those who enjoy watching volleyball, ﬁnd the probability that a randomly chosen person also enjoys watching basketball.

If a randomly selected person does not enjoy watching volleyball, then ﬁnd the probability that they dislike watching basketball too.

120

In a group of 100 people, 60 said they went swimming, 40 said they went hiking, and 10 people said they neither swam nor hiked. a

Complete the table:

Went hiking

Did not hike

Went swimming Did not swim Total b

3 8

1 2

Total

100

Find the probability that a randomly selected person: i

Went swimming and hiking

Went swimming given they went hiking

iii

Went hiking knowing they went swimming

For breakfast, Elijah has something to eat and something to drink. He will either eat an egg sandwich or a muﬃn and will drink orange juice or iced coffee. • The chance of Elijah eating an egg sandwich is 0.6 • The chance of Elijah drinking iced coffee is 0.5 a

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Construct a tree diagram showing all possible combinations of food and drink Elijah can have for breakfast and their associated probabilities.

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Find the probability Elijah drinks orange juice and eats an egg sandwich.

Find the probability Elijah drinks orange juice or eats an egg sandwich.

Find the probability Elijah drinks iced coffee given he ate a muﬃn.

In a city, 40% of people prefer tea over coffee, 30% of people enjoy reading newspapers, and 15% of people both prefer tea and read newspapers. A resident is chosen randomly. Find the probability that they: a

Enjoy reading newspapers given that they prefer tea.

Prefer tea, knowing that they read newspapers.

A box contains 20 candies, of which 6 are sour. If Jenna randomly chooses 3 candies to eat, ﬁnd the probability that: a

All are sour.

At least 1 is sour.

A small magazine asked people from different states to send in a vote on whether they supported Daylight Saving Time. The diagram shows the proportion of people that voted YES or NO in NSW, ACT and QLD:

T F

Find the probability that a randomly chosen person voted “YES”.

NSW YES NO

Of those that are not from NSW, ﬁnd the probability that a randomly chosen person voted “NO”.

YES

A R

Considering just the voters from NSW and ACT, ﬁnd the probability that randomly chosen person was from NSW and voted “YES”.

ACT YES NO

NSW

YES

NSW

YES

NSW

YES

ACT

QLD NO

YES

ACT

QLD NO

QLD

NSW

YES

QLD YES NO

YES

(Optional) A warehouse has a total storage volume of 75 000 m3. Within this warehouse, there is a section designated for fragile items that has a volume of 20 250 m3. If an item is placed at a random location in the warehouse, what is the probability that it ends up in the section designated for fragile items?

(Optional) A dart is thrown, and it hits the board shown. Find the probability that the dart lands in the shaded region. Round your answer to two decimal places.

4.5 cm

16 cm

(Optional) There is a sphere of unknown radius inside a cube with side lengths of 20 m. The sphere does not touch all sides of the cube. The probability of a random point in the cube being outside of the sphere is 0.82. Determine the radius of the sphere.

Chapter 15 review mathspace.co

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16 Univariate statistics Big ideas • Different representations of data highlight different characteristics of the data. • Comparing and interpreting data sets involves analysing statistical measures and visual representations to gain meaningful insights, make informed conclusions and understand the variation and relationships within the data.

Curriculum Outcomes VC2M10ST01 Compare data distributions for continuous numerical variables using quartiles and interquartile range and appropriate data displays including boxplots, histograms and dot plots; discuss the shapes of these distributions in terms of centre, spread, shape and outliers in the context of the data VC2M10ST04 Analyse claims, inferences and conclusions of statistical reports in the media and other places, by linking claims to displays, statistics and representative data, including ethical considerations and identification of potential sources of bias VC2M10AST01 Calculate and interpret the mean and standard deviation of data and use these to compare data sets; investigate the effect of individual data values, including outliers, on the standard deviation VC2M10AST02 Identify measures of spread, and understand their interpretation and usefulness with respect to different data distributions

Chapter outline 16.01 16.02 16.03 16.04 16.05

Quartiles Boxplots Cumulative frequency Compare data sets Samples and populations Chapter 16 review

832 845 859 871 887 897

In medicine, scientists use standard deviation to assist in understanding drug effectiveness and side effects!

16.01 Quartiles After this lesson, you will be able to… • identify the lower quartile, median and upper quartile in a data set. • calculate the interquartile range in a data set. • analyse the advantages of using IQR over the range in the presence of outliers. • apply the concept of quartiles and IQR to solve problems.

Quartiles The range is a measure of spread based on the minimum and maximum in a data set, but does not tell us about the spread of data within these two values.

Range = Maximum − Minimum

Maximum Minimum

The highest value in the data set The lowest value in the data set

The median is a measure of centre, and tells us where the middle of the data set is. For a data set with n items, Median is the

value

To get a better picture of the internal spread in a data set, we can ﬁnd the quartiles of a data set, which can be used for a measure of spread called interquartile range (IQR). First Second quartile quartile

Third quartile

Forth quartile

2, 3, 5, 5, 7, 9, 10, 11, 13, 16, 18, 22, 39, Minimum

Median Q1 = 5

Maximum Q3 = 17

Quartiles are scores at particular locations in the data set-similar to the median, but instead of dividing a data set into halves, they divide a data set into quarters. • Minimum: Lowest value • Lower quartile, Q1: At most 25% of the data is below this value • Median: 50% of the data lies on either side of this value • Upper quartile, Q3: At most 25% of the data is above this value • Maximum: highest value Let’s look at how we would divide up some data sets into quarters now. Make sure the data set is ordered before ﬁnding the quartiles or the median. Median

832

First locate the median, between the 4th and 5th scores:

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Now there are four scores in each half of the data set, so split each of the four scores in half to ﬁnd the quartiles. Q1

Median

We can see the ﬁrst quartile, Q1 is between the 2nd and 3rd scores. Similarly, the third quartile, Q3 is between the 6th and 7th scores.

For 9 scores, the 5th term is the median. There are four terms on either side of the median. So Q1 is still between the 2nd and 3rd scores and Q3 is between the 6th and 7th scores.

Median

For 10 scores, the median is between the 5th and 6th scores. This time, however, there are 5 scores on either side of the median. So Q1 is the 3rd term and Q3 is the 8th term.

Median

Each quartile represents approximately 25% of the data set. For a data set with n items, Q1 is the

value

and Median is the

value

Q is the

value

and

Example 1 Here are Ray’s scores from his last 13 rounds of golf played: 66, 66, 68, 68, 70, 78, 80, 84, 106, 116, 126, 130, 132 a What is his median?

Create a strategy Use the formula:

th where n is the total number of scores, to ﬁnd the position of the median.

Apply the idea There are n = 13 scores in the list. Position of median = = 7th Median = 80

Substitute n = 13 Evaluate Choose the 7th score

16.01 Quartiles mathspace.co

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b What is the lower quartile?

Create a strategy Find the median of the lower half of the scores excluding the median.

Apply the idea The lower half of the scores are: 66, 66, 68, 68, 70, 78. Lower quartile =

Average the middle scores

= 68

Evaluate

c What is the upper quartile?

Create a strategy Find the median of the upper half of the scores excluding the median.

Apply the idea The upper half of the scores are: 84, 106, 116, 126, 130, 132. Upper quartile =

Average the middle scores

= 121

Evaluate

Idea summary • • •

Q1 is the ﬁrst quartile (sometimes called the lower quartile). It is the middle score in the bottom half of data. The median, sometimes called Q2, is the second quartile. Q3 is the third quartile (sometimes called the upper quartile). It is the middle score in the top half of the data set.

Interquartile range The interquartile range (IQR) is the difference between the third quartile and the ﬁrst quartile. 50% of scores lie within the IQR. Since it focuses on the middle 50% of the data set, the interquartile range often gives a better indication of the internal spread than the range does, and it is less affected by outliers.

IQR = Q3 − Q1

834

Lower quartile

Upper quartile

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 16.01 to answer these questions. 1.

What do you notice about the number of values in each quartile?

Choose an n value which is divisible by 4. What do you notice?

Choose an n value which is divisible by 2. What do you notice?

Choose an n value which is divisible by 3. What do you notice?

Example 2 Use this frequency table to: Score 5 14 18 24 32 38 50

Frequency 1 1 3 2 1 5 5

a Find the number of scores.

Create a strategy Add all the frequencies.

Apply the idea Number of scores = 1 + 1 + 3 + 2 + 1 + 2 + 5 = 15

Find the sum Evaluate the addition

b Find the median.

Create a strategy Use the formula:

th where n is the total number of scores, to ﬁnd the position of the median.

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Apply the idea There are 15 scores in the list. Position of median = = 8th

Substitute n = 15 Evaluate

Using the frequency table we can see that the 8th score is 32. Median = 32

c Find the lower quartile of the set of scores.

Create a strategy

Apply the idea

Find the median of the lower half of the scores.

There are 7 scores in the lower half excluding the median. The lower quartile will be the 4th score from the frequency table. Lower quartile = 18

d Find the upper quartile of the set of scores.

Create a strategy

Apply the idea

Find the median of the upper half of the scores.

There are 7 scores in the upper half excluding the median. The upper quartile will be the 4th in the upper half, which is the 4 + 8 = 12th score overall. Upper quartile = 50

e Find the interquartile range.

Create a strategy Use the interquartile range formula: IQR = Q3 − Q1

Apply the idea IQR = 50 − 18 = 32

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Substitute the quartiles Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 Answer the questions using the dot plot shown:

a Find the total number of scores.

Create a strategy

Apply the idea Number of scores = 15

Count the number of dots in the dot plot.

Count the dots

b Find the median.

Create a strategy Use the formula:

th where n is the total number of scores, to ﬁnd the position of the median.

Apply the idea Position of median = = 8th

Substitute n = 15 Evaluate the division

The 8th score in the dot plot is on 15. Median = 15 c Find the lower quartile of the set of scores.

Create a strategy

Apply the idea

Find the median of the lower half of the scores.

There are 7 scores in the ﬁrst half excluding the median. The lower quartile will be the 4th score in the dot plot. Lower quartile = 9

d Find the upper quartile of the set of scores.

Create a strategy Find the median of the upper half of the scores.

Apply the idea There are 7 scores in the second half excluding the median. The upper quartile will be the 8 + 4 = 12th score in the dot plot. Upper quartile = 16

16.01 Quartiles mathspace.co

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e Find the interquartile range.

Create a strategy Use the interquartile range formula: IQR = Q3 − Q1

Apply the idea IQR = 16 − 9

Substitute the quartiles

Evaluate

Idea summary To calculate the interquartile range:

IQR = Q3 − Q1 IQR is the interquartile range Q1

is the ﬁrst quartile

is the third quartile

Practice questions Choose your question path Explorer 1aceg, 2–6, 7ace, 8, 9, 11–15, 17, 19, 22, 24, 25a-f

Adventurer 1bdfh, 2, 4, 5, 7ace, 8, 10, 11, 13, 16, 18, 20–22, 24, 25

Trailblazer 1bh, 2cd, 4, 5, 7bdh, 8, 10, 11, 15, 16, 18, 20–26

Understanding 1

Find the median for each data set of scores: a

31, 32, 34, 42, 49, 50, 51

21, 23, 26, 31, 34, 38, 42, 47, 48

2, 10, 28, 35, 50

41, 42, 43, 44, 49, 50, 51, 53, 58

8, 9, 11, 12, 13, 15

3, 4, 6, 7, 7, 8, 12, 14

5, 7, 12, 14, 15, 22, 24, 30

7, 8, 9, 12, 18, 21, 31, 33

For each data set, the median is shown. Find the upper and lower quartiles. a

0, 1, 4, 6 8, 9, 11

22, 24, 25, 29, 32, 34

Median Median: 27

5, 11, 19, 20, 22 24, 26, 27, 27

14, 15, 16, 16, 17, 18, 19, 20, 24, 25, 30, 36

Median Median: 18.5 838

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

What is the interquartile range for this data set? Q1 = 4

Q3 = 14.5

1, 2, 3, 5, 5, 7, 11, 14, 15, 16 Minimum Median = 6

Maximum

Match the term with the deﬁnition. i

Highest value

50% of the data lies on either side of this value

iii

At most 25% of the data is above this value

The lowest value

At most 25% of the data is below this value

Minimum

Lower quartile, Q1

Median

Upper quartile, Q3

Maximum

Fluency Ex 1

The data set shows Rachelle’s scores from her last 13 rounds of golf played: 56, 58, 60, 62, 64, 72, 74, 78, 98, 108, 120, 124, 128

Find her median score.

Find the lower quartile score.

Find the upper quartile score.

Find the interquartile range.

The data set shows Luke’s scores from his last 17 exams: 42, 46, 48, 51, 52, 54, 56, 68, 72, 76, 78, 82, 85, 86, 88, 92, 96

Find his median score.

Find the lower quartile score.

Find the upper quartile score.

Find the interquartile range.

Find the number of scores.

For each set of scores: i

Sort the scores in ascending order.

iii

Find the median.

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

40, 39, 15, 17, 10, 6, 24

−4, −6, −1, 7, 9, 7, 9

8, 20, 19, 4, 15, 14, 10

42, 28, 22, 40, 20, 54, 32, 43

84, 85, 79, 71, 69, 88, 82, 78

102, 115, 110, 113, 100

228, 205, 198, 202, 207, 197

19, 29, 55, 22, 15, 46, 35, 9, 27, 40

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8 Ex 2

The table shows the luggage weight, in kilograms, of 30 passengers: a

Find the mean check in weight, rounded to two decimal places.

In kilograms, ﬁnd the:

c 9

Ex 3

Median

Lower quartile

iii

Upper quartile

Frequency 4 5 5 3 4 6 3

Matches 45 46 47 48 49 50 51 52

Frequency 1 2 5 3 7 25 6 2

Phone calls 0 1 2 3 4 5 6 7

Frequency 8 5 10 6 6 7 3 2

In which quartile does the mean lie?

A sample of matchboxes were selected for quality control and the number of matches in each box recorded in a given frequency table. a

How many matchboxes were sampled?

Find the range of the data.

Find the interquartile range.

A group of students were asked how many phone calls they had made the previous day. The information was collected in a given frequency table. a

How many students were surveyed?

Find the range of the data.

Find the interquartile range.

For this dot plot:

840

Weight 16 17 18 19 20 21 22

9 10 11 12 13 14 15 16 17 18

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

Find the range.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The dot plot shows the ages of customers in a mobile phone store in one day:

Age of customers

Find the lower quartile.

Find the interquartile range.

Find the upper quartile.

The stem-and-leaf plot shows the number of hours students spent studying during an entire semester: a

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

Hours of study 6

2 7

0 1

2 2 4 7 9 2 5 7

Key: 5|2 = 52 For this bar chart: a

Organise the data into a frequency table.

Find the median score.

Find the lower quartile score.

Find the upper quartile score.

Find the interquartile range.

5 4 Frequency

3 2 1 0

The column graph shows the number of pets that each student in a class owns. a

Find the lower quartile.

Find the upper quartile of the set of scores.

Find the interquartile range.

Frequency

10 9 8 7 6 5 4 3 2 1 0

8 7 Score

2 3 4 5 6 Pets

16.01 Quartiles mathspace.co

841

Dylan records the number of ﬁsh he catches for each ﬁshing trip over a period of time:

Frequency

20 18 16 14 12 10 8 6 4 2 0

Number of fish caught

How many ﬁshing trips did he go on?

Find the range in the number of ﬁsh caught.

Find the interquartile range.

This list shows the number of points scored by a basketball team in each game of their previous season: 59, 67, 73, 82, 91, 58, 79, 88, 69, 84, 55, 80, 98, 64, 82

842

Find the median value.

Find the lower quartile score.

Find the upper quartile score.

What percentage of games did the team score more in than the upper quartile?

What percentage of games did the team score less in than the lower quartile?

The stem-and-leaf plot shows the batting scores of two cricket teams, the Emerald Eagles and the Sapphire Sharks. Cricket team batting scores a Find the median score for the Emerald Emerald Eagles Sapphire Sharks Eagles. b

Find the median score for the Sapphire Sharks.

Find the range for the Emerald Eagles.

Find the range for the Sapphire Sharks.

Find the interquartile range for the Emerald Eagles.

Find the interquartile range for the Sapphire Sharks.

What are the differences between the distribution of scores for the Emerald Eagles and the Sapphire Sharks?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Key: 6|1|2 = 16 and 12

Reasoning The bar graph shows the marks (out of 10) that students received on a spelling test: a

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

Explain the effect scores less than 7 have on the summary statistics.

28 24 Frequency

20 16 12 8 4 0

7 8 Mark

These data sets each contain 10 scores:

Set 1: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28

Set 2: 10, 12, 14, 16, 18, 40, 42, 44, 46, 48

Set 3: 10, 42, 44, 46, 48, 50, 52, 54, 56, 58

Which data set has the largest interquartile range and what does this tell you about the spread of data in this data set?

Identify if each set is symmetrical, skewed-left or skewed-right and justify your response.

There is a test to measure the Emotional Quotient (EQ) of an individual. Here are the EQ results for two groups of 21 people listed in ascending order: Group 1: 90, 90, 91, 92, 93, 94, 95, 95, 95, 97, 99, 100, 108, 114, 116, 116, 117, 118, 118, 122, 129 Group 2: 85, 85, 85, 86, 86, 88, 90, 95, 97, 98, 100, 100, 100, 100, 102, 105, 105, 105, 110, 110, 115

Calculate the range, interquartile range, median, and mean for both data sets.

Compare the mean of both data sets. What does this tell you about the EQ scores of the groups?

Compare the spread of both data sets. What does this tell you about the variability in EQ scores within the groups?

The dot plot shows the number of pets students have, for 25 students.

Number of pets

Find the lower quartile.

Find the percentage of students who have more pets than the upper quartile. Explain why it is not exactly 25%.

Find the upper quartile.

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Is the percentage of of data values between the upper and lower quartiles always exactly 50%? Explain why or why not, including an example to justify your conclusion.

Problem-solving 24

In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the ﬁrst 11 challenges are: 38, 43, 45, 66, 67, 82, 92, 102, 105, 108, 119 Determine her score in the 12th round if the lower quartile of all of her 12 scores is 55.

For a week, the airline Fly Air decided to keep track of how many minutes behind schedule each ﬂight departed. 100 results are shown in the dot plot. a

Find the median. What is the meaning of the median in this scenario?

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

If a ﬂight is delayed for 10 minutes or more, the airline incurs a fee. Based on the dot plot, how many ﬂights did the airline incurred a fee for?

What percentage of ﬂights did the airline incur a fee for?

A rival airline, Fly High, had a median delay time during the same week of 45 minutes.

5 10 15 20 25 30 35 40 45 50 55 60 minutes

How many delayed ﬂights did Fly Air have that were longer than 45 minutes? h

What percentage of Fly Air’s ﬂights had delay times that were longer than Fly High’s median delay time?

To gain a place in the main race of a car rally, teams must compete in a qualifying round. The median time in the qualifying round determines the cut off time to make it through to the main race. Here are some results from the qualifying round: • 75% of teams finished in 159 minutes or less. • 25% of teams finished in 132 minutes or less. • 25% of teams finished between with a time between 132 and 142 minutes. a

Find the cut-off time required in the qualifying round to make it through to the main race.

Find the interquartile range in the qualifying round.

In the qualifying round, the ground was wet, while in the main race, the ground was dry. To make the times more comparable, the ﬁnishing time of each team from the qualifying round is reduced by 5 minutes. Find the new median time from the qualifying round.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

16.02 Boxplots After this lesson, you will be able to… • identify the five-number summary from a boxplot. • construct a boxplot using given data. • interpret outliers and their impact on data representation. • describe the shape of data distribution through a boxplot.

Boxplots Boxplots (also known as box-and-whisker plots) are a way of showing the five-number summary for a data set. The five-number summary consists of the following five statistics: Minimum Lower quartile Median Upper quartile Maximum

17 52 69 87 100

The boxplot shows a nice summary of all this information:

Minimum value

Median

Lower Quartile

Maximum value Upper Quartile

100

As you can see the boxplot consists of a number line, a rectangle with a line inside (the box), and two horizontal lines (the whiskers). The box represents the middle 50% of the scores and its size tells us the interquartile range. About 25% of the scores lie in each quartile. So for this example, 25% of the scores would lie between 17 and 52, 52 and 69, 69 and 87, and 87 and 100. To ﬁnd the percentage of scores between 52 and 100, we would multiply 25% by 3 since there are 3 quartiles in this range: 3 × 25% = 75%. Minimum and outlier

Upper quartile Q3

Median Q2

Lower quartile Q1

Maximum ∼25%

∼25%

Interquartile range 16.02 Boxplots mathspace.co

845

Outliers are data points which are very far away from the rest of the data set. On a boxplot, the outliers are shown as points or crosses. When there are outliers, the end of the whiskers may not be the minimum or maximum. Boxplot with outliers

Similar to other data displays, we can describe the shape by saying it is symmetrical, skewed right or left, or uniform. Symmetrical

10 15 20 25 30 35 Left skew

846

A right-skewed data set is where the majority of the data points have lower values, with some data points at higher values. It is sometimes called a positive skew.

10 15 20 25 30 35 Uniform

A left-skewed data set is where the majority of the data points have higher values, with some data points at lower values. It is sometimes called a negative skew.

10 15 20 25 30 35 Right skew

A symmetrical data set is distributed around the centre with a similar frequency on the left and right.

A uniform data set is evenly distributed across all values.

10 15 20 25 30 35

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Keep in mind: • when describing skewed distributions, it’s better to use the median and interquartile range because they are less impacted by outliers. • when describing symmetrical or uniform distributions, it’s better to use mean and range because they take the values of all data points into account.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 16.02 to answer these questions. 1.

Describe the shape of the two different displays for the same data set. Are they the same or different?

Identify the median using the histogram.

Identify the median using the boxplot.

Compare the diﬃculty of using both methods to identify the median.

Example 1 Create a boxplot to represent the data in the table. Minimum Lower quartile Median Upper quartile Maximum

10 20 40 55 75

Create a strategy

Apply the idea

Construct a boxplot using the data from the table.

0 10 20 30 40 50 60 70 80 90 100

16.02 Boxplots mathspace.co

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Example 2 For the boxplot shown, fill in the table with the corresponding values: Minimum Maximum Outliers Range IQR

0 4 8 12 16 20 24 28 32 36 40 44

Create a strategy Use the illustration shown to determine the values for the table. Minimum and outlier

Upper quartile Q3

Median Q2

Lower quartile Q1

Maximum ∼25%

∼25%

Interquartile range

The range is the difference between the minimum and maximum. For the outliers, determine the points beyond the whiskers. For the IQR, measure the distance from the lower to the upper quartile, or the length of the box.

Apply the idea Here’s the complete table:

848

Minimum Maximum Outliers Range IQR

1 43 1, 41, 43 42 7

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 The stem-and-leaf plot displays the scores of students in a class on an exam.

Leaf 6

Key: 6|1 = 61

a Construct the five-number summary.

Create a strategy

Apply the idea

• To find the minimum value, locate the smallest score in the lowest stem. • To find the maximum value, locate the highest score in the highest stem. • Calculate the median by finding the middle score if the number of scores is odd, or by averaging the two middle scores if even. • Determine the lower quartile by identifying the middle score of the values below the median. • Identify the upper quartile as the middle score of the values above the median.

Here’s the five-number summary: Minimum Lower quartile Median Upper quartile Maximum

61 69 73 80 85

b Construct a boxplot for the data.

Create a strategy

Apply the idea

Use the five-number summary in part (a) to construct the boxplot.

Exam scores

c Describe the shape of the boxplot.

Create a strategy

Apply the idea

Observe the distribution of data.

The boxplot shows a slight right skew.

16.02 Boxplots mathspace.co

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Example 4 The boxplot shows the mathematics test results for Mr. Smith’s and Ms. Johnson’s classes. Mr. Smith’s class

Ms. Johnson's class

100

a Calculate the range of Mr. Smith’s class.

Create a strategy

Apply the idea

Use the whiskers of the boxplot to calculate the range.

95 − 70 = 25 Mr. Smith’s class has range of 25.

b Calculate the interquartile range of Ms. Johnson’s class.

Create a strategy

Apply the idea

Use the edge of the boxes to calculate the interquartile range.

85 − 70 = 15 Ms. Johnson’s class has an interquartile range of 15.

c How much higher was the median of Mr. Smith’s class than Ms. Johnson’s class?

Create a strategy

Apply the idea

Use the line inside the box to determine the median of both classes

85 − 80 = 5 Mr. Smith’s class had a median of 5 marks higher.

d Which class performed better overall? Explain your answer.

Create a strategy

Apply the idea

Use the measures of spread and central tendency to compare the class marks.

Mr. Smith’s class had a higher maximum, median and minimum score than Ms. Johnson’s. Mr. Smith’s class also had a smaller range, indicating more consistent results. Thus, Mr. Smith’s class performed better overall.

850

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary Boxplots are a graphical representation showing the ﬁve-number summary for a data set.

Minimum value

Lower Quartile

Maximum value

Median

Upper Quartile

100

Outliers can be shown as dots or crosses and are far away from the rest of the data set. We can describe the shape based on the distribution of the data set. • • • •

Symmetrical boxplots are symmetrical about the median. Left-skewed boxplots have the majority of data points with higher values. Right-skewed boxplots have the majority of the data points with lower values. Uniform boxplots have all quartiles the same width.

Practice questions Choose your question path Explorer 1–4, 5ac, 6–8, 9acegi, 11a, 12, 15, 17–19, 23

Adventurer 1bc, 2b, 3, 4, 5bc, 7b, 8, 9bdhj, 10, 11b, 13–20, 23, 25

Trailblazer 1c, 3, 4, 5cd, 8, 9egj, 10, 13, 14, 16, 19–25

Understanding 1

Fill in the missing numbers for each boxplot. a

Minimum 20 Lower quartile 28 Median

Upper quartile

Maximum

16.02 Boxplots mathspace.co

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Minimum 16 Lower quartile 33 Median

Upper quartile

Maximum

Upper quartile

Maximum

For each five-number summary, find the: i

Range

Interquartile range

Minimum 26 Lower quartile 33 Median 41 Upper quartile 44 Maximum 45

Minimum Lower quartile Median Upper quartile Maximum

16 27 35 42 50

Is each statement true or false for the data set shown in the boxplot? a

The maximum is 16

The minimum is 8

The median is 11

There is an error as the median is not exactly in the middle of Q1 and Q3

The interquartile range is 3

The value 1 is an outlier

2 4 6 8 10 12 14 16 18 20

For the boxplot shown: a

852

Minimum 26 Lower quartile 33 Median

Approximately what percentage of scores lie between each of these values? i

Q1 and Q3

Minimum and Q1

iii

Median and Maximum

Q1 and Maximum

Minimum and Q3

In which quartile(s) is the data the least spread out?

In which quartile(s) is the data the most spread out?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fluency 5

Construct a five-number summary for each boxplot: a

Scores

8 10 12 14 16 18 20

Scores

8 10 12 14 16 18 20

Scores

38 40 42 44 46 48 50 52 54 56 58 60

For the boxplot shown, ﬁnd the: a

Minimum

Maximum

Range

Median

Interquartile range

8 10 12 14 16 18 20

Construct a boxplot to represent the information in the tables: a

Ex 2

8 10 12 14 16 18 20

0 Ex 1

Minimum 5 Lower quartile 25 Median 40 Upper quartile 55 Maximum 65

Minimum Lower quartile Median Upper quartile Maximum

30 31 47 53 58

For the boxplot shown, ﬁnd: a

Minimum

Maximum

Outliers

Range

Interquartile range 0 4 8 12 16 20 24 28 32 36 40 44

16.02 Boxplots mathspace.co

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Ex 3

For each of the data sets: i

Construct the ﬁve-number summary.

Draw a boxplot.

iii

Describe the shape of the boxplot.

14, 15, 20, 21, 23, 24, 27

21, 25, 29, 31, 32, 34, 39, 40

40, 12, 8, 20, 12, 28, 48, 36, 60

28, 4, 40, 16, 20, 48, 20, 44, 56

42, 43, 48, 54, 59, 51, 53, 50, 57, 45

60, 66, 67, 68, 69, 72, 73, 76, 77, 78, 86

Leaf

2 3 5 8 9

0 4 4 5 7 7

0 2 3 4 6 7

3 3 5 9

2 4 5 Key: 1|2 = 12

Key: 1|2 = 12

20 21 22 23 24 25 26 27 28 29 30 31

2 3 5 5

10 11 12 13 14 15 16 17 18 19 20 21

A cyclist measured her heart rate immediately after ﬁnishing each event in which she competed. The results are recorded in the boxplot. a

Find and interpret the range.

Find and interpret the interquartile range.

Heart Rate

155 160 165 170 175 180 185 190 195 200

Use the summary statistics to determine: i

The shape of each data set

The data set with the higher measure of centre

iii

The data set with the higher measure of spread

a Data set A Data set B

Mean 10 15

Median 12 15

IQR 2 5

Data set A Data set B

Minimum 1 10

Q1 4 18

Median 5 20

854

Range 5 10 Q3 10 21

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Maximum 13 22

IQR 6 3

Range 12 12

A mathematics test is given to two classes. The marks out of 20 received by students in each class are represented in the boxplots. a

Class 10P

Complete the table: Class 10P

Class 10Q

Median Lower quartile Upper quartile Range Interquartile range Outliers

0 2 4 6 8 10 12 14 16 18 20 22

Class 10Q

Which class tended to score better marks?

Would the range or interquartile range be better for describing the spread of Class 10Q?

0 2 4 6 8 10 12 14 16 18 20 22

Ex 4

The boxplot shows the age at which a group of people obtained their driving licences: a

What is the oldest age?

What is the youngest age?

What percentage of people were aged from 18 to 22?

The middle 50% of responders were within how many years of one another?

In which quartile are the ages least spread out?

The bottom 50% of responders were within how many years of one another?

Age

The boxplots represent the daily sales made by Carl and Angelina over the course of one month. Angelina's sales

16.02 Boxplots mathspace.co

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Carl's sales

Calculate the range for Angelina’s sales.

Calculate the range for Carl’s sales.

By how much did Carl’s median sales exceed Angelina’s?

By comparing the middle 50% of sales for each person, identify which salesperson had the more consistent number of sales.

Which salesperson had a more successful sales month? Explain your answer.

Reasoning 15

Ten participants had their pulse measured before and after exercise with results shown in different boxplots. Compare the pulse rates before and after exercise for these participants. Pulse before exercise

Pulse after exercise

45 50 55 60 65 70 75 80 85 90

80 85 90 95 100 105 110 115 120 125

How does a boxplot help us to understand the centre and spread within a data set?

The test scores of 11 students in Science and Music are listed: • Science: 52, 59, 87, 66, 74, 76, 73, 63, 91, 86, 82 • Music: 87, 63, 60, 43, 74, 82, 66, 61, 88, 60, 84

856

Construct a boxplot for the Science scores.

Construct a boxplot for the Music scores.

In which subject did students perform better overall? Explain your answer with reference to measures of central tendency and spread.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

These boxplots shows the number of points scored by two basketball teams in each of their matches. Gamma Geckos

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70

Delta Dragons

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70

Compare the median score of the Gamma Geckos and the Delta Dragons.

Compare the range of scores for the Gamma Geckos and the Delta Dragons.

Compare the interquartile range for the Gamma Geckos and the Delta Dragons.

If the two teams were to play against each other in the next game, determine which team you expect to win. Justify your answer.

The boxplots summarise results from a medical study. The treatment group received an experimental drug to relieve cold symptoms, and the control group received a placebo. The boxplots show the number of days each group continued to report symptoms: Control group

8 10 12 14 16 18 20

Treatment group

8 10 12 14 16 18 20

Describe the shape of the data from the control group.

Describe the shape of the data from the treatement group.

Does the drug have a positive effect on patient recovery? Explain your reasoning.

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A boxplot was created for a data set. The value 32 was a typo and should have been 23. a

Can we draw a new boxplot with this information? Explain.

How does this change the range?

How does this change the interquartile range? 10

Create a context with a data set might meet the given conditions. Justify your answer. a

A range between 1 and 4.

A small interquartile range and a large range.

The marks in an end-of-year exam for a class of students are shown: 52, 56, 59, 64, 66, 77, 78, 80, 80, 80, 81, 84, 86, 90, 95, 96

Construct a boxplot for the data.

Calculate the interquartile range.

What percentage of marks lie in the range 85 to 96?

Which values do the lowest 75% of scores lie between?

If the score 91 was added to the data set, what percentage of the marks are below Q3? Why isn’t this 75%?

Construct a boxplot for the data set: • Range = 14 • Lowest score = 2 • IQR = 8

• 25% of scores are below 4 • 50% of scores are above 8

For each of the conditions, use ﬁve of the digits from 1–9 with no repetition to complete a ﬁve-number summary: min: ⬚, lower quartile: ⬚, median: ⬚, upper quartile: ⬚, max: ⬚

The smallest possible interquartile range with the largest possible range.

The range is three times the length of the interquartile range.

Match the histograms on the left to the corresponding boxplots on the right: Histogram A

0 1 1

858

2 3 4 5 6 7 8 9

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boxplot 1

9 10

Histogram B

boxplot 2

0 1

9 10

boxplot 3

2 3 4 5 6 7 8 9

Histogram D

0 1

2 3 4 5 6 7 8 9

Histogram C

boxplot 4

9 10

2 3 4 5 6 7 8 9

16.03 Cumulative frequency After this lesson, you will be able to… • calculate cumulative frequencies from a frequency table. • construct a cumulative frequency table from a given frequency distribution table. • interpret and solve problems involving cumulative frequencies. • construct and interpret cumulative frequency polygons. • apply knowledge of cumulative frequencies and their graphical representations to solve problems.

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Cumulative frequency We have seen that a frequency table allows us to group data into intervals and tells us how many data values are in that interval. Cumulative frequency is a ‘running total’ of the frequencies. To calculate it, we add an additional column to the frequency distribution table: Class interval 50 ≤ t < 55 55 ≤ t < 60 60 ≤ t < 65 65 ≤ t < 70 70 ≤ t < 75 75 ≤ t < 80 80 ≤ t < 85 Total

Frequency 5 10 25 26 40 49 28 183

Cumulative frequency 5 5 + 10 = 15 15 + 25 = 40 40 + 26 = 66 66 + 40 = 106 106 + 49 = 155 155 + 28 = 183

• The first value in the cumulative frequency column will always be the same as the first value in the frequency column. • To get the second cumulative frequency value we add the second frequency to the first cumulative frequency value, 50 + 10 = 15. • The second cumulative frequency value tells us there are 15 values in the interval 50 ≤ p < 60. • The third cumulative frequency value tells us there are 40 values in the interval 50 ≤ p < 65, and so on. • The final cumulative frequency value is always equal to the sum of the frequencies. In this case, there are 183 values in the entire data set, represented by 50 ≤ p < 85.

Example 1 A principal wants to investigate the performance of students at his school in Performing Arts. To do this, he has the marks of each student studying Performing Arts collected into groups and put into a frequency table. Each group of marks is assigned a grade. The frequency table for this is shown. a Complete the cumulative frequency column. Grade E D C B A

860

Score (x) 0 ≤ x < 20 20 ≤ x < 40 40 ≤ x < 60 60 ≤ x < 80 80 ≤ x < 100

Frequency ( f ) 7 14 32 97 62

Cumulative frequency (cf )

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Create a strategy The cumulative frequency of the ﬁrst row is the frequency of that row. For the following rows, add the frequency of each row to the cumulative frequency of the previous row.

Apply the idea The ﬁrst value in the cumulative frequency column is the same as the ﬁrst value in the frequency column, so we have 7. For the second cumulative frequency value, we have 7 + 14 = 15. Repeating the process to get the next cumulative frequency value, we have the completed table: Grade E D C B A

Score (x) 0 ≤ x < 20 20 ≤ x < 40 40 ≤ x < 60 60 ≤ x < 80 80 ≤ x < 100

Frequency ( f ) 7 14 32 97 62

Cumulative frequency (cf ) 7 21 53 150 212

b How many marks were recorded?

Create a strategy

Apply the idea

We can ﬁnd the total frequency in the last entry of the cumulative frequency column.

This value is the same as adding all the entries in the frequency column. Total frequency = 212

c Complete the sentence. ‘Approximately three quarters of the scores recorded are greater than ⬚’.

Create a strategy

This can be reworded to one quarter of the scores are below this value: ⬚’.

Apply the idea

There are 212 scores, so one quarter of that is

= 53.

The cumulative frequency reaches 53 in the interval 40 ≤ x < 60, so 60 is the corresponding upper boundary score. ‘Approximately three quarters of the scores recorded are greater than 60’.

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Example 2 For this table: Score (x) 10 11 12 13 14

Cumulative frequency (cf ) 7 15 18 20 26

a How many scores were there in total?

Create a strategy

Apply the idea

We can ﬁnd the total number of scores in the last entry of the cumulative frequency column.

This value is the same as adding all the entries in the frequency column. Number of scores = 26

b How many scores of 14 were there?

Create a strategy Find the difference between the cumulative frequency at 14 and the cumulative frequency at 13.

Apply the idea There were 26 scores after the 14s were counted and 20 before. Number of 14s = 26 − 20 =6

Subtract the values Evaluate

c How many scores of less than 13 were there?

Create a strategy

Apply the idea

Find the corresponding cumulative frequency of The corresponding cumulative frequency of the the score 12. score 12 is 18, so there were 18 scores which are less than 13.

Idea summary Cumulative frequency is a ‘running total’ of the frequencies. To calculate it, we add an additional column to the frequency distribution table.

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Cumulative frequency graphs

Cumulative frequency

Cumulative frequency distribution Global life expectancy, 2016 200 180 160 140 120 100 80 60 40 20 0 50 55 60 65 70 75 80 85 Life expectancy (years)

A cumulative frequency histogram will always increase in size from left to right. The frequency represented by any particular column will be equal to the difference in height between that column and the one before it.

Cumulative frequency

Cumulative frequency distribution Global life expectancy, 2016 200 180 160 140 120 100 80 60 40 20 0 50 55 60 65 70 75 80 85 Life expectancy (years)

A cumulative frequency polygon, also known as an ogive, is a line graph connecting cumulative frequencies at the upper endpoint of each class interval. Here is a cumulative frequency histogram and polygon displayed together.

Cumulative frequency distribution Global life expectancy, 2016 200 180 160 140 120 100 80 60 40 20 0 50 55 60 65 70 75 80 85 Life expectancy (years)

Here is a cumulative frequency polygon displayed on its own.

Cumulative frequency

We have already seen how frequency tables can be used to create a histogram. The cumulative frequencies can also be plotted to create another type of chart, called a cumulative frequency graph. This graph is made up of a cumulative frequency histogram and polygon.

We can use a cumulative frequency polygon to find values such as the median and quartiles which are at 25%, 50% and 75% of the cumulative frequencies.

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Example 3

Cumulative frequency

For this cumulative frequency histogram:

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 30

35 40 45 50 55 Score

a How many scores are there in total?

Create a strategy For a cumulative frequency histogram the total will be the cumulative frequency for the last bar.

Apply the idea The 50– < 55 bar has the cumulative frequency of 14. So there 14 scores in total.

Reﬂect and check Notice this is different from a normal histogram where we would add up the frequencies of all of the bars. Since these bars represent cumulative frequency, this is already done.

b How many scores between 50– < 55 occured?

Create a strategy Find the difference between the cumulative frequency at 50– < 55 and at 45– < 50.

Apply the idea There were 14 scores after the 50– < 55s were counted and 12 before. Number of 50– < 55s = 14 − 12 =2

Subtract the values Evaluate

Idea summary A cumulative frequency polygon, also known as an ogive, is a line graph connecting cumulative frequencies at the upper endpoint of each class interval.

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Practice questions Choose your question path Explorer 1–3, 5–9, 12–15, 17

Adventurer 2–9, 11–14, 15, 17–19

Trailblazer 2–5, 7, 8, 10–13, 15–19

Understanding 1

Complete the frequency table based on the data set: 2.1, 5, 4, 5, 5.15, 5, 3.8, 4, 5.32, 5.5, 1.25, 3, 3.75, 3, 5, 2.82, 4.6, 1, 6, 5, 6.56, 3, 1, 1.6, 6 Score 1≤x<2 2≤x<3 3≤x<4 4≤x<5 5≤x<6 6≤x<7

Complete the cumulative frequency column. Score (x) 1≤x<5 5≤x<9 9 ≤ x < 13 13 ≤ x < 16 17 ≤ x < 20

Frequency ( f ) 4 5 9 5 4

Cumulative frequency (cf )

Complete the cumulative frequency table: Score (x) 12 ≤ x < 13 13 ≤ x < 14 14 ≤ x < 15 15 ≤ x < 16

Frequency

Frequency ( f ) 8 5 8

Cumulative frequency (cf ) 8 21 26

Is each statement true or false? a

The height of the last bar tells us the total.

The cumulative frequency must always add up to 100.

To find the cumulative frequency we add the frequency to the previous cumulative frequency.

To find the frequency we subtract the previous cumulative frequency from the cumulative frequency.

A cumulative frequency polygon may have segments that are decreasing.

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Fluency Ex 1

Ex 2

866

For the frequency table: a

Complete the cumulative frequency column.

Calculate the total frequency.

What is the class size?

Approximately half of the scores recorded are greater than what score?

For the given frequency table: a

Complete the cumulative frequency column.

Calculate the total frequency.

Approximately one third of the scores recorded are greater than what score?

Score (x)

Frequency ( f )

1–5 6–10 11–15 16–20 21–25 26–30

15 26 18 14 7 2

Score (x)

Frequency ( f )

20–24 25–29 30–34 35–39 40–44 45–49 50–54

7 18 25 12 8 4 1

For the table: a

Find the total number of scores recorded.

Find the number of times a score of 13 occurred.

Find the number of times a score less than 15 occurred.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Score (x) 11 12 13 14 15

Cumulative frequency (cf )

Cumulative frequency (cf ) 9 13 19 27 30

Ex 3

For this cumulative frequency histogram: Score (x)

Frequency ( f )

20–24 25–29 30–34 35–39 40–44 45–49 50–54 55–59 60–64 65–69 70–74

3 3 3 1 2 4 13 13 12 4 2

Class marks

Cumulative frequency (cf ) 3 6 9 10 12 16 29 42 54 58 60

70 60 50 40 30 20 10 0

20 25 30 35 40 45 50 55 60 65 70 75 Marks

Find the total number of scores recorded.

Find the number of times a score between 25 and 30 occurred.

Find the number of times a score between 50 and 55 occurred.

Find the percentage of scores that were 30 or less.

For this cumulative frequency histogram: a

How many scores are there in total?

How many scores between 40– < 50 occurred?

The graph shows the cumulative frequency polygon of the mass (in grams) of fish caught in a fishing competition: Construct a frequency histogram to depict the distribution of the mass of the fish.

Frequency

170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10

20 18 16 14 12 10 8 6 4 2 0

20 30 40 50 60 70 80 Score

Cumulative frequency

Mass (g) 50 100 150 200 250 300 350 16.03 Cumulative frequency mathspace.co

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868

Construct a frequency table for the data represented in the cumulative frequency histogram.

A pair of dice are rolled 50 times and the numbers appearing on the uppermost face are added to give a score. The results are recorded in the given table. a

What is the lowest possible score when a single pair of dice are rolled?

What is the highest possible score when a single pair of dice are rolled?

Complete the table by ﬁnding the cumulative frequency values.

Find the number of times a score of 8 occurred.

Find the number of times a score more than 9 occurred.

Find the number of times a score of at most 6 occurred.

Cumulative frequency

20 18 16 14 12 10 8 6 4 2 0

Score (x)

Frequency ( f )

2 3 4 5 6 7 8 9 10 11 12

1 2 5 5 5 9 7 5 8 1 2

80 81 Score

Cumulative frequency (cf )

The number of sightings of the Northern Lights were recorded across various Canadian locations over a period of 1 month. The list represents the number of sightings at each location: 11, 10, 10, 9, 7, 8, 8, 12, 12, 12, 12, 12, 12, 9, 9, 12, 9, 9, 8, 8 a

Complete the table.

In how many locations were there at least 8 sightings?

In how many locations were there less than 11 sightings?

Number of sightings 7 8 9 10 11 12

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Number of locations ( f )

Cumulative frequency (cf )

Reasoning 14

A jump rope competition has routines which are judged on a scale of 0 to 10. The average of the scores from the middle three judges makes the score for the routine. 50

Cumulative frequency

45 40 35 30 25 20 15 10 5 0

Score

Explain how the frequency of a certain score can be determined from a cumulative frequency histogram.

What is the most frequent range of scores?

Families were asked how many times they contracted the ﬂu during winter. The information has been partially ﬁlled out in the table. Explain your reasoning for each part of the problem. a

Complete the frequency and cumulative frequency values in the given table.

Find the total number of families that responded.

Find the median number of times a family contracted the ﬂu.

Find the mean number of times a family contracted the ﬂu. Round your answer to two decimal places.

Construct a cumulative frequency histogram for the data. Explain the signiﬁcance of the histogram.

Find the number of families who contracted the ﬂu more than twice.

Score (x) 0 1 2 3 4

Frequency ( f ) 0

Cumulative frequency (cf ) 2

3 2 9

To find each key feature, would you rather use a histogram or cumulative frequency histogram? Explain your reasoning. a

Total

Frequency for one interval or value

Shape of distribution

Number of data points below a particular value

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A 1500 m swimmer records her time (min:sec) over several training sessions. Her times are recorded in the histogram. a

Construct a cumulative frequency table for the data using the given intervals.

Find the total number of training sessions she completed.

Find the number of times she recorded a swim time faster than 16 : 40.

Find the percentage of swims that were less than 16 : 30.

10 9 8 7 6 5 4 3 2 1 0

16 :0 0 16 :10 16 :2 0 16 :3 0 16 :4 0 16 :5 0 17 :0 0

Cumulative frequency

Problem-solving

Time Recorded

The heights (cm) of 22 students in a class are listed: 164, 167, 158, 159, 150, 166, 150, 146, 149, 161, 164, 163, 152, 161, 157, 157, 153, 157, 156, 165, 162, 161

Construct a cumulative frequency histogram for the data. Use the discrete intervals of 146–150, 151–155, etc.

How many students are taller than 155 cm?

How many students are at most 155 cm tall?

How many students are taller than 150 cm but shorter than 156 cm?

Calculate the centre of the class 151–155.

For each graph use the cumulative frequency curve to estimate: i

The median score

iii

The upper quartile

100 90 80 70 60 50 40 30 20 10

Score

The lower quartile

14 13 12 11 10 9 8 7 6 5 4 3 2 1

1500 1900 2300 2700

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Score 550 600 650 700 750

16.04 Compare data sets After this lesson, you will be able to… • interpret the five-number summary in a boxplot. • compare the range and interquartile range between two boxplots. • draw conclusions from real-world examples involving parallel boxplots. • compare data sets effectively. • understand the relationship between different graphical representations of the same data set.

Parallel boxplots We have seen that the range and interquartile range can be used to measure the spread of data and both can be seen on a boxplot. Parallel boxplots are used to compare two sets of data visually. When comparing boxplots, we compare the five-number summary and also the range and interquartile range. The two boxplots are drawn parallel to each other. It is important to clearly label each boxplot. Below are parallel boxplots comparing the time it took two different groups of people to complete an online task. 5

Under 30s

Over 30s

14 16 18 Seconds

20 22 24 26 28 30

The boxplots must be drawn on the same scale to properly compare them.

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Example 1 The two boxplots show the data collected by the manufacturers on the lifespan of light bulbs, measured in thousands of hours. Manufacturer A

Manufacturer B

3 4 5 Thousands of hours

a Complete this table using the two boxplots. Write each answer in terms of hours, remembering to multiply the values on the data display by 1000. Manufacturer A

Manufacturer B

Median Lower quartile Upper quartile Range Interquartile range

Create a strategy • To ﬁnd the lower quartile, median, and upper quartile, ﬁnd the corresponding values of the vertical lines of the box, respectively. • For the range, use the formula: Range = Highest score − Lowest score • For the interquartile range, use the formula: IQR = Q3 − Q1

Apply the idea Since the scale is in thousands of hours, we can multiply the numbers on the scale by a thousand to ﬁnd the number of hours. Median Lower quartile Upper quartile Range Interquartile range

872

Manufacturer A 4 × 1000 = 4000 2.5 × 1000 = 2500 4.5 × 1000 = 4500 (5 − 1) × 1000 = 4000 (4.5 − 2.5) × 1000 = 2000

Manufacturer B 5 × 1000 = 5000 3.5 × 1000 = 3500 6 × 1000 = 6000 (8 − 1.5) × 1000 = 6500 (6 − 3.5) × 1000 = 2500

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

b Which manufacturer produces light bulbs with the best lifespan?

Create a strategy Choose the manufacturer with the greater median.

Apply the idea The data set for Manufacturer B has a median of 5000 hours, while the median of the data set for Manufacturer A is 4000 hours. So Manufacturer B produces light bulb with the best lifespan.

Reflect and check In fact, the best lightbulb produced by Manufacturer A has a lifespan of 5000 hours, which is the same as the median of Manufacturer B. This means that about half of the lightbulbs produced by Manufacturer B have a greater lifespan than all of the lightbulbs produced by Manufacturer A.

Example 2 This boxplots show the number of goals scored by two football players in each season. Sophie

Holly

15 Goals

a Who scored the most goals in a season?

Create a strategy Compare the maximum value of both boxplots.

Apply the idea By looking at the endpoints of the right whiskers, Sophie scored 18 goals, while Holly scored 19 goals. So Holly score the most goals in a season.

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b How many more goals did Holly score in her best season compared to Sophie in her best season?

Create a strategy Subtract Sophie’s maximum from Holly’s maximum.

Apply the idea Number of goals = 19 − 18 =1

Subtract 18 from 19 Evaluate the subtraction

Holly scored 1 more goal in her best season.

c What is the difference between the median number of goals scored in a season by each player?

Create a strategy Find the difference of the medians.

Apply the idea Sophie’s median is 11 and Holly’s median is 10. Difference = 11 − 10 =1

Subtract 10 from 11 Evaluate the subtraction

d What is the difference between the interquartile range for both players?

Create a strategy Use the formula: IQR = Q3 − Q1

Apply the idea Sophie’s IQR = 14 − 7 =7 Holly’s IQR = 15 − 6 =9 Difference = 9 − 7 =2

Substitute the quartiles Evaluate Substitute the quartiles Evaluate Subtract the IQRs Evaluate

Idea summary When comparing boxplots, we compare the ﬁve-number summary values of each data set (the quartiles and minimum and maximum values) and the range and interquartile range.

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Different graphs of the same data We can identify the centre and spread using key features. Measures of centre summarise the data set with a single value. We often use this to generalise which group performed better. • Mean • Median • Mode Measures of spread summarise the spread of a data set. We often use this to generalise which group was more consistent. The lower the spread the more consistent the data. • Range • Interquartile range Different displays help us to identify different key features. From frequency displays like histograms, dot plots, and stem-and-leaf plots we can identify: • Shape • Frequencies, grouped or individual • Mode • Outliers From boxplots, we can identify: • Shape • Range and interquartile range • Median • Outliers We should expect then that the shape of data would be the same whether it is represented in a polygon, boxplot or histogram.

Frequency

All representations of symmetric data: Histogram of approximately symmetric data

10 9 8 7 6 5 4 3 2 1 8 9 10 11 12 13 14 15 16 17 Score

Boxplot of symmetric data

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Frequency

All representations of positively skewed data: Histogram of positively skewed data

10 9 8 7 6 5 4 3 2 1 8 9 10 11 12 13 14 15 16 17 Score

Boxplot of positively skewed data

Frequency

All representations of negatively skewed data: Histogram of negatively skewed data

10 9 8 7 6 5 4 3 2 1 8 9 10 11 12 13 14 15 16 17 Score

Boxplot of negatively skewed data

Looking at the diagrams above, can you see the similarities in the representations? We can see the skewed tails, where the bulk of the data sits and general shape. These are some of the features you can use to match histograms and boxplots. We can also look at the data range.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 16.04 to answer these questions. 1.

Create a uniform distribution. How do the boxplot and cumulative frequency graph compare visually?

Create a positively skewed distribution. How do the boxplot and frequency graph compare visually?

Create a negatively skewed distribution. How do the boxplot and frequency graph compare visually?

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Example 3 Match the boxplot shown to the correct histogram.

9 10

2 3 4 5 6 7 8 9

Create a strategy Look for characteristics of skewed and symmetrical distributions of data.

Apply the idea The given boxplot shows symmetrical data. Only option B shows symmetrical data. The correct answer is option B.

Idea summary We should expect that the shape of data would be the same whether it is represented in a boxplot or histogram. Remember that the shape of data can be symmetrical, left skewed or right skewed.

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Practice questions Choose your question path Explorer 1, 2, 4, 5, 7, 9, 11, 12, 14a-c, 15

Adventurer 1–8, 11–13, 15, 16

Trailblazer 1–3, 5–8, 10, 12–16

Understanding 1

The histogram and the boxplot display the same data. 10

0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90

Which data display shows the mode of the data set?

Which data display shows the median of the data set?

Which data display shows the range of the data set?

Which data display shows the number of values between 10 and 20?

Each pair displays the same data set. What is the skew for each pair of histograms and boxplots? a

10 20 30 40 50 60 70 80 90 0

878

10 2030405060708090

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

10 20 30 40 50 60 70 80 90 0

10 2030405060708090

1 2 3 4 5 6 7 8 9

Which key features can be read off each display? a

Histogram

Boxplot

Dot plot

Cumulative frequency histogram and polygon

Fluency Ex 1

A class took an English test and a Mathematics test. Both tests had a maximum possible mark of 20. The parallel boxplots show the results of the tests:

English

Mathematics

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Complete the table using the two boxplots. English

Mathematics

Median Lower quartile Upper quartile Range Interquartile range b Ex 2

In which test did the class tend to score better marks?

The parallel boxplots show the prices, in dollars, of the items on the menu of an upmarket restaurant and the menu of a fast food restaurant. Upmarket Restaurant

Fast Food

Price

Which restaurant has the higher median price for the items they sell?

What is the difference between the median prices of the items sold by each restaurant?

Which restaurant has a greater price range for the items on their menu?

What is the price difference between the most expensive items sold by each restaurant?

What amount of the cheapest item at the fast food restaurant could be bought for the same price as the most expensive item at the upmarket restaurant?

Two weightlifters both record their number of repetitions with a 70 kg bar over 30 days. The results are displayed in the boxplots:

880

Which weightlifter has the more consistent results?

Which statistics determine which is more consistent?

Which statistic is the same for each weightlifter?

Which weightlifter can do the most repetitions of 70 kg?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Repetitions

Ex 3

Match the histograms to the corresponding boxplots. Histogram A

Boxplot 1

10 20 30 40 50 60 70 80 90

Histogram B

Boxplot 2

10 20 30 40 50 60 70 80 90

Histogram C

Boxplot 3

10 20 30 40 50 60 70 80 90

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Histogram D

Boxplot 4

10 20 30 40 50 60 70 80 90

The residents of two blocks of townhouses were asked the number of pets they own. The frequency of various responses are presented in these dot plots. a

Is the pet ownership a little lower or higher in Block A than Block B?

Block A

In Block A, how many pets do most households have?

In Block B, how many pets do most households have?

What is the shape of the data for Block A?

Find the range of the number of pets in Block A.

Which block has more variability in the the number of pets?

Do either sets of scores have an outlier?

Block B

The parallel boxplots show the distances, in centimetres, jumped by two high jumpers.

Matt

Paul

140

150

160

170

Distance (cm)

882

Who had a higher median jump?

Who made the highest jump?

Who made the lowest jump?

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Match each cumulative frequency graph to its corresponding boxplot: Cumulative Frequency Graph A

boxplot 1

Cumulative Frequency 80 70 60 50

5 10 15 20 25 30 35 40 45 50

40 30 20 10 Mass (g) 5 10 15 20 25 30 35 40 45 50 55 60

Cumulative Frequency Graph B

boxplot 2

Cumulative Frequency 80 70 60 50

5 10 15 20 25 30 35 40 45 50

40 30 20 10 Mass (g) 5 10 15 20 25 30 35 40 45 50 55 60

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Cumulative Frequency Graph C

boxplot 3

Cumulative Frequency 80 70 60 50

5 10 15 20 25 30 35 40 45 50

40 30 20 10 Mass (g) 5 10 15 20 25 30 35 40 45 50 55 60

Cumulative Frequency Graph D

boxplot 4

Cumulative Frequency 80 70 60 50

40 30 20 10 Mass (g) 5 10 15 20 25 30 35 40 45 50 55 60

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5 10 15 20 25 30 35 40 45 50

Reasoning 11

Two History classes, each with 20 students, were given a 10 question True/False test. The results for each class are shown. Class 1

Class 2

10 Number of Students

Number of Students

4 5 6 7 Questions Correct

7 8 9 10 Questions Correct

Calculate the mean for Class 1 rounded to one decimal place.

Calculate the mean for Class 2 rounded to one decimal place.

Find the range for Class 1.

Find the range for Class 2.

Do you think Class 1 studied for their test? Justify your reasoning.

Do you think Class 2 studied for their test? Justify your reasoning.

Two bookstores recorded the selling price of all their books. The results are presented in the parallel boxplots. Bookstore A prices ($)

Bookstore B prices ($) x 50

100

110

120

130

140

150

Which bookstore had the more consistent prices? Explain your answer.

Comparing the most expensive books in each store, how much more expensive is the one in store B?

Is each statement true or false? i

5% of the books in Bookstore B are at least as expensive than the most expensive 2 book in Bookstore A.

5% of the books in Bookstore B are cheaper than the cheapest book in 2 Bookstore A.

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Explain why each pair of histograms and boxplots do not match: a

Salaries earned by employees at a software company are given in a histogram.

Frequency

10 9 8 7 6 5 4 3 2 1 0

65 000

75 000

85 000 Salaries ($)

95 000

105 000

Using the centres, construct a boxplot of the data.

Find an estimate for the interquartile range.

Using the boxplot, approximately what percentage of salaries lie in the range $90 000 to $100 000?

Using the boxplot, what can be said about the highest 25% of salaries? Would this conclusion be different from using the histogram? Explain your answer.

Problem-solving 15

Cooper and Marion are racing go-karts. The times (in seconds) for the 12 laps of their qualifying race are shown: • Cooper: 58.9, 46.5, 52.6, 66.6, 58.4, 53.1, 45.0, 52.1, 52.4, 52.7, 44.8, 51.7 • Marion: 47.8, 54.6, 68.5, 68.0, 62.8, 57.2, 54.8, 63.4, 58.1, 64.3, 66.2, 47.1 a

Construct the ﬁve-number summary for each set.

Are there any outliers?

Create a parallel boxplot of the two sets of times with the outlier displayed separately.

Which racer will be in pole position for the ﬁnal race, if it is given to the racer with the fastest qualifying lap time?

Does spinning out on a lap, causing a high outlier, impact the selection for pole position? Explain your answer.

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Two friends compete in hammer throw competitions and train together over a season. They compete in 15 competitions and their ﬁnal throw for each competition is shown: • Tim: 29.8, 37.4, 33.9, 38.8, 34.3, 36.5, 34.5, 30.0, 35.2, 38.4, 33.0, 33.2, 39.6, 35.0, 36.9 • Odi: 32.2, 35.4, 34.8, 33.0, 38.4, 26.0, 40.0, 37.2, 39.5, 42.4, 38.6, 42.3, 38.4, 42.8, 37.2 a

Complete the table of statistics, rounded to one decimal place: Minimum Q1 Median Q3 Maximum Mean Range Interquartile range

Tim 29.8 33.2 35.0 37.4 39.6

Odi

37.2

Which competitor throws more consistently? Explain your answer.

Identify any outliers.

Who is the better hammer thrower? Explain your answer.

When considering Odi’s average throw is it reasonable to remove the outlier before calculating the mean? Explain your answer.

16.05 Samples and populations After this lesson, you will be able to… • distinguish between a population and a sample in statistics. • identify the key factors that make a sample representative of a population. • understand the differences between a parameter and a statistic. • recognise the importance of sample size in data collection. • distinguish between primary and secondary data, and assess the reliability of secondary data sources. • apply the concepts of samples and populations in practical scenarios.

Samples and populations In statistics, a population refers to every member within any particular group of interest. A survey conducted on every member of a population is called a census. Collecting data from every member of a population is the most accurate way of gathering information, but it is not always possible and can be very expensive. Collecting data from a sample, can be quicker and less expensive. A parameter is a number that summarises data from a population. • Population size: N • Population mean: m

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A statistic is a number that summarises data from a sample. • Sample size: n • Sample mean: A sample is representative if the characteristics of the sample are close to the population. Key factors that affect if a sample is representative include: • Demographic factors: If the population is diverse in terms of age, gender, ethnicity, or other factors, the sample should reflect this diversity. • Characteristics relevant to the study: If you’re studying a particular trait or behavior, individuals in the sample should exhibit a range of these characteristics. • Random selection: This minimises bias and increases the likelihood of the sample being representative of the population from which it was drawn. If a sample is unbiased and represents the population well, we can use sample survey data to make conclusions about the population. The size of the sample is important. If the sample is too large, it may be too expensive or timeconsuming to collect the data. If it is too small, the sample may not be representative of the population.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 16.05 to answer these questions. 1.

What features of the frequency histogram help you identify which population the sample belongs to?

How does changing the sample size change the difficulty of identifying which population it belongs to? What minimum sample size would you suggest for these populations so that the histograms looks similar to the bell curves?

How does changing bin width change the difficulty of identifying which population it belongs to?

How does changing the means change the difficulty of identifying which population it belongs to?

Primary data is data that has been collected firsthand. If you are the researcher, primary data is data that you gather yourself, directly from observation, surveys, experiments, or interviews. Secondary data is data that has already been collected and analysed by someone else. This could be in the form of books, articles, reports, or statistics that you find online or in the library. When using secondary data: • Make sure the population from which the data was collected matches your target population. • Check whether a census or sample was used. If a sample was used, make sure it is representative of the population. • Analyse whether there is any source of bias.

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Example 1 A sample of 25 people is drawn from a population. In this sample, the youngest is 18 years old, and the oldest is 64. Which two of these might be populations that this sample was drawn from? A Residents of a retirement home

Employees at a bank

C Students from a primary school

People stuck in traﬃc due to road works

Create a strategy Remember that a sample is a smaller group than the entire population but every member of the sample must be a member of the overall population.

Apply the idea

Reﬂect and check

Which of these populations would be likely to include people of this age group?

To check our answer we can conﬁrm that the other options are incorrect. Option A, residents The answers are options B Employees at a bank of the retirement home must be of at least a retirement age (usually 65). While for Option C, and D People stuck in traﬃc due to roadworks the students attending a primary school will only range from ages 4 to 12

Example 2 A researcher captures 400 fish in a lake, tags them, then releases them. The next day, he captures 1200 fish, of which 100 have tags attached to them. Estimate the size of the fish population.

Create a strategy The proportion of tagged ﬁsh in the second sample should be equal to the proportion of 400 ﬁsh out of the total population.

Apply the idea

Therefore, there is estimated to be 4800 ﬁsh in the lake.

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Reflect and check To check our answer, we can multiply the total population by the proportion of tagged fish to see if it gives us the 400 fish originally tagged.

Example 3 Maryke is researching the monthly spending habits of students on school lunches at a local high school. She found two secondary sources of data: • Source A: This data set provides the average monthly spending of students on lunch for the entire school year. • Source B: This data set provides raw data for the monthly spending of each student for each month of the school year, broken down by demographics like age, sex, and postal code. a If Maryke wants to ﬁnd out in which month the spending was the lowest and in which month the spending was the highest, for year 12s, which source should she refer to? Explain.

Create a strategy Decide which source provides data on individual months of the school year and where she can separate out the year 12s.

Apply the idea Maryke should refer to Source B. Firstly, she can separate out the year 12 data. Also, she can determine in which month the spending was the lowest and the highest because it provides the monthly spending of each student for every month of the school year, allowing her to calculate the average for each speciﬁc month.

Reﬂect and check We some times call data which is all grouped together and summarised aggregate data and raw data disaggregated data.

b If Source A reports that the average monthly spending is $50 and there are 10 months in the school year, can Maryke determine if there was a month where students spent more than $100, on average? Why or why not?

Create a strategy Decide whether the monthly average can be used to ﬁnd the spending habits from an individual month.

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Apply the idea

Reﬂect and check

No, Maryke cannot determine if there was a month where students, on average, spent more than

Many times, it is best to ﬁnd secondary sources that report raw data values rather than data that has already been summarised.

$100 just from the average monthly spending If you need to use the data for a different reason data of Source A. The overall average does not than the researcher who collected the data, the provide details of spending in individual months. information in their report may not be useful for your statistical study.

Idea summary • •

• •

In statistics, data can be gathered from every member of a group, known as a population, or from a subset of the population, known as a sample. Gathering data from a sample instead of the entire population is often more practical and less expensive, however, the sample needs to be representative of the population for the data to be accurate. A representative sample has the same demographic diversity as the population, relevant characteristics to the study, and random selection. The size of the sample is crucial; a sample that’s too large may be too costly or timeconsuming, while a sample that’s too small may not represent the population and is not useful to make conclusions.

Practice questions Choose your question path Explorer 1–5, 7, 8, 9ab, 10, 11ab, 12, 15, 16–18, 19, 21, 22, 25, 26

Adventurer 1–5, 7, 8, 9bc, 10, 11ab, 13, 15–17, 19, 21, 22, 24–26

Trailblazer 1–4, 6–8, 9cd, 11cd, 13–17, 20, 21, 23–27

Understanding 1

True or false: a

In a sample survey, information is obtained from the entire population.

A census of the whole population is always possible.

Using a sample can save time and money for the researcher.

A sample should have the same diversity as the population.

A convenience sample is a good sampling method to use.

The population and sample means will always be the exactly the same.

If the population has 42 000 members, would this sample size be large enough? a

4000

40 000

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A study is being done on physical activity in a small town. Would this characteristic be important to consider when selecting the sample? a

Favourite colour

Gender

Age

Favourite sport

Preferred method of transportation

Name

Identify the population and sample in each scenario: a

The mayor of Anytown wants to determine the opinion of his residents on the new mall planned to be built. He surveys 500 residents at random.

A bookstore owner wants to know what genres of books are most popular amongst her customers. She decides to pull up the receipts for 20 purchases out of the 650 sales made over the weekend.

Antony wants to know what teams the students at his school support. He ﬁnds that 30 students support the Sydney Swans, and 55 students support the Melbourne Demons.

For a statistical survey, the population is deemed to be all the students that attend the local secondary school. Are these samples of that population? a

All of the seniors

The ﬁrst 30 students to arrive at school that day

All of the students that were in the library at lunch time

All of the teachers

Some of the teachers

For a statistical survey, the population is deemed to be people aged under 18 who live in Sydney, Australia. Are these samples of that population? a

40 professors chosen from University of Sydney

Every person who lives in Bondi

All elementary students who are visiting Taronga Zoo in a single day

Students enrolled in the on-campus preschools of high schools in Parramatta

100 children chosen from a Sydney junior rugby league team

Fluency 7

Match each sample histogram to the population it was taken from. i

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10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0

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0 1 2 3 4 5 6 7 8

iii

10 9 8 7 6 5 4 3 2 1 0

Ex 1

0 1

2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0

9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0

iv 10

2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8

0 1

2 3 4 5 6 7 8

0 1

2 3 4 5 6 7 8

A sample of 50 people is drawn from a population. In this sample, the youngest person is 18 years old and the oldest person is 64 years old. Are these populations that the sample could have been chosen from? a

People in a grocery store

People who got tickets for speeding

People at a drop-in centre for seniors

Employees at a grocery store

For each population, give an example of a biased sample of the population. a

The population is all the students that attend the local high school.

The population is all people aged under 18, who live in Sydney.

The population is all people in a city who play in any organised sporting competition.

The population is all people who own a pet dog.

Sean wants to know what year 10s think of their English class, so he polls 70 random year 10s. Is the sample chosen biased or fair?

A school principal wAnts to estimate the number of students who ride a bicycle to school. Is each sample biased? a

All students who are in the school band.

Eight students in the hallway.

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15 Ex 2

Ten students from each year, chosen at random.

130 students during the lunch periods.

At a certain chocolate factory, 30% of products contain nuts. 400 chocolates are tested to check if they meet the required quality for sale. Of those tested, 61% contained nuts. a

What is the population?

What proportion of the population contains nuts?

What proportion of the sample contains nuts?

Do the 400 chocolates tested represent a random sample?

At a popular orchard, 65% of apples are Gala apples. 500 apples are picked for a tastetesting event. Of those picked, 310 were Gala apples. a

What is the population?

What proportion of the sample is of the “Gala” variety?

Do the 500 apples picked represent a random sample?

Lachlan asks 120 year 12 students at his school how much time they spend on homework per night. 78 year 12 students say they do more than 3 hours. At a meeting of the student council, Lachlan reports “65% of students at this school do too much homework.” For each statement, decide if it explains why Lachlan’s report is misleading. a

The survey does not represent the population of the school.

The question should have been multiple choice.

The question was biased.

The sample size was too small.

The conclusion does not align with the survey question.

An ecologist collects 250 of a particular bee species in a meadow, marks them, and then releases them. In a follow-up collection, the ecologist gathers 200 bees, and 82 of them are found to be marked. Estimate the total bee population in the meadow. A television station wants to estimate the number of viewers it had for a new show. They know that the country’s population is 13 620 000. When they randomly selected 5000 people and asked them if they had watched the show, they found that 400 of them had. Estimate the number of people in the entire population who watched the show.

Ex 3

Yakov is studying the performance of students in two different classes over a term. He found two secondary sources of data: • Source A: This data set provides the individual grades of every student from both classes. • Source B: This data set provides the average final grade of students from both classes.

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If Yakov wants to know how many students from one class scored above 90%, which source should he use?

Which source should Yakov use if he wants to calculate the median grade for only one class?

Source B indicates that the average grade for one class is 85%. Can Yakov determine how many students scored exactly 85% using Source B?

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Reasoning 18

Pablo has concluded that about 36% of people would drive an electric car based on his survey results: Would drive an electric car Would not drive an electric car

58 104

Jenny has concluded that about 44% of people would drive an electric car based on her survey results: Would drive an electric car Would not drive an electric car

224 285

Which claim do you think is more valid? Explain your reasoning. 19

For a statistical survey, the population is deemed to be all people who own a pet dog. Ariel decides to ﬁnd her sample by talking to every person at her local dog park. Describe the advantages and disadvantages of her sampling method.

Explain why these samples are biased:

Hannah is surveying customers at a shopping precinct. She wants to know which stores customers shop at the most. She walks around an entertainment store and chooses 30 customers from the store for the survey.

A TV station wants to know what the most popular type of music is, so they ask listeners to contact them and vote for their favourite type of music.

The community health nurse wants to survey the students in a school about their eating habits. At lunchtime, she stands by a vending machine and surveys every student who purchases something from the machine.

A political polling company calls 1000 people at home between 4:00 p.m. and 7:00 p.m. on week nights to ﬁnd out who they are most likely to vote for in an upcoming election. They publish their numbers based on the responses of only the 410 people who answered their call. a

How could they have used random sampling to choose the 1000 people to call?

Explain why the company’s published results will not be accurate.

A researcher captures 300 dingoes, tags them, then releases them. The next day, they capture 750 dingoes, of which 90 have tags. The day after that, they capture 100 dingoes, of which 6 have tags. a

Compare the sample proportions from the second and third day of research.

Explain why the sample proportions are different.

Explain which sample statistic you would use.

An environmentalist is interested in tracking the population of koalas in Australia. She begins by tagging 480 koalas. In a follow-up collection, she catches 820 koalas, of which 62 are tagged. In a final round of collection, she catches 820 koalas, of which 58 have tags. a

Explain why the sample proportions are different.

Explain which sample statistic they would use.

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Problem-solving 24

You are writing an essay on Australia’s role in World War II. While researching, you come across two sources of information: • Source A: A school textbook published in 1990 detailing Australia’s military engagements, key figures, and notable battles during the war. • Source B: A documentary released in 2015 that focuses on the personal stories of Australian soldiers, interviews with war historians, and unseen archival footage.

If you find conflicting information between the two sources, how might you determine which one is more accurate?

Why is it important to use multiple secondary sources when trying to get a comprehensive understanding of a historical event?

Marine biologists want to determine if a local species of ﬁsh is growing to a smaller size than it used to. They collect and measure 150 of the 450 ﬁsh known to exist in the area. a

Did they catch a large enough proportion of the species to be able to determine if the ﬁsh are growing to a smaller size?

What additional information would make us more conﬁdent that the sample is representative of the population?

For a statistical survey, the population is considered to be all of the athletes at an Australian Football League (AFL) game. a

Describe two different samples that could be taken at the game to represent this population.

Describe two samples that could be taken at this game that do not represent this population. Explain why these samples aren’t appropriate.

Farma, a newly established pharmaceutical company in Australia, has recently conducted clinical trials for a new medication. The trials involved 1000 participants across Sydney. According to the data they released, it appears that 10% of the trial participants experienced mild side effects such as headaches and fatigue. Farma plans to roll out this medication across the country, making it available to the 5 million residents of Sydney first. Consider the potential bias, accuracy, and statistical validity in your analysis.

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Based on the trial data, estimate the number of Sydney residents who might experience side effects if the medication were given to the entire target population.

Discuss the ethical considerations of releasing this prediction to the public. How might the way the data is reported influence public opinion about the medication?

Do you think the sample size used in the initial trial is representative of the entire target population of Sydney? Why or why not?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Chapter 16 review

For the surveys, should the data be grouped or ungrouped when constructing a frequency table? a

A survey conducted of 3500 people, asking them how tall they are in centimetres.

A survey conducted of 1000 people, asking them how many books they read in the past year.

The scores for a recent mathematics test are displayed in the stem-and-leaf plot. The scores are out of 110.

Stem

Leaf

How many students took the test?

5 7 7 9 9

What test score occurred most frequently?

0 3 5 7

How many marks separate the highest score and the lowest score?

T F

A R

0 3 3 3 3 5 8

0 2 2 4 5 7 7

0 2 5 7 9

Key: 10|0 = 100

The histogram displays amount of time that Aaliyah spent watching television on her streaming platform over different days. Minutes Frequency 0 ≤ Minutes ≤ 10 10 < Minutes ≤ 20 20 < Minutes ≤ 30 30 < Minutes ≤ 40 40 < Minutes ≤ 50 50 < Minutes ≤ 60 60 < Minutes ≤ 70 70 < Minutes ≤ 80

Number of days

Time spent watching television

0 10 20 30 40 50 60 70 80 Time spent (in minutes)

Complete the frequency table based on the histogram.

Aaliyah’s subscription offers 1450 minutes of free streaming time, then charges $0.50 per minute for any additional time. How much was she charged for watching these episodes?

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A musical has a large cast of students with various ages. The column graph shows the ages of all the students in the cast. 25

Ages of students in a musical

Frequency

20 15 10 5 0

10 11 12 13 14 15 16 17 18 Age

How many students are in the cast?

What age is most common?

Find the range of the students’ ages.

Find the interquartile range.

T F

These data sets each contain 10 movie ratings out of 10:

Set A: 2, 3, 4, 5, 6, 7, 8, 9, 10, 10

A R

Set B: 1, 2, 3, 4, 5, 9, 9, 9, 10, 10 Set C: 1, 1, 3, 4, 5, 7, 8, 8, 9, 10

Set B has the largest interquartile range. What does this tell you about the spread of ratings in this data set?

Based on the ﬁve-number summaries and the shapes of the data sets, identify which set is symmetric, which is skewed left, and which is skewed right.

In a game show, a contestant must complete 8 different mini games, and the time (in seconds) taken to complete each is recorded. One contestant’s times for the ﬁrst 7 games are: 47, 50, 52, 60, 66, 80, 82

Find their time in the 8th segment if the upper quartile of all 8 times is 75 seconds. 7

For the data set shown in the dot plot: a

Construct the ﬁve-number summary.

Draw a box plot.

Describe the shape of the box plot.

Stem

Leaf

5 8 8

0 2 5 5 6 8

0 1

2 3 3 5

Key: 1|2 = 12

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The box plot shows the number of books a group of students read in a year: a

Find and interpret the range.

What percentage of students read from 12 to 20 books?

The middle 50% of students read within how many books of each other?

In which quartile is the number of books read the most spread out?

The bottom 50% of students read within how many books of each other?

Number of Books

The box plots summarise the results from a botanical study. The treatment group received an experimental fertilizer to boost plant growth, and the control group received regular water. The boxplots show the number of weeks each group of plants took to reach maturity:

T F

Control group

Treatment group

A R

12 15 18 21 24

Compare the medians of the control group and the treatment group.

Compare the ranges of the control group and the treatment group.

Compare the interquartile ranges of the control group and the treatment group.

Does the fertilizer have a positive effect on plant growth speed? Explain your answer.

For the cumulative frequency histogram: a

Find the total number of scores recorded.

Find the number of times a score between 150 and 200 occurred.

Find the percentage of scores that were 150 or more. Round your answer to one decimal place.

300 Cumulative frequency

250 200 150 100 50 0

50 100 150 200 250 Scores

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The number of attendees at various workshops during a tech conference was noted. The list represents the number of attendees in each workshop: 24, 23, 23, 22, 20, 21, 21, 25, 25, 25, 25, 25, 25, 22, 22, 25, 22, 22, 21, 21

Complete the table.

In how many workshops were there at least 24 attendees?

In how many workshops were there fewer than 23 attendees?

Approximately half of the data values recorded are greater than what number of attendees?

Number of attendees 20 21 22 23 24 25

Use the cumulative frequency histogram to estimate: a

The median.

The lower quartile.

The upper quartile.

Cumulative frequency (cf )

Number of workshops (f )

T F

A R

Match the box plot shown to the correct histogram.

100 90 80 70 60 50 40 30 20 10

Score 48

Score

7 8 9 10 11 12 13 14 15 16 17 18

10 9 8 7 6 5 4 3 2 1

B Frequency

Frequency

8 9 10 11 12 13 14 15 16 17 Score

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

10 9 8 7 6 5 4 3 2 1 8 9 10 11 12 13 14 15 16 17 Score

9 8 7 6 5 4 3 2 1

Frequency

10 9 8 7 6 5 4 3 2 1

8 9 10 11 12 13 14 15 16 17 Score

The residents of two apartment buildings were asked the number of indoor plants they own. The frequency of various responses are presented in the dot plots: a

Which building has more plants?

By how many plants does the mode of each building differ?

Which building has more variability in the number of plants?

Do either sets have an outlier?

A R

2 3 4 5 Building X

2 3 4 5 Building Y

Two athletes compete in long jump events and train together over a season. They compete in 10 competitions and their ﬁnal jump (in metres) for each competition is shown: • Antoine: 6.8, 7.4, 6.9, 7.8, 7.3, 7.5, 7.1, 6.5, 7.2, 7.6 • Benji: 7.0, 7.2, 6.8, 6.7, 7.9, 6.0, 8.1, 7.5, 8.0, 7.7

T F 0

Complete the table of statistics:

Which athlete jumps more consistently?

When considering Benji’s average jump, is it reasonable to remove the outlier before calculating the mean? Explain your answer.

Would removing the outlier from Benji’s data set change your answer to part (b)? Explain your answer.

Minimum Q1 Median Q3 Maximum Mean (1 d.p.) Range Interquartile range

Antoine 6.5 6.9 7.3 7.5 7.8

Benji

7.4

A fruit farm has 52 000 cherries of different varieties. During a day of harvest, 800 cherries are randomly picked. Of those picked, 468 were Bing cherries. a

What is the population?

What proportion of the sample is of the “Bing” variety?

Do the 800 cherries picked represent a random sample?

Estimate the number of cherries in the entire population that are Bing cherries.

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A gym manager wants to estimate the number of members who attend yoga classes. Is each sample biased? a

All members who participate in the morning spin class.

Twelve members near the water reﬁll station.

Every 10th member that enters the gym.

Entomologists are tracking the habits of a local beetle species. At the beginning of their research project, they collect and tag 600 beetles. The next week, they collect 350 beetles, of which 98 had tags. The week after, they collect 200 beetles, of which 42 had tags. a

Why are the sample proportions different?

Explain which sample statistics you would use to estimate the population.

(Optional) For each data set, calculate the statistics rounded to two decimal places. i

Mean

iii

Standard deviation

Mean absolute deviation

T F

Stem 1

5 9 9

4 6 9

2 3

2 4

0 2 4 6

A R

71 72 73 74 75 76 77 78 79 80 81 Score

Leaf

8 8

Key: 1|5 = 15

(Optional) Use technology to ﬁnd the standard deviation for the data represented by the frequency table, rounded to two decimal places.

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Score 80 81 82 83 84 85

Frequency 4 5 10 7 2 2

(Optional) Macy is tracking the amount of data (in MBs) she uses on her phone each day of the month to see if she can save money by purchasing a smaller data package. a

Use technology to ﬁnd the mean.

Use technology to ﬁnd the standard deviation using the class centres, rounded to two decimal places.

Calculate the standard deviation if 50 MBs are subtracted from each data value. How is the standard deviation affected?

Should Macy use the mean or the standard deviation to determine how much data she uses in one day, on average?

Megabytes of data used over time

Frequency

10 9 8 7 6 5 4 3 2 1 0

300 400 500 600 700 800 900 Data (in MBs)

T F

(Optional) A city records the amount of rainfall (in centimetres) they got for the same 10 days in two different years. The results are shown: • Year 1: 120, 132, 124, 120, 125, 131, 130, 128, 127, 131 • Year 2: 130, 133, 132, 123, 124, 132, 128, 130, 131, 133

What does a comparison of the means tell us?

What does a comparison of the standard deviations tell us?

A R

(Optional) The houses in a neighbourhood have an average value of $450 000 with a standard deviation of $7000. A new house, estimated to be worth $800 000, is then built in the neighbourhood.

Will the new house increase or decrease the mean?

Will the new mean absolute deviation be higher, lower or unchanged from before?

(Optional) The table illustrates the marks obtained by a student in two different classes: a

How many standard deviations above the mean was the student’s score in History?

What mark in Literature is equivalent to a mark of 32 in History?

History Literature

Mark

Mean

96 95

48 70

Standard Deviation 16 10

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17 Bivariate statistics Big ideas • Different representations of data highlight different characteristics of the data. • Many sets of bivariate data can be modelled using familiar functions. • Analysing data patterns and trends enables making informed estimations and predictions about values within or beyond the known data range. • Effective data representation requires the use of accurate and unbiased techniques to provide an accurate and unbiased understanding of the information being conveyed.

Curriculum Outcomes VC2M10AP02 Investigate reports of studies in digital media and elsewhere for information on their planning and implementation VC2M10ST02 Construct scatterplots and consider a line of good fit; comment on the association between the 2 numerical variables in terms of strength, direction and linearity VC2M10ST03 Construct two-way tables and discuss possible relationship between categorical variables VC2M10ST04 Analyse claims, inferences and conclusions of statistical reports in the media and other places, by linking claims to displays, statistics and representative data, including ethical considerations and identification of potential sources of bias VC2M10ST05 Plan and conduct statistical investigations of situations that involve bivariate data, including where the independent variable is time; evaluate and report findings with consideration of limitations of any inferences VC2M10AST03 Use digital tools to investigate bivariate numerical data sets; where appropriate use a straight line to describe the relationship allowing for variation, make predictions based on this straight line and discuss limitations

Chapter outline 17.01 17.02 17.03 17.04 17.05 17.06 17.07

Two-way tables Associations between categorical variables Bivariate numerical data Associations between numerical variables Line of good fit Interpolation and extrapolation Misrepresentations of data Chapter 17 review

906 917 935 945 959 970 983 998

There’s a correlation between the number of honeybees and crop production!

17.01 Two-way tables After this lesson, you will be able to… • construct a two-way table using the given data. • identify and explain joint and marginal frequencies in a two-way table. • apply basic calculations to find percentages and totals in a two-way table. • interpret data presented in a two-way table.

Two-way tables Exploration Students were asked whether they prefer high-top sneakers or low-top sneakers, and they were also asked whether they prefer to shop online or shop in a store. Some of the results from the survey are: • The survey included 75 students. • 30 students said they prefer high-top sneakers. • 35 students said they prefer to shop online. • 7 students said they prefer to shop in a store and high-top sneakers. • 33 students said they prefer low-top sneakers and shopping in a store. 1.

How many students prefer low-top sneakers?

How many students prefer low-top sneakers and shopping online?

Is there an easier way to display the data in order to answer the ﬁrst two questions?

A two-way table can be used to summarise two categorical variables. The cells in the table represent the intersections of the categories, displaying the frequency count for each combination of categories. This table contains information about the preference of art medium, and age. If we read across each row, we can tell how many people surveyed prefer drawing, painting, or graphics and how many of those are children or adults. If we read down each column, we can tell how many children or adults surveyed prefer drawing, painting, or graphics.

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Child

Adult

Total

Drawing

Painting

Graphics

Total

100

Joint frequencies

Add across

Add down

Marginal frequencies

Each joint frequency gives us speciﬁc combinations of conditions. Meanwhile, each marginal frequency gives us overall totals. Both types of information are important in understanding the data in a two-way table and performing calculations to draw conclusions.

Example 1 Some students in Ms. Rochelle’s class were polled on how many siblings they have. The results are provided in the table: 1st period 2nd period 3rd period

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 7 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 7 Period 1

Period 2

Period 3

Total

Less than two siblings Two or more siblings Total a Fill in the two-way table showing how many students have less than two siblings, and how many have two or more siblings.

Create a strategy Identify the joint frequencies of the number of students from each period who have less than two siblings or two or more siblings.

Apply the idea Here’s the complete two-way table: Less than two siblings Two or more siblings Total

Period 1 10 10 20

Period 2 11 9 20

Period 3 13 7 20

Total 34 26 60

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b How many of the students in Period 1 have two or more siblings?

Create a strategy

Apply the idea

Look at the ‘Period 1’ column and the row for ‘Two or more siblings’ in the table.

Number of students = 10

c What proportion of the students have less than two siblings?

Create a strategy

Apply the idea

Divide the marginal frequency of the number of students with less than two siblings by the total number of students polled.

Proportion of students =

Example 2 A university conducted a survey asking students to rate the effectiveness of online classes and traditional in-person classes. Students were divided into two categories: undergraduate and postgraduate students. The data from the survey is shown in the two-way tables. Undergraduate students

Online classes In-person classes

Not effective 27 32

Slightly effective 11 12

Somewhat effective 2 5

Effective 13 6

Very effective 31 29

Total 84 84

Postgraduate students

Online classes In-person classes

Not effective 9 3

Slightly effective 14 0

Somewhat effective 4 8

Effective 3 13

Very effective 5 11

Total 35 35

a Calculate the percentage of undergraduate students who found classes to be Effective or Very effective, for both online and traditional in-person classes. Round your answer to one decimal place.

Create a strategy Add up the joint frequencies for undergraduate students who rated classes as Effective or Very effective separately for both online and in-person classes, then divide each total by the respective class’s total number of undergraduate students.

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Apply the idea Start with the equation

Add the numerator

Convert fraction to percentage and round Start with the equation

Add the numerator

Convert fraction to percentage and round

For undergraduate students, the percentages for Effective or Very Effective ratings are 52.4% for online classes and 41.7% for traditional in-person classes. b Calculate the percentage of postgraduate students who found classes to be Effective or Very effective, for both online and traditional in-person classes. Round your answer to one decimal place.

Create a strategy Add up the joint frequencies for postgraduate students who rated classes as Effective or Very effective separately for both online and in-person classes. Then, divide each total by the respective class’s total number of postgraduate students.

Apply the idea Start with the equation

Add the numerator

Convert fraction to percentage and round Start with the equation

Add the numerator

Convert fraction to percentage and round

For postgraduate students the percentages for Effective or Very effective ratings are 22.9% for online classes and 68.6% for traditional in-person classes. c What does this data suggest about the perception of online classes among different student levels? A

ndergraduate students slightly prefer online classes to in-person classes, while U postgraduate students strongly prefer in-person classes.

Both student groups greatly prefer online classes to in-person classes.

ndergraduate students have about equal preferences, while postgraduate students U strongly prefer online classes to in-person classes. 17.01 Two-way tables mathspace.co

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ndergraduate students strongly prefer online classes to in-person classes, while U postgraduate students have about equal preferences.

Create a strategy Look at the percentages that we calculated in parts (a) and (b).

Apply the idea Undergraduate students have a slightly higher percentage favoring online classes (52.4%) over in- person classes (41.7%). On the other hand, postgraduate students have a signiﬁcantly higher percentage preferring in-person classes (68.6%) to online classes (22.9%), so the correct answer is option A.

Idea summary Types of frequencies to consider in a two-way table are: • •

Joint frequency: How often a combination of two conditions occur Marginal frequency: How often a particular category of data occurs (or the total row and column data)

Both types of information are important in understanding the data in a two-way table and performing calculations to draw conclusions.

Practice questions Choose your question path Explorer 1–3, 5–10, 13, 15, 16

Adventurer 2–5, 7, 9, 10, 12, 14, 16, 17

Trailblazer 2–5, 7, 11, 12, 14, 16–18

Understanding 1

This table gives information on the treatment plans prescribed to cats with feline diabetes. Depending on the severity of the case, cats are either prescribed a special diet or insulin injections. Venetian Pet Clinic Mohawk Animal Clinic Outback Animal Care Centre Total

910

Special diet 14 11 15 40

Insulin injections 14 9 16 39

Total 28 20 31 79

How many cats from Venetian Pet Clinic were prescribed insulin injections?

How many cats from Outback Animal Care Centre were prescribed insulin injections?

Which animal clinic prescribed a special diet less frequently than insulin injections?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

This two-way frequency table displays the number of deaf-blind children in each state and their age group. Victoria New South Wales Queensland Total

0–5 year-olds 63 52 56 171

6–17 year-olds 388 143 193 724

Total 451 195 249 895

For each of the groups:

Classify as either a joint frequency or a marginal frequency.

Find the value.

The number of deaf-blind children in Victoria.

The number of deaf-blind 0–5 year-olds.

The number of deaf-blind 6–17 year-olds in New South Wales.

The number of deaf-blind children in Victoria who are more than 5 years old.

This table shows how much homework students do each night in different year levels. Use the information in the table to determine whether the statements are true or false. Year 8 students 7 18 25

1.5 hours or more Less than 1.5 hours Total

Year 9 students 28 6 34

Total 35 24 59

Most students in general do homework for less than 1.5 hours per night.

Most Year 8 students do homework for 1.5 hours or more per night.

There were more students surveyed who do homework for 1.5 hours or more per night than there were who do less than 1.5 hours per night.

There were more Year 8 students surveyed than Year 9 students.

This table shows the preferred flavour of milkshake for children and adults. Is each statement true or false based on the table? Child Chocolate Vanilla Strawberry Total

16 39 98

Adult 50 71 11

Total 93 50 230

43 children prefer chocolate.

A total of 50 adults were surveyed.

There were more adults surveyed than children.

87 people prefer vanilla.

230 people were surveyed.

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Fluency 5

Use the given information to complete each two-way frequency table. a

This table gives information about the number of sales of each type of shoe at Foot Swag, a shoe store in a shopping centre. Running shoes

Basketball slides

Work boots

43 85

13 36

29 48

Teenagers Adults Total b

Total 84 85 169

Students were asked if they prefer chocolate (C) or vanilla (V ) ice cream. Each of the students was also asked if they prefer sprinkles (S) or hot fudge (F ) as an ice cream topping. The results are displayed in the tree diagram: 12

Sprinkles

C 18

Hot fudge

Total

Chocolate Vanilla Total

V 12

Laura asked some people in her community what their preference was between beef, chicken, and ﬁsh.

20 responders said they preferred beef, of which 8 were children. 15 children said they preferred chicken, and 15 adults said they preferred ﬁsh. 35 children were surveryed, and there were 73 responders overall. Beef

Chicken

Fish

Total

Children Adults Total 6

912

Special diet 14 11 15 40

What does 28 mean in this context?

What does 11 mean in this context?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Insulin injections 14 9 16 39

Total 28 20 31 79

Ex 1

Ex 2

Sophia asked 38 children and 54 adults in her community about their dietary preferences. The options given were: vegetarian, vegan, or neither. 40 responders said they were vegetarian, of which 15 were children. 18 children said they were vegan, and 22 adults said they were neither vegetarian nor vegan. a

Construct a two-way table based on the results of Sophia’s survey.

What is the proportion of responders that are vegetarian?

What is the proportion of adults that are vegan?

Mr. Tobit asked the students on football teams to pick their favourite subject. He displayed the results in a two-way table. a

How many defence players did not pick maths as their favourite subject?

How many players picked music?

How many offence players are on the football teams?

At a local high school, a survey is conducted to gauge student satisfaction with canteen food. The responses are measured on a five-point Likert scale, with 1 being “very unsatisfied” and 5 being “very satisfied”. The survey also takes into account whether the student is a junior student or senior student.

Maths Music Science English

Offence 11 8 13 15

Defence 12 18 11 10

Rating 1 Rating 2 Rating 3 Rating 4 Rating 5

Junior students 23 35 50 70 22

Senior students 30 20 45 40 15

What percentage of juniors were very satisﬁed with the canteen food? What about the seniors?

Determine which rating is most common among junior students. What about senior students?

Comment on any patterns or discrepancies you observe between the two groups of students.

This table shows the number of trains arriving either on time or late at a particular station: a

How many trains were late on Friday?

How many trains passed through the station on Wednesday?

How many trains were on time for the entire week?

What fraction of trains were on time for the whole week?

Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Arriving on time 23 12 22 30 11 15 14

Arriving late 38 25 29 27 14 11 10

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At a local university students were asked for their favourite subject at high school, and what they had chosen to study at university. The results are shown in a table. Maths Science Music Art Total

Maths favourite 76 64 64 6 210

Science favourite 20 46 11 19 96

Music favourite 61 53 67 38 219

Art favourite 43 59 59 74 235

Total 200 222 201 137 760

Based on this data: a

What percentage of students surveyed said their favourite subject was mathematics at high school? Round your answer to two decimal places.

What percentage of students are studying either music or arts at university? Round your answer to two decimal places.

What fraction of students had music as their favourite subject in high school but decided to study a different ﬁeld at university?

What percentage of students have chosen to study in the same ﬁeld as their favourite subject from high school? Round your answer to two decimal places.

Reasoning 12

Determine if each statement is true or false. Justify your response. a

The marginal frequencies for the columns must have the same sum as the marginal frequencies for the rows.

Each of the joint frequencies in a single row can be equal to the marginal frequency of that row.

A scientist recorded some data on the diet of two types of possums observed on Phillip Island: • 310 possums were observed. • 45 brushtail possums were observed to eat a diet of fruit. • 180 possums observed were honey possums. • 120 possums were observed to eat insects. • Of the 110 possums that were observed to eat nuts, 75 were honey possums. a

Complete the table shown based on the given information about possums: Brushtail possum

Honey possum

Total

Fruit Insects Nuts Total

914

Compare the total number of brushtail possums to the number of honey possums.

Compare the number of possums who were observed eating fruit to the total number of possums eating insects and nuts.

Compare the number of brushtail possums observed eating nuts to the number of honey possums observed eating insects.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A poll was given where adults and teenagers were asked if they like pineapple on their pizza. The possible responses were “yes”, “no”, and “sometimes”. The results of the poll show that: • 60 people responded to the poll. • 18 adults responded “yes”. • 24 of the people who responded were adults. • The number of people who responded “yes” was equal to the number of people who responded “no” plus the number of people who responded “sometimes”. Find and explain the error made in each statement: a

Since 18 adults responded yes, there were 60 − 18 = 42 adults who responded either no or sometimes.

There were 18 adults who said yes and 24 of the people were adults so the total of adults was 42.

The number of people who responded “yes” was equal to the number of people who responded “no” plus the number of people who responded “sometimes”. So we can take the total number of people, 60, and divide by 3 to ﬁnd the number of people who said each response.

Problem-solving Yuri surveyed a group of people about the type of employment they had. He recorded the data in the graph. 40 35 30 Number of People

25 Men

20 15

Women

10 5 0

None

Part-time Casual Type of Employment

Full-time

Complete the two-way table with the information from the graph. No job

Casual

Part-time

Full-time

Men Women b

How many people are unemployed?

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What percentage of men are employed full-time? Round your answer to one decimal place.

What fraction of people employed casually are male?

If 111 people from the group are randomly selected, how many would you expect to be employed part-time?

In a biology study, the appearance of an animal carrying a certain gene is recorded. Assume that all animals must carry the MC1R gene or the Agouti gene but not both. a

Light fur Dark fur

Agouti gene 37 x

If the proportion of all the animals recorded carrying the MC1R gene and having light fur is

MC1R gene 36 28

, ﬁnd the value of x.

What does the value of x mean in this context?

Lucinda sets out to see how many of her classmates use the term “soccer” or “football” to describe her favourite sport. Lucinda surveys her classmates from her ﬁrst three periods. Football

Soccer

Total

Period 1 Period 2 Period 3 Total Complete the frequency table so that it meets the conditions: • None of her class periods have the same number of students. • Overall, most students use the term “soccer”. • One of Lucinda’s class periods is equally divided between “soccer” and “football”. • The number of students who prefer “football” in one class period was equal to the number of students who prefer “soccer” in a different class period. 18

At a high school with 5000 students, a survey was conducted to investigate the participation of junior and senior students in school sports activities. Out of a sample of 100 students, a two-way table was produced. Always Often Sometimes Rarely Never Total

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Junior students 12 9 10 7 11 49

Senior students 8 7 11 12 13 51

Total 20 16 21 19 24 100

Estimate the number of senior students in the entire school population who would say they “Never” participate in school sports.

On a particularly hot week, the school canteen had a sale on ice blocks. The sports coach believes that the sale might influence sports participation due to students preferring to sit out with an ice block instead of playing. How might you design a followup survey or data collection method to investigate this hypothesis using two-way tables?

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17.02 Associations between categorical variables After this lesson, you will be able to… • calculate and interpret row and column percentages from given data in a two-way table. • analyse data using row and column percentage tables to determine associations. • use 100% stacked column graphs to visually represent and analyse data. • describe the association between variables, if any, and provide reasoning based on the data. • differentiate between an association and causation between variables.

Row and column percentages Two-way tables display relationships between two categorical data sets. Categories label the table’s top and side, while frequencies ﬁll the table. Row and column totals are usually included. This two-way table displays the statistics of the passengers and crew who sailed on the Titanic on its fateful maiden voyage in 1912: Survived Died Total

First class 202 123 325

Second class 118 167 285

Third class 178 528 706

Crew 212 696 908

Total 710 1514 2224

Although it is interesting to note that 202 First Class passengers survived, it is far more useful to know the percentage break up of survivors from each class. Row percentage tables and column percentage tables give us more useful information than the raw data alone. To know the proportional break up of survivors from each class, we need to calculate row percentages. To ﬁnd the percentage divide the value in the table by the row total and multiply by 100%. The row percentages are shown in this table with working shown in some of the cells: First class

Second class

Survived Died

11%

Third class

Crew

Total

25%

30%

100%

35%

46%

100%

This percentage frequency table now gives us more useful information than the raw data. The ﬁrst row gives us the percentage breakdown of survivors by class. We can now easily read that 46% of the people who died were crew members whereas only 8% of the people who died were in ﬁrst class. We can also calculate the percentage in each class type that survived or died.

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To do this we calculate column percentages. To ﬁnd the percentage divide the value in the table by the column total and multiply by 100%. The column percentages are shown in this table with working shown in some of the cells: First class

Second class

Third class

Crew

25%

23%

59%

75%

77%

100%

Survived Died Total

100%

The first column gives us the percentage breakdown of survivals and deaths in first class. From the raw data we can see that a similar number of first class (202) and third class (178) passengers survived, but this can be misleading. The percentage frequency table shows us that 62% of first class passengers survived whereas only 25% of third class passengers survived.

Example 1 Maria surveyed a group of people according to their age about the type of job they had. She recorded the data in this graph: 40

Number of people

35 30 25 Under 50

Over 50

15 10 5 0

None

Casual Part-time Type of employment

Full-time

a Complete the two-way table displaying the row percentages, rounding your answers to one decimal place. Under 50 Over 50

None 17.6% 22.2%

Casual

Part-time 29.4%

Full-time 41.2%

27.8%

Total 100% 100%

Create a strategy Create a table for the data. Then for each missing value, divide the value in the table by the row total and multiply by 100%.

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Apply the idea From the graph we can make the following two-way table: Under 50 Over 50 Total

None 15 20 35

Casual 10 25 35

Part-time 25 15 40

Full-time 35 30 65

Total 85 90

From the table we can see that 10 of the casual employees are under 50, and the row total for under 50 is 85. Find the row percentage

Evaluate and round

From the table we can see that 15 of the part-time employees are over 50, and the row total for over 50 is 90. Find the row percentage

Evaluate and round

From the table we can see that there were 30 full-time employees that are over 50. Find the row percentage

Evaluate and round

The row percentage table is: Under 50 Over 50

None 17.6% 22.2%

Casual 11.8% 27.8%

Part-time 29.4% 16.7%

Full-time 41.2% 33.3%

Total 100% 100%

Reflect and check For casual employees that are over 50, we also could have subtracted the other values in the row from 100. Casual over 50 Percent = 100 − 17.6 − 29.4 – 41.2 = 11.8

b Complete the two-way table displaying the column percentages, rounding your answers to one decimal place. None Under 50 Over 50 Total

57.1% 100%

Casual 28.6% 100%

Part-time 62.5% 37.5% 100%

Full-time

100%

Create a strategy Use the table from (a), for each missing value divide the value in the table by the column total and multiply by 100%.

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Apply the idea Under 50 Over 50 Total

None 15 20 35

Casual 10 25 35

Part-time 25 15 40

Full-time 35 30 65

Total 85 90

From the table we can see that there were 15 under 50 in the None category, and the None column total is 35. Find the column percentage

Evaluate and round

From the above we can see that there were 35 full-time under 50, and the full-time column total is 65. Find the column percentage

Evaluate and round

From the table we can see that there were 25 casual employees that are over 50, and the casual column total is 35. Find the column percentage

Evaluate and round

From the table we can see that there were 30 full-time employees that are over 50, and the fulltime column total is 65. Find the column percentage

Evaluate and round

The column percentage table is shown: Under 50 Over 50 Total

None 42.9% 57.1% 100%

Casual 28.6% 71.4% 100%

Part-time 62.5% 37.5% 100%

Full-time 53.8% 46.2% 100%

Reflect and check Notice that when have row percentage tables or column percentage tables that the values only add up to one of the margins. In this case, adding across the rows we would not get 100%.

Idea summary To find the row percentage for a particular value, divide the value in the table by the row total and multiply by 100%. To find the column percentage for a particular value, divide the value in the table by the column total and multiply by 100%.

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Associations between categorical variables To find if there is an association between the variables in the Titanic table we can ask the question “Is the survival rate dependent on the class of the passenger?” In order to find this we must first identify the independent variable. This is also known as the explanatory variable, which is the factor that is manipulated or controlled in an experiment or study to test its effects on the dependent variable. It explains, influences, or predicts changes in the response variable. In this problem it is the class of passenger. Which percentage frequency table should we use? When to use a row percentage table Independent variable forms row headings

When to use a column percentage table Independent variable forms column headings

How do we tell if there is an association? For a row percentage table Look at values in the columns

For a column percentage table Look at values across the rows

The independent variable forms the heading of each column, therefore the column percentage frequency table will best indicate any patterns. Survived Died Total

First class 62% 38% 100%

Second class 41% 59% 100%

Third class 25% 75% 100%

Crew 23% 77% 100%

When we read across the first row in the column percentage frequency table and look at the numbers we can see a clear difference in the percentages of passengers that lived or died in each class. This suggests there is an association between the class of passenger and the rate of survival. If the values are similar (±5%), we say there is no clear association. How do we describe the association? 1. State if there appears to be an association apparent or not. In this example there appears to be an association between the variables. 2. Describe the association. In this example we could suggest class of passenger affects the survival rate of the passengers. 3. Provide reasoning. We could say that based on the data, the higher the class of passenger, the more likely a passenger was to survive. Two-way tables show potential links between variables, with percentages making associations clear. Subtler links need deeper statistical study. Note: ‘association’ means one variable’s change seems to impact the other, not necessarily causing the change.

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Example 2 Glen surveyed all the students in year 12 at his school and summarised the results in a table: Height ≥ 170 cm Height < 170 cm Total

Play netball 33 13 46

Do not play netball 72 30 102

Total 105 43 148

a Which variable is the independent variable? A

Play netball

Height

Create a strategy

Apply the idea

Choose a variable which, when changed, would affect the other variable.

The decision of a student to play or not may depend on their actual height, whereas their height won’t change depending on their decision to play netball or not. So the height is the independent variable, option B.

b To examine if there is an association between height and playing netball, should Glen use a column or row percentage frequency table?

Create a strategy

Apply the idea

In a percentage frequency table, we want to sum We found in part (a) that the height is the the percentages of the independent variable. independent variable, which are the rows. To sum the percentage of height we should use a row frequency table, option B. c Complete the row percentage frequency table for this data. Round your answers to the nearest percentage. Height ≥ 170 cm Height < 170 cm

Create a strategy

Play netball ⬚% ⬚%

Do not play netball ⬚% ⬚%

Total ⬚% ⬚%

Divide each number of students by the total number of students in that row and multiply the answer by 100%.

Apply the idea Here is the original table with the row totals: Height ≥ 170 cm Height < 170 cm

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Play netball 33 13

Do not play netball 72 30

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Total 105 43

Here is the table with the calculations for the row percentages: Play netball

Do not play netball

Total

Height ≥ 170 cm

100%

Height < 170 cm

100%

d Looking at the columns of the completed table, does there appear to be an association between height and playing netball? A

No, there does not appear to be any association as the numbers are similar.

es, there appears to be an association as the numbers are quite different. It seems that Y taller people like to play netball.

Create a strategy In a row percentage table, if the values are similar then we say there is no clear association.

Apply the idea The completed row percentage table is shown: Height ≥ 170 cm Height < 170 cm

Play netball 31% 30%

Do not play netball 69% 70%

Total 100% 100%

The numbers in each column are quite similar quite similar with just 1% difference in values. So we can say that there does not appear to be any association between height and playing netball, option A.

Idea summary When analysing associations between categorical variables, consider the type of percentage frequency table and how to identify and describe associations: •

Which percentage frequency table should we use? When to use a row percentage table Independent variable forms row headings

•

How do we tell if there is an association? For a row percentage table Look at values in the columns

•

When to use a column percentage table Independent variable forms column headings

For a column percentage table Look at values across the rows

If the values are similar then we say there is no clear association. How do we describe the association?

1. State if there is or is not an association apparent. For example: There appears to be an association between the variables.

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2. Describe the association. For example: The class of passenger affects the survival rate of the passengers. 3. Provide reasoning. For example: Based on the data, the higher the class of passenger, the more likely the passenger were to survive.

100% stacked column graphs Association between variables can often be seen more clearly in a stacked column graph. Shown is a stacked column graph for the data from the Titanic table earlier. When we look at each column we can see the proportion of death to survival in each column is different. This indicates there is an association between the variables. Death vs. survival: Titanic 1912 100% 90% 80%

Died, 38 Died, 59

70% 60%

Died, 75

Died, 77

Survived, 25

Survived, 23

Third class

Crew

50% 40% 30%

Survived, 62 Survived, 41

20% 10% 0% First class

Second class

To draw a stacked column graph: • Label the horizontal axis with the independent variables. • Label the vertical axis with percentages from 0% to 100%. • Draw a column for each independent variable that reaches the height of 100% on the vertical axis. • To divide each column into the percentages as shown in the frequency table start from the bottom of the column, count to the ﬁrst percentage and draw a horizontal line to mark it off, then count up to the second percentage from the horizontal line and then mark off again, until all sections are complete. • Write the label and percentage in each section of the columns which indicates the dependent variables displayed or provide a key.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Example 3 A group of year 12 students surveyed their class and recorded the hair colour and eye colour for each student. The results are displayed in the 100% stacked column chart shown: Hair colour for various eye colours 100% 80% Blonde

60%

Red Brown

40%

Black

20% 0% Blue

Green

Brown

Hazel

Eye colour

a What is the independent variable for this chart? A

Eye colour

Hair colour

Create a strategy In a stacked column graph, the independent variable is placed along the horizontal axis.

Apply the idea The stacked column graph has the eye colour placed along the horizontal axis. So it is the independent variable, option A.

b Does the chart suggest an association between eye colour and hair colour? A

Yes, as the corresponding segments are similar in size.

Yes, as the corresponding segments are of different sizes.

No, as the corresponding segments are similar in size.

No, as the corresponding segments are of different sizes.

Create a strategy Check whether the proportion of the segments in each column are similar sizes.

Apply the idea The stacked column graph segments differ in size, which suggests an association between eye colour and hair colour, option B.

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c Can we say that having blue eyes causes a high chance of having blonde hair? A

Yes. The data shows that students with blue eyes are more likely to have blonde hair.

No. There appears to be an association, but we cannot say that one causes the other.

Create a strategy Consider the difference between association and causation.

Apply the idea An association does not mean one variable causes the other variable to change but that a change in one variable appears to affect the other. The chance of having blonde hair can be caused by many factors. Having blue eyes may affect the chance of having blonde hair, but we cannot say it is a cause. The correct answer is option B.

Idea summary When analysing a stacked column graph, we can determine if there is an association between the categorical variables by comparing the proportions of the segments in each column: • •

If the segments vary in size, we may say there is an association between the variables. If the segments are similar in size, then there is no clear association between the variables.

Association differs from causation: An association does not mean one variable causes the other variable to change but that a change in one variable appears to affect the other.

Practice questions Choose your question path Explorer 1–13, 15, 16, 18, 20, 21

Adventurer 1–9, 11, 12, 14–16, 18, 20, 22

Trailblazer 1–4, 6, 8, 9, 11, 12, 14, 17, 19, 21, 22

Understanding 1

This table gives information on the treatment plans prescribed to dogs with canine diabetes. Depending on the severity of the case, dogs are either prescribed a special diet or insulin injections. Parkside Veterinary Clinic Riverbend Animal Hospital

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Special diet 23 16

Insulin injections 15 12

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Total 38 28

This table gives the row percentages on the treatment plans prescribed based on the previous table. Parkside Veterinary Clinic Riverbend Animal Hospital

Special diet 61% 57%

Insulin injections 39% 43%

Total 100% 100%

What percentage of dogs from the Riverbend Animal Hospital were prescribed with insulin injections?

Will the percentages remain the same when calculating the column percentages?

Is there an association between the pet clinic and the type of prescriptions for dogs with canine diabetes?

What type of percentage frequency table should be used when the explanatory variable forms the column headings?

When constructing a stacked column graph, is the explanatory variable labelled on the horizontal axis or the vertical axis?

Is there an association between household composition and distribution of money based on the data in the stacked column graph? Household expenditure by domestic situation 100% 90% 80% 70% 60% 50% 40% 30% 20% 10%

Housing

Single

Married

Transportation

Food

Married with kids Insurance

Single with kids

Entertainment

Other

Fluency 5

In a survey people were asked about their favourite subjects. The results are shown in the table. Calculate the row percentages for each subject rounded to two decimal places. Math Science Total

Adults 30 45 75

Children 40 55 95

Total 70 100 170

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6 Ex 1

A teacher recorded the hair type and colour of their students. The results are shown in a column graph. 10 9

Number of students

8 7 6

Straight

Curly

4 3 2 1 0

Red 5.9% 10%

Blonde

Black

Brown

Blonde 5.9%

Black 41.2%

40%

Total 100% 100%

Complete the two-way table displaying the column percentages rounded to one decimal place. Red Straight Curly Total

Brown

Complete the two-way table displaying the row percentages rounded to one decimal place. Straight Curly

Red

50% 100%

Brown 66.7% 100%

Blonde 25% 75% 100%

Black

100%

A survey was conducted about people’s favourite movie genre. The results are shown in a column graph. 50 45

Number of people

40 35 30

Teens

Children

20 15 10 5 0

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Comedy

Action Drama Movie genre

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Horror

Complete the two-way table displaying the row percentages rounded to one decimal place. Teens Children

Comedy 30% 30%

Teens Children Total

Ex 2

50% 100%

Horror 5%

40%

Total 100% 100%

Action 33.3% 100%

Drama 81.8% 18.2% 100%

Horror

100%

Isaac surveyed all of the students in Year 12 at his school and summarised the results in a table. Height ≥ 170 cm Height < 170 cm

Drama 45%

Complete the two-way table displaying the column percentages rounded to one decimal place. Comedy

Action

Play tennis 62 60

Do not play tennis 138 140

Which variable is the explanatory variable?

Create a row percentage frequency table for this data. Round the values to the nearest percentage.

Members of a gym who are either currently pregnant or postpartum were asked what kind of training they do. Each of them only did one kind of training. The table shows the results.

Currently pregnant Currently postpartum

Cardio

Weight

Which variable is the explanatory variable?

Create a row percentage frequency table for this data. Round the values to the nearest percentage.

Does there appear to be an association between the type of training and the pregnancy status of gym members?

Does a person’s pregnancy status cause them to choose a certain type of training?

Some students were asked if they are left-handed or right-handed. The results are provided in the table: Not female Female Total

Left-handed 8 7 15

Right-handed 47 65 112

Total 55 72 127

Construct a row percentage two-way table. Round your answers to one decimal place.

Construct a column percentage two-way table. Round your answers to one decimal place.

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The results of a survey of 279 people on their employment status and parenting status is shown in a row percentage table. People without kids People with kids

Ex 3

None 8% 10%

Part-time 17% 47%

Full-time 75% 43%

Total 100% 100%

Is employment status or parenting status the explanatory variable?

Does there appear to be an association between parenting status and employment status?

A stacked column chart shows the ﬁtness levels for various categories of smokers. Fitness of smokers 100% 80% 60% 40% 20% 0%

Very heavy (1-5)

Heavy (16-25)

Below average fitness

930

Moderate (6-15) Normal fitness

Light (1-5)

Non-smoker (0)

Above average fitness

What is the response variable for this chart?

Does there appear to be an association between ﬁtness level and category of smoker?

Can we say that being a non-smoker causes a high chance of an above average ﬁtness level?

These tables and 100% stacked column graph shows the results of a survey on smoking: Introverts Extroverts

Smokers 37 51

Non-smokers 69 123

Introverts Extroverts

Smokers 35% 29%

Non-smokers 65% 71%

Total 100% 100%

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Personality type of smokers 100

Percentage

80 60

Smokers Non-smokers

40 20 0

Extroverts Introverts Personality type

Is smoking status or personality type the explanatory variable?

Does the 100% stacked column graph indicate an association between the variables?

170 people were surveyed about their music preference. The results have been recorded in a table. Music Preference Rock and Roll Classical Pop Rap Country and Western R and B Punk Total

Vegetarian 24 8 17 6 17 6 4 82

Non-vegetarian 19 15 17 2 24 9 2 88

Total 43 23 34 8 41 15 6 170

What is the explanatory variable in this data set?

Which of the 100% stacked column charts should be used to look for an association between the variables? A

Music preferences by dietary restriction 100% 80% 60%

Non-vegetarian Vegetarian

40% 20% 0%

Rock & Classical roll

Pop

Rap Country & R&B western

Punk

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Music preferences by dietary restriction 100% 80%

Punk R&B

60%

Country & western Rap Pop

40%

Classical Rock & roll

20% 0% Vegetarian

Non-vegetarian

Dietary restriction

Does this stacked column chart suggest that there is an association between music preference and dietary restriction?

Reasoning 15

What does it mean if the sections in each column of a stacked column graph are of similar size?

In Australia, 130 random people were surveyed, examining their carbon footprint and the city they lived in. The people were then categorised as living in either an urban or regional location, and whether they caused high carbon emissions or low carbon emissions. The results are displayed in a table. High emissions Low emissions Total

Urban 48 30 78

Total 68 62 130

Is location or emission level the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Construct this table for the data, rounding your answers to the nearest whole number.

Does there appear to be an association between location and emission level? Explain you answer.

Mr. Tranor asked his class to pick their favourite subject. He displayed the results in a twoway table. Right-handed children Left-handed children a

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Regional 20 32 52

Maths 13 20

Music 9 12

Science 11 18

English 11 11

How many Left-handed children did not pick maths as their favourite subject?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

How many students picked music as their favourite subject?

Is favourite subject or dominant hand the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Create this frequency table, rounding your answers to the nearest percentage.

Does there appear to be an association between a child’s dominant hand and their favourite subject? Explain your answer.

In a study of car accidents, data was found on the number of passengers in the car and whether or not the car rolled over. No. passengers: Roll over No roll over Total

<5 335 1622 1957

5—9 18 42 60

10—15 17 42 59

> 15 2 5 7

Total 372 1711 2083

Which variable is the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Create this percentage frequency table for the data. Round your answers to one decimal place.

Does there appear to be an association between passengers and rolling over? Explain your answer.

At a local university, students were asked what their favourite subject at high school was and what they have decided to major in at university. The results are shown in the following table. Favourite subject

Maths major Science major Music major Art major Total

Maths 66 51 38 12 167

Science 56 40 33 17 146

Music 21 43 68 48 180

Art 42 25 37 76 180

Total 185 159 176 153 673

Which variable is the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Create this percentage frequency table for this data. Round your answers to two decimal places.

Does there appear to be an association between favourite subject at school and university major? Explain your answer.

A lecturer commented that “having mathematics as their favourite subject at high school causes many students to take mathematics at university”. Is he correct? Explain your answer.

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Problem-solving 20

The local library constructed divided bar graphs showing the last six months of book loans by category. Assume the total number of loans was exactly the same for each month.

Jul Aug

In which month were business books the most borrowed type of book?

Sep

In which month were children’s books the most popular?

Oct

Which variable is the response variable?

Nov

Does the divided bar graph suggest that there is an association between the variables?

Fiction Design Environmental

Jun

Children’s Business

A group of people were asked if they are employed and if they have a smartphone. The results are shown in the table: Owns a smartphone Does not own a smartphone Total

Employed 34 71 105

Unemployed 18 87 105

Total 52 158 210

Is employment status or smartphone ownership the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Create this percentage frequency table rounding your answers to the nearest percentage.

Does there appear to be an association between employment and owning a smartphone? Explain your answer.

Does a person’s employment status cause them to own a smartphone? Explain your answer.

Create a 100% stacked column graph for your percentage frequency table.

Does the 100% stacked column graph indicate an association between the variables? Explain your answer.

118 people were surveyed about their age and how many hours they watch television each week. The results are shown in a table.

Less than 3 hours 4 to 8 hours 9 to 15 hours Over 15 hours Total

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10 years and under 9 2 7 10 28

11 to 20 years 8 10 8 9 35

21 to 30 years 2 2 8 9 21

31 years or older 5 9 10 10 34

Total 24 23 33 38 118

Is age or time spent watching TV the explanatory variable?

To examine if there is an association between the variables should we use a column or row percentage table?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Create this percentage frequency table rounding your answers to two decimal places.

Create a 100% stacked column graph for this percentage frequency table.

Does the 100% stacked column graph indicate an association between the variables? Explain your answer.

17.03 Bivariate numerical data After this lesson, you will be able to… • identify what bivariate numerical data is. • identify the independent and dependent variables in a pair of bivariate data. • plot and interpret scatterplots to analyse bivariate data. • apply understanding of bivariate numerical data to solve real-world problems.

Bivariate numerical data Bivariate numerical data is the name for numerical data consisting of pairs of values. Each study participant will have two pieces of data collected. We use these pairs to find out whether there is a simple relation between variables.

Person 1 Person 2 Person 3 Person 4 Person 5

Age ( years) 30 40 50 60 70

Bone density (g/cm³) 1.35 1.28 1.22 1.10 0.97

For a study about bone health, a person’s bone density is measured against their age. Their age is the independent variable and can be any value. Their bone density is the dependent variable that is recorded against their age.

Each person’s age and bone density make a pair of values in the bivariate data set. A single data point in a bivariate data set is written in the form (x, y), with x being the independent variable and y being the dependent variable. We display bivariate data graphically by plotting the data points Bone density with the value of the independent variable on the horizontal Bone density ( g/cm3) axis and the value of the dependent variable on the vertical 1.4 axis. This is known as a scatterplot. We may then check whether age is related to bone density by analysing the relationship between the variables graphically. It’s important to note that the existence of a relationship between two variables in a scatterplot does not necessarily imply that one causes the other.

1.2 1

0.8 0.6 0.4 0.2

Age (years)

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Example 1 Consider the following variables: • Number of ice cream cones sold • Temperature (°C) a Which of the following statements makes sense? A

The temperature affects the number of ice cream cones sold.

The number of ice cream cones sold affects the temperature.

Create a strategy

Apply the idea

Determine which variable has an effect on the other.

When hot weather occurs, more ice cream cones will be sold. Less ice cream cones will be sold in cold weather. So, the correct answer is option A.

b Which is the dependent variable and which is the independent variable?

Create a strategy

Apply the idea

Recall that the independent variable is not affected by the other variable. In contrast, the dependent variable is affected or changed by the other variable.

Based on the previous problem, we can see that the independent variable is the temperature and the dependent variable is the number of ice cream cones sold, since the temperature affects the number of ice creams sold.

Example 2 The price of ten houses are graphed against the house’s land area on the following graph:

Price (thousands) 1200 1000 800 600 400 200

Describe the relationship between land area and house price in the data.

Area (m2) 200 400 600 800 1000 1200

Create a strategy Describe what happens to the dependent variable ( price) as the independent variable (area) increases.

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Apply the idea The area increases from left to right. We can see from the graph that the price increases (rises) from left to right. So as the land area increases, the house price increases.

Example 3 Scientists were looking for a relationship between the number of hours of sleep we receive and the effect it has on our motor and process skills. Subjects were asked to sleep for different amounts of time, and were all asked to undergo the same driving challenge in which their reaction time was measured. The table shows the results, which are to be presented as a scatterplot. Amount of sleep (hours) 9 6 4 10 3 7 2 5

Reaction time (seconds) 3 3.3 3.5 3 3.7 3.2 3.85 3.55

a Create a scatterplot for the observations in the table.

Create a strategy Plot the points on a Cartesian plane.

Apply the idea 3.9

Reaction time (seconds)

The scatterplot of the data is shown. Reaction time is dependent on how much sleep the person gets. So the values on the horizontal axis represent the number of hours of sleep and the values on the vertical axis represent the reaction time.

3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1

Sleep (hours) 1 2 3 4 5 6 7 8 9 10

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b According to the results, describe the relationship between amount of sleep and reaction time.

Create a strategy Describe what happens to the dependent variable (reaction time) as the independent variable (sleep) increases.

Apply the idea The sleep hours increases from left to right. We can see from the graph that the reaction time decreases ( falls) from left to right. So as the hours of sleep increases, the reaction time decreases.

Idea summary In statistics we often study pairs of values to understand potential relationships between them. This is called bivariate numerical data. Each pair of values in bivariate data is known as a data point. These data points consist of an independent variable, which could be any value, and a dependent variable, which is the outcome we measure against the independent variable. To visualise these pairs of values, we plot the data on a scatterplot. By plotting these data points, we can observe any potential relationships between the variables.

Practice questions Choose your question path Explorer 1, 2ac, 3, 4, 6–9, 11–13, 15, 17–19, 21, 22

Adventurer 1, 2bc, 3, 4, 6, 8, 9, 11–13, 15, 17–19, 21–23

Trailblazer 1, 2cd, 3–5, 8, 10–12, 14, 16, 18, 20–23

Understanding 1

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Is each an example of univariate data or bivariate data? a

A school collects data on the height of each student in two classes in order to compare the classes.

A scientist collects data on iron levels in soil and growth of a type of weed in order to investigate the relationship between them.

A school collects data on the shoe size of each student in the school.

A hockey club collects data on time travelled to get to training from members of three teams in order to compare them.

A political group collects data on the taxable income and addresses of the local electoral constituents in order to investigate if there is an association between them.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A psychologist collects data on the number of days in daycare and NAPLAN results in grade three in order to investigate the relationship between them.

John wishes to investigate whether there is an association between the amount of natural sunlight in a classroom and student’s exam results.

Emma wishes to investigate whether players in the A hockey team have higher average mathematics test results than players in the B hockey team.

Create a scatterplot for each set of data: a

x 1 3 5 7 y 2 7 11 15

9 17

x y

1 3

3 7

5 11

7 13

9 17

x 1 2 4 7 y 20 18 17 15

8 11

x y

1 7

3 3

5 11

9 15

12 19

Is there likely to be a relationship between each pair of variables? a

Study time and test score

Length of hair and number of pets

Typing speed and dancing ability

Driving time and petrol used

Identify whether the lines are increasing, decreasing, or neither. a

1 −4 −3 −2 −1 −1

x 1

−2

−3

−4

1 −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

−2

y 4

x 1

1 −4 −3 −2 −1 −1

−2

−3

−4

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Fluency 5

Ex 1

For each of the pairs of variables: i

What is the independent variable?

Number of dogs owned and number of destroyed shoes

Size of a pizza and cost of a pizza

Time spent studying and exam mark

What is the dependent variable?

Consider the variables: • Ticket sales

• Revenue from a show

Which variable affects the other?

What are the dependent and independent variables?

A kite can rise to a height of 161 m when the wind is blowing at 6 kilometres per hour and to a height of 322 m when the wind is blowing at 12 kilometres per hour. a

What are the two variables?

From the two variables, state the dependent and independent variables.

Does one variable have a direct effect on the other? If yes, which variable affects the other? a

Temperature (°C), Distance from the equator (km)

Weekly income, Number of friends

Time of travel (minutes), Distance covered (m)

Age (years), Time spent watching television in a day

Time spent working, Wages earned

Time spent to ﬁnish a novel, Number of pages

Height (cm), Number of children

Number of pets, Time spent caring for pets

Time spent training, Athletic performance

A study was performed to ﬁnd the relationship between the number of soft drinks consumed per week and the waist circumference of students. a

Which variable is the independent variable?

Which variable is the dependent variable?

Which variable should be put on the x-axis?

Which variable should be put on the y-axis?

The graph shows the height of a ball after it is dropped off the side of a building. a

Which variable is the dependent variable?

Which variable is the independent variable?

Height

Time

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Ex 2

A cafe records the number of soups sold and the daily maximum temperature on a graph. What is the relationship between temperature and soups sold in the data?

Soups sold 60 48 36 24 12 4

A company records the height and annual salary ($’000) of ten employees on a graph. What is the relationship between height and salary in the data?

Temperature 16 20

120

160 200

Annual salary 100 80 60 40 20 Height (cm) 40

Ex 3

The table presents data on the number of immigrants and the unemployment rate in Australia over a 10-year period: Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10

Number of immigrants (thousands) 250 265 275 280 290 295 300 305 310 320

Unemployment rate (%) 5.2 5.0 4.8 4.6 4.5 4.3 4.1 4.0 3.9 3.7

What are the independent and dependent variables?

Create a scatterplot for the set of data.

According to the results, what is the relationship between the number of immigrants and the unemployment rate?

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The table presents data on the global annual average carbon dioxide emissions and the average global temperature anomalies over a 10-year period: Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10

CO2 emissions (million metric tonnes) 35 000 35 500 36 000 36 500 37 000 37 500 38 000 38 500 39 000 39 500

Which are the independent and dependent variables?

Create a scatterplot for the set of data.

Based on the scatterplot, how do the CO2 emissions relate to the global temperature anomalies?

This table shows the number of apples eaten per week and the number of visits to the doctor per year by six students: Number of apples eaten per week Number of visits to the doctor per year

3 14

6 11

7 6

9 3

11 0

Create a scatterplot for the set of data.

Vanessa wants to visit the doctor as few times as possible each year. According to the plot, should she eat 0 or 11 apples per week?

The height of the tide at a beach is recorded at ten different hours in one day: 13 −0.21

24 −0.05

7 0.92

2 0.55

4 0.84

What are the dependent and independent variables?

Create a scatterplot for the set of data.

15 −0.76

6 1.01

11 0.31

21 −0.68

This table shows the age and height of six children in a family: Age ( years) Height (cm)

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0 17

Time (hours) Height (metres)

Global temperature anomaly (°C) 0.6 0.65 0.68 0.7 0.75 0.8 0.83 0.85 0.88 0.9

3 90

4 105

9 130

11 145

13 160

16 165

Create a scatterplot for the set of data.

Bob is the second youngest child in the family. Use the plot from part (a) to ﬁnd out how tall he is.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Reasoning 18

Six students were asked how many people there are in their family, and the number of trees planted in their garden. Their responses were plotted on a graph. Each dot represents one student.

One student, Jack, has 7 people in his family. How many trees are planted in his garden?

Do you think there is a connection between the number of people in a student’s family and the number of trees in their garden? Explain your answer.

This table shows the average time spent on a website (in minutes) and the average amount spent (in dollars) for a group of online shoppers in a particular month: Week 1 Week 2 Week 3 Week 4

30 Number of trees in the garden 28 26 24 22 20 18 16 14 12 10 8 6 4 2 Number of people in the family 3 4 5 6 7 8 9 10 11 12 13 14 15

Average time spent (minutes) 15 30 45 60

Average amount spent (dollars) 75 130 185 250

Create a scatterplot for the set of data.

A statement is made: “The more time a shopper spends on a website, the less they spend.” Critique this claim based on your scatterplot.

The table shows the number of books read per year by six students and the mark they received in their yearly maths exam. Number of books read per year Mark in yearly maths exam

5 50

15 65

20 70

30 80

40 90

50 100

Create a scatterplot for the set of data.

In this group of students, Valentina achieved the highest mark in the yearly maths exam. Use the plot from part (a) to ﬁnd out how many books she read throughout the year.

The teacher told the class: “Students who read more books also tend to score higher on maths exams.” According to the plot in part (a), is the teacher correct? Explain your answer.

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Problem-solving Georgia has a high income but does not like her job. Which dot would represent her? Explain your answer.

3 2

Job Satisfaction

4 1

0 0

The scatterplot represents the social media usage (in minutes per day) and the respective grades of ﬁve students: A, B, C, D, and E. a

Which student seems to spend the least amount of time on social media?

Which student has the lowest grades despite not spending the most time on social media?

Which student seems to manage their time best on social media while maintaining good grades?

Income

Student survey Grades

B B A A A E E E C C D D Social media usage (minutes per day)

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I dentify a pair of variables in everyday life that you believe may have a relationship. Explain what the variables are and why you think there may be a relationship.

Identify which variable is the independent variable and which is the dependent variable in your chosen pair from part (a). Explain your choice.

Identify a pair of variables in everyday life that you believe do not have a relationship. Explain what the variables are and why you think they do not have a relationship.

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17.04 Associations between numerical variables After this lesson, you will be able to… • interpret scatterplots to describe the association between two numerical variables. • identify the form of a data set as linear or non-linear. • examine scatterplots to determine the form, direction and strength of data points. • recognise outliers in a data set and understand their impact on correlation. • distinguish between correlation and causation using real-world examples.

Associations between numerical variables We analyse how the independent variable (x) inﬂuences the dependent variable ( y) when determining associations between numerical bivariate variables. To identify an association, also called correlation, between variables, we examine: • Form: The overall pattern or shape of the data points—linear or non-linear. • Direction (if linear): The trend of the data—positive (increasing y with increasing x, similar to a positive gradient) or negative (decreasing y with increasing x, similar to a negative gradient). • Strength (if linear): The extent to which points follow a straight line—strong (close to a line), moderate, weak, or no correlation. While data points might display non-linear forms like curves, we’ll focus on linear models. A perfect correlation means all points lie exactly on a straight line, and no correlation signifies a random scatter of points. Most plots fall between these extremes, showing weak, moderate, or strong correlation.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 17.04 to answer these questions. 1.

What do you think a graph with a positive correlation would look like? What do you think a graph with a negative correlation would look like?

For a particular correlation, if you change the number of data points, does this change the general shape of the scatterplot?

Create a scatterplot you would say is linear, positive and strong. Compare with a peer. Do your scatterplots look similar? How do they compare to the examples later in the lesson?

Create a scatterplot you would say is linear, negative and weak. Compare with a peer. Do your scatterplots look similar? How do they compare to the examples later in the lesson?

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Perfect positive correlation

Perfect negative correlation

Strong positive correlation

Strong negative correlation

Weak positive correlation

Weak negative correlation

No correlation

We may also see outliers, which are data points drastically out of the pattern. Correlation means that the variables are related, and often does not mean that the changes in the dependent variable are caused by the independent variable. Causation means there is a cause and effect and must be shown with carefully designed studies.

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Two variables may be strongly correlated, but not necessarily causation for a few reasons. • Third (contributing) variable: A third variable is contributing to both x and y, so the relationship between x and y is not causal. • For example: There is a correlation between number of cats and number of street lights in a city, but cats don’t cause street lights. There is a third variable of population size. • Coincidence: It is purely coincidence that the two variables are related. • For example: There is a correlation between butter sales and number of female CEOs. • Unproven: In some cases, to be causation, it must be proven using proper and repeatable statistical studies. • For example: Cigarette smoking was not proven to cause cancer until the 1950-60s, though it was suspected and correlated before that.

Example 1 From the scatterplot given, we can conclude that:

Vision

x 130 140 150 160 170 180 Height (cm)

A There is no direct relationship between someone’s vision and their height. B Those who are taller have better vision. C The worse someone’s vision, the taller they are.

Create a strategy Examine the scatterplot and determine whether there is a discernable relationship between height and vision.

Apply the idea Based on the given scatterplot, there is no direct relationship between someone’s vision and their height. So, the correct answer is option A.

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Example 2 The table shows the marks of 12 students in English and French. Student 1 2 3 4 5 6 7 8 9 10 11 12

Mark in English 85 71 57 60 79 76 71 91 50 49 66 92

Mark in French 89 71 56 62 86 76 77 86 90 47 67 92

a Which of the following scatterplot correctly represents the above data? 100

100

90 80 70 60 50

40 40 50 60 70 80 90 100

Mark in English

100 Mark in French

B Mark in French

Mark in French

90 80 70 60 50 40 40 50 60 70 80 90 100 Mark in English

Create a strategy

Apply the idea

Draw your own scatterplot and plot each given point to identify the correct graph.

The correct answer is A.

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b Is the correlation between students’ English and French marks positive or negative? A

Positive

Negative

Create a strategy

Apply the idea

Think of what happen to the students’s mark in French as their marks in Engiish increase.

As the students’ marks in English increase, so too do their marks in French. So the answer is A: Positive

c Is the correlation between students’ English and French marks strong or weak? A

Strong

Weak

Create a strategy

Apply the idea

Look at how scattered the points are on the scatterplot.

When the points on a scatterplot tend to follow a single line, the correlation is strong. When the points on a scatterplot are scattered greatly around a single line, the correlation is weak. So the answer is A: Strong

d Which student’s scores represent an outlier? A

Student 10

Student 9

Student 12

Create a strategy

Apply the idea

Which point, if any, on the scatterplot is positioned away from the rest of the data?

The answer is B: Student 9

None of them

Example 3 This table of values that show four excerpts from a database comparing the income per capita of a country and the child mortality rate of the country. If a scatterplot was created from the entire database, what relationship would you expect it to have? A Strongly positive

No relationship

Income per capita 1465 11 428 2621 32 468 C

Child mortality rate 67 16 35 9

Strongly negative

Create a strategy

Apply the idea

Look at how the mortality rate changes as the income increases.

As the income per capita increases, the child mortality rate decreases, so there is a negative correlation between the two sets of data. The correct answer is C.

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Example 4 A study found a strong positive association between the temperature and the number of beach drownings. a Does this mean that the temperature causes people to drown? A

Yes

Create a strategy To better understand the relationship between temperature and beach drownings, let’s explore other variables that may directly impact both temperature and drowning incidents.

Apply the idea The statement that there’s a strong positive association between temperature and beach drownings does not mean temperature causes people to drown. This simply indicates a correlation, not causation. So, the correct answer is B: No.

b Which of the following is the most likely explanation for the strong positive association? A

ontributing variables - there are other causal relationships and variables that come in to C play and these may lead to an indirect positive association between the temperature and the number of beach drownings.

oincidence - there are no other contributing factors or reasonable arguments to be made C for the strong positive association between the temperature and the number of beach drownings.

Create a strategy Is there a common factor that has a direct relationship with both temperature and the number of beach drownings?

Apply the idea A plausible argument would be that the increase in temperature could lead to more people visiting the beach and therefore more chances of drowning incidents. There might be other variables at play, like stronger currents during warmer periods or increased alcohol consumption. So, the correct answer is A.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary To identify correlation between variables, we examine: • •

• •

Form: The overall pattern or shape of the data points—linear or non-linear. Direction (if linear): The trend of the data—positive (increasing y with increasing x, similar to a positive gradient) or negative (decreasing y with increasing x, similar to a negative gradient). Strength (if linear): The extent to which points follow a straight line—strong (close to a line), moderate, weak, or no correlation. Presence of outliers

However, correlation doesn’t always imply causation. Apparent relationships between x and y don’t necessarily mean changes in one cause changes in the other. There could be an unobserved variable indirectly linking x and y.

Practice questions Choose your question path Explorer 1–6, 8, 9, 12–14, 16–19, 21, 23

Adventurer 1–3, 5–8, 10, 12, 14, 16, 18, 20–24

Trailblazer 1–3, 5–8, 10, 11, 15, 16, 18, 20–23, 24

Understanding Four different classes with four different professors had the same final exam. The exam results and number of classes attended by each student is displayed for each class. Does each graph suggest that there is an association between number of classes attended and final exam mark? y

b 100

100

Final exam mark

x 0

10 20 Number of class attended

Number of classes attended

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d 100 Final exam mark

Final exam mark

100

x 0

Number of classes attended

Each point on the scatterplot shows the time (in weeks) Sumon spent training for a half marathon and the corresponding kilometres they were able to run: Using the scatterplot, are these statements true or false?

Number of classes attended Running distance (km) 20 15

The number of weeks that Sumon trained for the half marathon is the independent variable.

The y-coordinates of the points represent the time spent by Sumon training.

There is evidence to suggest that the longer Sumon trains, the further they can run.

No. of weeks 5

Training for a half marathon causes further running distance over time.

The correlation between the number of weeks training and the number of kilometres Sumon is able to run is positive.

Complete each sentence with one of the words from this list. • association • causation • coincidence • contributing (third) variables • correlation a b c d

952

The strength and direction of a scatterplot describe the ⬚. When two variables are related, we say there is an ⬚. To show ⬚, we must do a carefully designed study.

When correlation is not causation, some possible explanations are ⬚ and ⬚.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Fluency 4

Does each scatterplot show a linear or non-linear relationship? a

5 x

x 5

5 x

x 5

5 x 5

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Describe the strength and direction for each linear association. a

5 x 5

5 x

x 5

This scatterplot shows the relationship between air and sea temperature: a

Describe the relationship between air and sea temperature.

Describe the correlation between air and sea temperature.

35 Sea temperature

Ex 1

30 25

Air temperature

Scientists conducted a study where each person was asked to read a paragraph then recount as much information as they could remember. They found that the longer the paragraph, the less information each person could retain. If the length of the paragraph were plotted (on the horizontal axis) against the amount of information retained (on the vertical axis), would the correlation be negative or positive?

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 2

This table shows the marks of 12 students in Maths and Sports: a

Construct a scatterplot for the students’ marks in Maths versus their marks in Sports.

Describe the correlation between students’ marks in Maths and marks in Sports.

Which student’s scores appear to represent an outlier?

Sports 44 74 52 70 67 60 73 86 84 93 57 92

20 41

25 44

30 39

35 34

40 30

45 25

50 22

55 18

60 19

Construct a scatterplot for this data.

Describe the correlation between a person’s age and the number of accidents.

Does this data set contain any outliers?

65 17

This table shows the number of hours of sleep a person gets per week associated with a sample of persons of different age groups. Age Hours (per week)

Maths 63 92 60 79 88 81 61 91 72 42 66 92

This table shows the number of traﬃc accidents associated with a sample of drivers of different age groups. Age Accidents

Student 1 2 3 4 5 6 7 8 9 10 11 12

20 42

25 39

30 45

35 37

40 38

45 41

50 36

55 34

60 33

Construct a scatterplot for this data.

What is the correlation between a person’s age and the hours of sleep per week?

Does this data set contain any outliers?

The market price of bananas varies throughout the year. Each month, a consumer group compared the average quantity of bananas supplied by each producer to the average market price ( per unit). The results are shown in a table. a

Construct a scatterplot using the data from the table.

Describe the correlation between the supply quantity and the market price of bananas.

According to this data, when will a supplier of bananas receive a higher price per banana?

Supply (kg) 550 600 650 700 750 800 850 900 950 1000

65 30

Price (dollars) 16.25 15.75 15.75 14.75 14.50 14.00 13.75 12.75 12.50 12.00

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This table lists the time taken to sprint 400 m by runners who all ran in different temperatures as part of a study. a

Construct a scatterplot to represent the data in the table.

How many runners were tested in the study?

Describe the correlation between temperature and sprint time for the data.

Which data point represents an outlier?

Time (sec) 60 67 48 69 65 49 57 53

A researcher is studying the relationship between the number of passers-by in an emergency, and the time taken (in seconds) before a passer-by helps a stranger during an emergency. The data is recorded in this table. Number of passers-by (n) Time until help is offered (t)

Ex 3

Temperature (°C) 5 2 10 8 1 7 6 4

1 8

2 19

3 26

4 37

5 51

6 65

Construct a scatterplot using the data from the table.

As more passers-by are present what happens to the time taken until help is offered?

Describe the correlation between the number of passers-by and the time until assistance is offered.

Does the scatterplot contain any outliers?

This table of values shows the time taken to finish a lap on a race track for ten average speeds. If a scatterplot was created from the entire database, would you expect the relationship to be strongly positive, strongly negative or that there was no relationship? Speed (x) Time ( y)

956

25 87

30 75

35 82

40 69

45 73

50 60

55 57

60 45

65 49

This table of values shows four excerpts from a database comparing the literacy rate of a country and the gross domestic product (GDP) per capita of the country. If a scatterplot was created from the entire database would you expect the relationship to be strongly positive, strongly negative or that there was no relationship? Literacy rate 85% 92% 78% 95%

Ex 4

20 85

GDP per capita $5 000 $10 000 $3 000 $15 000

For each data example, is there likely a causal relationship between the variables? a

Number of times a coin lands on heads and the likelihood that it lands on heads on the next ﬂip

Amount of weight training a person does and their strength

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Hair length and gender

Marital status and ﬁlm preference

Height and number of children

Number of pets and time spent caring for pets (hours)

Time spent to ﬁnish novel and number of pages in the novel

Time spent training and athletic performance

Reasoning 17

A student collects some data that shows there is a positive increase between height and the ability to play soccer. Does that mean if you are tall, you are better at soccer? Justify your answer.

As preparation for a Science test a group of 10 students was given a revision sheet containing 60 questions. The table shows the number of questions from the revision sheet successfully completed by each student and the mark, out of 100 of that student on the test.

Which variable is independent and which variable is dependent?

Construct a scatterplot of the data.

Describe the correlation between the two variables suggested by the scatterplot. Explain your reasoning.

Explain why the relationship is not perfectly linear.

Number of questions 11 14 36 60 57 42 20 27 50 59

For the scatterplot:

Test result 20 23 62 97 100 66 35 52 87 99

Explain why the correlation between the variables is weak.

Determine whether the pairs of variables could be represented by the data set. i

arks in an English examination and distance M travelled from home to school

Cost of cars and cost of petrol

iii

istance travelled in a car and the cost of a D driver’s licence

Height and weight of students at school

A researcher is studying the relationship between the average temperature of a city and the number of people migrating to that city. She collected data from 10 different cities and plotted the data on a scatterplot. Describe a possible pattern that might be observed in the scatterplot if there is a relationship between the average temperature and migration.

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Problem-solving 21

A chemical company is testing the effect of different chemicals on slowing the the melting of snow. They are currently testing four different chemicals with the results showing how much ice is remaining after 3 hours at a certain temperature. A

B 100 Snow remainint (%)

Snow remainint (%)

100

x 0

Temperature outside (°C)

100

100 Snow remainint (%)

Snow remainint (%)

Temperature outside (°C)

x 0

Temperature outside (°C)

A snow sculpture company wants a chemical that will keep snow from melting.

Which chemical should they choose to ensure that at least half of the snow is still there after 3 hours?

If the company operates at a temperature of 3 °C to 4 °C, which chemical should they choose?

After a Maths exam, Uther commented that he left out Question 10 because it was not relevant to the topics that were assessed, but that the other questions were ﬁne. The teacher felt that Question 10 and Question 7 were assessing the same skills, but decided to look at the results of students across all 8 classes. She plotted the percentage who had answered Question 10 correctly against the percentage who had answered Question 7 correctly. a

% that got Q7 correct

There is no relationship between students’ performance in Q7 and Q10.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

% that got Q10 correct

Determine whether the statements describe the data. i

958

Students performed similarly in Q7 and Q10.

iii

Students who performed poorly in Q7 performed well in Q10.

Let’s assume that Uther performed exceptionally well in every other question. Which region should the teacher expect Uther’s data point to be in?

Determine whether each statement is true or false. Provide an example to support your claim. a

If event A has a causal relationship with event B, then there is a correlation between event A and event B.

If there is a correlation between event A and event B, then there is a causal relationship between event A and event B.

If event A is positively associated with event B, then there is a causal relationship between event A and event B.

Construct a scenario with data that suggests a weak positive relationship between two variables. Justify why the relationship may be weak.

17.05 Line of good fit After this lesson, you will be able to… • draw a line of good fit for a given data set. • interpret the gradient and y-intercept of a line of good fit. • use the equation of the line of good fit to make predictions about one variable based on the other. • evaluate the correlation between two variables using a line of good fit.

Line of good fit Exploration This scatterplot shows the car value in thousands of dollars and the number of years since the car was purchased for 20 different cars. 1.

Is there a correlation between the years since purchased and the value in thousands of dollars?

Which of the lines on the graph ﬁts the data best dashed or solid?

Car value Value (in thousands of dollars) y 30 25 20 15 10 5

Time since purchase (years) x 1

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A line of good fit is a straight line drawn by eye that best represents the data on a scatterplot. It is sometimes called a trend line or a line of best fit. We can use lines of good fit to help us make predictions or conclusions about the data. To draw a line of good fit, we want: • a line that passes through the centre of the points. Unless the correlation is perfect, it cannot pass through all of the points. • approximately the same number of points above and below the line • to minimise the vertical distances from the points to the line Given a set of data relating two variables x and y, it may be possible to form a linear model. This model can then be used to understand the relationship between the variables. A line of good fit has an equation of the form y = mx + c. The m value shows the gradient, identifying whether the correlation is positive or negative. • If the gradient is positive, when the independent variable increases by 1 unit the dependent variable increases by m units. The correlation in this case is positive. • If the gradient is negative, when the independent variable increases by 1 unit the dependent variable decreases by m units. The correlation in this case is negative.

Height (cm) 14 12 10 8 6 4 2

Weekly growth 1

2 3 4 5 6

8 9

The c value shows the vertical intercept (also known as the y-intercept): • When the independent variable is 0, the value of the dependent variable is c. When we interpret the vertical intercept at (0, c), we need to consider if it makes sense for the independent variable to be zero and the dependent variable to have a value of c.

Example 1 The scatterplot shows the data for two variables, x and y:

10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10

Draw a line of good ﬁt for the data.

Create a strategy Draw a line that follows the trend of the points, and has the same number of points above and below the line.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea 10 9 8 7 6 5 4 3 2 1

Here is an example of a line of good ﬁt that follows the trend of the data and has the same number of points above and below the line.

x 1 2 3 4 5 6 7 8 9 10

Example 2 The number of hours spent watching TV each evening, h, is measured against the percentage results, m, achieved in the Economics exam. The following linear regression model is ﬁtted to the data: m = −10h + 97 a The vertical intercept indicates that when a student watched no TV, the mark in the exam, on average, was ⬚%.

Create a strategy

The vertical intercept occurs when x = 0. In this case, the x-variable is the number of hours spent watching TV, h.

Apply the idea When a student watched no TV, h = 0. m = −10h + 97

Given equation

m = −10(0) + 97

Set h = 0

m = 97

Evaluate

So, the vertical intercept indicates that when a student watched no TV, the mark in the exam, on average, was 97%.

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b Does the interpretation in the previous part make sense in this context? A

es, it is reasonable that a student who spends no time watching TV could get an exam Y mark of 97%.

o, it does not make sense because the exam mark is not within the valid range for the N dependent variable.

Create a strategy In this case, the independent variable is “Hours Watching TV”. Is watching no TV possible in the context? Is earning a mark of 97 possible in this context?

Apply the idea It is possible to watch no TV and earn a mark of 97, so the answer is option A.

c State the gradient of the line.

Create a strategy When an equation is of the form y = mx + c, m is the gradient of the line, and c is the value of the y-intercept.

Apply the idea m is the number in front of the x-variable so the gradient is −10.

d If a student increases the amount of TV they watch by 3.5 hours, how much can we expect their Economics exam mark to drop by?

Create a strategy The gradient indicates how much their exam mark will decrease by for each additional hour they spend watching TV. The gradient of −10 indicates that their exam mark will decrease by 10% for each additional hour they spend watching TV.

Apply the idea Exam mark drop = 10% × 3.5 = 35%

962

Multiply the decrease percentage by hours Evaluate

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Idea summary To draw a line of good ﬁt, we want to minimise the vertical distances from the points to the line. This will roughly create a line that passes through the centre of the points. The number of points above and below the line should be equal. A line of good ﬁt has an equation of the form y = mx + c. The m value shows the gradient, and whether the correlation is positive or negative. • •

If the gradient is positive, when the independent variable increases by 1 unit the dependent variable increases by m units. The correlation in this case is positive. If the gradient is negative, when the independent variable increases by 1 unit the dependent variable decreases by m units. The correlation in this case is negative.

The c value shows the vertical intercept (also known as the y-intercept): •

When the independent variable is 0, the value of the dependent variable is c.

Practice questions Choose your question path Explorer 1–5, 6a-c, 7, 9ac, 10–14, 16–18

Adventurer 1–5, 6c-e, 8, 9bd, 10–14, 16–19

Trailblazer 1, 2, 4, 6d-f, 8, 9cd, 10–15, 17–19

Understanding 1

The equation for the line of good ﬁt is given by P = −4t + 116, where t is time. Would the correlation be increasing, decreasing, or neither?

Choose the line of good fit for this scatterplot. 9

8 7

Line 2

Line 3

6 5 4

Line 1

3 2 1 1

2 3 4 5 6

8 9

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Does there appear to be a linear relationship between the variables? a

5 x 5

Looking at the scatterplot:

Is the relationship between the x and y-variables positive or negative?

Which could be the equation for the line of good ﬁt?

y = −4x − 4

y = 44 + 4x

y = −4x + 44

y = 4x − 4

35 25 20 15 10 5

x 1

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2 3 4 5 6

8 9

Fluency Ex 1

This scatterplot shows the data for two variables, x and y:

19 y 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Draw a line of good ﬁt for the data.

1 2 3 4 5 6 7 8 9 10 11 1213 14 1516 17 1819

For each scatterplot: i

Draw a line of good fit by hand.

Write the equation of the line of good fit.

5 x 5

x 5

5 x 5

x 12

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2 x 12

This scatterplot graphs data for the number of people in a room and the room temperature. Draw a line of good ﬁt for this scatterplot.

38 36 34 32 30 28 26 24 22 20

Number of people

Room temperature 10 20 30 40 50 60 70 80 90

This scatterplot graphs data for the number of copies of a particular book sold at various prices. Draw a line of good ﬁt for this scatterplot.

190 180 170 160 150 140 130 120 110 100 90

Copies sold

Price 18 20 22 24 26 28 30 32 34 36

966

For each line of good fit: i

What is the gradient of the line?

Describe the correlation of the data set as positive or negative.

iii

Describe the change in y as x increases by 1 unit.

What is the value of the y-intercept?

y = 4.62x + 7.58

y = −7.68x + 5.49

y = −5.69x + 7.84

y = 3.16x − 2.72

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Ex 2

The amount of money households spend on dining out each week, D, is measured against their weekly income, I. This linear model D = 0.3I + 27 is fit to the data.

Explain the meaning of the vertical intercept.

Does the interpretation in the previous part make sense in this context?

State the gradient of the line.

If the weekly income of a family increases by $200, by how much can we expect their spending on dining out to increase?

This table shows the price of various used cars. Age ( years) Price (thousands of dollars)

1 16

2 13

0 21.99

Draw an approximate line of good ﬁt.

Find the y-intercept of the line of good ﬁt.

What does the y-intercept represent?

At what age would a car be worth $0?

5 10

7 8.6

4 12.5

3 11

4 11

8 4.5

2 14.5

Price (thousands of dollars) 24 20 16 12 8 4 Age (years) 2

Scientists record the number of aphids in areas with different numbers of ladybirds, and graph the data in the scatterplot shown.

A 3200

A = −3.82L + 3865. 21 represents the line of good fit. a

How much does the average aphid population change by with each extra ladybird? Round your answer to the nearest aphid. Find the average aphid population of a region with no ladybirds. Round your answer to the nearest aphid.

2400 1600 800 L 200 400 600 800 1000

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Reasoning 13

The heights (in centimetres) and the weights (in kilograms) of 8 primary school children is shown on the scatterplot:

Weight (kg) 60 50

State the y-value of the y-intercept.

Describe the meaning of the y-intercept in context.

Does the interpretation in the previous part make sense in this context? Explain your answer.

(0, 27.7)

20 10 Height (cm) 20 40 60 80 100 120 140

Concern over student use of the social media app LookAtMe leads to a study of student grades in Maths versus minutes spent using the app. The results are shown in the table: Minutes, M Grade, P %

292 26

153 63

354 13

253 37

11 97

42 89

195 51

7 98

162 59

254 36

Predict the grade of a student who spends no time on the LookAtMe app. Use a scatterplot and its line of good ﬁt to justify your response.

Use your model to explain and interpret the relationship between minutes spent on the LookAtMe app and Maths grades.

Lorena and Frasier each draw a line of good fit. a

Frasier said he noticed that the there was a point that was far away, so he moved his line closer to it. Does Frasier’s line meet the requirements for a line of good fit? Explain. Lorena said she noticed that the there was a point that was far away, but decided to ignore it. Does Lorena’s line meet the requirements for a line of good fit? Explain.

y 7

Frasier

6 5 4 3

Lorena

2 1

x 2 4 6 8 10 12 14 16 18

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Problem-solving 16

This scatterplot graphs data for the number of ice blocks sold at a shop on days with different temperatures. a

Sketch the line of good ﬁt for this data.

Use your line of good ﬁt to estimate the number of ice blocks that will be sold on a:

31 °C day

42 °C day

Does the number of ice blocks sold increase or decrease as the temperature increases?

Ice blocks 140 120 100 80 60 40 20 Temperature (°C) 14 16 18 20 22 24 26 28 30 32 34 36 38 40

The distance of several locations from the equator and their temperature on a particular day is measured and their values are presented on the scatterplot: a

Which equation could represent the line relating distance (x) and temperature ( y)? A

y = −0.005x + 49

y = −0.005x – 49

y = 0.005x + 49

y = 0.005x − 49

Temperature (°C)

20 Distance (1000 km)

Estimate the distance from the equator, x, if the temperature is 30.59 °C.

For the scatterplot shown: a

Find the equation of the line of good ﬁt.

Use the line of good ﬁt to approximate the value of y for x = 6.9.

11 10 9 8 7 6 5 4 3 2 1

x 1

2 3

4 5 6

8 9

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Beekeepers and scientists study a phenomenon known as colony collapse disorder (CCD), where the majority of worker bees in a hive disappear, leaving behind the queen and immature bees. The percentage of beehive losses that can be attributed to CCD each year, since 2005, is shown in a table. a b

Construct a scatterplot and find a line of good fit for this data.

Year 2005 2006 2007 2008 2009 2010

Y 0 1 2 3 4 5

Hives lost to CCD, (H)% 18 13 23 26 28 30

More data has been collected, and the results are listed: • In 2011, 30% of hives were lost to CCD. • In 2012, 33% of hives were lost to CCD. • In 2013, 36% of hives were lost to CCD. • In 2014, 40% of hives were lost to CCD. How would this new data affect the line of good fit?

17.06 Interpolation and extrapolation After this lesson, you will be able to… • understand and differentiate between interpolation and extrapolation. • use technology to find a regression line. • predict values using both the equation of a line and a graph of a line of good fit. • evaluate the reliability of predictions. • apply their understanding of interpolation and extrapolation to solve real-world problems.

Interpolation and extrapolation We can draw lines of good fit by hand, but by using technology we can find a regression line which is the best line that summarises the data set. The least-squares regression line is a line of fit that is calculated using technology.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 17.06 to answer these questions. 1.

What do you notice about the squares as your line gets further from the actual regression line?

What do you think is the relationship between the total area of the squares and the regression line?

Without knowing the meaning of the words interpolation and extrapolation, would you consider the prediction of the value of y when x = 13 interpolation or extrapolation? How about the value of y when x = 6?

Once we have the equation of a line of fit, we can use it to make predictions. We can: • Substitute the x-value into the equation of the line to calculate the predicted y-value. • Evaluate whether the prediction is reliable. Predictions can also be made using the graph of the line of good fit. We go up from the x-axis to the line of fit and then across to the y-axis to find the predicted value. S

Ice Cream Sales

400

300

200

100

T 10

20 Temperature (°C)

These terms describe the range in which we make predictions: • Interpolation: Prediction within the range of x-values in the data • Extrapolation: Prediction outside the range of x-values in the data

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Interpolations

Extrapolations

The reliability of predictions depends on the correlation strength, whether the data is interpolated or extrapolated, and the number of points in the data set. • Strong correlation + interpolation = reliable prediction • Moderate/weak correlation + interpolation = less reliable prediction • Extrapolation = generally unreliable prediction • More data points increase reliability

Example 1 The equation y = −8.71x + 6.79 is a line of good fit for a bivariate data set. When x = 3.49 predict the value of y.

Create a strategy Substitute the value of x into the equation for the line of good ﬁt.

Apply the idea y = −8.71x + 6.79

Write the equation

y = −8.71 × 3.49 + 6.79

Substitute x = 3.49

y = −23.6079

Evaluate

Example 2 Chirping crickets can be an excellent indication on how hot or cool it is outside. Different species of crickets have different chirping rates, but for a particular species the following data was recorded. Number of chirps per minute Temperature (°C)

77 14

115 17

150 21

176 24

32 Temperature °C 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

Chirps per minute

20 40 60 80 100 120 140 160 180200220

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

a According to the graph, what is the temperature when the crickets make 140 chirps each minute?

Create a strategy If we move from the frequency of chirps to the temperature, we find the y-value for a given x-value. Start with x = 140 and find the y-value.

Apply the idea We can draw a line from our x-value, 140, vertically to the line of the relationship, then another line horizontally left to the y-axis. What is the y-value we end up at?

32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

The answer is 20 °C.

Temperature °C

Chirps per minute 20 40 60 80 100 120 140 160 180 200 220

b According to the graph, how many chirps per minute will the crickets make if the temperature is 27 °C?

Create a strategy When y = 27, find the value of x.

Apply the idea We can draw a line from our y-value, 27, horizontally to the line of the relationship, then another line vertically down to the x-axis. What is the x-value we end up at?

The answer is 210 chirps per minute.

32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

Temperature °C

Chirps per minute 20 40 60 80 100 120 140 160 180200220

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c According to the graph, how many chirps are the crickets making each minute if the temperature is 19 °C?

Create a strategy When y = 19, find the value of x.

Apply the idea We can draw a line from our y-value, 19, horizontally to the line of the relationship, then another line vertically down to the x-axis. What is the x-value we end up at?

The answer is 130 chirps per minute.

32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

Temperature °C

Chirps per minute 20 40 60 80 100 120 140 160 180200220

d Which prediction is more reliable, part (b) or (c)?

Create a strategy Remember that for reliability of predictions: • Strong correlation + interpolation = reliable prediction • Moderate/weak correlation + interpolation = less reliable prediction • Extrapolation = generally unreliable prediction • More data points increase reliability

Apply the idea On the initial scatterplot, the minimum temperature is 14 °C and the maximum temperature is 24 °C. This means that part (b) uses extrapolation and part (c) uses interpolation. Part (c) is more reliable than part (b).

Reflect and check There are only four points, so none of the predictions are incredibly reliable. However, the correlation is strong, so that improves the reliability somewhat.

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Idea summary Predictions can be made either using a graph of the line of good ﬁt or substituting into the equation of the least-squares regression line. The reliability of predictions depends on correlation strength, interpolation or extrapolation, and the number of data points. Interpolation: Prediction within the range of x-values in the data Extrapolation: Prediction outside the range of x-values in the data Reliability of predictions: • • • •

Strong correlation + interpolation = reliable prediction Moderate/weak correlation + interpolation = less reliable prediction Extrapolation = generally unreliable prediction More data points increase reliability

Practice questions Choose your question path Explorer 1, 2, 3a, 4a, 5, 6, 8, 9, 11, 13, 15–17, 19–22

Adventurer 1, 2, 3b, 4b, 5, 6, 8, 9, 11, 13, 16, 17, 19, 21–23

Trailblazer 1, 2, 3c, 4c, 5–7, 10, 12, 14, 16–18, 20, 22–24

Understanding 1

Is each statement true or false? a

The line of good ﬁt can only be used for interpolation, not extrapolation.

Extrapolation using the line of good ﬁt is generally more reliable than interpolation.

Interpolation can lead to inaccurate predictions if the relationship changes beyond the range of known data points.

Scatterplots may not accurately represent the relationship between variables if there are outliers, non-linear relationships, or if the data is too sparse.

It is not important to consider data validity and limitations when using interpolation or extrapolation.

Considering data validity and limitations helps ensure that the predictions made are accurate and reliable, and that they do not lead to incorrect conclusions or decisions based on the data.

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Does each graph show interpolation or extrapolation? a

90 80 70 60 50 40 30 20 10

80 70 60 50 40 30 20 10

10 20 30 40 50 60 70 80 90

Find the y-value for the given line of good fit and x-value. a

y = 0.45x + 7.1, at x = 3

y = 1.75x - 3.2, at x = -2.4

y = -2x - 7, at x = 5

Fluency 4

Are these predictions an extrapolation or an interpolation? a

A prediction for the y-value when x = 5 is made from the data set: x y

976

7 2

8 4

11 7

12 6

13 4

17 8

18 8

19 11

20 8

A prediction for the y-value when x = 33 is made from the data set: x y

4 0

37 72

54 53

58 26

59 21

43 73

55 47

60 12

38 102

64 10

35 112

A prediction for the x-value when y = 95.69 is made from the data set: x

94.4

96.4

94.4

97.8

94.9

95.9

94.4

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

5 Ex 1

A line of good fit for this data set is y = 1.26x − 57.01: When x = 3.49, predict the value of y. Is this an example of interpolation or extrapolation? Round your answer to two decimal places. x y

93 51.2

57 25.4

86 38.9

97 58.6

78 38.2

96 60.8

68 26.3

69 28.5

54 5.4

92 92

Scientists conducted a study to see people’s reaction times after having different amounts of sleep. The results are recorded in the table: Number of hours of sleep (x) Reaction time in seconds ( y)

1.1 4.66

1.5 4.1

2.1 4.66

The data has been graphed along with a line of good fit found by eye. a b

Predict the reaction time for someone who has slept for 5 hours. Predict the number of hours someone sleeps if they have a reaction time of 4 seconds.

2.5 3.7

3.5 3.6

4 3.4

Reaction time 5 4 3 2 1 1

Number of hours of sleep 4 5 2 3

The number of ﬁsh in a river is measured over a ﬁve year period: Time in years (t) Number of fish (F )

0 1903

1 1994

The data has been graphed along with a line of good fit found by eye. a b c

2 1995

3 1602

2000 1800 Predict the number of years until there are no 1600 fish left in the river. 1400 Predict the number of fish remaining in the river 1200 after 7 years. 1000 According to the line of good fit, how many years 800 600 are there until there are 900 fish left in the river? 400 200

4 1695

5 1311

t 2 4 6 8 10 12 14 16 18 20

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Ex 2

The number of people at the beach can be affected by the temperature. After observing a particular beach, the data was recorded in a table. Temperature (°C) Number of people at the beach

15 80

18 120

Sketch a line of good fit by eye.

Approximately, how many people are at the beach when the temperature is 20 °C?

c d e

22 180

25 230 225

People

200

Approximately, what is the temperature when there are 200 people at the beach?

175 150 125

Estimate how many people are at the beach when the temperature is 19 °C?

100

Is the prediction for 30 °C or 19 °C more reliable?

75 50 25

Temperature °C 12 14 16 18 20 22 24 26 28

One litre of gas is raised to various temperatures and its pressure is measured is shown in a table. Temperature (K) Pressure (Pa)

300 2400

302 2416

304 2434

308 2462

310 2478

312 2496

314 2512

316 2526

318 2546

The data has been graphed along with a line of good ﬁt:

Use the line of good ﬁt to predict the pressure when the temperature is 306 K.

Is the prediction in part (a) an example of interpolation or extrapolation?

Is the prediction in part (a) reliable?

State whether it is reasonable to use the line of good ﬁt to predict pressure for the following ranges of temperatures: i

300 ≤ Temp ≤ 320

300 ≤ Temp ≤ 600

iii

0 ≤ Temp ≤ 320

280 ≤ Temp ≤ 340

Pressure (Pa)

Temperature (K) 296 300 304 308 312 316

An ice cream shop records the number of ice creams sold and the maximum temperature of each day for two weeks: Temp (°C) Sales

978

2580 2560 2540 2520 2500 2480 2460 2440 2420 2400

35 70

29 45

34 71

33 67

25 30

25 28

34 65

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

34 75

32 60

35 74

33 63

28 35

33 62

32 65

The data has been graphed along with a line of good ﬁt:

Predict the number of ice creams sold if the maximum temperature for a day is 38 °C.

Predict the maximum temperature if 10 ice creams were sold.

Is it reasonable to use the line of good ﬁt to predict ice cream sales within each of these range of temperatures? i 0 ≤ Temp ≤ 40 ii 20 ≤ Temp ≤ 40 iii 20 ≤ Temp ≤ 40 iv 0 ≤ Temp ≤ 60

100 90 80 70 60 50 40 30 20 10

Sales

Temp (°C) 22 24 26 28 30 32 34 36 38 40

A plane’s altitude (A) is measured at several times (t) during its descent. Time (t seconds) Altitude (A metres)

0 9000

200 7815

400 7092

The data has been graphed along with a line of good ﬁt:

A 9000

Predict the altitude of the plane 100 seconds into the descent.

Predict the altitude of the plane 500 seconds into the descent.

For how many seconds has the plane been descending when it is at an altitude of 7500 metres?

How many seconds did the plane take to descend to the ground?

1700 593

8000 7000 6000 5000 4000 3000 2000 1000 t 200 400 600 800 1000 1200 1400 1600

A car company looked at the relationship between how much it had spent on advertising and the amount of sales each month over several months. The data has been plotted on the scatter graph and a line of good ﬁt drawn. a

Two points on the line are (3200, 300) and (5600, 450). Find the gradient of the line of good ﬁt. The line of good ﬁt can be written in the form S = mA + c, where m is the gradient, c is the vertical intercept, S is the money spent on sales in thousands of dollars, and A is the advertising costs.

Find the value of c, the vertical intercept of the line. c

S 800 700 600 500 400 300 200 100 A 1000 2000 3000 4000 5000 6000 7000 8000

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The depth a diver, x, has descended below the surface of the water is plotted against her lung capacity, y.

y 105

Does the line of good ﬁt have a positive or negative gradient?

100

Find the gradient of the line.

Find the equation of the line of good ﬁt.

Use the line of good ﬁt to estimate the lung capacity, y, at a depth of 4 metres.

85 80 x

Reasoning 14

Several cars underwent a brake test and their age, x, in years was measured against their stopping distance, y, in metres. The scatterplot shows the results and a line of good ﬁt of y = 3x + 30 that approximates the positive correlation.

1 2 3 4 5 6 7 8 9 10 y 70 60 50

Use the line of good ﬁt to estimate the stopping distance of a car that is 7.5 years old.

Is the predicted value in part (c) reliable? Explain your answer.

The land size of a sample of properties and their selling price were graphed on the scatterplot: Another property that is not represented in the scatterplot is sold for $670 000 and measured 3400 square metres in size. Is this consistent with the relationship shown in the graph? Explain your answer.

Price (100 000s of dollars)

1 2 3 4 5 6 7 8 9 10 11 12

12 10 8 6 4 2

00 00 00 00 00 00 00 10 20 30 40 50 60 70 Size (square metres)

Are these predictions examples of interpolations or extrapolations? Justify your answer. a

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Using the provided table that shows a car’s fuel eﬃciency at various speeds, a prediction for the fuel eﬃciency at 70 km/h is made from the data set.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Speed (km/h) 40 50 60 80 90 100

Fuel efficiency (L/100 km) 6.5 5.8 5.2 4.9 5.4 6.1

Using the table of a company’s annual revenue over the past 5 years, a prediction for the next year is made from the data set.

Using this scatterplot that shows a student’s test scores at various hours of study, a prediction for the test score at 15 hours of study is made from the data set.

Year 2015 2016 2017 2018 2019

Revenue (in millions) 10 12 15 18 22

Test score

80 70 60 50 40 30 20 10

Hours of study 2 4 6 8 10 12 14 16 18

Discuss the limitations of using interpolation and extrapolation for making predictions in the context of climate change.

Problem-solving 18

The distance in kilometres, x, of several locations from the equator and their temperature, y in °C, is measured and graphed on a scatterplot. 50 45 40 35 30 25 20 15 10 5

x 1000

2000

3000

4000

5000

6000

7000

8000

9000

Find the equation of the line of good ﬁt shown.

If your distance from the equator is 4000 km, estimate the temperature, y, using the equation in the previous part.

Is the predicted value in part (b) reliable? Explain your answer.

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A line of good ﬁt has been drawn to approximate the relationship between sea temperature, T in °C, and the area of healthy coral, A in hectares, in a particular location. Two particular points, (2, 700) and (23, 175), lie on the line. a

Find the equation of the line of good ﬁt.

Using the line of good ﬁt, ﬁnd the area A of coral expected when the sea temperature is 26 °C.

Is the predicted value in part (b) reliable? Explain your answer.

1 3.9

2 3.7

500 400 300 200 100

T 5

3 3.5

4 3.1

5 3.1

6 2.9

7 2.7

Construct a scatterplot for this data.

Sketch a line of good ﬁt on your scatterplot.

Find the equation for your line of good ﬁt.

Use your equation of the line of good ﬁt to ﬁnd the number of kilograms of litter collected 8 weeks after the start of the anti-littering campaign.

Is the predicted value in part (d) reliable? Explain your answer.

The ﬂeet manager for the Australian Automotive C Association wants to estimate how car 550 maintenance costs, C (in hundreds of dollars), 500 are related to the distance, K (in thousands of kilometres), driven each year. 450 The data collected and the line of good fit are shown on a scatterplot. a

The equation of the least-squares line for the graph is C = 9.9K + 146.2

Interpret what the coeﬃcient of K represents in this context.

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700 600

The table shows data on the number of kilograms of litter collected each week in a national park x weeks after the park managers started an anti-littering campaign: Weeks (x) Kilograms of litter collected ( y)

A 800

Explain what the constant term in the equation represents in this context.

Is this interpretation of the constant term reasonable given the context? Explain your answer.

400 350 300 250 200 150 100 50 K

Use the least-squares equation C = 9.9K + 146.2 to predict the annual maintenance cost for a car that is driven 36 000 km per year.

Is this predicted value reliable? Explain your answer.

Is there any reason to believe that there is a causal relationship between distance driven and maintenance costs? Explain your answer.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The height of a particular projectile y in metres was measured at different times t in seconds and the curve of good ﬁt was drawn. a

y 100 80

Using the curve of good ﬁt, what is the predicted height of the projectile after 1 second?

Using the curve of good ﬁt, what is the predicted height of the projectile after 4 seconds?

The height at a given time has the following relationship y = −5t2 + 100. Which of these statements about interpolation and extrapolation are true?

60 40 20 t 1

xtrapolation is less reliable since we should use a E line of good ﬁt instead.

xtrapolation is less reliable since we make assumptions about the relationship E outside of the observed data.

Draw a scatterplot with a line of good fit which would not be suitable for interpolation. Explain.

Based on the given scatterplot and line of best fit, are either interpolation or extrapolation reliable? Explain.

8 7 6 5 4 3 2 1

x 1

2 3

4 5 6

8 9

17.07 Misrepresentations of data After this lesson, you will be able to… • understand what data misrepresentation is and its implications. • explore common strategies of data manipulation and identify them in real-world examples. • develop skills to critically evaluate data displays and graphs for potential misrepresentation.

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Data manipulation strategies Statistics are often used in the media to support facts, strengthen arguments, or provide insights. However, they can also be manipulated to inﬂuence opinions. It’s vital to interpret statistics with caution as they can be skewed intentionally or unintentionally. Common ways of data manipulation include: • Cherry-picking data: selectively using favourable results • Strategic choice of measures of centre and spread: measures of centre and measures of spread may give different impressions if outliers are present • Biased surveys: question phrasing and interviewer influence can affect responses • Sampling technique: poor methods can introduce bias; small or non-random samples are typical culprits • Over-generalisation: asserting conclusions for a larger population based on a nonrepresentative group • Overstating significance of findings: unclear sample sizes or error margins can mislead • Correlation and causation: mistaking observed relationships for causality • Comparing apples and oranges: meaninglessly comparing dissimilar data sets • Misinterpreting statistics or probability: common in media and legal contexts Misrepresentation may be accidental or deliberate, and critical evaluation of ﬁgures is essential in our information-rich society.

Example 1 A company that sells a weight-loss supplement and their website says that “this product led to an average weight loss of kilograms over a month”. This is based on the results of two studies of 10 people whose weight loss/gain is shown in this table. The results of the second study were excluded as they said there were too many outliers. Study 1 Study 2

+2 +2

−4 +1

−6 −1

−8 −3

−4 −2

−5 −5

−6 −2

−7 +2

−6 +3

−6 +1

Which option(s) explains why this is misleading? A Cherry-picking data

Comparing apples and oranges

C Biased survey question

Poor sampling technique

Create a strategy Decide which one most accurately describes the scenario: • Cherry-picking data: selectively using favourable results • Comparing apples and oranges: meaninglessly comparing dissimilar data sets • Biased surveys: question phrasing attempts to inﬂuence responses • Poor sampling technique: poor methods can introduce bias; small or non-random samples are typical culprits

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea

Reﬂect and check

The company only chose to use a study that showed favourable results. They failed to reference other studies that could provide a more complete and accurate picture of the supplement’s effectiveness.

Using the process of elimination: • The answer is not B because the study is relevant to the claim. • The answer is not C because they are not using a survey to collect data.

Both studies only had 10 participants which is too small to draw any conclusions from. The answer is A: Cherry-picking data and D: Poor sampling technique.

Idea summary Common ways that data may be manipulated during reporting: • • • •

Cherry-picking data Strategic choice of measures of centre and spread Biased surveys Sampling technique

• • • • •

Over-generalisation Overstating significance of findings Correlation and causation Comparing apples and oranges Misinterpreting statistics or probability

Misleading graphs Charts are often used in the media to show comparisons and describe trends.

Exploration

78 77 76 75 74 73 72 71 70 69 68

In favour of bill X

80 70 Percent in favour

Percent in favour

Graph 1 and Graph 2 show which political party is in favour of a particular bill.

60 50 40 30 20 10

Liberal

Labor

Other

Liberal

Labor

Political party

Graph 1

Graph 2

Other

Are they showing the same or different data? Are they conveying the same message? Explain.

What do you notice that is different about the graphs? Explain.

When would each graph be useful for a media source to show to viewers? Why or why not?

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Sometimes, graphs are also used to misrepresent information and promote a biased view. Some of the ways in which a chart can be misleading include: • Misleading axes or scale: • Biased labelling • Missing a scale on one of the axes • Using a scale that isn’t uniform • Having an inappropriate scale for the given data (making variations seem less important or more important than they are) • Using inappropriate scaling on chart elements like bars or sectors • Using a logarithmic scale without specifying • Incorrect display creation • Using unconventional methods or using a display incorrectly • Only displaying a select portion of the data • Using the wrong type of chart With any set of data, it is important to analyse the data within the correct context.

Example 2

Women in cabinet around the world

12 10

Tony Abbott’s new cabinet

8 6 4

Cuba

Uganda

Indonesia

New Zealand

Liberia

Zimbabwe

Afghanistan

Egypt

India

United States

China

Malaysia

Australia

Rwanda

2 Canada

Authorised by G. Wright, Australian labor 5/9 Sydney Ave, Barton ACT 2600

The Australian Labour Party released this graph after Tony Abbot was elected as Prime Minister:

Source:https://alp.org.au/our-history/labor-governments-and-women

Which of the following comments apply? A This graph is misleading because the scale on the vertical axis is not uniform. B This graph is misleading because it assumes that there are always an equal number of male and female cabinet members to choose from. C This graph is misleading because it assumes all cabinet sizes are the same. D The graph is misleading because there are no European countries included.

Create a strategy Check if each statement matches what the graph shows.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Apply the idea The vertical axis has a uniform value of 2, so option A is incorrect. There is no assumption on male cabinet members since their data is not shown, so option B is incorrect. There is no assumption on cabinet sizes since their data is not shown, so option C is incorrect. The title is “Women in cabinet around the world” and there are no mentioned European countries, so option D is correct.

Reflect and check Instead of having the total number, a proportion would be even more appropriate. If a country has 10 members in the parliment and one is female, then that is more representative then a country that has 60 members in the parliment and only three are female.

Example 3 Which feature of this graph is misleading? Risk of dying from COVID-19 versus normal annual risk

50% 10% 1% 0.5% 0.1% 0.05% 0.01% 0.005% 0.001%

Normal male death rate Normal female death rate Death rate with COVID-19

10 20 30 40 50 60 70 80 90 100 Age

Create a strategy Many of the issues relate to the scale and type of graph. Analyse the axes to see what might be misleading.

Apply the idea The y-axis does not use a linear scale. Instead, a logarithmic scale was used but it was not specified.

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Reflect and check Even if you specify that a logarithmic scale was used, like the image shown, it can still mislead someone who does not know how to interpret a logarithmic scale. It would be best to provide an additional explanation with the graph that explains how the data changes. Risk of dying from COVID-19 versus normal annual risk 10% 1%

Normal male death rate Normal female death rate Death rate with COVID-19

0.1% 0.01% 0

10 20 30 40 50 60 70 80 90 100 Age

*Log scale used to see differences in rates at younger ages

Idea summary Some of the ways in which a chart can be misleading include: • • • • • •

Biased labelling Missing a scale on one of the axes Using a scale that isn’t uniform Having an inappropriate scale for the given data (making variations seem less important or more important than they are) Using inappropriate scaling on chart elements like bars or sectors Using a logarithmic scale without specifying

Some of the ways in which a display can be misleading include: • • •

Using unconventional methods or using a display incorrectly Only displaying a select portion of the data Using the wrong type of chart

Practice questions Choose your question path Explorer 1–3, 5, 6, 9, 10, 11, 13–18, 20, 22, 23

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Adventurer 1–6, 8, 10, 11, 13, 14–17, 19, 20, 22, 23

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Trailblazer 2–5, 7, 8, 11, 12, 14–16, 19–21, 23, 24

Understanding 1

300 people were asked if they would support any of the three candidates for an election. a

What should the percentages in a proper pie chart add up to?

What do the percentages in this pie chart add up to? Supports Huckabee Is this an appropriate data display?

Presidential run 2012

Supports Palin 70% 63% 60% Supports Romney

A survey is being conducted to make a conclusion about mobile phone use in Australia. Which samples would likely be representative of the population? a

All the people who respond to a survey sent out on a popular social media platform.

All researchers at all universities in Australia.

Adding a question to the census.

Every tenth person going into a shopping centre in a randomly selected suburb or area for each town or city.

Determine whether each statement, that describes how this graph might misrepresent the statistics, is true or false. a

The wrong graph type was used.

The graph doesn’t start at 0.

The scale on the vertical axis is not uniform.

Line graphs are often helpful to show trends over time.

Line graph 16 8 4 2 1 0

The data in the table is based on a base 10 logarithmic scale. It shows the number of influenza cases at the beginning of a flu season. Convert the data to a normal scale. Round each answer to the nearest whole number. Date October 15 October 18 October 21 October 24 October 27 October 30

Cases on log10 scale 1.5 2.3 2.24 2.53 2.4 2.6

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Fluency Ex 1

State the type of data manipulation described in each example. a

There is a strong correlation between temperature and ice-cream consumption. Therefore rising temperatures cause higher ice-cream consumption.

75% of a high school class surveyed said they like sport. Therefore all high school students like sport.

Asking an air traﬃc controller about his stress levels compared to a four year old child.

A politician cites a single year’s crime statistics to argue that crime has dramatically decreased under their 4-year tenure.

A newspaper states: “Oil prices have risen by 33% over the last two years”. The article states that in 2018, the oil prices increased by 10% and in 2019, the oil price increased by 23%. Are these statements true or false?

The headline is true and accurate.

We need to know the prices to determine whether this statement is accurate.

An increase of 10% followed by an increase of 23% does not give an increase of 33% overall.

A graph is required for more complete analysis.

In NSW, data collected on assaults occurring in hospitals showed that from 2006 to 2011, the number of assaults increased, from 2011 to 2014, the number of assaults remained constant and in 2015, there was a drop in the number of assaults. Based on this, a newspaper reported “Assaults in hospitals have risen dramatically from 2006 to 2015”. Are these statements accurate?

The reporter has not correctly included the data since 2011 in their representation.

The reporter has represented the trend correctly.

The reporter has misinterpreted the data, as the number of assaults has decreased from 2006 to 2015.

The reporter has misrepresented the ﬁndings by not reporting a drop in assaults in 2015.

A new affordable housing initiative requires 10% of units in new buildings to be $250 000 or less, so the units available for purchase are: $250 000, $250 000, $250 000, $600 000, $605 000, $620 000, $630 000, $640 500, $645 000, $650 000, $660 000, $662 500, $668 000, $675 000, $680 000, $695 000, $700 000, $710 000, $715 000, $725 000, $740 000, $744 000, $750 000, $755 000, $759 000, $760 500, $765 000, $770 000, $772 000, $775 000 The marketer for that building says “Most units cost $250 000.”

990

Do the prices of the units meet the affordable housing initiative requirement?

Is the marketer’s statement accurate?

Which data manipulation strategy is the marketer using?

The marketer decides to change his statement to “Units starting at $250 000.” Is this new statement accurate?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

A survey sent to dentists and hygienists asked for their preference on brands of toothpaste. The participants were allowed to choose more than one brand if they wished. The results of the survey are shown in the table. Brand A 65%

Percentage

Brand B 75%

Brand C 72%

Brand D 79%

Brand B advertises the claim “Most dentists recommend Brand B.”

Which data manipulation strategy is the toothpaste coming using?

Which statement describes why the claim is misleading? A

The statement is inaccurate.

The survey was conducted by the company that owes toothpaste Brand B.

The statement implies that dentists recommend Brand B over the other brands.

The advertisement does not include the hygienists that were surveyed.

A statement on the box of an anti-wrinkle face cream reads: Noticeably firmer, younger feeling skin in just 4 weeks. Below the statement in small print, it says: *Reported by 25 women. Why is this claim misleading?

Ex 3

A restaurant is encouraging its staff to promote a new menu item, mushroom risotto. The servers are encouraged to ask each customer this question to improve sales: “Do you want a nutritious risotto for lunch or the standard sandwich?” a

Which manipulation strategy are the servers using?

Suggest a way for the servers to promote the new item menu that is less manipulative.

What feature is misleading about the line graph? 60 000 50 000

Turnover million

40 000 30 000 20 000 10 000 0 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Year

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The ﬁgure shows the number of tourists from each country to the United States:

CN CH

Brazil: 613 347

France: 930 265

Germany: 1 263 344

Japan: 2 169 716

United Kingdom: 2 905 909

Mexico: 4 295 124

M X

Canada: 14 043 658

Top 20 tourist-generating countries to United States of America

South Korea: 560 405 Italy: 558 594

India: 447 079

Colombia: 288 439

Australia: 526 441

Netherlands: 416 870

Argentina: 271 737

China: 488 484

Venezuela: 344 802

Switzerland: 262 595

Spain: 451 530

Ireland: 296 771

Israel: 237 818

What is the issue with the length of the line segments representing each country? Ex 2

Consider the graph:

Average house price ($)

Massive increase in house prices this year 82 000 81 000 80 000 79 000

2008

Year

2009

What two features of this graph might cause it to misrepresent the statistics? Ex 3

When evaluating new drugs, health officials assess the drug’s potential for addiction, among other effects. One aspect of this evaluation involves studying how the drug levels change in the blood over time. Drugs that quickly peak and then decrease rapidly in the blood could be at a higher risk for abuse. Note: while drugs that maintain relatively stable blood levels might have a reduced risk of causing peaks and valleys associated with abuse patterns, this alone does not ensure that the drug won’t be addictive. The line graph shows the amount of a drug that remains in the blood over time.

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Oxycodone concentration (ng/mL) log scale

Plasma Oxycodone by time 100 10 mg 20 mg 40 mg 80 mg single dose 10 mg q12h steady-state

0 1 2 3 4 5 6 7 8 9 10 11 12 Hours from dosing

The manufacturer claims this drug has a very slow decrease, so it won’t be addictive. This is not an accurate claim. Which aspects of the graph allows for this misleading claim? 16

The pie chart shows the number of nuclear weapons possessed by each country. Estimated nuclear stockpiles, 2005 Russia U.S. 7,200 10,315 deployed 8,800 inactive Russia has about 8,800 warheads in reserve or awaiting dismantling

In addition to known nuclear powers, North Korea says it has nuclear weapons. Iran could be next. Est. warheads China

410

India

75-110

France

350

Pakistan

150-110

Britain

200

N. Korea

7-15?

Israel

100-170

Source: Time Magazine a

Considering only the size of the slices (i.e. ignoring the numbers), which country has the largest nuclear stockpile?

Looking only at the numbers listed for each country, which country has the largest nuclear stockpile?

Has the pie chart been drawn accurately?

Reasoning 17

In the polls for an upcoming election, various polls are analysed to see who is likely to win the election. Most polls put Kodos ahead of Kang by between 10% and 12%. A few polls have Kang ahead of Kodos by a few percentage points. Kang tells his followers that they are sure to win the election based on the latest polls. Explain why this statement misrepresents the statistics.

17.07 Misrepresentations of data mathspace.co

993

Consider the line graphs of the same set of sales ﬁgures: 9000

Sales ($)

8000 7000 6000 5000 4000 3000 2000 1000

Month 1

Describe how the difference in scale between the two Sales ($) 9000 graphs might be misleading. 8000 7000 6000 5000 4000 3000 2000 1000 1

Consider the given column graph: a

Why is this a particularly poor example of a graph with broken axes?

Explain why it is not generally good practice to use a vertical scale with a broken axis for column graphs.

Describe a graphical display that would be better suited to this situation.

Month 5

Quarterly operating deficit 55 50 $ (million)

45 40 35 0

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Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Sept - 94 Dec - 94 Mar - 95

The same information is displayed in two different ways. 15

Average global temperatures Degrees (Celsius)

18 16 14 12

10 8 6 4 13 1920

2 Years

2020

1920

Years

2020

Which graph could be used to support the claim that the global temperature is increasing significantly?

Which graph could be used to support the claim that the global temperature is not increasing significantly?

At the United Nation’s climate summit, it was stated that limiting the planet’s temperature to a 1.5 °C overall increase by 2100 could avoid catastrophic events. How does this claim help you understand what is considered a “significant increase” in temperature?

Read the hypothetical article: Your mobile phone is ageing you - Research indicates that smartphones are causing premature skin ageing in over half of all users. ‘Smooth as Silk’, an anti-aging skin product company, does a survey which reveals that more than 50% of frequent smartphone users display early signs of ageing such as ‘tech-neck’ (wrinkles from device use). The study also identifies ‘screen face’ ( pigmentation from screen’s blue light) among significant users. Dr. Louise Fairweather’s survey involving more than 2000 participants suggests blue light exposure accelerates ageing and recommends reducing screen time or using blue light skin products for skin protection. a

Was the survey’s sample selection well done or not?

Based on the company doing the study, do these results surprise you?

Do you find the article’s headline fitting? Why or why not?

17.07 Misrepresentations of data mathspace.co

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Problem-solving 22

Compare the two graphs and explain which graph is misleading and why. Graph 1

Graph 2 100 Days of freezing temps

Days of freezing temps

85 84 83 82 81 2017

80 60 40 20

2018

2017

This graph was shown on an Argentinian news service with the comment that ‘Argentina’s testing rate for Covid-19 is almost as high as that of the United States’. COVID-19 testing by country

14 100 330

258

Brazil

Argentina

15 700

22 300

7000

USA

Italy

Germany

Number of COVID-19 tests per million people

Source: Reddit May 2020

996

2018

Explain why the graph and comment are misleading.

Redraw the graph correctly.

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Norway

Consider the shown line graph: The Main NAEP shows dramatic gains in math. From 1990 to 2000, fourth grade math scores increased fifteen points - equivalent to about 1.2 years of learning

Scale score

240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 1990

1992

1996

2000

Source: Brookings report on American education a

The labels on the horizontal axis are not evenly spaced. Explain why this may be misleading.

Is the scale on vertical axis reasonable or misleading? Justify your reasoning.

17.07 Misrepresentations of data mathspace.co

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Chapter 17 review 1

Emrik conducted a survey in his neighborhood about the type of car people preferred. The options given were: Truck, Sedan, or SUV. 36 responders said they preferred trucks, of which 20 were under 30 years old. Among 22 responders who preferred sedans, there were 14 people older than 30 years old, and 18 people were under 30 years old among 28 responders who preferred SUVs.

Construct a two-way table based on the results of Emrik’s survey.

What is the proportion of respondents that prefer trucks?

What is the proportion of people older than 30 that prefer sedans?

At a university with 32 000 students, a survey was conducted, asking students what they were majoring in and what activities they are currently engaged in. The results are shown in the table. Physical therapy Education Software engineering Graphic design Total

Sports team 90 58 60 10 218

A R

Dance club 55 60 47 35 197

Art club 27 45 38 47 157

Total 190 225 158 107 680

How many students are members of either the dance or art club?

What fraction of students studying software engineering chose an activity other than dancing?

What percentage of students surveyed are majoring in physical therapy? Round your answer to two decimal places.

Estimate the number of students in the entire university population who are majoring in physical therapy.

The results of a survey of 379 people on their transportation preference based on their living area is shown in the row percentage table: People living in Urban areas People living in Suburban areas

998

T F

Book club 18 62 13 15 108

Public Transport 25% 5%

Bike 20% 15%

Private Car 55% 80%

Total 100% 100%

Which variable is the explanatory variable, living area or transportation preference?

Does there appear to be an association between living area and transportation preference?

Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

The 100% stacked column graph show the results of Beverage preferences a survey on beverage preferences: 100% a Which is the explanatory variable, beverage 75% preference or preference on time of day? b

25% 0%

Morning Evening person person

210 people were surveyed about their education level and how many hours they read each week. The results are shown in the table:

Less than 2 hours 3 to 6 hours 7 to 12 hours Over 12 hours Total

Tea Coffee

50%

Does the 100% stacked column graph indicate an association between the variables?

High school or less 25 20 7 0 52

Some university 20 20 12 3 55

Bachelor’s degree 3 10 25 8 46

T F

A R

Graduate degree 2 12 31 12 57

Total 50 62 75 23 210

Which variable is the explanatory variable, education level or time spent reading?

To examine if there is an association between the variables, should we use a column or row percentage table?

Create this percentage frequency table. Round your answers to two decimal places.

Does there appear to be an association between the variables?

The local music store constructed divided bar graphs showing the last six months of instrument sales by category (there are ﬁve categories: S P W K B Jan String, Percussion, Wind, Keyboard, and Brass). Assume that S P W K B Feb the total number of sales was consistent for each month. a

In which month were bass instruments the most purchased type of instrument?

In which month were string instruments the highest selling type of instrument?

Which variable is the response variable?

Does the divided bar graph suggest that there is an association between the variables?

Mar

Apr

May

Jun

B B

Percussion Wind String Keyboard Brass

Does one variable have a direct effect on the other? If yes, identify the independent and dependent variables. a

Time spent studying a subject, Number of chapters covered

Shoe size, Number of siblings

Duration of sunlight exposure, Plant growth

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999

The table shows the average number of days of exercise and the average number of sick days of students over a six month period: Number of days of exercise 2 7 12 15 20 24

Month 1 Month 2 Month 3 Month 4 Month 5 Month 6

Create a scatterplot for the set of data.

A statement is made: “As the number of days a student exercises increases, the number of days they are sick decreases.” Is this claim correct? Explain your answer.

T F

Describe the strength and direction for each linear association. a

A R 20

5 5

Number of sick days 7 5 4 3 1 0

x 5

The table represents the monthly sales ﬁgures of a new software product in its ﬁrst year in the market. a

Construct a scatterplot for this data.

Describe the correlation between the month and the sales ﬁgures.

Does this data set contain any outliers?

Does the data show that the month of the year causes an increase in sales? Explain your answer.

1000 Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Sales (in thousands) 15 18 20 21 22 22 23 22 23 24 24 26

For each scatterplot: i

Draw a line of good ﬁt by hand.

iii

Describe the correlation of the data set as positive or negative.

Write the equation of the line of good ﬁt.

16 14

10 8

6 4

x 5

2 4 6 8 10 12 14 16 18

Describe the meaning of the y-intercept in context.

Does the interpretation in the previous part make sense in this context? Explain your answer.

After how many days will the sprout be 16 cm tall?

A R

T F

The height above the ground (in centimetres) of a radish sprout over time is shown on the scatterplot:

Height (cm)

20 15

10 5

Age (days) 5

20 25

A line of good ﬁt for this data set is y = 2.5x + 20.5 x y a

2 25.5

2.7 28.1

3 28.4

3.5 29.2

Predict the value of y when x = 5.3

4 30.0

4.2 30.6 b

5 32.45

6 35.5

6.7 38.0

Is this interpolation or extrapolation?

A line of good ﬁt has been drawn to approximate the relationship between the number of days Charlee cooks at home, C and the amount of money she spends at restaurants in a month, M in dollars. A line of good ﬁt, M = −28.6C + 970, approximates the correlation. a

Using the line of good ﬁt, ﬁnd the amount of money Charlee would spend if she only cooks at home 1 day of the month.

Is the predicted value in part (a) reliable? Explain your answer.

Predict the number of days Charlee would cook at home if she spends $250 at restaurants in a month.

Is the predicted value in part (c) reliable? Explain your answer.

900 800 700 600 500 400 300 200 100

M (in dollars)

C (in days) 4

12 16 20 24 28 Chapter 17 review 1001 mathspace.co

State the type of data manipulation described in each example: a

Chaysten’s car has broken down exactly 3 times, and all 3 times were during rainy days. He claims that the rain caused his car to break down.

68% of university students surveyed said they would not sign up for a drama class. The university removed the drama class from the next term’s schedule, stating the reason as “No student interest”.

A company received the ratings shown during their opening day of business. 5.0

4.5

4.8

4.0

0.0

4.5

5.0

4.4

4.7

5.0

4.4

The next day, they posted a sign in their window that said “Customer rating: 5 stars”. 16

400 people were surveyed on new models of smartphones, and which ones they preferred. Participants were allowed to choose more than one smartphone. a

The company marketing GammaDevice claims “Most people prefer GammaDevice”. Which data manipulation strategy are they using?

Why are the sizes of the sectors misleading?

Describe a graphical display that would be more appropriate for this data.

T F

Prefers BetaPhone

A R 70%

Prefers GammaDevice

(Optional) For each scatterplot:

70%

67%

Prefers AlphaModel

Describe the type of linear relationship between the variables.

Estimate the value of the correlation coeﬃcient.

y 21

y 20

18 15

12 10

9 6

3 1 2 3 4 5 6 7 8 9 10

x 3

(Optional) A linear association between two data sets is such that the coeﬃcient of determination, r2 is 0.78. a

Calculate the correlation coeﬃcient if the relationship is positive. Round your answer to four decimal places.

Describe the strength of the linear relationship between the data sets.

1002 Mathspace Victorian Curriculum – Year 10 Methods pathway mathspace.co

(Optional) Smartphone manufacturers are researching the optimal battery size to ensure consumer satisfaction. The results in the table alongside show the battery size of the smartphone (in mAh), and the proportion of consumers who reported satisfaction: a

Calculate and round your answers to two decimal places. i

Describe the strength of the relationship between the two variables.

Explain the meaning of the value of r2 in context.

Is this claim valid: “The customer satisfaction rating is primarily inﬂuenced by battery capacity”?

Battery size Satisfaction (mAh) proportion 2500 0.9 3000 0.8 1500 0.6 2800 0.7 2000 0.6 3500 0.9 2600 0.7 2100 0.65 3200 0.85 2400 0.7 2900 0.8 1700 0.55 2300 0.7 2700 0.8 3400 0.85

T F

A R

Chapter 17 review 1003 mathspace.co