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Year 10 Western Australia Curriculum

© 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.au For our full digital offering, visit mathspace.co Mathspace Pty Ltd Level 11, 477 Pitt St Sydney 2000 School Curriculum and Standards Authority. Mathematics Scope and Sequence of the mandated curriculum content: Pre-primary – Year 10 | Revised Curriculum for familiarisation in 2025. The School Curriculum and Standards Authority does not endorse this publication or product. Title: Mathspace WA Curriculum - Year 10: 2026 ISBN: 978-1-963022-80-3 Editors and lead authors: Erin Gallagher, Ronnel Jeppsen, Julia Reid, Sarah Butterworth, Anjelica Sieger, Jennifer George, Stephen George, Dot Yung Further reviewers: Melissa Archey, Warren Bardsley, Lianna Beeching, Kathryn Bennett, Benjamin Bui, Eric Chan, Cory Duker, Bahija Elmir, Wayne Gillie, Jeremy Hunter, Geoff Jamieson, Shahad Odeh, Natasha Pollard, Helen Soar, Patricia Toutzaris, Paul Unkles, Jie Yang Writing and development team: Adam Humphries, Neil Lopez, Ma. Lourdes Ona, Bernadette Barillo, Jelly Candido, Sheramie Castillo, Dhave Fernandez, Abegail Fontanilla, Regine Jara, Albert Jara, Shaira Llanita, Lesley Lopez, Kerby Mendoza, Mikaela Nicolas, Rolly Apdo, Valerie Baja, Marielle Belen, Michaela Belen, Christine Bibon, Allen Calabia, Adrian Doctolero, Dharent Fernandez, Janine Francisco, Rose Francisco, Rona Josue, Crystel Lontoc, Mae Lucid, Grace Manzano, Julie Manzano, Monisa Tacud, Paul Platero Images and Design team: Alex Duckmanton, Chastine Marquez, Keith Gimeno ii

Mathspace WA Curriculum – Year 10 mathspace.co

Contents 1

Indices and surds 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

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

4 13 20 27 33

Algebra 44 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09

Factorise algebraic expressions Expand binomial products Factorise binomial products Factorise monic quadratic trinomials Optional: Factorise non-monic quadratic trinomials Optional: Complete the square Optional: Add and subtract algebraic fractions Optional: Multiply and divide algebraic fractions Optional: Simplify algebraic fractions Chapter 2 review

46 51 57 62

Finance 70 3.01 3.02 3.03

Simple interest Compound interest Modelling investigation: Home battery storage system PAYG and tax Investigation: Superannuation (Optional) Chapter 3 review

72 81 96 98 112

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iii

Linear relationships 4.01 4.02 4.03 4.04 4.05

5.04 5.05

6.04 6.05 6.06 6.07 6.08

Exponential relationships Characteristics of exponential graphs Exponential models Investigation: Half-life Transformations of exponentials Optional: Exponential equations Optional: Logarithms Optional: Order of magnitude and logarithmic scales Optional: Connect exponentials and logarithms Chapter 6 review

Simultaneous equations 7.01 7.02 7.03 7.04 7.05

Graphs and features of quadratics Vertical translations Horizontal translations Investigation: Transformations of quadratics (Optional) Optional: Quadratic graphs Optional: Quadratic equations Chapter 5 review

Exponentials and logarithms 6.01 6.02 6.03

116 126 138

147

Quadratics 150 5.01 5.02 5.03

Linear graphs Linear models Linear inequalities Optional: Graph linear inequalities Investigation: Experiment with linear inequalities (Optional) Optional: Parallel and perpendicular lines Chapter 4 review

114

Solve graphically Optional: Solve using substitution Optional: Solve using elimination Optional: Simultaneous inequalities Optional: Applications of simultaneous equations Chapter 7 review

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152 162 169 175 198 204

210 212 222 236 251 275 280 288 297

302 304 312 320

328

Scale factor 8.01 8.02 8.03

332 339 344 351

Pythagoras’ theorem Trigonometric ratios Angles of elevation and depression Optional: Bearings Optional: Sine and cosine rules Chapter 9 review

356 365 379

389

Measurement 392 10.01 10.02 10.03

10.04 10.05 10.06 10.07 10.08

Scale and perimeter Scale and area Scale and volume Investigation: Volume versus surface area Chapter 8 review

Trigonometry 354 9.01 9.02 9.03 9.04 9.05

330

Surface area of prisms and cylinders Volume of prisms and cylinders Surface area and volume of composite solids Investigation: Rainfall collection Modelling investigation: Wind turbine design Estimation and accuracy Optional: Pyramids Optional: Spheres Optional: Cones Optional: Composite solids Investigation: Instruments and methods Chapter 10 review

Probability 11.01 11.02

11.03 11.04

394 404 414 426 428

437

440

eview Venn diagrams and two-way tables R 442 Conditional probability 455 Modelling investigation: Animal tagging for environmental monitoring 466 Mixed probability concepts 468 Investigation: Simulation Optional: Geometric probability Chapter 11 review 485

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Univariate statistics 12.01 12.02 12.03 12.04 12.05 12.06 12.07 12.08

Bivariate statistics 13.01 13.02 13.03 13.04 13.05 13.06 13.07 13.08

Two-way tables Associations between categorical variables Bivariate numerical data Associations between numerical variables Line of good fit Interpolation and extrapolation Misrepresentations of data Investigation: Misrepresentations in the media Correlation coefficient Investigation: Conduct a statistical investigation Chapter 13 review

Optional: Further quadratics 14.01 14.02

14.03 14.04 14.05

epresentations of data R Quartiles Boxplots Compare data sets Modelling investigation: Ferry schedules Samples and populations Optional: Cumulative frequency Optional: Standard deviation Optional: Comparisons using standard deviation Chapter 12 review

ptional: Solve by factorisation O Optional: Solve by completing the square Investigation: Use bisection to solve quadratics (Optional) Investigation: Derive the quadratic formula (Optional) Optional: The quadratic formula Optional: The discriminant Optional: Further quadratic equations Chapter 14 review

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488 490 508 520 534 547 549

559

564 566 578 598 608 622 633 647 660 674

Optional: Cubic functions 15.01 15.02 15.03

Optional: Circles 16.01 16.02 16.03

Investigation: Hyperbolas (Optional) Optional: Hyperbolas Optional: Transformations of hyperbolas Chapter 17 review

Optional: Geometry and proofs 18.01 18.02 18.03 18.04 18.05 18.06 18.07 18.08

Investigation: Circles (Optional) Optional: Circles at origin Optional: Translations of circles Investigation: Equation of any circle (Optional) Optional: Transform circle equations Chapter 16 review

Optional: Hyperbolas 17.01 17.02

Optional: Cubic function Optional: Factored form of cubic functions Optional: Cubic equations Chapter 15 review

Optional: Deductive reasoning Optional: Proofs with parallel lines Optional: Congruent triangles Optional: Similar triangles Optional: Further geometric proofs Optional: Circles and chords Optional: Angles in circles Optional: Tangents Chapter 18 review

Optional: Further trigonometry 19.01 19.02 19.03 19.04

Optional: Angles with exact values Optional: The unit circle Optional: Graphs of sine and cosine Optional: Trigonometric equations Chapter 19 review Answers

680 Contents mathspace.co

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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 • Extend and apply index laws with positive-integer indices and variable bases, to include negative-integer indices • (Optional) Establish the connection between fractional indices and surds. Perform the four operations with surds and rationalise the denominator if required • (Optional) Simplify algebraic products and quotients involving indices with integer and fractional indices

1 Indices and surds Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

4 13 20 27 33

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.

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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

Apply the idea

Use the division law.

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 To see how this law works, let’s work through an example of (52)3. (52)3 = (5 × 5)3

= (5 × 5) × (5 × 5) × (5 × 5)

= 56

This gets generalised to (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

Mathspace WA Curriculum – Year 10 mathspace.co

Zero power law We also have the zero power law which says that for any non-zero base a, a0 = 1 = a m – n.

We know using the division law that = an – n = a0.

If m = n, then Also

= 1.

This means that 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.

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

1.01 Index laws mathspace.co

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

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

Mathspace WA Curriculum – Year 10 mathspace.co

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

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, a ≠ 0 The power of a power law: (am)n = am×n The zero power law: a0 = 1, a ≠ 0

1.01 Practice questions Understanding

Explorer 3, 4

Adventurer 1–4

Trailblazer 1–4

When simplifying a division of terms with powers of 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.

1.01 Index laws mathspace.co

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: ( p3)2 = p3+2

i c

( p3)2 = p3×2

Complete the equation: ( p3)2 = p⬚

Complete each statement: a

x⬚ ÷ x3 = x7

(b8 × b2) ⬚ = b20

( y⬚ × y17)1 = y17

(3m2 × ⬚)4 = 81m24

5a-f, 7a-g, 8a-c, 9a-g, 10a-c, 11a-d, 12a-h, 13a-c

Fluency Ex 1

Ex 2

a 3 × a5

6y2 × y5 2

u ×u ×u

a 2 × a8

3y6 × 4y

2w × w × w

b × b4

a3 × −2a8 5

5z × 3z × z

6bc4 × 8b6a × 0

5g4h2 × 8g3h4

8m2n4 × 7m6n

y3 × y7

ama n

4y8 × 6y6 × 3y3

Simplify:

b a10 ÷ a5

c m5 ÷ m2

d j11 ÷ j4

h l

Simplify: a

Simplify:

5g-l, 6, 7e-l, 8c-f, 9g-l, 10d-h, 11d-h, 12g-t, 13e-h

Simplify:

i 6

5c-j, 7c-i, 8b-e, 9d-j, 10b-f, 11b-e, 12e-n, 13d-f

4y4 × (−2y6) × 4y5 2

2 3

3 4

(−4y7) × (−6y2) × (−4y2)

2s t × (−5r s ) × 3r t

9y9 × 8 (−y)8 × 7y7

(−20b4) ÷ 2b4

(−9y9) ÷ (−3y5)

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9r8 ÷ (4r2)

Ex 3

Simplify: a

(j4)6

(w2)11

(5h7)2

(xy2)5

(−x9)4

(−5x4)3

Ex 4

Ex 5

(3a2)4

(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

Simplify: c

(x7 y2)4 40

(11x ) − (12x )

p9 ÷ p5 ÷ p2

(3a2 b5 c)4

(15x3)0 + 150 − 15y0

(4u5 v2)2

m8 ÷ m2 × m5

(4h0)2 + 18 ÷ (3g0)

m ((x ) )

(10x2 × 10x5)0 − 10x0

p5 × ( p4)3

265

(4y4)3

(3y5)2 × (5y2)3

(c10)11 ÷ (c8)3

(−240u32) ÷ (−8u9) ÷ (−5u12)

o q

(−2x )

s 13

Simplify:

Ex 6

( y4)3

Simplify:

1.01 Index laws mathspace.co

Reasoning 14

15, 16ace

When simplifying the expression

15, 16

14, 16

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

Is Clarice’s work correct? Explain. 15

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. 16

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 17–19

18, 19, 21

18–20, 22

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

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

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

Rewrite 16p with a prime base.

Simplify: a

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−xn = (−x)n for all x always

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

By simplifying

we see this is the same as

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.

1.02 Negative indices mathspace.co

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

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

1. How could we rewrite

using positive indices?

2. Are there any values that a or b cannot have?

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

1.02 Negative indices mathspace.co

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

Idea summary Negative index law with fractions:

where a, b ≠ 0.

1.02 Practice questions Understanding 1

Explorer 1–3

1–3

Trailblazer 1–3

True or false: a

Is 4−2 less than or greater than 1?

Write

Adventurer

using a negative index?

Mathspace WA Curriculum – Year 10 mathspace.co

a−4 = −a4

4b, 5a-h, 6a-h, 7ac, 8ab, 9a-c, 11a-c, 12a-f, 13ab

Fluency

4a, 5c-k, 6g-n, 7bd, 8bc, 9d-g, 10a-d, 11c-e, 12f-k, 13b-d

4c, 5d-l, 6i-p, 7cd, 8cd, 9gh, 10e-h, 11d-h, 12k-p, 13d-f

Complete each table: a

2−4

2−3

10−4

10−3

2−2

2−1

10−2

10−1

100

101

102

103

104

100 c

3−4

3−3

3−2

3−1

27 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

p−2 q3

m−5 n−4 p4

Simplify: a

Rewrite as terms with negative indices: a

Ex 2

7x−9

Simplify: a e

5y9 × 4y−3 2h

−4

× 4h

b f

7a4 × 4a−2 3y

−2

−5

× 2y

c g

5x4 × (−3x−8) 2

−5

−4y × (−4y )

3y−2 × 4y−3

(5mp)2 × mp−2

1.02 Negative indices mathspace.co

Evaluate: a

Ex 3

Ex 4

−1

2 × 3−2

10 ÷ 2 + 3

20 × 2−2 + 6

3−1 + 4−1

3 × 4−2 + 5 × 2−4

102 × 5−2

(7 + 32) × 2−2

Simplify: a

2y6 × 4y7 × 4y−5

6y7 × 2y−5 × 5y3

2x−2 × 5x5 × 4x−5

12x−9 × 2x4 × 3x−2

4y × 3y ÷ 2y

8y10 ÷ 2y−4 ÷ y3

40x−2 ÷ 5x8 ÷ 4x−9

14x−8 ÷ 2x × 3x6

(4m−6)4

−1

Simplify: (2m)−3

(3p−4)−2

(3y2)−2

Simplify: a

(10x3 y−4)2

(3x−4 × 2x2)2

(4x3)−1 × ( y−2)−3

(10x3)3 × (2x4 y3)−2

Reasoning 14

2−2 × 24

14, 16

14, 16, 18

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

15–18

as a single term.

and Donovan writes down x−10.

Who is correct and why? 15

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.

Mathspace WA Curriculum – Year 10 mathspace.co

. 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 21 19

Write

Complete:

Solve for n:

19, 21, 23

19–24

at least three different ways.

(x3 y−5)n = x−12y20

(a−5 b3)n = a15 b−9

Why is 2−2 < 2 but

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

? Give another example where this pattern applies.

, where C is the capacitance measured in farads, Q is the charge on the capacitor measured in coulombs 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 coulombs. A capacitor has a charge of 20 microcoulombs, 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?

1.02 Negative indices mathspace.co

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 Exploration x

1. Fill in the table of values. 2. Plot the points on a Cartesian plane using happens for values of x less than 0?

as the y-value. What do you think

for the following values of x: 0, 0.01, 0.0001

3. Use a calculator to evaluate

4. What does the calculator display for

5. Is it possible to take the square root of a negative number? Why? Why not?

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. Real Numbers Rational numbers Integers 4

Whole numbers 1 0 1 2 3 4

Irrational numbers

Mathspace WA Curriculum – Year 10 mathspace.co

A particular type of irrational number is a surd. A surd is a root which cannot be simpliﬁed to a rational number. A surd is an irrational expression of the form , where x is a rational number and n is an integer such that n ≥ 2, and x > 0 when n is even. root symbol

index

radicand

surd as the nth root of x.

We would read Radicand

The radicand is the number or expression inside the radical symbol. It is the value from which we are extracting the root. Example: In

, the radicands are 16 and 27, respectively.

Radical The radical symbol (root symbol) is used to denote roots. The entire expression, including the radical symbol and the radicand, is often referred to as the radical. Example:

and

are radicals.

Surd A surd is an irrational number that can be expressed in root form but cannot be simplified to remove the root. and are surds because they cannot be simplified to a rational number. A surd is an Example: exact value that can be approximated by a rounded decimal. We find the approximations using a calculator.

A number can have an exact value, or it could have an approximation. An approximation is a value that is close to, but not exactly the same as the exact value. Approximations are typically found by rounding the decimal form of a number. For instance, if the exact value is , one possible approximation would be 0.667. When dealing with surds, we could have exact values like = 2.924, rounded to 3 decimal places.

= 5, or approximate values like

1.03 Optional: Simplify surds mathspace.co

Example 1 Is

rational or irrational?

Create a strategy Put the number in your calculator.

Apply the idea

Reﬂect and check

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

For this to be rational, the radicand would need to be a cube number. The ﬁrst four perfect cubes are: • 13 = 1 • 23 = 8 • 33 = 27 • 43 = 64

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

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 an irrational expression of the form , where x is a rational number and n is an integer such that n ≥ 2, and x > 0 when n is even. A surd is an exact value that can be approximated by a rounded decimal. We find the approximations using a calculator.

Simplify surds We simplify surds to make them easier to understand, compare, and work with. It also helps to have a common way to express 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:

Mathspace WA Curriculum – Year 10 mathspace.co

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 squares.

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

30 10

3 2

1. How can you identify if a number has a perfect square factor after drawing a factor tree? 2. Is it possible to get two different factor trees for the same number? 3. Is it possible to get two different prime factorisations? 4. Can every surd be simpliﬁed? 5. How would you rewrite 180 as a product of a perfect square and a remainder? 6. 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,

Example 2 Simplify

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

1.03 Optional: Simplify surds mathspace.co

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

into two factors.

Apply the idea

Write the surd as a product of its factors Rewrite 8 as 23

Simplify

Evaluate the product

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:

Mathspace WA Curriculum – Year 10 mathspace.co

1.03 Practice questions Understanding 1

1–3

Fill in the boxes:

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. 4, 6, 7a-h, 8bef, 9ac

4, 5abd, 6cd, 7e-l, 8acfg, 9bc

4, 5bc, 6d, 7i-p, 8dgh, 9cd

Is each number rational, irrational, or neither? a

i m

∞

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

Simplify: a

Ex 2

1, 2

a 6

Trailblazer

Fluency Ex 1

1, 2

Adventurer

Are these surds? a

Explorer

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

1.03 Optional: Simplify surds mathspace.co

Ex 3

Simplify: a

Reasoning 10

10, 12

11–13

A calculator states that

11, 13, 14

an exact value? 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. is negative

is rational

Problem-solving 16, 17

is positive

15, 16, 18a-d

d 15–17, 18e-h

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 make a the subject.

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

Write the expression

Simplify

Simplify:

as a single surd.

Mathspace WA Curriculum – Year 10 mathspace.co

is positive

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 with the same index and radicand is a like surd. For example,

and

are like surds.

Exploration Sometimes terms might not appear to be 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 simplify to , just as 2p + 3p = 5p.

can be rewritten as

. This would

1. Can all surds be simplified? 2. What are two surds that don’t appear to be like, but can be simplified to be like? 3. 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?

1.04 Optional: Add and subtract surds mathspace.co

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.

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

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

. For example: Evaluate the roots

Evaluate the addition

A common mistake that is made when adding or subtracting surds is to combine them in the radicand. Incorrectly combining like terms

Evaluate the addition

Evaluate the root

See how we arrived at a different and incorrect answer. 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.

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.

1.04 Optional: Add and subtract surds mathspace.co

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 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.

Example 5 Simplify

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

Apply the idea Write the surds as products of their factors

Simplify the cube root using 83 = 512

Subtract the like surds

Mathspace WA Curriculum – Year 10 mathspace.co

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.

1.04 Practice questions Understanding 1

b 2

Ex 3

Trailblazer 1–3

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

Are these equations true or false? a

Fluency Ex 2

Adventurer 1–3

Fill in the blanks: a

Ex 1

Explorer 1–3

4a-eh, 5a-c, 6a-e, 7a-d

4d-i, 5b-e, 6d-h, 7d-i

4g-l, 5c-f, 6f-l, 7df-j

Simplify: a

1.04 Optional: Add and subtract surds mathspace.co

Ex 4

Ex 5

Simplify: a

Simplify:

Reasoning 8

8, 9

8–10

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:

Explain why we simplify surds before adding or subtracting.

Problem-solving 12, 13 12

9–11

a 11

13, 14

13–15

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

125

Mathspace WA Curriculum – Year 10 mathspace.co

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 each surd to the nearest whole number.

4− 7

4+ 7

. Katie estimates the perimeter by rounding

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

1.05 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 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

1.05 Optional: Multiply and divide surds mathspace.co

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.

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

Mathspace WA Curriculum – Year 10 mathspace.co

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 known 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.

Example 4 Simplify

Create a strategy Use the rule:

Apply the idea

Apply the rule Perform the division

1.05 Optional: Multiply and divide surds mathspace.co

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

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

Mathspace WA Curriculum – Year 10 mathspace.co

1.05 Practice questions Understanding 1

Ex 1

Ex 3

Trailblazer 1e-j, 2

Complete these statements using this example:

d 3a-df, 4ab, 5a-d, 6a-d, 7ab, 9a-e, 10ab

3c-g, 4bc, 5c-f, 6d-h, 7a-d, 9d-h, 10b-f, 11a-d

3d-h, 4cd, 5d-h, 6dh-l, 7d-h, 8, 9dh-l, 10e-h, 11

Simplify: a

Ex 2

Adventurer 1c-h, 2

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

Fluency 3

Explorer 1a-f, 2

Simplify:

Expand:

1.05 Optional: Multiply and divide surds mathspace.co

Simplify: a

Expand: a

Ex 4

Ex 5

Simplify:

Reasoning 12

13, 15

12, 14, 16, 17

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.

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

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

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.

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 18, 20, 22, 24a 18

19, 20, 22–24

19, 21, 23–25

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

125 cm

10 m

2 20 cm

Find the area of the trapezium in simpliﬁed surd form:

8 1.05 Optional: Multiply and divide surds mathspace.co

The surface area of a human body 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 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.

Round 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 25

10x

x2 + 5

Vaughan needs 40 lengths of rope, each

50x − 2x2

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 WA Curriculum – Year 10 mathspace.co

1 Chapter review 1

Simplify: a

x3 × 6x2

a4 × −3a5

−2y7 × 2y2 × −2y6

6m3 n7 × 9m6 n

(−2x3)4

Simplify: a

Find the value of a and b in the equation:

Simplify:

a 5

4y10 × 5y−4

3y−2 × 4y−3

(5mp)2 × mp−2

(6x−2 × 2x3)4

Given a = m3n2, b = m−1n4, and c = m5n−3, find: a×b

a÷c

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

(Optional) Simplify fully: a

5x4 × (−3x−8)

a 11

Fill in the blanks:

a 10

(4a)−3

a 9

Simplify, giving your answers with positive exponents: a

Fill in each box. Evaluate if possible. a

(Optional) Simplify fully, given the variable represents a positive value: Chapter 1 review mathspace.co

(Optional) Simplify fully: a

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

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

(Optional) Simplify fully: a

Did you know?

Square roots play an important role in genetics! For example, geneticists use them when calculating allele frequencies or building probability trees to model how traits are passed from one generation to the next. These values often involve surds, which help simplify complex expressions for easier analysis and comparison. Even if the term “surd” isn’t commonly used in reports, the skill of simplifying square roots supports accurate calculations and helps researchers interpret genetic data more efficiently!

Mathspace WA Curriculum – Year 10 mathspace.co

“The power of mathematics is often to make difficult things simpler.” Edward Frenkel

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 • Extend and apply knowledge of the expansion of binomial products to explore the factorisation of monic quadratics • (Optional) Factorise monic and non-monic quadratic expressions using techniques, such as completing the square, perfect squares, difference of squares and grouping in pairs for four-term expressions • (Optional) Explore efficient strategies to simplify expressions that involve addition, subtraction, multiplication or division of algebraic fractions with an algebraic expression in the numerator and/or denominator, including the use of factorisation

2 Algebra Chapter outline 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09

46 51 57 62

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

2.01 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

Mathspace WA Curriculum – Year 10 mathspace.co

Select the common factors Simplify

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

Apply the idea We can factor out the highest common factor of 4. 8 + 16x − 4x2 = 4 × 2 + 4 × 4x − 4 × x2 2

= 4(2 + 4x − x )

Write the terms as products of 4 Factorise

2.01 Factorise algebraic expressions mathspace.co

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.

Mathspace WA Curriculum – Year 10 mathspace.co

2.01 Practice questions Explorer

Understanding Ex 1

15n2m and 8nm2 16abnx and 12abmy

6abmn and 8bnxy

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: 2x3 + 4x2 + 2x = ⬚(x2 + 2x + 1)

ax2 − 3ax + 6a = ⬚(x2 − 3x + 6)

4kx2 + 12kx + 16k = 4k(⬚ + ⬚ + 4)

4a-f, 5a-d, 6, 7

4c-h, 5c-h, 6, 7

4e-j, 5g-l, 6, 7

Factorise: b

2x3 − 16x2 + 2x

2ax + 2ax + 4ax

5xy2 + 10xy + 15x

4b3 − 16b2 + 20b

6qr2 − 36qr − 12q

3ax2 + 6ax − 6a 3

3xyz + 21xyz + 9xz

−4abc2 + 8abc + 4ac

−6efgh2 − 30efgh − 12ehg

10rstu2 + 70rstu − 40rtu

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

Factorise: a

Ex 4

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

Ex 3

20x and 15x2n

16a2bx3y6 and 12b4m2n6x

Fluency 4

Ex 2

1–3

Trailblazer

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

1–3

Adventurer

2y2(x + 2) − 3(x + 2) 2

4m (n − 3) + 6(n − 3)

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

2p3q(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 an expression for the dimension of the width of the field.

Factorise: a

20j − 15jk − 55j2k

3p2q2 + 4p4q4 + 5p6q6

2yz − 10xy + 12xy2z

pqr + p2q2r + p3q3r

−hf − hj + hg

25m − 15mn − 10m2n

2.01 Factorise algebraic expressions mathspace.co

Reasoning

8, 9, 11

9–11

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, 14, 15 12

13–15

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

Factorise this expression by taking out a common factor.

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 each, place a digit in each box to create a true statement.

Mathspace WA Curriculum – Year 10 mathspace.co

⬚x + ⬚⬚ = ⬚(⬚x + ⬚⬚)

2.02 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. • expand perfect squares. • understand and apply the difference of two squares identity. • solve problems requiring the expansion of binomial products.

Expansion of binomial products A binomial is a type of algebraic expression that consists of two terms. A binomial product refers to the result obtained when multiplying two binomials. A C

+ D

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 × x = x2

5 × x = 5x

(x + 3) (x + 5) = x2 + 3x + 5x + 15 We can simplify this to: x2 + 8x + 15

+ 3

3 × x = 3x

5 × 3 = 15

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

Apply the idea Use the rule (A + B) (C + D) = AC + AD + BC + BD. (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

2.02 Expand binomial products mathspace.co

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

Expand 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? 2. What do you notice about the constant of the product? 3. 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

+ 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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

Apply the idea Use the perfect square expansion: (A + B)2 = A2 + 2AB + B2 (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? 2. What do you notice about the constant of the product? 3. 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 As 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.

2.02 Expand binomial products mathspace.co

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 2

= m – 9

Apply the identity Evaluate the square

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

2.02 Practice questions Understanding 1

(4y) (5y)

(−3z) (7z)

(x + 2) (x + 1) = x2 + ⬚ + 2

(x − 3) (x + 2) = x2 − ⬚x − 6

(x + 2) (x − 5) = x2 − 3x − ⬚

(x + 3)2 = x2 + ⬚ + 9

(5p) (−2p)

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

(x + ⬚)2 = x2 + ⬚x + 36

(3x − 8)2 = 9x2 − ⬚ + 64

(x + 5) (x − 5) = x2 − ⬚

(x + 2) (x − 2) = ⬚2 − 4

Fill in the missing boxes. a

(2x) (3x)

Fill in the missing boxes. a

Trailblazer 2–4

Fill in the missing boxes. a

Adventurer 2–4

Multiply the terms. a

Explorer 1–3

(x − ⬚) (x + ⬚) = x2⬚64

Mathspace WA Curriculum – Year 10 mathspace.co

(x + 14) (x − ⬚) = x2 − ⬚

5a-d, 6a-ce, 8a-d, 9a-d, 11ab, 12a

Fluency Ex 1

(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)

Expand and simplify: a

(2x + 3) (3x − 4)

(3x − 2) (4x + 5)

(4x + 1) (x − 2)

(3x − 4) (4x + 5)

2(x + 3) (x−2)

3(2x−5) (x + 4)

5(2x + 1) (x − 2)

−3(3x + 2) (2x − 1)

Expand and simplify: b

(x + 7)2

(7 − z)

(e + 5)2

( y − 5)2

(−6 − a)

(a + b)

(a − b)2

(3x + 7)2

(2y − 5)2

(2y − 3)2

(x2 + 1)2

Expand and simplify: 5(m − 5)2

6(n + 6)2

8(q + 3)2

3x(7x − 5y)2

Expand and simplify: a

(x − 7) (x + 7)

(8 − m) (m + 8)

(7 − 6p) (7 + 6p)

(−9y − 1) (−9y + 1)

(9x − 2y) (9x + 2y)

(0.2x + 3) (0.2x − 3)

Expand and simplify: a

3x(5x − 6y) (5x + 6y) 2

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)

Reasoning 14

(c + 6)2

Expand and simplify:

a 11

Ex 3

Expand and simplify:

5f-h, 6df-h, 7, 8c-h, 9d-h, 10, 11c-f, 12, 13

Expand and simplify:

a Ex 2

5d-f, 6d-f, 7, 8c-g, 9b-f, 10, 11b-d, 12ab, 13ab

14, 17

16, 17

15–18

Is (3x + 4)2 equal to 9x2 + 16? Explain your answer. 2.02 Expand binomial products mathspace.co

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 a

Explain how to expand and simplify the expression.

Is the result a trinomial?

Problem-solving 19, 20 19

19–22

21–24

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 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

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.

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

b x

5x − 7

x 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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

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

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.

y = −10

y = 20

y=0

iii

y=2

y y+8

12 − y

y = 12

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

2.03 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 have 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, then factorise using the rule: A (B + C) + D (B + C) = (A + D) (B + C)

2.03 Factorise binomial products mathspace.co

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

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 use a model to visualise it. (A + B)2 = A2 + 2AB + B2 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 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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

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 We know that the difference of two squares looks like this: 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)

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)

2.03 Factorise binomial products mathspace.co

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

2.03 Practice questions Understanding 1

1–5

2a + 18b

10xy + 25x2

32(c + 5) − x(c + 5)

6d4 + 8d2

⬚ = 42

Fill in the box to make the equation true: 81 = ⬚2

100 = ⬚2

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?

Are these expressions a difference of two squares? a2 – 16

4w2 – 25a2

6b-e, 7c-g, 8a-f, 9c-g, 10a-d

25 = ⬚2

100 + 25y2

6e-h, 7e-j, 8f-j, 9f-l, 10

Factorise: a

m(x + y) − 6(x + y)

x( y − z) − w( y − z)

5y(4w + 3x) − z(4w + 3x)

8y( y − 4) + 3(4 − y)

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

2y(2x2 + 3z) − (2x2 + 3z)

(2c − d) (c + 5d) − 3(d − 2c)

3x2(x + 4y) − 5y(x + 4y)

x2 − 3x + 8x − 24

Factorise: a

b2 – 9b – 81

6a-d, 7a-d, 8a-d, 9a-d

Fluency 6

Trailblazer 1–5

Ex 2

Adventurer

Identify the highest common factor:

Ex 1

Explorer 1–3, 5

x2 + 2x + 5x + 10 3

a + 8a + a + 8

8x + xz − 16y − 2yz

24 + 3y + 8x + xy

7xy + wx + 7yz + wz

2mp + 6 + 3p + 4m

4x + 24yz + 32xy + 3z

2x + 18yz + 12xy + 3z

3df 2 − 11ydf + 3rzf − 11yrz

Mathspace WA Curriculum – Year 10 mathspace.co

Ex 3

Factorise: b

64 + 16e + e2

49 − 14p + p

u2 − 2uq + q2

6c2 − 24c + 24

5q2 + 10qt + 5t2

16w2 − 40w + 25

64r2 + 48r + 9

b2 − 2br + r2

v2 − 1

121 − v2

16 − 9y2

x2y2 − 49

7x2 − 63

45t2 − 20

x2 + 6x + 9

i Ex 4

Factorise: a

x2 − y2

e i 10

n2 − 25

f 25m2 − 49

81x2 − 16y2

Factorise: a

3x2y − 6y4 + 9x2y3

16x2y2 + 24xyz + 9z2

121c2 − 49d2

3x2y + x2z + 3y2 + yz

8a b + 12bc

(b + 4)2 − (3 − b)2

14h3p2r + 21h5p2 + 49hp4

16w2 − 64wt2 + 6wt − 24t3

16q4 + 24q2r + 9r2

48df 2 − 3dh2

2 4

Reasoning

11, 12

12–14

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

Identify and correct the error in each statement. a

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

x2 − 4x + 4 = (x + 2) (x + 2)

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 2

n3 − 121n

2x − 6x + 5x − 15

9x2 − 12xy + 4y2

(x + 16)2 − y2

4x3 + 16x2 − x − 4

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

Problem-solving 15ac, 16 15

15bd, 16–18

15d, 16–20

Using the difference of two squares, evaluate: a

652 − 632

992 − 972

452 − 422

582 − 522

2.03 Factorise binomial products mathspace.co

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

Find the side length of the square.

Find the perimeter of the square.

Using four of the digits 1 to 9 with no repeats, ﬁll in the blanks to create a perfect square trinomial. x2 + ⬚⬚x + ⬚⬚

Factorise:

x4 + 8x2 + 16

a4 − 18a2 + 81

Factorise x4 − 1.

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

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

Factorise monic quadratic trinomials A monic quadratic trinomial is an expression with three terms (trinomial), whose highest power is 2 (quadratic) and whose coefficient of the x2 term is a 1 (monic). This means that quadratics of the form ax2 + bx + c, when a = 1 are monic quadratic trinomials. 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

Mathspace WA Curriculum – Year 10 mathspace.co

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) To factorise a monic quadratic trinomial of the form x2 + bx + c, we want two numbers p and q so that when you add them you get the coefficient of the middle term (b) and when you multiply them you get the constant term (c). For example: • To factorise x2 + 8x + 12, we want two numbers that sum to 8 and multiply together to give 12. The two numbers would be 6 and 2. • To factorise x2 + 4x + 3, we want two numbers that sum to 4 and multiply together to give 3. The two numbers would be 3 and 1. • To factorise x2 − x − 6, we want two numbers that sum to −1, and multiply together to give −6. The two numbers would be −3 and 2.

Example 1 Factorise: x2 + 6x + 8

Create a strategy

Apply the idea

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

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)

Factorise

Example 2 Factorise: x2 − 17x + 60

Create a strategy

Apply the idea

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

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

2.04 Factorise monic quadratic trinomials mathspace.co

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).

2.04 Practice questions Understanding 1

Explorer 1–3

1–3

Sum: 19

Product: 90

Sum: −10

Sum: 3

Product: −63 Product: 24 Product: −88

x2 + 11x + 18

x2 + 16x + 64

x2 + x − 30

(m + 8) (⬚ + ⬚) = m2 + 18m + 80

Complete the missing values in each equation. a

Sum: −2

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

Trailblazer 1, 3, 4

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

Adventurer

(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).

Mathspace WA Curriculum – Year 10 mathspace.co

5ab, 6a-f, 7a-f, 8a-f, 9ab

Fluency 5

Ex 1

Ex 2

Ex 3

5cd, 6g-l, 7g-l, 8g-l, 9e-l

Factorise: a

x2 + 3x + 2x + 6

x2 + 4x + 5x + 20

y2 + 6y + 12y + 72

t + 7t + 8t + 56

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

y + 9y + 14

40 + 13x + x

36 + 12x + x2

x + 7x + 12

Factorise: a

x2 − 4x + 3

x2 − 16x + 60

x2 − 19x + 84

x2 − 12x + 35

x − 8x + 7

y − 10y + 16

t − 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

x − 3x − 70

x + 4x − 117

y − 13y − 48

t2 + 8t − 84

y2 − 12y − 45

x2 + 10x − 56

x2 + 10x − 24

x2 + 3x − 28

Factorise: a

2x2 + 14x + 20

4x2 + 24x + 32

4x2 − 4x − 288

3x2 − 21x + 30

5x2 + 5x − 30

−2x2 − 18x + 44

−4x2 + 12x + 40

−3x2 + 12x − 12

2x3 + 16x2 + 30x

−8 − 6x − x

Reasoning 10

5bc, 6c-h, 7c-h, 8c-h, 9a-h

−12 + 7x − x

10–12

x + 7x + 12x

10–12, 14

12–16

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. 2.04 Factorise monic quadratic trinomials mathspace.co

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.

Area I + Area II

x2− Area II

Area I

x2− Area I

Maximises the value of b.

(x + ⬚) (x + ⬚)

Area II

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. x

Area I

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

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.

Mathspace WA Curriculum – Year 10 mathspace.co

Problem-solving 17–19 17

19, 20, 21bd, 22, 23

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. m

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

17–20, 21ac

Factorise u2 − 11u + 24.

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

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?

Factorise: a

x4 − x2 − 12

y4 − 5y2 + 4

(a + 3)2 + 3(a + 3) − 10

(d − 1)2 − 12(d − 1) + 36

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.

2.04 Factorise monic quadratic trinomials mathspace.co

2 Chapter review 1

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

(8 − 5p) (8 + 5p)

Expand and simplify: a

(m + 4) (m − 4)

(3x − 0.7) (3x + 0.7)

Expand and simplify: a

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

(x − 2) (x − 8)

−2 (3x − 4) (4x + 3)

Expand and simplify: a

(m + 5)2

(n − 4)2

(2p + 3q)2

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 each expression by taking out the highest common factor: a

−3cd2e + 9cde − 6ce

Factorise by grouping: a

a2 + 3a + 4a + 12

p2 − 2p + 6p − 12

xy + 5x + 3y + 15

mk − 4m − 3k + 12

5r2 + 10rs + 5s2

Factorise: a

k3 + 8k2 + k + 8

35m2 − 49

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

5x2 (x + 4y) − 5y(x + 4y)

1 − 2x + x2

x2 + 17x + 72

x + 10x − 24

x2 + x − 20

Factorise: a

x2 − 10x + 25

x + 4x − 117

x2 + 16x + 64

y − 13y − 48

Factorise each trinomial: a

y2 + 10y + 25

a2 − 12a + 36

b2 + 9b + 14

c2 − 11c + 18

e2 − 5e − 24

5by2 + 10by − 15b

d + 3d − 10

Mathspace WA Curriculum – Year 10 mathspace.co

The area of a rectangular plot of land is given by the expression A = x2 + 12x + 35 square units. a

If the length and width are of the form (x + p) and (x + q), find the values of p and q.

If x = 5 units, calculate the perimeter of the plot.

The area of a rectangular banner is 20xy − 15x square metres. If its length is 5x metres, find an expression for its width.

The area of a square tile is x2 − 14x + 49 square centimetres.

Find an expression for the side length of the tile.

Find an expression for the perimeter of the tile.

A large rectangular garden measures (3x + 4) metres by (x + 5) metres. A square flower bed of side length (x − 1) metres is placed in the corner of the garden. a

Write an expression for the area of the large garden in expanded form.

Write an expression for the area of the square flower bed in expanded form.

Find an expression for the remaining area of the garden (the part not covered by the flower bed) in its simplest expanded form.

Did you know?

Expanding and factorising algebraic expressions helps scientists predict climate change by modelling how factors like carbon emissions and sea levels are connected. These simplified equations allow researchers to test scenarios and support decisions that protect the planet.

Chapter 2 review mathspace.co

Big ideas Accurate calculation and interpretation of interest, earnings and profit are crucial for effective money management in earnings, investments and loans.

Curriculum Outcomes • Explore, explain and calculate income tax, including the use of tax tables • Apply repeated simple interest to develop the compound interest formula and solve problems that relate to saving and borrowing • Analyse, represent and interpret problems in real-world situations involving real numbers, absolute and percentage error, linear inequalities, simultaneous equations, real-world formulae, quadratic or exponential functions, taxation, and/or compound interest • (Optional) Use authenticated websites to investigate how changes to the principal, rate of return, voluntary contributions and time can affect superannuation balances or compare characteristics of insurance, such as young driver car insurance or holiday insurance and recognise that the cost is higher when the risk is higher

3 Finance Chapter outline 3.01 3.02 3.03

Simple interest Compound interest Modelling investigation: Home battery storage system PAYG and tax Investigation: Superannuation (Optional) Chapter 3 review

72 81 96 98 112

Long-term saving is a great way to set yourself up in the future. If you saved and invested $1 each week from your 5th birthday until you turned 65, how much would you have?

3.01

Simple interest After this lesson, you will be able to… • define what simple interest is and how it is calculated. • convert different time periods for calculating interest. • understand loan repayments and the formula used to calculate them. • solve problems involving calculating simple interest and determining loan repayments.

Simple interest Simple interest is calculated as a ﬂat percentage of the amount that was initially borrowed or invested, called the principal. We can calculate simple interest using the formula:

I = PRT

I is the interest accumulated P is the principal R is the rate of interest per time period, as a decimal T is the number of time periods (the duration of the loan/investment)

The units of time for R and T must be the same. For example, if an annual interest rate is used, then the time must be in years. Note that “p.a.” is an abbreviation of “per annum” and means per year. Remember that converting a rate per time period uses the inverse operation to converting a time period. For example, to convert a time period from months into years, we divide by 12. However, to convert a monthly interest rate to an annual rate, we multiply by 12. We can use the following conversions:

1 year = 4 quarters = 12 months = 26 fortnights = 52 weeks = 365 days

When you have borrowed money, you must pay back both the principal and the interest. When you have invested money, you own both the principal and the interest. In both cases, the total amount is: A=P+I

Mathspace WA Curriculum – Year 10 mathspace.co

Example 1 $906 is invested at 5% p.a. simple interest. Dave wants to know the number of years it will take the investment to grow to $1132.50 a Calculate the interest that will be earned on the investment.

Create a strategy Use the formula for the total amount.

Apply the idea A = P + IWrite the formula 1132.50 = 906 + I

Substitute the values of A and P

I = 1132.50 – 906 Subtract 906 from both sides to solve for I = $226.50

Evaluate

b Calculate the number of years it will take the investment to grow to $1132.50

Create a strategy Use the simple interest formula.

Apply the idea Write the formula

Substitute I, P, and R

Convert the percentage to a decimal

Evaluate the multiplication

Divide both sides by 45.3 to solve for T

Evaluate

Reﬂect and check Each year the investment of approximately $900 is earning 5% interest. 10% of $900 is $90, so 5% is $45. After 5 years the investment has earned 5 × 45 = $225 interest, bringing the total amount to about $1125, which is close to $1132.50

3.01 Simple interest mathspace.co

Example 2 Calculate the simple interest earned on an investment of $6050 at 0.7% per quarter for 3 years.

Create a strategy To use the simple interest formula the interest rate and time period both need to be in the same unit. In this case, quarters. Three years has 3 × 4 = 12 quarters. So now the interest rate of 0.7 % per quarter matches the time period.

Apply the idea I = Prn

Write the simple interest formula

= 6050 × 0.7% × 12

Substitute the values

= 6050 × 0.007 × 12 Convert the percentage to a decimal = $508.20

Evaluate

Reﬂect and check We could have left the time in years and converted the quarterly rate to an annual rate. To ﬁnd the annual rate, we multiply the quarterly rate of 0.7% by 4 to get 2.8%. Then I = 6050 × 2.8% × 3 which gives $508.20 again.

Example 3 The simple interest on a loan of $6600 over 33 months is $1252.35 Find the annual interest rate as a percentage, rounded to one decimal place.

Create a strategy Use the simple interest formula.

Apply the idea Since we are looking for the annual interest rate, we should convert the time to years (by dividing by 12 months). Write the formula

Substitute I, P, and T (in years)

Evaluate the multiplication

Divide both sides by 18 150

Evaluate the division

Convert to a percentage

Mathspace WA Curriculum – Year 10 mathspace.co

Reﬂect and check We could have left the time in months. Then the rate found would be a monthly value which we could convert to an annual rate by multiplying by 12: 1252.35 = 6600 × R × 33 R = 0.575% per month = 6.9% p.a.

Idea summary We can calculate the simple interest for a loan or investment using the formula:

I = PRT I P R T

is the interest is the principal (initial amount) is the rate of interest per time period (as a decimal) is the number of time periods

The time periods for R and T must be consistent. The total amount owed or in the account is: A=P+I

Loan repayments Loans are usually repaid in a number of instalments (repayments), rather than all at once. We can use the following formula to calculate loan repayments:

Repayment amount =

A is the total amount (principal plus interest) n is the number of repayments

Example 4 Katrina takes out a loan to purchase a surround sound system. The total amount to be repaid is $95 000. a If Katrina makes 19 equal loan repayments, what is the value of each repayment?

Create a strategy Use the loan repayments formula.

3.01 Simple interest mathspace.co

Apply the idea Write the formula

Substitute the values

Evaluate

b If Katrina pays the loan back in 2 years with repayments of $11 875, how often did she make repayments?

Create a strategy Use the loan repayments formula.

Apply the idea Write the formula Substitute the values Multiply both sides by n Divide both sides by 11 875 Evaluate Divide by 2 years to ﬁnd repayments per year Katrina made quarterly repayments (1 repayment every 3 months).

Idea summary

Repayment amount A n

is the total amount (principal plus interest) is the number of repayments

Mathspace WA Curriculum – Year 10 mathspace.co

3.01 Practice questions Understanding 1

Adventurer

1–3

Trailblazer

1–3

Calculate the simple interest on each scenario: a

An investment of $45 000 at 6% p.a. for 2 years

A 3-year loan of $3000 at 12%. p.a.

How can we convert: a

A time period from months to years?

A monthly interest rate to an annual rate?

True or false: a

A higher interest rate will result in more interest being paid on a loan.

A higher principal will result in less interest being paid on a loan.

A larger number of repayments over the same time period will result in a lower repayment amount.

Fluency 4

Explorer

4, 6–7, 9–11, 12ace, 13, 15, 16a, 17–18

5–7, 9–11, 12bdf, 14–15, 16b, 17–18

5–6, 8–11, 12bdf, 14–15, 16c, 17, 19

Homer borrowed $7600 from a bank at a simple interest rate of 4.5% for one year. a

How much interest will Homer need to pay after 1 year?

Calculate the total amount Homer will have to repay the bank at the end of 1 year.

Find the total amount to be paid on a 4-year, $7000 loan at 11% p.a. ﬂat interest.

Toshko invests $12 500 at 4% p.a. simple interest and wants to know the number of years it will take for the investment to grow to $15 000.

Ex 1

Calculate the interest that will be earned on the investment.

Calculate the number of years it will take the investment to grow to $15 000.

For how many years does $7050 need to be invested for it to grow to $7950, if simple interest is paid at 7% p.a.? Round your answer correct to two decimal places.

Ajaz invests $5680 at a simple interest rate of 5% p.a. and earns $426 in interest.

Calculate the number of years taken to earn the interest.

Find the number of quarters taken to earn the interest.

Find the equivalent annual simple interest rate. a

0.63% per month

0.03% per day

1.96% per quarter

The simple interest rate for an investment is 6% p.a. Find the percentage rate, rounded to two decimal places where necessary, used when calculating interest: a

Quarterly

Monthly

Weekly 3.01 Simple interest mathspace.co

11 Ex 2

Calculate the simple interest earned on these investments, rounded to two decimal places. a

$5720 is invested at 10% p.a. for 63 weeks.

$9780 is invested at a rate of 5% p.a. for 17 months.

$9020 is invested at 1% per month for 2 years.

$5440 is invested at 0.3% per fortnight for 566 days.

Calculate the ﬁnal value of these investments over the given time period rounded to two decimal places: a

$4570 at 4% p.a. for 28 months

$9980 at 7% p.a. for 26 weeks

$1430 at 8% p.a. for 130 days

$6170 at 1.2% per month for 3 years

$9060 at 1% per quarter for 4 years

$4620 at 0.1% per fortnight for 3 years

Maria is given three investment account options by her bank: • Option 1: Interest is earned weekly at a simple interest rate of 0.14% per week. • Option 2: Interest is earned monthly at a simple interest rate of 0.63% per month. • Option 3: Interest is earned yearly at a simple interest rate of 7.3% p.a. Which option should Maria choose if she wants to earn as much interest as possible?

Sean is offered three different loan plans when borrowing money from the bank: • Option 1: Interest is earned daily at a simple interest rate of 0.02% per day. • Option 2: Interest is earned fortnightly at a simple interest rate of 0.34% per fortnight. • Option 3: Interest is earned quarterly at a simple interest rate of 1.98% per quarter. Which option should Sean go for if he wants to pay as little interest as possible?

For his investment into government bonds, Buzz was paid simple interest of 9% p.a. Calculate the size of Buzz’s initial investment if he earned $294.12 interest after 24 months.

Find the annual interest rate as a percentage. Round your answer to the nearest whole percent.

Ex 3

17 Ex 4

The simple interest on an investment of $5670 over 36 months is $1360.80

The simple interest on a loan of $25 100 over 39 weeks is $1694.25

The simple interest on a loan of $7500 over 50 months is $1250

Gitte takes out a loan to purchase a new fridge. She makes 16 equal loan repayments. The total loan amount paid is $1280. Find the value of each repayment.

Iain takes out a loan to purchase a jet ski. He makes 15 equal loan repayments of $5239. Calculate the total amount paid back on the loan.

Summer borrows money to pay for a vet bill and wants to pay it back in a year. If the total loan amount is $2470 and her repayments are $95, how often must Summer make repayments?

Mathspace WA Curriculum – Year 10 mathspace.co

Reasoning

20–21

20–22

Yarel borrows $1200 at a simple interest rate of 8% p.a. To calculate the interest he will owe over 22 months, Yarel uses the calculation I = 1200 × 0.08 × 22. Explain why Yarel’s calculation is not correct, and how he can ﬁnd the correct value.

Ariela wants to invest her savings. She is considering two options: • Option 1: Simple interest rate of 0.5% per month for two years • Option 2: Annual rate of 2.5% simple interest, for 42 months Which option will result in Ariela earning the most interest? Justify your choice.

Explain why a shorter loan term (T value) will result in less interest paid, even though the monthly repayments may be higher.

Problem-solving 23–26 23

24–27

24, 26–30

Tara takes out a car loan of $6000 at a simple interest rate of 2% per quarter. She plans to repay the loan over 2 years through regular monthly repayments. a

Calculate the total interest that Tara will accrue over the duration of the repayment.

Calculate the value of each repayment of the loan.

The graph shows the amount of simple interest charged each year by a particular bank, on a 8-year loan. a

Find the total amount of simple interest charged on a loan of $5000.

Calculate the simple interest rate per annum charged by the bank on these 8-year loans. Give your answer as a percentage.

350

Interest ($)

300 250 200 150 100 50

Principal ($) 1000 2000 3000 4000 5000

Mohcin and Emily are looking to invest $11 600 and $18 300 respectively for 6 years. The simple interest rate applied is determined by the following table: Principal Between $4800 and $14 800 More than $14 800 to $24 800 Greater than $24 800

Quarterly rate 1% 1.25% 1.5%

How much interest will Mohcin earn on his investment?

How much interest will Emily earn on her investment?

How much will they earn if they invest their combined principal amounts?

How much more interest will they earn by combining their principal amounts?

3.01 Simple interest mathspace.co

The graph shows the amount of money in a savings account earning simple interest. a

Find the principal amount.

Calculate the interest earned after 10 months.

1400

Calculate the monthly rate of simple interest.

1200

Amount ($)

1000 800 600 400 200

Time (months) 5

An amount is invested at 0.5% per quarter, simple interest, for 8 years. What loan term would earn the same amount of interest if the interest rate was 0.8% per quarter?

A borrower received a loan of $5500 which was repaid in monthly instalments of $269 over 36 months. Calculate the flat rate of interest per annum as a percentage, rounded to two decimal places.

At the start of a month, Lana had $7000. She invested some of it in a friend’s startup and the rest in a savings account with a simple interest rate of 3% per month. By the end of the month, her friend’s startup gave her a 10% return on her investment and her total worth was $7410. Use equations to calculate how much she invested in her friend’s startup.

Shawn is going to purchase a car. He has two options: • Option 1: Using dealer finance which charges simple interest of 5.6% over 3 years for the full purchase amount of $5599. • Option 2: By paying an initial deposit of $1000, and then financing the rest at a higher rate of 7.2% for 3 years. Which option should Shawn take if he wants to pay the least amount in total for the car?

Mathspace WA Curriculum – Year 10 mathspace.co

3.02 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 = Prn I P r n

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 $1000 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

3.02 Compound interest mathspace.co

Exploration (Continued) 2. 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

3. 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? 4. 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) 0

a Which investment has a higher principal amount?

Create a strategy Choose the graph with the higher y-intercept.

Mathspace WA Curriculum – Year 10 mathspace.co

10 12

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.

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 Investment A.

Investment ($1000) 10 8

So, Investment B has a higher ﬁnal amount after 10 years.

Investment A

4 Investment B

Time (years) 0

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 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) 0

10 12

3.02 Compound interest mathspace.co

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

Balance + interest − $3200 + $192 $3392 + ⬚ ⬚

Total balance $3200 $3392 ⬚ ⬚

Interest earned $192 $203.52 ⬚ −

Create a strategy 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

Balance + interest − $3200 + $192 $3392 + $203.52 $3595.52 + $215.73

Mathspace WA Curriculum – Year 10 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 P r n

I = Prn interest accumulated principal amount rate of interest, as a decimal time in years

Simple interest pays the same amount of interest in Investment value each period because it is based on a percentage of the 10 principal only. 8 The growth in the balance of a simple interest account is linear. 6 Compound interest pays increasing amounts of interest Simple interest 4 in each period because it is based on a percentage of the Compound interest balance in the account, which grows as interest is added. 2 The growth in the balance of a compound interest account is Time 0 2 4 6 8 10 12 14 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

3.02 Compound interest mathspace.co

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. 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. $1331 = $1000 × 1.1 × 1.1 × 1.1 = $1000 × 1.13 Generalising this, we can write the compound interest formula as:

A = P (1 + r)t A P r t

future value of the investment present value of the investment interest rate per time period, as a decimal number of time periods

If we want to ﬁnd the amount of interest earned, we can subtract the present value from the future value of the investment. Interest = Future value − Present value

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

Create a strategy Use the compound interest formula: A = P (1 + r)t

Mathspace WA Curriculum – Year 10 mathspace.co

1250 × (1 + 0.076)15

Apply the idea • P = 1250 • • t = 15 A = P (1 + r)t

Write the formula 15

= 1250 (1 + 0.076)

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 present value 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 Evaluate the future value = $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)t A future value of the investment P present value of the investment r interest rate per time period, as a decimal t number of time periods If we only want to ﬁnd the amount of interest earned, we can simply subtract the present value from the future value of the investment. Interest = Future value − Present value

3.02 Compound interest mathspace.co

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:

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)t

Apply the idea We are given that: P = 9450

t = 14 × 12 months = 168 months

Substitute the values The future value is $13 593.90

Mathspace WA Curriculum – Year 10 mathspace.co

Evaluate

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)t

Apply the idea We are given that: P = 35 000 t = 4 months

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

3.02 Compound interest mathspace.co

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.

3.02 Practice questions Understanding 1

Explorer 1–3, 5

Adventurer 1–3, 5

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

According to the data, identify which account is Xuan’s.

Trailblazer 1, 2, 4, 5

Mathspace WA Curriculum – Year 10 mathspace.co

Year 3 $6125.22 $6050.00

$1000 is invested for 10 years with an interest rate of 10% p.a., compounded annually. Which graph represents the growth of the investment? Graph 1 Graph 2 Investment ($)

Investment ($)

3000

2500

2000

1500

1000

500

500 T (years)

T (years) 0

2 3 4 5 6 7 8 9 10

Determine if each graph can represent a savings account that earns compound interest over 50 years. a

Balance ($)

Time (years)

Balance ($)

Time (years)

Balance ($)

Time (years)

Time (years) 0

3.02 Compound interest mathspace.co

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

(

6–12, 14, 15

6, 7, 9–11, 13–15

6, 8, 10, 11, 14, 15

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) 0

Balance at beginning of year $3200 $3360

Interest earned $160 $168 −

Calculate the total interest earned over the three years.

$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. First year Second year Third year Fourth year

Ex 2

$3200 is invested for three years at a rate of 5% p.a., compounded annually. a

Balance at beginning of year $3500 $3633

Interest earned $133 $138.05

Calculate the total interest earned over the three years.

Mathspace WA Curriculum – Year 10 mathspace.co

−

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

10 Ex 3

11 Ex 4

14 Ex 5

$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

Balance + interest −

Balance + interest − $2090 + $88

Value $2090 $2178

Interest earned $88 $91

−

What is the difference in the two values after 5 years?

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 24 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?

3.02 Compound interest mathspace.co

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

16, 17

16, 18, 19

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

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.

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.

Mathspace WA Curriculum – Year 10 mathspace.co

Problem-solving 20–24

21–25

21–23, 25–27

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?

David places $800 into an investment fund where interest is compounded annually at a rate of 6% per annum. Calculate the number of years required for his investment to reach $3200.

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.

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 equivalent interest rate for an account which compounds yearly. Write your answer as a percentage, rounded to three decimal places.

Mia invests $1000 in a savings account with interest compounded quarterly at an annual rate of 5%. Determine how many years it will take for her investment to grow to $2500.

3.02 Compound interest mathspace.co

MODELLING INVESTIGATION

Home battery storage system Plan a home battery storage system for a WA household to store solar energy. Choose the battery’s capacity, installation costs, loan interest rate and tax rebate, then model costs, energy savings and repayment time within budget.

Objectives • Analyse the battery storage scenario and identify key financial variables and constraints • Use real numbers, linear inequalities, compound interest and taxation to model costs, savings and repayment • Interpret and communicate financial findings in the context of the installation

Analyse the battery storage system You are planning a home battery storage system for a WA household to store solar energy. Choose the battery’s capacity, installation costs, loan interest rate and tax rebate to meet energy needs and budget constraints. Select a battery capacity (between 10 kWh and 20 kWh), fixed installation cost (between $1000 and $2000), battery cost (between $500 and $800 per kWh), loan interest rate (between 3% and 6% per annum) and tax rebate (between $1000 and $2000). The net cost (after rebate) must be less than $10 000 and the battery must store at least 8 kWh daily. Sketch the system layout or note your choices.

Investigate

Record your chosen capacity, costs, interest rate and rebate. Why did you select these values?

List one assumption (for example, consistent energy use) and one constraint (for example, budget).

Mathspace WA Curriculum – Year 10 mathspace.co

Represent with mathematics Use real numbers, linear inequalities, compound interest and taxation to model the battery system’s total cost, daily energy savings and loan repayment time.

Interactive exploration

Discover this concept in action online

mathspace.co

Calculate total cost (for example, 15 × 600 + 1500 − 1500 = $9000), daily savings (for example, 15 × 0.30 = $4.50) and repayment time using A = 9000 × (1 + 0.04)t with annual savings (for example, 4.50 × 365 = $1642.50). Check cost and capacity constraints. Use GeoGebra or manual calculations.

Investigate 1.

Calculate the total cost and daily savings using your values. Show your work.

Use the compound interest formula to find the time to repay the loan with annual savings. Verify capacity meets 8 kWh daily.

Use an inequality to check if the net cost is within $10 000.

Interpret your plan Analyse your model to ensure the battery system meets energy needs and is financially viable. Confirm capacity exceeds 8 kWh daily and repayment is under 5 years. Calculate a 5% cost increase error. Adjust one variable (for example, capacity) and recalculate.

Investigate 1.

Does your battery meet energy needs and repayment goals? Is the plan realistic? Why?

Calculate the absolute and percentage error for a 5% cost increase. How does adjusting capacity affect repayment time?

Present your battery plan to the class. How do assumptions (for example, fixed energy prices) affect it?

Discussion 1.

What was the most challenging part of modelling the battery system’s finances? Why?

How could you improve your plan with more data or a larger budget?

Compare your plan with a classmate’s. How do your costs and repayment times differ?

Modelling investigation: Home batteryChapter storage system review mathspace.co

3.03 PAYG and tax After this lesson, you will be able to… • understand what is PAYG tax and how it is deducted from gross pay. • identify different types of allowable deductions and how they reduce taxable income. • calculate taxable income based on gross income and allowable deductions. • understand how to use tax brackets and rates to calculate income tax. • calculate total tax payable and net income.

PAYG tax Each pay period, whether weekly, fortnightly or monthly, an employer will deduct tax from their employee’s gross pay, and send it to the Australian Taxation Oﬃce (ATO). This is known as Pay As You Go (PAYG). Deducting or withholding tax in this way means that income earners are not faced with a single large tax bill at the end of the ﬁnancial year. The correct amount of tax to withhold will depend on the employee’s income. Net pay = Gross pay − Income tax − Other deductions • Net pay is the amount of pay the employee takes home • Gross pay is the amount of pay the employee earns • Income tax is the amount of money that goes to the government to pay for roads, schools, etc. • Other deductions include money that is deducted voluntarily for expenses like union fees

Example 1 The PAYG tax amount, y, that an employer must withhold each week, in January 2023, was given by the formula y = ax − b where x is the number of whole dollars of gross weekly income, plus 99 cents, and a and b are values obtained from this table: Weekly earnings (x) less than $355 $422 $528 $711 $1282 $1730 $3461 $3461 and over

0 0.19 0.29 0.21 0.3477 0.345 0.39 0.47

0 67.4635 109.7327 67.4635 165.4423 161.9808 239.8654 516.7885

Note: This table is based on tax withheld with the tax-free threshold. Jeremy’s gross weekly earnings are $959.50. 98

Mathspace WA Curriculum – Year 10 mathspace.co

a Calculate the amount of PAYG tax his employer will need to withhold from his wage each week.

Create a strategy To calculate the amount of PAYG tax, substitute the values of a and b into the formula.

Apply the idea We notice that the number of whole dollars in Jeremy’s weekly income is 959. Adding 99 cents gives x = $959.99. This amount is greater than $711, but less than $1282, so from the ﬁfth row of the table, we use the values a = 0.3477 and b = 165.4423. y = ax – b

Rewrite the formula

= 0.3477 × 959.99 − 165.4423 Substitute a, x, b = 168.346

Evaluate

= $168

Round to the nearest dollar

Tax withholding amounts are always rounded to the nearest dollar, so Jeremy’s employer would deduct $168 in income tax from his gross weekly pay.

b Calculate Jeremy’s net pay, if there are no other deductions.

Create a strategy To calculate Jeremy’s net pay, subtract his income tax and any other deductions from his gross pay.

Apply the idea In this case, we have no other deductions. Net pay = 959.50 – 168 = $791.50

Subtract $168 from $959.50 Evaluate

So $791.50 is the amount Jeremy would receive each week from his employer. It is his ‘take-home’ pay.

Reﬂect and check If Jeremy’s employer didn’t deduct tax from his pay automatically, $8736.00 would not have been allocated at the ATO towards his actual tax bill. This represents 52 weeks at $168.00 per week.

Idea summary An employer will take tax from their employee’s gross salary each pay period, whether weekly, fortnightly, or monthly, and transmit it to the Australian Taxation Office (ATO). This is referred to as Pay As You Go (PAYG). Net pay = Gross pay − Income tax − Other deductions • Net pay is the amount of pay the employee takes home • Gross pay is the amount of pay the employee earns • Income tax is the amount of money that goes to the government to pay for roads, schools, etc

3.03 PAYG and tax mathspace.co

Allowable deductions Allowable deductions are a particular type of expense that can reduce the amount of income on which tax is calculated. This reduces the overall tax liability. To qualify as an allowable deduction, an expense must be directly related to earning assessable income. Here are some common allowable deductions for Australian income tax: • Work-related travel expenses: Costs incurred for travel between different work locations; this does not include the travel between work and home. • Uniform and protective clothing: Expenses for purchasing and maintaining required uniforms or protective clothing. • Self-education expenses: Costs associated with courses and training directly related to your current job. • Home office expenses: A portion of costs for utilities, phone, and internet if you work from home. • Union fees and subscriptions: Fees paid to unions and professional associations directly related to your employment. Taxable income = gross income − allowable deductions

Example 2 Paul has a job in a retail store where he earned $45 768 during the last financial year. He also did some weekend work as a DJ in a local club and received an additional $12 250 during the same period. Paul has the following allowable tax deductions: • $510 in union fees • $240 for the purchase of new DJ equipment • $55 in fees for his tax agent Calculate his taxable income for the financial year.

Create a strategy Use the formula: Taxable income = gross income − allowable deductions

Apply the idea First, calculate Paul’s gross income by adding his two incomes together. Gross income = 45 768 + 12 250 = $58 018 Next, we add up all of Paul’s allowable tax deductions. Allowable tax deductions = 510 + 240 + 55 = $805 Finally, we calculate his taxable income. Taxable income = gross income − allowable tax deductions = 58 018 − 805 = $57 213

100

Mathspace WA Curriculum – Year 10 mathspace.co

Idea summary Allowable deductions are expenses directly related to employment. Taxable income = gross income − allowable deductions

Taxable income The total tax payable by most individuals will be made up of income tax and the Medicare levy, both of which are based on their taxable income for the ﬁnancial year. The correct amount of income tax for Australian residents, not including the Medicare levy, is determined from the following table: Resident tax rates (2022-2023) Taxable income 0 − $18 200 $18 201 − $45 000 $45 001 − $135 000 $135 001 − $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

The use of this table is best explained using an example. To calculate the tax on a taxable income of $50 000, we complete the following steps. 1. Determine the tax bracket the income falls into using the left-hand column of the table. $50 000 falls in the $45 001 − $135 000 bracket. Reading across the row, we can see that the tax payable is: $4288 plus 30c for each $1 over $45 000 2. Calculate additional tax The $4288 covers tax for the first $45 000 of the income. $50 000 is $5000 over $45 000. On this $5000 we pay 30c for each $1. This is calculated as: 0.30 × $5000 = $1500 3. Calculate total income tax Total income tax = $4288 + $1500 = $5788 4. Calculate Medicare levy The Medicare levy is an additional tax that is paid to help fund Medicare, Australia’s public health care system. It is calculated as: Medicare levy = 2% of taxable income $50 000 × 0.02 = $1000

3.03 PAYG and tax mathspace.co

101

5. Calculate total tax payable The total tax payable is found by adding together the income tax and the Medicare levy. Total tax payable = income tax + Medicare levy = $5788 + $1000 = $6788 An individual’s net income can be calculated by subtracting their total tax payable from their taxable income. net income = taxable income − total tax payable Therefore, the net income of someone who has a taxable income of $50 000 is: $50 000 − $6788 = $43 212

Example 3 Last financial year, Roxanne earned $157 872. Using the income tax rates shown in the table, evaluate the following. Resident tax rates (2024-2025) Taxable income 0–$18 200 $18 201–$45 000 $45 001–$135 000 $135 001–$190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

a Calculate the income tax payable on Roxanne’s income.

Create a strategy Use the income bracket where Roxanne’s income lies, and use the necessary rule.

Apply the idea Income tax payable = 31 288 + (157 872 − 135 000) × 0.37

Substitute the values

= 31 288 + 22 872 × 0.37 Evaluate the subtraction = 31 288 + 8462.64 Evaluate the multiplication = $39 750.64

102

Mathspace WA Curriculum – Year 10 mathspace.co

Evaluate

b Calculate the Medicare levy payable, given that it is 2% of taxable income. Round to the nearest cent.

Create a strategy Multiply Roxanne’s income by 2%.

Apply the idea Medicare levy payable = 157 872 × 0.02 = $3157.44

Substitute Income = 157 872 Evaluate

c Calculate the total amount Roxanne needs to pay in income tax and Medicare levy.

Create a strategy Add the income tax payable from part (a) and the Medicare levy payable from part (b).

Apply the idea Total amount payable = 39 750.64 + 3157.44 = $42 908.08

Add the payables altogether Evaluate

Idea summary Most working people in Australia pay income tax and the Medicare levy. Taxable income is the gross pay minus any deductions Medicare levy = 2% of taxable income Total tax payable = Income tax + Medicare levy Net income = Taxable income − Total tax payable To find the amount of income tax, we can follow these steps: 1. Determine the tax bracket the taxable income falls into using the left-hand column of the table. 2. Find the amount above the threshold. 3. Find the variable amount of income tax by multiplying the amount above the threshold by the tax rate. 4. Find the total amount of income tax by adding the fixed amount and the variable amount of tax for their bracket.

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103

Personal income tax The key ideas when it comes to personal income tax are: • Your hourly wage or annual salary is a gross income. • Your employer will generally withhold tax, called PAYG. Take home pay = Gross income – PAYG • The income you receive from your job is your net income. Net income = Gross income – Tax • You may incur costs associated with employment. These are called allowable deductions. Keep a record of these throughout the year. Taxable income = Gross income – Allowable deductions • Tax payable is calculated using your taxable income and the rates stated in the tax tables. Medicare levy of 2% Total liability = Tax payable + Medicare levy

= Tax payable + 2% × Taxable income

• The total tax payable is compared to the PAYG tax paid by your employer. PAYG − Tax payable = Tax refund If the tax payable is higher than the PAYG then you will have a tax debt.

Example 4 Jeremy has a gross income of $45 000, with allowable deductions totalling $1850. Total PAYG withholding was $4920. It was calculated that the total tax liability is $4740.50. a Will Jeremy get a tax refund? If so, what is the value of the refund?

Create a strategy Use the formula: Tax refund = PAYG − tax payable

Apply the idea Tax refund = PAYG − tax payable

104

Write the formula

= 4920 − 4740.50

Substitute the amounts

= $179.50

Evaluate

Mathspace WA Curriculum – Year 10 mathspace.co

b Jeremy forgot to mention a further $820 in deductions. Calculate the new taxable income and tax payable on that amount.

Create a strategy Use the formula: New taxable income = gross income − allowable deductions

Apply the idea New taxable income = gross income − allowable deductions

Write the formula

= 45 000 − (1850 + 820)

Substitute the amounts

= 45 000 − 2670

Sum up allowable deductions

= $42 330

Evaluate

Tax payable = (42 330 − 18 200) × 0.19

Write the equation

= 24 130 × 0.19

Subtract amounts on the bracket

= $4584.7

Multiply tax rate of 19%

c Does this alter the value of the refund?

Create a strategy Use the formula: Tax refund = PAYG − tax payable

Apply the idea Tax refund = PAYG − tax payable

Write the formula

= 4920 − 4584.5

Substitute the amounts

= $335.30

Evaluate

Idea summary Personal income tax is a tax levied on an individual’s earnings, such as wages, salaries, and other income sources.

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105

3.03 Practice questions Understanding

Explorer 1–4

Adventurer 1–4

Trailblazer 1–4

What percentage of their taxable income is the Medicare levy for most Australian residents?

What is personal income tax?

For PAYG:

What does PAYG stand for?

What is PAYG?

Why is PAYG helpful?

What are deductions, and how do they affect an individual’s taxable income?

Fluency

5–11, 14–16, 18, 19a-d, 20, 21

6–10, 12, 14–21

6, 7, 9, 12–21

Each week, Jack has $195 in PAYG tax withheld from his pay. How much does Jack have withheld in a full ﬁnancial year? Assume that there are 52 weeks in a year.

In the last ﬁnancial year, Carl was paid a wage every month, and each week there was $61 deducted as PAYG tax by Carl’s employer.

7 Ex 1

How much did Carl’s employer deduct in total in PAYG tax last ﬁnancial year?

At the end of the ﬁnancial year, he calculates his total tax payable to be $3049. Will he receive a refund or have tax owing?

Tina wants to calculate how much PAYG tax to withhold from her employees’ gross weekly income. She uses the formula y = ax − b where y is the PAYG tax to withhold, x is the number of whole dollars of gross weekly income plus 99 cents, and a and b are values obtained from the table shown. Calculate how much PAYG tax Tina would withhold from her employee who has gross weekly earnings of $833.

8 Ex 2

106

Weekly earnings (x) less than $355 $422 $528 $711 $1282 $1730 $3461 $3461 and over

−

0.19 0.29 0.21 0.3477 0.345 0.39 0.47

67.4635 109.7327 67.4635 165.4423 161.9808 239.8654 516.7885

Emma’s total allowable deductions for the year are $493. If her gross annual income is $25 000, calculate her taxable income for the year. Hannah earns a salary of $86 220 and claims $1293 for work-related expenses. She also contributes $190 to charities throughout the year. Calculate her taxable income.

Mathspace WA Curriculum – Year 10 mathspace.co

This year, Victoria had a taxable income of $71 250. Her gross income is $74 000. a

State how much she claimed in allowable tax deductions.

What percentage of Victoria’s gross income were these tax deductions? Round your answer to one decimal place.

Yuri’s gross annual income is $51 120, but he wants to make superannuation contributions throughout the year so that his taxable income reduces to $50 820 (assuming no other deductions). Calculate the monthly superannuation contribution he will need to make.

Dylan has a gross income of $92 749 and can claim $877 in allowable tax deductions. Dylan wants to reduce his taxable income to $90 000 by making voluntary weekly contributions to his superannuation. Calculate how much he must contribute each week.

Luigi has a $49 per month mobile phone plan. He regularly uses his phone for work purposes and can claim his work-related phone expenses as an allowable tax deduction. Using his itemised bill each month, Luigi calculates that his work-related phone usage is consistently around 20% of his total phone usage. Luigi works for 48 weeks (11.1 months) of the year. Calculate the amount he can claim as a tax deduction for his mobile phone.

For a gross weekly income less than $528 but greater than or equal to $422, the amount of PAYG tax to withhold is given by the formula PAYG tax to withhold = 0.19x − 67.4635 where x is the number of whole dollars of gross weekly income plus 99 cents.

Calculate how much PAYG tax is withheld from a gross weekly income of $450.

Calculate how much PAYG tax is withheld from a gross weekly income of $511.40.

For gross weekly incomes that are $3461 or more, the amount of PAYG tax to withhold is given by the formula PAYG tax to withhold = 0.47x − 516.7885 where x is the number of whole dollars of gross weekly income plus 99 cents. Calculate how much PAYG tax is withheld from a gross weekly income of $3797.70.

The Medicare levy is charged at 2% of taxable income. Calculate the Medicare levy on a taxable income of $60 000.

For the last ﬁnancial year, Patricia claimed $1857 in total deductions and paid a Medicare levy of $1744. If her gross income for the year was $x, write an equation and solve for x. Note that the Medicare levy is charged at 2% of taxable income.

3.03 PAYG and tax mathspace.co

107

Isabelle is a working holiday maker. She has a job in Australia and her taxable income for the year is $47 600. Use the table to ﬁnd how much income tax she is required to pay: Working holiday makers income tax rates for 2024 to 2025 Taxable income $0 - $45 000 $45 001 - $135 000 $135 001 - $190 000 $190 001 and over

Tax on this income 15c for each $1 $6750 plus 30c for each $1 over $45 000 $33 750 plus 37c for each $1 over $135 000 $54 100 plus 45c for each $1 over $190 000

Questions 19 – 24 will use the following table of tax rates for 2024-2025. Resident tax rates (2024-2025) Taxable income $0 – $18 200 $18 201 – $45 000 $45 001 – $135 000 $135 001 – $190 000 $190 001 and over 19

20 Ex 3

21 Ex 4

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

Calculate the income tax payable for these taxable incomes: a

$8326

$36 810

$194 388

$225 810

$93 395

$146 664

Last financial year, Carmen’s taxable income was $197 955. Based on the Resident Tax Rate (2024-2025) table, calculate: a

The income tax payable on Carmen’s income.

The Medicare levy payable, given that it is 2% of taxable income.

The total amount Carmen needs to pay in income tax and Medicare levy.

Anna has a gross income of $52 000, with allowable deductions totalling $2400. Her PAYG withholding for the year was $5600. It was calculated that her total tax liability is $5550. a

Will Anna get a tax refund, and if so, what is the value of the refund?

Anna forgot to claim an additional $900 in deductions. Calculate her new taxable income and the tax payable on that amount.

Calculate the new refund based on the revised tax payable.

Reasoning

22, 23

23, 24

Most people pay income tax and the Medicare levy, which is charged at 2% of taxable income. Janet has a taxable income of $17 000 and says that her total payment to the tax department is 2% of her income. Is Janet correct? Explain your answer.

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Stewie computed his tax using the tax rates for the financial year 2024-2025: Here is his work:

Tax = $4288 + 0.30 × ($65 000 − $45 000) Apply the corresponding tax formula

= $4288 + 0.30 × $20 000

Evaluate the brackets

= $4288 + $7400

ultiply 0.30 by $20 000 to find the M additional tax amount

= $11 688

dd the base tax amount and the A additional tax amount

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

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

A proposal suggested that to help those on low incomes the tax-free upper bound be increased from $18 200 to $20 000. Part of the proposal is that the tax paid by people earning more than $45 001 should remain the same. In order to achieve this, it is proposed that those earning between $20 000 and $45 000 should pay 18c for each $1 instead of the current 16c for each $1. a

Will this proposal achieve its stated aim? Explain.

An alternative proposal is to raise the rate for those earning between $20 000 and $45 000 to pay 18c for each $1 over $20 000. What effect would this have on people earning more than $45 000?

If the 18c for each $1 over $20 000 is adopted, what change would it make to the instructions for ‘Tax on this income’ for those earning between $135 001 and $190 000?

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Problem-solving 25–27 25

110

Tobias works as a chef. Because he wears a uniform for his job, he is able to claim the cost and cleaning of his uniform as an allowable tax deduction. Tobias’s work-specific uniform expenses are shown in the table:

25–29

Uniform item Chef’s checked trousers Chef’s jacket Chef’s hat Apron Pair of protective shoes

25, 26, 28–30

Number of items 2 2 1 2 1

Price per item (dollars) 21.50 26.00 7.85 23.50 60.00

For his uniform, Tobias’s total cleaning expenses for the year come to $52. Calculate Tobias’s total clothing and cleaning expenses for the year.

Tobias also spent $295 on a set of knives and $17.25 on a knife pouch. He can claim these as a tools and equipment expense because he uses them only for work. Calculate his total tools and equipment expenses.

Calculate Tobias’s total allowable tax deductions.

Tobias earns $19.50 per hour and works 40 hours per week. Calculate his annual gross income if he works 48 weeks in a year.

Calculate Tobias’s taxable income.

Mathspace WA Curriculum – Year 10 mathspace.co

Questions 26 – 30 will use the following table of tax rates for 2024-2025. Resident tax rates (2024-2025) Taxable income $0 - $18 200 $18 201 - $45 000 $45 001 - $135 000 $135 001 - $190 000 $190 001 and over 26

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

Emma is a researcher with a taxable income of $97 589. Throughout the year, her employer has deducted $560 per week in pay-as-you-go (PAYG). a

Calculate the income tax payable using the Resident Tax Rate (2024-2025) table.

Calculate the Medicare levy, which is 2% of taxable income.

Since Emma’s employer has deducted more than is needed from her income, Emma is entitled to a refund for the year. Find the amount of this refund.

Jasmine, a marketing manager, has a gross income of $72 000 and allowable deductions of $3500. Her total PAYG withholding amounts to $9200. Will Jasmine receive a tax refund? If so, how much? (Don’t forget to account for the Medicare levy in the calculations.)

Michael, a teacher, receives a gross monthly salary of $5416.67 with PAYG withholding each month of $1842.25. He has allowable deductions totalling $1600 annually. What is Michael’s tax refund or liability? (Don’t forget to account for the Medicare levy in the calculations.)

Elias has just been hired in a new role where he will earn a fortnightly income of $5697 for an entire year.

Calculate how much Elias earns in a year. Assume 26 fortnights in a year.

Calculate the income tax payable on Elias’s annual income using the Resident Tax Rate (2024-2025) table.

Elias also needs to pay $113.94 alongside his income tax. How much PAYG tax would his employer need to withhold from his fortnightly pay so that Elias’s total PAYG tax instalments for the year exactly cover his total tax payable?

Cinta has a gross weekly income of $1023. Each week, her employer withholds $193 in PAYG tax. At the end of the financial year, Cinta has to submit a tax return. She has $842 in allowable tax deductions for the year. Assuming there are 52 weeks in a year, calculate: a

Her taxable income.

Her income tax payable using the Resident Tax Rate (2024-2025) table.

How much Cinta pays towards the Medicare levy, which is 2% of her taxable income.

How much Cinta has already paid in PAYG tax instalments throughout the year.

How much Cinta is entitled to as a tax refund.

Investigation: Superannuation (Optional)

Investigate online

mathspace.co

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3 Chapter review 1

Calculate the simple interest earned on these investments, rounded to two decimal places. a

$6320 is invested at 9% p.a. for 72 weeks.

$8570 is invested at a rate of 4.5% p.a. for 19 months.

$10 020 is invested at 0.9% per month for 1.5 years.

Leia took out a loan of $9000 from a credit union at a simple interest rate of 6% p.a. for two years. a

How much interest will Leia need to pay after 2 years?

Calculate the total amount Leia will have to repay the credit union at the end of 2 years.

Ivy purchased a TV with a payment plan over two years. The store charges simple interest at 9% per year. The price tag on the TV was $1299.99. How much would her monthly payment be? Round your answer to the nearest cent.

For each representation, identify whether it shows simple interest or compound interest. Explain how you know. (Assume A is the total amount in dollars after n time periods, such as months or years.) Opening Closing Month Interest Balance Balance 1 $300 $15 $315 2 $315 $15 $330 3 $330 $15 $345

A = 1000(1.03)n

A ($)

1800

A = 1000 + 30n

1400

1200

1000

800

600

400

Opening Closing Interest Balance Balance $300 $15 $315 $315 $15.75 $330.75 $330.75 $16.54 $347.29

A ($)

1600

200

112

1 2 3

1800

1600

Month

n (years) 2 4 6 8 10 12 14 16 18

200 0

n (years) 2 4 6 8 10 12 14 16 18

Brad borrows $6000 at a rate of 5.5% p.a., compounded annually. a

After 3 years, Brad 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 three years?

Mathspace WA Curriculum – Year 10 mathspace.co

Calculate the interest that each investment accumulates: a

A $6500 investment earns interest at 2.5% p.a., compounded annually over 18 years.

A $6000 investment earns interest at 3.0% p.a., compounded monthly over 11 years.

A $6800 investment earns interest at 2.4% p.a., compounded weekly over 7 years.

Jasmine is planning to make an investment which accumulates monthly compound interest at a rate of 5.2% p.a. If she aims to have her investment reach $120 000 after 10 years, find the principal amount that she should invest.

For a gross weekly income less than $528 but greater than or equal to $422, the amount of PAYG tax to withhold is given by the formula, where x is the number of whole dollars of gross weekly income plus 99 cents. PAYG tax to withhold = 0.19x − 67.4635 Calculate how much PAYG tax is withheld from each gross weekly income. Round your answer to the nearest dollar. a

$460

$510.50

For the last ﬁnancial year, Robert claimed $2000 in total deductions and paid a Medicare levy of $1800. The Medicare levy is charged at 2% of taxable income. What was his gross income?

Lucy is a working holiday maker. She has a job in Australia and her taxable income for the year is $50 000. Use the table to ﬁnd how much income tax she is required to pay: Working holiday makers income tax rates for 2024 to 2025 Taxable income $0 – $45 000 $45 001 – $135 000 $135 001 – $190 000 $190 001 and over

Tax on this income 15c for each $1 $6750 plus 30c for each $1 over $45 000 $33 750 plus 37c for each $1 over $135 000 $54 100 plus 45c for each $1 over $190 000

The table shows the tax rates for 2024 to 2025: Resident tax rates (2024-2025) Taxable income 0 − $18 200 $18 201 − $45 000 $45 001 − $135 000 $135 001 − $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

Calculate the income tax payable for these taxable incomes: a

$12 800

$58 662

$185 495

$285 965

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Big ideas Equations can be solved more easily by manipulating algebraic expressions and applying a series of organised steps and techniques.

Curriculum Outcomes • Solve one-variable linear inequalities involving brackets and/or a variable on each side. Represent the solution on a number line and verify the solution by substitution • (Optional) Identify the region on the Cartesian plane defined by linear inequalities • (Optional) Use gradient and/or point/s to graphically and algebraically determine equations of parallel and perpendicular lines • Move flexibly between real number inequalities expressed as a worded statement, algebraically or on a number line

4 Linear relationships Chapter outline 4.01 4.02 4.03 4.04 4.05

Linear graphs Linear models Linear inequalities Optional: Graph linear inequalities Investigation - Experiment with linear inequalities (Optional) Optional: Parallel and perpendicular lines Chapter 4 review

116 126 138

147

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

4.01 Linear graphs After this lesson, you will be able to… • plot a series of points from a table of values on the Cartesian plane and connect them to form a linear graph. • plot linear graphs using the x-intercept and y-intercept as primary points. • plot a linear graph using its gradient and a specified point on the line.

Sketch from a table of values Each pair of values in a table of values represents the x-coordinate and y-coordinate for a point on the line. To plot from a table of values we identify the points from the table. x y

1 −2

2 1

3 4

4 7

The x and y-value in each column of the table can be grouped together to form the coordinates of a single point, (x, y).

(1, -2), (2, 1), (3, 4), (4, 7)

Example 1 For y = 2x − 4: a Complete the table of values. x y

Create a strategy

Apply the idea

Substitute each x-value into the given equation.

For x = 0: y=2×0−4 = −4

Substitute x = 0 Evaluate

y=2×1−4 = −2

Substitute x = 1 Evaluate

y=2×2−4 =0

Substitute x = 2 Evaluate

y=2×3−4 =2

Substitute x = 3 Evaluate

For x = 1:

For x = 2:

For x = 3:

x y

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0 −4

1 −2

2 0

3 2

b List the 4 coordinates from the table of values.

Apply the idea The coordinates are (0, −4), (1, −2), (2, 0) and (3, 2).

c By choosing any two coordinates from part (b), plot the line on the Cartesian plane.

Create a strategy Connect the plotted points.

Apply the idea Choosing points (0, −4) and (3, 2), we plot them on the Cartesian plane and connect them with a straight line. y 4 3

y = 2x − 4

2 1 −2 −1

−1

x 1

−2 −3 −4

Idea summary To plot a point, (a, b), on a Cartesian plane, we ﬁrst identify where x = a lies along the x-axis, and where y = b lies along the y-axis. If a series of points lie on a straight line, we can join them to represent all points that satisfy the relationship y = mx + c.

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Sketch using the gradient and a point We can graph a line by identifying the gradient and any point on the line.

Example 2 Sketch the line that has a gradient of −5 and an x-intercept of −4.

Create a strategy Use the gradient to ﬁnd another point starting from the x-intercept.

Apply the idea 6 5 4 3 2 1 −5 −4 −3 −2 −1

−5 −4 −3 −2 −1

The gradient is

−1 −2 −3 −4 −5 −6

so the rise is −5 and the run

is 1. We can use this to ﬁnd another point on the line. x 1

−1 −2 −3 −4 −5 −6

6 5 4 3 2 1

The x-intercept has coordinates of (−4, 0).

By moving right 1 unit and down 5 units we get to the point (−3, −5).

So now we can draw our line through these two points.

x 1

Example 3 Sketch the line that has a gradient of 2 and passes through the point (3, −1).

Create a strategy Plot the point (3, −1).

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A gradient of 2 means that for every 1 unit we go to the right, we go 2 units up. A positive gradient means that the line will go upwards from left to right.

Apply the idea 8 7 6 5 4 3 2 1 −1

−1

−1 −2 −3 −4

Produce a series of points from (3, −1) using the gradient.

A gradient of 2 means that for each 1 unit we move to the right, we also move 2 units up. Alternatively, it means that for each 1 unit we move to the left, we also move 2 units down. x 1

−1 −2 −3 −4

8 7 6 5 4 3 2 1 −1

x 1

−1 −2 −3 −4

8 7 6 5 4 3 2 1

Plot the point (3, −1).

Draw the line through the points.

x 1

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Idea summary To sketch a line given the gradient and a point on the line: 1. Plot the point on the Cartesian plane. 2. Use the gradient to find one other point that lies on the line. 3. Draw a line through the two points that spans the plane. If we have the choice, the y-intercept is a good point to use.

Sketch by using the intercepts The x-intercept of a graph occurs where y = 0. To ﬁnd the x-intercept, set y = 0 and then solve the equation. The y-intercept of a graph occurs where x = 0. To ﬁnd the y-intercept, set x = 0 and then solve the equation. This method is often useful when the equation is not in gradient-intercept ( y = mx + c) form but in the two-intercept form (ax + by = c).

Example 4 Sketch the graph of 2x + 3y = 12 by finding the x and y-intercepts.

Create a strategy First, ﬁnd the x-intercept by setting y to zero and solving for x. Next, ﬁnd the y-intercept by setting x to zero and solving for y. Then, plot the intercepts and sketch the line.

Apply the idea Solve for the x-intercept: 2x + 3y = 12 2x + 3(0) = 12 2x = 12 x=6

Write the equation Substitute y = 0 into the equation Simplify Divide both sides by 2 7 6 5 4 3 2 1

−7−6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6 −7

120

Plot the coordinate (6, 0).

x 1 2 3 4 5 6 7

Mathspace WA Curriculum – Year 10 mathspace.co

Solve for the y-intercept: 2x + 3y = 12

Write the equation

2(0) + 3y = 12

Substitute x = 0 into the equation

3y = 12

Simplify

y=4

Divide both sides by 3

7 6 5 4 3 2 1

−7−6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1 −7−6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6 −7

Plot the coordinate (0, 4).

1 2 3 4 5 6 7

Sketch the line.

x 1 2 3 4 5 6 7

Idea summary The x-intercept occurs at the point where y = 0. The y-intercept occurs at the point where x = 0. We can use the x and y-intercepts to sketch a line.

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4.01 Practice questions Understanding 1

Explorer

Adventurer

1–8

Trailblazer 1–8

Complete the table: y-intercept

Gradient y = 3x − 4 y = −5x + 2 y=4 x=3 2

To sketch a linear graph using its intercepts, you need to: A

Find the maximum and minimum points on the graph

Locate where the line crosses the x-axis and y-axis

Determine the gradient of the line

Calculate the midpoint of the line

What is the equation of the line parallel to the y-axis and passing through the point (6, 2)?

What is the equation of the line parallel to the x-axis and passing through the point (−2, −1)?

What are the equations of the x and y-axes?

Identify the x-intercept and y-intercept: a b

(2, 3), (0, −2), (5, −1), (6, 0), (3, 3) x y

−7 0

−2 5

3 10

Find the gradient, m, and y-intercept, c, for each line: a

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

122

0 7

x 1

2 3 4 5

Mathspace WA Curriculum – Year 10 mathspace.co

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

y = 2x + 9

y=5

Complete the table of values:

Sketch the line on a Cartesian plane.

9d-h, 11, 13b-d, 14, 15b-d, 16e-h, 17b-d, 19, 20ab, 21

For each equation:

2 3 4 5

x y

x=2

9d-h, 10, 12, 13c-e, 14cd, 15d-f, 16e-h, 17cd, 19–21

−1

y=x−3

y=x+2

y = 2x − 7

y = 3x + 2

y = −x + 4

y = 2x − 1

y=3−x

y = −3x + 5

For

Complete the table of values:

Sketch the line.

x y

−2

For y = 2x + 3: a

Complete the table of values. x y

4x + 2y = 3

9a-e, 11, 13a-c, 14, 15a-c, 16a-d, 17a-c, 18, 19, 21

Fluency 9

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

Is the point (−2, 5) on the line? a

Ex 1

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

−1

Sketch the line on a Cartesian plane.

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercept.

Find the y-value when x = 3.

Find the x-value when y = −4.

Sketch the line on a Cartesian plane.

For 2x − 3y − 6 = 0: a

Complete the table of values. x y

−1

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercept.

Find the gradient.

Find the y-value when x = 2.

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Ex 2

Ex 3

Ex 4

Sketch the lines on a Cartesian plane: a

A line that has an x-intercept of 4 and whose gradient is 3.

A line that has a y-intercept of 7 and whose gradient is −3.

A line that has a y-intercept of 4 and whose gradient is .

A line that has a y-intercept of −3 and whose gradient is 6.

A line that has an x-intercept of −6 and whose gradient is −3.

Sketch each linear relationship: a

A line that passes through the point (2, 3) and has a gradient of 1.

A line that passes through the point (0, −3) and has a gradient of 2.

A line that passes through the point (−1, 4) and has a gradient of −1.

A line that passes through the point (3, 0) and has a gradient of 3.

For each linear equation: i

Find the gradient.

Find the coordinates of the y-intercept.

iii

Use the gradient and y-intercept to sketch the line.

y = 2x + 4

y = −x + 5

3x − 2y + 6 = 0

4x − y + 4 = 0

Find the x-intercept.

Find the y-intercept.

iii

Use the intercepts to sketch the line.

y = 2x − 3

y = 2x + 3

y = −2x + 3

y = −2x + 5

y = 3x − 2

y = −3x + 1

y = −3x − 4

y = 4x + 3

2x + y = 4

Sketch the graph by ﬁnding any two points on the line: y = −2x − 1

y = 4x − 2

Sketch the line using the y-intercept and any other point on the line: y = 2x + 7

y = 4x − 3

Find the x-intercept of the line 3x − 7y = 21.

For each linear equation: i

Find the x-intercept.

Find the y-intercept.

iii

Sketch the line.

3x + 2y = 6

4x − 7y = 28

y = −2x − 4

y = −3x + 6

4x − 3y = −24

10x + 9y = −90

Sketch the lines using the gradient and y-intercept: a

124

y = 5x − 2

For each linear equation:

a 18

y = 6x − 3

y = −6x + 2

Mathspace WA Curriculum – Year 10 mathspace.co

Reasoning

22, 23

23, 24

23–25

Explain how the gradient of a line affects the direction in which the line slants. Why does a positive gradient slant upwards from left to right, while a negative gradient slants downwards?

Justify why knowing the x-intercept and y-intercept of a linear equation is suﬃcient to sketch its graph.

Complete the table: Gradient

y-intercept

y = kx + d y = −px + r y=k 25

Consider the linear equation y = mx + c. If we change m to 2m, how will the graph change? Explain your reasoning.

Problem-solving 26, 27 26

27–29

28, 30, 31

A toy manufacturer estimates that the number of toys they sell each week will decrease linearly as the price increases. Their prediction is shown in the table: Price Sales

$10 500

$20 400

$30 300

Sketch a graph using the table of values and estimate how many toys they would sell if the price was $25. 27

A car rental company charges a ﬂat fee of $60 per day plus $0.60 per kilometre driven. If a customer drives 100 kilometres in a day, they will be charged $120. How much would it cost for a customer to rent a car for a day and drive 250 kilometres?

Suppose a company’s profit (P ) in thousands of dollars is linearly related to the amount of goods (x) sold in thousands. The company breaks even (P = 0) when they sell 3000 goods, and they make a profit of $5000 when they sell 6000 goods. Determine the equation of the line that represents the company’s profit.

Sandro has been given the task of designing a straight path from the fountain to the library in a park. He knows the coordinates of the fountain (3, 5) and the library (7, 9). The gradient of the path needs to be 1. Write the equation of the path.

The cost (C) of manufacturing a certain product is a linear equation of the quantity (q) produced. The cost of producing 20 units is $250 and the cost of producing 60 units is $450. Find the equation that describes the cost as a function of quantity.

A runner is planning a straight route for their daily run. They want to start at point A and ﬁnish at point B. They know A has coordinates (2, 3) and B has coordinates (6, 8). They want the route to have a constant gradient. Write the equation of the route. 4.01 Linear graphs mathspace.co

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4.02 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) 100 200 300 400 500 600 700 800 900

As we know already, the two key features of a linear equation, 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.

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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. 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.

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 passes through the points.

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127

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: Use the coordinates (0, 0) as (x1, y1) and (10, 16.40) as (x2, y2). We can find the gradient of the line: Write the gradient formula

Substitute the values

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.

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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

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 often displayed in the first row of the table, followed by the dependent variable. 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.

4.02 Practice questions Understanding 1

1–4

Adventurer 1–4

Trailblazer 1–4

Which of the equations represents a linear model? A

Explorer

y = 2x + 3

y = x2 + 2x + 1

y = 3x3 + 2x + 1

f (x) = 2x+1

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.

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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 initial value for these linear models: a

C = 250 + 7.55x

Fluency 5

T = 12.5 + 1.5m

5–11, 16, 20

What is the gradient of the line?

What does the gradient represent? A

The initial value

The speed

Cost of petrol

Time travelled

5, 7, 12–15, 17–19

The horizontal axis represents the time in hours and the vertical axis represents the distance travelled in kilometres. a

4x + 5y = 40

F = 0.6k − 5

5, 7, 12–15, 17–19

Distance (km) 575 460 345 230 115 t (hrs) 1

The following graphs represent the cost and revenue in dollars for x items produced and sold, respectively. C = 15x is the cost equation, indicating each item costs $15 to make. R = 0.5x2 + 50 is the revenue equation. a

Identify each equation on the graph.

Explain the meaning of the intersection points of the graphs.

$ 600 500 400 300 200 100 Items (x) 10

130

Identify whether the situation is represented by a linear or non-linear equation: 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 WA Curriculum – Year 10 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?

0 75

Complete the table. Graph the linear relationship represented in the table.

By how much is her heartbeat increasing each minute?

4 80

6 93

8 99

10 105

A diver starts at the surface of the water and begins to descend below the surface at a constant rate given by y = 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.

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)

4 87

Number of minutes passed (x) Depth of diver in metres ( y)

Ex 1

2 81

3 110

After Sally starts running, her heartbeat increases at a constant rate as shown in the table: Number of minutes passed (x) Heart rate ( y)

2 106

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.

Find the cost of petrol per litre.

Calculate the cost of 75 L of petrol.

The time in minutes and the temperature in degrees Celsius (°C) are given in the table: Time (x) Temperature ( y °C)

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.

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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 (T)

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 T.

Chirping crickets can be used as an indication of how hot or cool it is outside. For a particular species this data was recorded: 110 19

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?

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 an equation 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: Descent (t seconds) Altitude (A)

132

80 16

Day (d) Height of plant (h cm)

5 30

Number of chirps per minute (x) Temperature (T °C)

4 25

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 an equation of the number of seconds after commencing descent, t.

Mathspace WA Curriculum – Year 10 mathspace.co

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 $0.10 per kilometre travelled. The fares for particular distances are shown in the table: Distance travelled, d (km) Fare, F ($)

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 an equation 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

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)

6 0.90

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.

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Reasoning 21

22, 24

21, 23, 25

21, 25, 26

If a line were plotted on each of these graphs, what would the gradient represent? a

Petrol used (L)

Amount earned ($)

Hours worked (h)

Distance (km)

Heartbeat (b)

Altitude in metres (a)

Minutes running (m)

The graph shows the conversion between Country A and Country B’s currency: a b c

Use the graph to convert 8 of currency A to currency B.

Number of seconds flying (f)

16 14 12

Use the graph to convert 2 of currency B to currency A.

Discuss the usefulness and limitations of this model.

8 4 2

B 1

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2 3 4 5 6 7 8 9 10

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

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.

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:

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, in terms 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.

d e

Temp (°C)

5 3 1

t (mins) 1

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Problem-solving 27, 28, 32, 33 27

29–31, 33

The graph shows the conversion between miles and kilometres:

29–31, 34, 35

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

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

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

Calculate the cost of 47 L petrol.

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10 16.40

20 32.80

30 49.20

40 65.60

The graph shows the conversion between temperatures in Celsius and Fahrenheit: 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?

d e

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).

20 10

Convert 35 °C into Fahrenheit.

°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.

The table shows Peter’s earnings from ironing shirts: Shirts ironed Earnings ($)

°F

100

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 is it 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

8 12 16 20 24 28 32 36

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The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side.

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, in terms 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

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.

4.03 Linear inequalities After this lesson, you will be able to… • plot linear inequalities on a number line. • interpret worded problems into inequalities and solve them. • check whether a particular value satisfies a linear inequality. • solve real-world problems involving linear inequalities.

Solve linear inequalities Suppose we want to plot the solutions to the inequality 2 (3 + x) < 8 on a number line. 2 (3 + x) < 8 3+x<4 x<1

Divide both sides by 2 Subtract 3 from both sides

To test if x < 1 is correct, we can substitute some values into the original equation to check. Let’s check with 2 values: one less than 1 and one larger than 1. • When x = 0, we have 2 (3 + x) = 2 (3 + 0) = 6, which is less than 8. • When x = 2, we have 2 (3 + x) = 2 (3 + 2) = 10, which is not less than 8.

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To plot the solutions on a number line, we use a hollow circle for the endpoint (since the problem uses a “<” symbol, x = 1 is not included in the solution set): −8

−6

−4

−2

We solve inequalities just we solve equations, except that the inequality sign flips direction if multiplying or dividing by a negative. When plotting inequalities: Symbol < and > ≤ and ≥

Endpoint Doesn’t include Does include

On a plot

Key words “less/greater than” “less/greater than or equal to”

●

“at least/at most” “no less/more than”

Example 1 Consider the inequality 3x + 1 > 4. a Solve the inequality.

Create a strategy Use inverse operations to ﬁnd the values of x that will satisfy the inequality.

Apply the idea Subtract 1 from both sides

Evaluate the subtraction

Divide both sides by 3

Evaluate

Reﬂect and check Test a value which satisﬁes our solution by substituting it into the original inequality. For example, choose x = 3 3×3+1>4 10 > 4 This is true, suggesting the solution is correct.

b Plot the solutions to the inequality 3x + 1 > 4 on a number line.

Create a strategy Plot the solution from part (a).

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Apply the idea To graph the inequality x > 1, plot a hollow circle on point 1 with a ray towards positive inﬁnity. −5

−4

−3

−2

−1

Example 2 Solve: 5a > 15 (2 + a)

Create a strategy Use inverse operations to isolate a on one side of the inequality.

Apply the idea 5a > 30 + 15a –10a > 30 a < −3

Expand the brackets Subtract 15a from both sides Divide both sides by −10 (remembering to flip the inequality symbol)

Reﬂect and check Test a value which satisfies our solution by substituting it into the original inequality. For example, choose a = –4: 5 × (−4) > 15 (2 + (−4)) –20 > –30 This is true, suggesting the solution is correct.

Example 3 Consider this situation: “2 less than 4 groups of p is no more than 18”. a Construct and solve the inequality described above.

Create a strategy Translate the sentence into mathematical symbols, then solve the inequality by isolating x on one side.

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Apply the idea The phrase “is no more than” has the same meaning as “is less than or equal to” (≤), “less than” means substracted from, and “groups of” means a multiplication operation. Write the inequality

Add 2 to both sides

Simplify

Divide both sides by 4

Simplify

b What is the largest value of p that satisﬁes this condition? A

p = 5

p = −5

There is no largest value.

p=4

Create a strategy Determine the largest value of the inequality from part (a) by recalling the deﬁnition of inequality symbols.

Apply the idea Using a number line, we can plot p ≤ 5 as: −5

−4

−3

−2

−1

The inequality p ≤ 5 means that p can take any value that is less than or equal to 5. So the largest value of p that can satisfy the inequality is 5. The correct answer is option A.

Idea summary We solve inequalities just like we solve equations, except that the inequality sign flips direction if multiplying or dividing by a negative. When plotting inequalities: Symbol < and > ≤ and ≥

Endpoint Doesn’t include Does include

On a plot ●

Key words “less/greater than” “less/greater than or equal to” “at least/at most” “no less/more than”

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Check solutions to a linear inequality In some cases, we don’t need to solve for all of the possible solutions. We may just want to know if a particular value will satisfy an inequality (if it is in the solution set). We can do this by substituting the value into the inequality and evaluating to see if the statement is true.

Example 4 Does t = 6 satisfy the inequality 3t − 5 < 12?

Create a strategy Substitute t = 6 into the inequality, then evaluate.

Apply the idea Evaluate the left side of the inequality. 3t − 5 = 3 × 6 − 5

Substitute t = 6

= 18 − 5

Evaluate the multiplication

= 13

Evaluate

To test, we check if 3t − 5 < 12, when t = 6. This results to 13 < 12, which is false. The statement is false, so t = 6 does not satisfy the inequality.

Idea summary To decide if a particular value will satisfy an inequality (if it is in the solution set), substitute the value into the inequality and evaluate to see if the statement is true.

4.03 Practice questions Understanding 1

142

Adventurer 1–3

Trailblazer 1–3

For each inequality: i

x ≥ 29

Plot the inequality on a number line.

Determine if each value satisfies them (makes them true). i

Explorer 1–3

x = 27

x < −29

x = −28

iii

x = 29

x = −30

What happens to the inequality symbol when we multiply or divide both sides of an inequality by a negative number?

Mathspace WA Curriculum – Year 10 mathspace.co

Is each statement true or false? a

The symbols ≤ and ≥ don’t include the endpoint, which we show with a hollow circle.

The symbols < and > don’t include the endpoint, which we show with a hollow circle.

The symbols < and > do include the endpoint, which we show with a ﬁlled-in circle.

The symbols ≤ and ≥ do include the endpoint, which we show with a ﬁlled-in circle.

Fluency 4 Ex 1

5 Ex 2

5, 6a, 7d–l, 8–10, 12, 15, 17, 19, 20

6, 7g–p, 8, 9, 11, 14–16, 18–20

Solve: a

6x − 54 ≥ 30

d g

21 + 7v ≥ 21

8.4 < 1.9c + 2.7

3x + 27 > 3

4 (b + 3) ≤ 12

3.3g + 4.9 ≥ 34.6

24.15 > 1.5s − 5.55

19.8 > 33 (0.4y + 1.208)

0.25 (x − 8.2) + 1.13 ≤ 1.13

Solve: a

8 − y < 7y

4x ≤ 1 − x

−9 + e > 4e

3 − 2m < 5m

−7b − 7 ≥ 7b

−9h + 20 > −14h

2.4m − 34.33 > 7.19 + 3.6m

2.1 (0.37x − 4.63) ≤ 10.5x

−1.2(u + 3.7) ≥ −7.2

Solve: a

For each inequality: i

Solve.

Plot the solutions to the inequality on a number line.

3 < 4x − 1

7v + 4 > v + 16

0.75b − 1.3 ≤ 0.8

m Ex 3

4, 5a–c, 7a–e, 8–10, 12, 13, 17, 18, 20

−4y + 9 > 9

7.6 < 9.1 − 0.2s

3 (j − 1) < 9

−7a − 5 ≤ 2

2m + 6 > 30 − m

5.5a − 13.7 < −11.5

12.2y ≥ 15y − 14

Construct and solve an inequality for this situation: “The sum of 2 groups of x and 1 is at least 7.”

For “5 more than half of q is greater than 13”:

Construct and solve the inequality.

Find the greatest value of q that satisﬁes this condition (if it exists).

Sophia has a budget for school stationery of $32, but has already spent $15.25 on books and folders. Write an inequality that shows how much Sophia can spend on other stationery, where p represents the amount she can spend, and solve for p.

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Xavier struggles to save money and spent $33.40 last week. He needs $32.00 for a movie with friends on the weekend. Write an inequality representing the amount Xavier had at the beginning of the week, where A is the initial amount in his bank account, and solve for A.

To get a grade of C, John needs a total score of at least 300 over four exams. He scored 63, 73, and 98 on the ﬁrst three exams. If x represents what he must score on the last exam to get a C or better, write an inequality and solve for x.

Skye was given $72 for a birthday present. This present, along with earnings from a Saturday job, is being set aside for a mountain bike. The job pays $5.00 per hour, and the bike costs $379. a

Write an inequality to ﬁnd the minimum number of hours that Skye needs to work to afford to buy the mountain bike, where h represents the number of hours worked.

Solve for h. Round your answer to one decimal place.

If Skye can only work her job for a whole number of hours, ﬁnd the minimum number of hours she must work to afford her bike.

Lachlan is planning on going on holiday. He has saved $2118.40, and spends $488.30 on his airplane ticket. a

Let x represent the amount of money Lachlan spends on the rest of his holiday. Write an inequality to represent the situation.

Solve for x.

Find the highest amount that Lachlan could spend on the rest of his holiday.

James wants to order some books from an online bookstore. Each book costs $13.30 and shipping for the entire order is a ﬂat rate of $29.05. James can spend no more than $74.00. a

If B represents the possible number of books that James can buy, write an inequality and solve for B. Round your answer to one decimal place, if necessary.

Find the maximum number of books that James can afford to buy.

A farmer wants to construct a rectangular enclosure alongside a barn. One side of the enclosure will be the length of the barn, which measures 36 metres. The farmer has a maximum of 80 metres of fencing material available for the project. If the width of the roped area is W, form an inequality and solve for the range of possible widths.

Ex 4

For 21 + 7x > 21: Do these values of x satisfy the inequality 21 + 7x > 21? a

x=1

x = 14

x=7

x = −6

x=6

x=0

x = 12

x = 6.8

Do these values of x satisfy the inequality 3.3 − 5.2x < − 3.2x − 7.5? a

144

Do these values of x satisfy the inequality 6 (x + 2) ≤ 48? a

x=0

x = 5.4

Mathspace WA Curriculum – Year 10 mathspace.co

x = −5.5

When breeding certain types of fish it is recommended that the number of female fish is more than double the number of male fish. This can be expressed as the inequality f > 2m, where f is the number of female fish and m the number of male fish. Determine whether these values satisfy the recommendation. a

f = 10, m = 8

Reasoning

f = 23, m = 9

21, 22

f = 13, m = 10

f = 7, m = 4

21–23

21–24

Explain the differences between the inequalities x < 4 and 0 ≤ x ≤ 4.

Isabelle tried to solve the inequality but made a mistake in her work.

25 − 2x < 31

Step 1:

2x < 6

Step 2:

x<3

Determine which step is incorrect and explain the error.

Write the correct working to solve the inequality.

Percy tried to plot the solution to the inequality 4x + 28 ≥ − 8 on a number line, however his answer is incorrect. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Identify the error(s) and explain how to rectify them. 24

Ryan wants to save up enough money to buy a new sports equipment set, which costs $40.00. Ryan has $22.10 that he saved from his birthday. To make more money, he plans to wash neighbours’ windows for $2 per window. The expression which represents the number of windows, w, required to be washed is w ≥ 8.95. Determine whether each statement is correct. Explain your reasoning. a

Ryan must wash more than 9 windows to be able to afford the equipment.

Ryan must wash at least 8 windows to be able to afford the equipment.

If Ryan washed 8 windows and 95% of another window, he could afford the equipment.

The number of windows Ryan must wash to be able to afford the equipment must be greater than or equal to 9.

Problem-solving 25, 26 25

25–27

27–29

Jimmy spends $120 to produce some rockmelons. He is able to sell each one for $9. a

If n represents the number of rockmelons he must sell to make a proﬁt, write an inequality and solve for n.

Find the least number of rockmelons that Jimmy must sell to make a proﬁt.

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The table shows the times of six swimmers in a 1500 m race. Only those who achieve a time of 22 minutes or below qualify for the final. Use the inequality t ≤ 22, where t is the time of each swimmer, to ﬁnd all of the swimmers who qualify for the ﬁnal.

Swimmer James Harry Lachlan Adam Glen Kenneth

Time (minutes) 21 24 22 27 31 23

Neville is saving up to buy an LED TV that is selling for $950. He has $650 in his bank account and expects a nice sum of money for his birthday next month. a

If the amount he is to receive for his birthday is represented by x, write an inequality that models the situation where he is able to afford the LED TV.

How much money would he have in total if his parents were to give him $310 for his birthday?

Would he have enough to buy the plasma TV if his parents were to give him $310 for his birthday?

A particular bus can carry a maximum weight of 4300 kg on board. If the average adult passenger weighs 75 kg and the driver weighs 81 kg, then the number of passengers on the bus at any one time must satisfy 75a + 81 ≤ 4300, where a represents the number of adult passengers. a

There are 58 adults already on board and 8 more at the bus stop waiting to get on. Find the combined weight of everyone on board if the driver lets all of the new passengers on.

Can the driver safely let all the passengers waiting at the bus stop to get on board?

A particular beef mince brand must contain between 10% and 15% fat before it is shipped. To make 1500 grams of mince which is acceptable to be shipped, the processors mix w grams of 18% fat mince with mince containing 3% fat. Write an inequality for the number of grams of fat in the mix, and solve this to ﬁnd the least and greatest acceptable values of w.

Investigation: Experiment with linear inequalities (Optional) Investigate online

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mathspace.co

4 Chapter review 1

For the equation y = –3x + 7: a

Complete the table of values:

Sketch the line on a Cartesian plane.

For the linear equation

x y

−3

−2

−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, −4) and has a gradient of .

Sketch the graph of 2x + 3y = 12 by ﬁnding the x- and y-intercepts.

A water tank leaks at a constant rate. The volume of water in millilitres remaining after t minutes is shown below: Time (min) (t) Volume (mL) (V)

0 500

5 475

10 450

15 425

Plot the points on a Cartesian plane.

Write the linear equation relating t and V.

What does the gradient represent in this context?

A cyclist travels at a constant speed. The distance in kilometres after t hours is given in the table: Time (h) (t) Distance (km) (d)

0 0

1 15

2 30

3 45

Find the cyclist’s speed in km/h.

Write the linear equation relating t and d.

How far will the cyclist travel in 4.5 hours?

A phone plan charges a base fee plus a rate per minute. The total cost in dollars for m minutes is shown below: Minutes (m) Cost (C)

0 10

5 12.5

10 15

Find the cost per minute.

Determine the base fee.

Write the linear equation for the cost.

Calculate the cost for a 20-minute call.

15 17.5

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147

The graph shows the conversion between litres and gallons for a car’s fuel tank: L

55 50 45 40 35 30 25 20 15 10 5

gal 0

148

8 10 12 14 16

If the tank holds 10 gal, how many litres is this?

A trip requires 45 L. How many gallons is this?

Does the coordinate pair satisfy the inequality? a

(4, 2); 3x − y ≥ 7

(−3, 5); y < −x + 2

A student earns $12 per hour tutoring and $8 per hour babysitting. They need to earn at least $100 per week to cover expenses. a

Write an inequality relating tutoring hours t and babysitting hours b.

Will working 5 hours tutoring and 4 hours babysitting meet the goal?

A bakery sells cakes for $25 each and cookies for $2 each. They need to sell at least $150 worth of goods daily. a

Write an inequality for cakes c and cookies k.

Does selling 4 cakes and 20 cookies satisfy the requirement?

Find the minimum number of cakes needed if no cookies are sold.

Mathspace WA Curriculum – Year 10 mathspace.co

“Mathematics allows us to model real-world situations using abstract language.” Keith Devlin

Big ideas Understanding quadratic relationships requires identifying their key features which highlight various characteristics of the function.

Curriculum Outcomes • (Optional) Solve monic and non-monic quadratic equations graphically and algebraically, including the use of the quadratic formula, factorising techniques and digital tools and verify the solution/s by substitution • Use a table of values to plot points and graph quadratic functions of the form y = ax2 + c. Identify and relate key graphical and algebraic features and make connections to the graphical and algebraic solution/s of ax2 + c = k. Use digital tools to explore the shapes, features and related solutions to more complex quadratic functions • (Optional) Graph monic and non-monic quadratics of the form y = ax2 + bx + c or y = a(x − p)2 + q and their transformations, manually and using digital tools. Identify and connect key graphical and algebraic features and make connections to the algebraic solution/s

5 Quadratics Chapter outline 5.01 5.02 5.03 5.04 5.05

152 162 169 175 198 204

Quadratics can model real-life situations, like the projectile motion of a basketball flying through the air!

5.01 Graphs and features of quadratics After this lesson, you will be able to… • identify the key features of a parabola: intercepts, turning points, and the axis of symmetry. • graph quadratic relationships using a table of values.

Identify key features Intercepts are where the parabola crosses the horizontal (x) and vertical ( y) axes. • The x-intercepts are where the parabola crosses the x-axis. This occurs when y = 0. • The y-intercept is where the parabola crosses the y-axis. This occurs when x = 0. Note, that there can be 0, 1, or 2 x-intercepts. x-intercepts

3 2 1

−6 −4 −2

x 2

−1

−2 y-intercept

−3

For now, we will just identify the intercepts from a graph, later we will look at doing it algebraically.

−4

Parabolas can have a maximum or minimum value. We call this a turning point, or vertex. If a parabola opens upwards, we call this concave up and it will have a minimum value. If a parabola opens downwards, we call this concave down, and it will have a maximum value. 3

1 −3 −2 −1

−1

1 −3 −2 −1

Min value (Minimum turning point)

Concave up, with minimum value

152

−3 −4

−2

−5

Max value (Maximum turning point)

Mathspace WA Curriculum – Year 10 mathspace.co

−1

x 1

−2 −3

Concave down, with maximum value

Parabolas are symmetrical. The axis of symmetry is the dotted line pictured here. It has an equation of the form x = ⬚, and you can see that this x-value will be the same as the x-value for the maximum or minimum turning point.

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Example 1 For each graph, identify: 1. the turning point 2. whether the turning point is a maximum or minimum 3. the equation of the axis of symmetry 4. any intercepts that can be identiﬁed a 16

14 12 10 8 6 4 2 −4

−2

x 2

Apply the idea 1. The graph has a turning point at (0, −2). 2. This graph has a minimum turning point because it is concave up. 3. The equation of the axis of symmetry is the vertical line through the turning point. The equation is x = 0 4. The graph has x-intercepts at (−1, 0) and (1, 0) and a y-intercept at (0, −2).

5.01 Graphs and features of quadratics mathspace.co

153

y 50 25 x −10−8 −6 −4 −2 −25

2 4 6 8 10

−50 −75 −100

Apply the idea 1. The graph has a turning point at (0, 25). 2. This graph has a maximum turning point because it is concave down. 3. The equation of the axis of symmetry is the vertical line through the turning point. The equation is x = 0 4. The graph has x-intercept at (0, −5) and a y-intercept at (0, 5).

Idea summary The key features of a quadratic graph are: • Turning point: point where the graph has a maximum or minimum and it changes between increasing and decreasing. • The minimum or maximum value of the graph is the y-value of the turning point. • y-intercept: point where the graph crosses the y-axis. • x-intercept(s): point(s) where the graph touches or crosses the x-axis, there may not be any. • Axis of symmetry: vertical line that cuts the graph in half; it passes through the turning point.

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Plot from table of values If we have an equation for a quadratic function we can create a table of values to create a series of points to plot.

Example 2 Consider the quadratic function y = 2x2 + 1. a Complete the table of values from x = −2 to x = 2 and graph the parabola. −2

x y

−1

Create a strategy Substitute each x-value into y = 2x2 + 1 to complete the table. Plot the points and draw a smooth curve.

Apply the idea Calculate y for each x: y = 2(−2)2 + 1 Substitute x = −2 = 2 × 4 + 1 Evaluate the square = 9 Multiply Similarly, compute for x = −1, 0, 1, 2: x y

−2 9

−1 3

0 1

1 3

2 9

Plot these points and draw the parabola: 20 18 16 14 12 10 8 6 4 2 −4 −3 −2 −1

x 1

Key features: The vertex is at (0, 1), the parabola opens upward (since a = 2 > 0), and it is symmetric about the y-axis.

5.01 Graphs and features of quadratics mathspace.co

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b Solve 2x2 + 1 = 19 and show those solutions on the graph.

Create a strategy Solve the equation 2x2 + 1 = 19 algebraically to find the x-values, then plot these points on the graph from part a.

Apply the idea Start with the given equation

Subtract 1 from both sides

Divide both sides by 2

Take the square root of both sides

Evaluate the square root

The solutions are x = −3 and x = 3.

20 18 16 14 12 10 8 6 4 2 −4 −3 −2 −1

x 1

Reﬂect and check Our two solutions are x = −3 and x = 3. To check these, substitute each into the left-hand side of the equation and check if the resulting value will be the same value as the right-hand side of equation. For example, substitute x = −3: Left-hand side = 2x2 + 1 2

Write the left-hand side of the equation

= 2 × (−3) + 1

Substitute x = −3

= 2 × 9 + 1

Evaluate the power

= 19

Evaluate

This is the same as the right-hand side of the equation, so the value of x = −3 is correct.

Idea summary The graph of a quadratic relationship is called a parabola. To graph it, we can complete a table of values and plot the points on the Cartesian plane.

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Did you know?

Quadratics are used to design curved ramps that are safer and more comfortable to use! Engineers use parabolic shapes to make ramps that start flat, get slightly steeper in the middle, then flatten out again at the top. This smooth curve makes it easier for people using wheelchairs, prams, or walkers to move around safely in public spaces.

5.01 Practice questions Understanding 1

Explorer

Adventurer

1, 2

Trailblazer

1–3

Describe these curves as concave up or concave down. a

1 −1

−1 −2

x 1

−1

x 1

−2

−3

−4

5.01 Graphs and features of quadratics mathspace.co

157

Determine if the statements are true or false. a

The axis of symmetry is a vertical line passing through the turning point.

A minimum turning point is the highest point on the graph.

The x-intercepts are the points where the graph cuts the y-axis.

The y-intercept occurs where x = 0.

In quadratic equations, there are always two x-intercepts since the degree is 2.

Describe the relationship between the turning point and the axis of symmetry.

Fluency 4

4, 5, 6abeg, 7

4, 5, 6acfg, 7, 8

4, 6c-eh, 7, 8

For each graph: i

Describe the graph as concave up or concave down.

Is the turning point a maximum or minimum point?

x −4 −3 −2 −1

4 20

−5 15

−10

−15

5 x −2

2 −1

−2

x 1

−4 −6 −8

158

8 4

−3 −2

−1

Mathspace WA Curriculum – Year 10 mathspace.co

−2 −4 −6 −8 −10 −12 −14

y x 1

Ex 1

For each graph: i

Identify the turning point.

State whether the turning point is a maximum or minimum.

iii

Identify the equation of the axis of symmetry.

Identify any intercepts that can be identified.

10 8 6 4 2

25 20 15

−1.5 −1 −0.5 −2 −4 −6 −8 −10

10 5 x −1.5 −1 −0.5

1.5

Create a table of values from x = −2 to x = 2.

Plot the points and join with a parabolic shaped line.

y = 8x2 + 1

y = 6x − 5

y = −5x2 + 4

y = −15x − 1

y = −12x2 − 2 2

y=x −7

This diagram shows only part of the graph of a quadratic relationship. a

Complete the table of values for the graph. x y

x 0.5

1.5

For each equation:

e 7

0.5

−3

−2

−1

Does the vertex represent a minimum or a maximum point?

If we knew the equation of the curve, what would be the sign of the coeﬃcient of x2?

y = −7x2 + 10

y = 13x2 + 3

22 y 20 18 16 14 12 10 8 6 4 2 −2 −4 −6

x 1 2 3 4 5 6 7 8 9 10

For the equation y = −21x2 + 4: i

State whether the graph is concave up or down.

Find the turning point.

iii

Sketch the graph using at least five points.

5.01 Graphs and features of quadratics mathspace.co

159

Reasoning

9, 10, 12, 13

9–11, 13

Explain why the graph of y = ax2 + c has a minimum value of c when a > 0.

Determine whether this statement is always true, sometimes true, or never true. Justify your answer with examples or counter-examples. “The graph of y = ax2 + c has either a maximum or minimum value, but not both.”

Justify the signiﬁcance of the y-value of the turning point in determining the range of values of the y-values for the quadratic function y = ax2 + c.

Match each equation to its graph. A

y = 16x2 −36

40 y

y = −4x2 + 25

35 30 25 20 15 10 5

−2

iii

−1 −5 −10 −15 −20 −25 −30 −35 −40

160

−2

−1 −5 −10 −15 −20 −25 −30 −35 −40

x 1

40 y 35 30 25 20 15 10 5

x 1

y = −9x2 + 25

D 40 y 35 30 25 20 15 10 5

x 1

35 30 25 20 15 10 5

−1 −5 −10 −15 −20 −25 −30 −35 −40

y = 49x2 − 36

40 y

−2

−1 −5 −10 −15 −20 −25 −30 −35 −40

Explain how the values of a and c affect the solutions to ax2 + c = 20 and their positions on the graph of y = ax2 + c.

Mathspace WA Curriculum – Year 10 mathspace.co

Match the given quadratic equations with their corresponding graphs and features. Equations: A

y = −20x2 − 20

y = 8x2 + 4

y = 4x2 + 4

Parabola concave up, turning point at (0, 4), symmetric about the y-axis.

Parabola concave down, turning point at (0, −20), symmetric about the y-axis.

y = −12x2 − 20

Graphs and features: a

−2

−20

−30

−40

−50

x 1

−10

−2

−1

Parabola concave up, turning point at (0, 4), symmetric about the y-axis.

Parabola concave up, turning point at (0, −20), symmetric about the y-axis.

y −2

−1

x 1

−10 40 −20 30 −30 20

−40

−50

x −2

−1

Problem-solving 16 14

14–16

A small bakery’s weekly profit is modelled by P = 20x2 − 80, where P is the profit in dollars and x is the number of dozen cupcakes sold. a

Does the parabola open upward or downward? Does it have a minimum or maximum value?

Find the coordinates of the vertex of the parabola.

What is the minimum value, and what does it represent in this context?

Solve 20x2 − 80 = 420 and interpret the solutions.

5.01 Graphs and features of quadratics mathspace.co

161

A tutoring service charges a fee that increases with the square of the hours booked. It charges $25 for 1 hour and $100 for 2 hours. a

Complete the table for the fee for 2, 3, … 6, hours. Hours Fee

1 $25

2 $100

Plot the graph of the fee against the number of hours.

Find the equation for the fee y as a function of hours x.

How much would a 8-hour session cost?

If a client paid $2500, how many hours were booked?

The distance a toy rocket travels upward is modelled by D = 9x2 + 16, where D is the distance in meters and x is the time in seconds after launch. a

Create a table of values for x from 0 to 5 and sketch the graph.

Solve 9x2 + 16 = 160 and explain what the solutions mean. Verify using the graph below.

What does the vertex represent in this context?

D 150

100

50 x −5 −4 −3 −2 −1

1 2 3 4 5

5.02 Vertical translations After this lesson, you will be able to… • interpret and describe the effects of vertical translations on quadratic graphs. • identify the equation of a quadratic function given certain points or intercepts. • understand and apply combinations of transformations including vertical translations, compressions, enlargements, and reflections. • find the equation of a specific quadratic function using given transformations.

Vertical translations Interactive exploration

Discover this concept in action online

162

Mathspace WA Curriculum – Year 10 mathspace.co

mathspace.co

The features of y = x2 + q and y = −x2 + q are: y = x2 + q Turning point Minimum turning point at (0, q) Axis of symmetry x=0 For q > 0, no x-intercepts x-intercept(s), also called For q ≤ 0, the x-intercepts are the roots or solutions to the quadratic

y = −x2 + q Maximum turning point at (0, q) x=0 For q ≥ 0, the x-intercepts are

For q < 0, no x-intercepts This is the graph of y = −x2, that has been vertically translated by q units: up (for positive q) and down (for negative q)

Description

This is the graph of y = x , that has been vertically translated by q units: up (for positive q) and down (for negative q)

Example 1 Identify the vertical translation amount and direction from y = x2, for each of these: a y = x2 + 2

Create a strategy The vertical translation is the amount up or down that the graph has shifted from the base y = x2. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Apply the idea The graph has been vertically translated up by 2 units.

5.02 Vertical translations mathspace.co

163

Reﬂect and check Show the graph of y = x2 also on it. y 4 y = x2 + 2

3 2

y = x2

x −4 −3 −2 −1 −1

−2

We can see the turning point has moved up 2 units from (0, 0) to (0, 2).

b y = x2 − 3

Create a strategy The vertical translation is the amount up or down that the graph has shifted from the base y = x2.

Apply the idea In equations of the form y = ax2 + q, the q value is the vertical translation. This equation has a vertical translation of 3 units downwards.

Reﬂect and check Show the graph with the graph of y = x2 on it. y 4 3 2 1 −4 −3 −2 −1 −1 −2 −3

y = x2 2 3

x 4

y = x2 − 3

−4

We can see the turning point has moved down by 3 units from (0, 0) to (0, −3).

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Example 2 Find the equation of the quadratic in the form y = x2 + q that passes through the point (1, 7).

Create a strategy Using the point (1, 7) we can substitute the values of x and y into the equation y = x2 + q and ﬁnd the value of q.

Apply the idea y = x2 + q

Start with the equation

7 = (1) + q

Substitute in the values (1, 7) for (x, y)

7 = 1 + q

Evaluate the square

6 = q

Subtract 1 from both sides

The equation is y = x2 + 6.

Example 3 Find the equation of the quadratic in the form y = ax2 + q that has a y-intercept of −4 and passes through the point (2, −16).

Create a strategy Using the y-intercept provided, we know that the q value in the equation will be −4. We can then use the other point to ﬁnd the value of a.

Apply the idea y = ax2 – 4

Start with the equation

−16 = a(2) – 4

Substitute in the values (2, −16) for (x, y)

−12 = 4a

Add 4 to both sides, and evaluate the square

−3 = a

Divide both sides by 4 2

The equation is y = −3x − 4.

Idea summary In this lesson, we explored vertical translations of quadratic functions in the form of y = x2 + q and y = −x2 + q. The value of q determines the vertical translation: a positive q shifts the graph upwards, while a negative q shifts the graph downwards. The turning point of the quadratic is affected by the vertical translation, but the axis of symmetry remains the same. To ﬁnd the equation of a translated quadratic function, we can substitute the given point into the equation and solve for the unknown constant.

5.02 Vertical translations mathspace.co

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5.02 Practice questions Explorer

Understanding

Trailblazer

1–5

List two equations for a quadratic that has been vertically translated upwards from y = x2.

List two equations for a quadratic that has been vertically translated downwards from y = x2.

What changes and what stays the same, when comparing the graphs of y = x2 and y = x2 + 7?

What happens to the x-intercepts when the graph of y = −x2 + 1 is vertically translated downwards by 2 units?

What is the connection between the y-intercept and the value of q in y = x2 + q? 6, 7, 8ab, 9ab, 10ab, 12

Fluency Ex 1a

Adventurer

1–5

6, 7, 8bc, 9bc, 10bc, 11, 12

Identify the vertical translation amount and direction from y = x2, for these graphs: a

−6

−3

6 3

x 3

−3

−6

−3

y 4

1 −4 −3 −2 −1 −1

x 3

−6

x 1

−2

166

6, 7, 8cd, 9cd, 10cd, 11, 12

1 −4 −3 −2 −1 −1 −2

−3

−4

Mathspace WA Curriculum – Year 10 mathspace.co

x 1

Ex 1b

Identify the vertical translation amount and direction from y = −x2, for these equations: a

Ex 2

Ex 3

y = −x2 + 2

y = −x2 − 3

y = −x2 − 1

For each of these quadratic equations: i

Sketch the graph on a Cartesian plane alongside the graph of y = x2.

Find the value of the y-intercept.

iii

Find the turning point.

y = x2 − 5

y = x2 + 3

y = x2 − 12

y = 2x2 + 10

Find the equation of the quadratic in the form y = x2 + q that passes through the point: a

y = −x2 + 4

(1, 1)

(−4, 5)

(7, 0)

(2, −4)

Find the equation of the quadratic in the form y = ax2 + q that has the y-intercept provided and passes through the point given. Then describe the transformations that have occurred from the base graph of y = x2. a

y-intercept at y = 3, passes through (1, 5)

y-intercept at y = −1, passes through (3, 8)

y-intercept at y = −4, passes through (−6, −184)

y-intercept at

, passes through

A stone is thrown vertically into the air. Its height above the ground is given by the equation h = −5t2 + 20, where h is the height in metres, and t is the time in seconds. Determine the maximum height the stone reaches and the time it takes to reach the ground.

The height of a rocket above the ground is described by the equation h = 4t2 + 10, where h is the height in metres, and t is the time in seconds since the rocket was launched. a

Calculate the rocket’s height 3 seconds after it was launched.

Sketch the rocket’s trajectory for the ﬁrst 3 seconds of ﬂight.

What was the initial height of the rocket above the ground? What feature of the graph does this relate to?

Reasoning 13

13–15

y = −x2 + 3, y = −x2 − 3 2

y = −x − 1, y = −5x − 2

y = −2x2 + 8, y = −2x2 − 10

y = −x2, y = −x2 − 1

With each pair of quadratic equations, identify the quadratic that has the smallest minimum value: a c

13–15

With each pair of quadratic equations, identify the quadratic that has the largest maximum value: a

13, 14

y = x2, y = x2 + 1 2

y = x − 3, y = 5x + 1

y = 3x2, y = 3x2 − 4

y = x2 + 12, y = x2 − 12

Do vertical translations always result in a new turning point?

5.02 Vertical translations mathspace.co

167

Problem-solving 16, 17 16

168

16, 17, 19

16–19

A toy rocket is launched vertically from an elevated platform. The height of the rocket above the ground can be modelled by the equation h = −4t2 + 30, where h is the height in metres and t is the time in seconds. The rocket launches from a platform that is 10 metres above the ground. a

How does the vertical translation of 10 metres affect the equation modelling the toy rocket’s height? What is the new equation?

Using the new equation, ﬁnd the maximum height the toy rocket reaches above the ground.

Calculate the time it takes for the toy rocket to return to the ground. Round your answer to two decimal places.

Sketch a graph showing the movement of the toy rocket.

Water is sprayed into the air from a fountain, forming a parabolic arc. The height of the water above the fountain can be described by the equation h = −2t2 + 8, where h is the height in metres, and t is the time in seconds. a

What do the values of the coeﬃcient −2, and the constant term +8 reﬂect in the context of the problem?

Determine the maximum height the water reaches.

What is the time it takes for the water to hit the ground?

Another fountain nearby has a similar equation, h = −2t2 + 10. Describe the similarities and differences between the two fountain arcs.

The elevation of a hill can be modelled by the equation h = Ax2 + B, where h is the elevation in metres and x is the horizontal distance from the base of the hill in metres. a

5 metres from the base of the hill the elevation is 65 metres, and the elevation is 10 metres at the base, what is the equation?

As we move further and further away from the hill, what happens to the elevation? Is there a problem with the model?

A water tank is being drained, and the water level y in centimetres after x minutes is described by the quadratic equation y = −3x2 + 1500. a

What is the initial water level in the tank?

Find the time it takes for the tank to be completely drained.

What parameter in the equation (the dilation or the vertical translation) affects: i

The amount of water in the tank

The rate the water drains

Mathspace WA Curriculum – Year 10 mathspace.co

5.03 Horizontal translations After this lesson, you will be able to… • define horizontal translation and explain its effect on a graph. • understand and interpret the equation y = (x − p)2 to denote a quadratic function that has been horizontally translated by p units. • identify the direction of horizontal translation based on the sign of p. • explain how the turning point and axis of symmetry are affected by horizontal translation. • apply the concept of horizontal translation to find the equation of a translated quadratic function.

Horizontal translation Interactive exploration

Discover this concept in action online

mathspace.co

The features of y = (x − p)2 are: y = (x − p)2 Minimum turning point at ( p, 0) x=p

Turning point Axis of symmetry x-intercept(s), also called the roots or solutions to ( p, 0) the quadratic This is the graph of y = x2 which has been horizontally translated by p units: Description

– right for p > 0, (x − p)2 – left for p < 0, (x − p)2 – for p = 0, no horizontal translation

y = −(x − p)2 Maximum turning point at ( p, 0) x=p ( p, 0) This is the graph of y = x2 which has been vertically reflected across the x-axis and horizontally translated by p units: – right for p > 0, (x − p)2 – left for p < 0, (x − p)2 – for p = 0, no horizontal translation

5.03 Horizontal translations mathspace.co

169

Example 1 For this graph:

6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1

x 1 2 3

−2

a Identify the horizontal translation amount and direction from y = x2.

Create a strategy The horizontal translation is the amount left or right that the graph has shifted from the base y = x2. Count from the vertex. y 4 3 2 1 −6 −5 −4 −3 −2 −1 −1

x 1

2 3 4

−2 −3 −4

Apply the idea The graph has been horizontally translated left by 4 units.

b Identify the horizontal translation amount and direction from y = x2 to y = (x − 7)2.

Create a strategy In equation y = (x − p)2, the p value is the horizontal translation.

Apply the idea Notice that in the translated equation, the p term inside the brackets has been replaced by 7. This equation has a horizontal translation of 7 units to the right.

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Example 2 From the base y = x2, write the equation of the quadratic function in the form y = (x − p)2 using the description: a Horizontally translated by 9 units to the left

Create a strategy The p value in y = (x − p)2 is the horizontal translation. y 4 3 2 1 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −1

x 1 2

−2 −3 −4

Apply the idea The function has been translated left by 9 units so p = −9. So the equation is y = (x − (−(9))2 which becomes y = (x + 9)2.

b Horizontally translated 11 units to the right and has a vertical reflection across the x-axis

Create a strategy The p value in y = (x − p)2 is the horizontal translation. A vertical reflection means multiplying the entire function by −1.

Apply the idea The function has been translated right by 11 units, so p = 11. The equation is y = (x − 11)2 which is then multiplied by −1, giving us the final equation y = − (x − 11)2.

Idea summary The equation y = (x – p)2 represents a quadratic function that has been horizontally translated by p units. • If p > 0, the graph is translated right. • If p < 0, the graph is translated left.

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5.03 Practice questions Understanding

Adventurer 1–5

Trailblazer 1–5

List two equations for a quadratic that has been horizontally translated left from y = x2.

List two equations for a quadratic that has been horizontally translated right from y = x2.

What changes and what stays the same when comparing the graph of y = x2 and the graph of y = (x − 2)2?

What happens to the y-intercept when the graph of y = −x2 is horizontally translated left?

What is the connection between the x-value of the turning point and the value of p in the equation y = (x − p)2? 6, 7, 8a–d, 9ab, 10ab, 11, 12a–c

Fluency 6

6, 7, 8c–f, 9bc, 10bc, 11, 12b–d, 13, 14

y = (x − 8)2

y = (x + 9)2

y = −(12 − x)2

y = (0.75 + x)2

Identify the horizontal translation amount and direction from y = x2 for these graphs: a

7 6

7 5

2 1

−5 −4 −3 −2 −1 −1

1 2 3 4 5 6 7

−2

−3 −2 −1 −1

x 1

2 3 4

−2

7 6 5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3

172

6, 7, 8e–h, 9cd, 10bd, 11, 12cef, 13, 14

Has the given equation been translated up, down, left or right from y = x2? a

Ex 1a

Explorer 1–5

x 1

Mathspace WA Curriculum – Year 10 mathspace.co

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3

x 1 2 3

Ex 1b

Ex 2

Identify the horizontal translation amount and direction from y = x2 for these equations: a

y = (x − 1)2

y = (3 + x)2

y = (x − 10)2

y = (x + 5)2

y = (−6.5 + x)2

y = −(5 − x)2

For each of these quadratic equations: i

Sketch the graph on a Cartesian plane alongside the graph of y = x2.

Find the value of the y-intercept.

iii

Find the turning point.

y = (x + 1)2

y = (x − 1)2

y = (x + 3)2

y = −(x + 2)2

For each of these quadratic equations: i

Sketch the graph on a Cartesian plane.

Identify the axis of symmetry.

iii

Find the turning point.

y = −(x + 1)2

y = −(x − 5)2

y = −(6 + x)2

Write the equation of the quadratic function that has the form y = (x − p)2 using the description of the horizontal translation from the base y = x2. a

Horizontally translated left 7 units

Horizontally translated right 13 units

Horizontally translated left

Horizontally translated right 4.6 units

Horizontally translated 3 units left and has a vertical reﬂection across the x-axis

Horizontally translated 2 units right and has a vertical reﬂection across the x-axis

units

For these equations, describe the transformations that have taken place from the base y = x2. a e

y = −(x + 3)2

y = (x + 6)2

y = (x − 9)2

y = (x − 0.15)2

y = −(x − 3)2

y = (13 + x)2

Find the equation of the quadratic that: • Passes through the point (5, −1) • Is concave down

A photographer is taking a picture of a tall building, and the camera lens is positioned at H = (D − 1)2 metres above the ground. If the height of the building (H) is 40 metres, ﬁnd the horizontal distance from the photographer to the building.

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173

Reasoning

15, 16

16, 17

Explain why with a quadratic of the form y = (x − p)2, the y-values can never be negative.

What is the maximum y-value the quadratic y = −(x − p)2 + 5 can have? Explain why this is the maximum y-value for any value of p.

When subtracting 3 on a number line we move our starting point 3 units to the left. But when we go from y = x2 to y = (x − 3)2 we translate the graph 3 units to the right. Explain why these are different.

Problem-solving 19 18

18, 19

A marksman is practising at a shooting range with a target positioned at y = (x − 3)2 metres, where y is the height above the ground, and x is the horizontal distance from the shooter. a

If the bullseye is at a height of 16 metres, ﬁnd the horizontal distance from the shooter to the centre of the bullseye.

If the marksman wants to hit a target 25 metres away, at what height should the target be placed?

A hiker travels along a straight path in the forest for 6 hours at a speed modelled by s = (t − 2)2, where s represents the speed in km/hour and t is the time in hours. a

What is the hiker’s initial speed?

At what time is the hiker travelling at a speed of 9 kilometres/hour?

When does the hiker stop momentarily?

Draw a graph to determine the hiker’s fastest speed.

A company’s monthly sales data is modelled by the equation R = (m − 4)2, where R represents the sales revenue in thousands of dollars, and m is the month. The model is only accurate for months 1 through 12. a

Determine the month in which the company had its highest sales revenue

Calculate that maximum revenue.

Find the month when sales ﬁrst reached $6000.

What happens to the revenue at month 4?

Graph the model and describe what happens to revenue over the 12-month period.

Investigation: Transformations of quadratics (Optional) Investigate online

174

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mathspace.co

5.04 Optional: 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. • 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 knowledge of quadratic graphs to solve problems.

Transformations of parabolas We can graph a quadratic 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.

Interactive exploration

Discover this concept in action online

mathspace.co

A parabola can be vertically translated by increasing or decreasing the y-values by a constant number. A translation of y = x2 up by q units gives the equation y = x2 + q. y 4 y = x2 + 2

3 2

y = x2

x −4 −3 −2 −1

1 −1

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 = x2 − 2

−2

Similarly, a parabola can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value and the translation must be squared together. That is, to translate y = x2 to the left by p units we get y = (x + p)2.

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175

y 6 5 y = x2

4 3 y = (x + 2)2 2

y = (x − 2)2

1 −4 −3 −2 −1

x 1

−1

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.

−2

A parabola can be dilated by multiplying every y-value by a constant number greater than 1. So to enlarge 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. y 6 5 y = 2x2 4

y = x2

3 2 1 −4 −3 −2 −1

x2 2

This graph shows y = x2 vertically enlarged by a scale factor of 2 to get y = 2x2 and compressed by x

−1

a scale factor of 2 to get

−2

y 4 3 2 y = x2

1 −4 −3 −2 −1

−1

x 3

−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).

−3 −4

y = −x2

All of these transformations can be performed on a quadratic and are represented by the equation y = a(x − p)2 + q.

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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

Apply the idea

Identify the values of a, p, and q in turning point form where: • a represents a reflection and/or dilation • p represents a horizontal translation • q represents a vertical translation

y = (x + 2)2 shifts y = x2, 2 units to the left. y = −(x + 2)2 reﬂects y = (x + 2)2 across the x-axis. 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 Apply the transformations from part (a) to the turning point.

Apply the idea y 2 1 −4 −3 −2 −1

−1

−2 −3

x 1

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

Reﬂect and check Because the graph has been reflected across the x-axis, the parabola is concave down. This means the turning point will be a maximum.

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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.

Idea summary The parabola y = x2 can be transformed by: • Vertical translation by q units: y = x2 + q • If q > 0, like y = x2 + 3, then it is translated upwards. • If q < 0, like y = x2 − 1, then it is translated downwards. • Horizontal translation by p units: y = (x − p)2 • If p > 0, like y = (x − 2)2, then it is translated to the right. • If p < 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 vertically enlarged (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 − p)2 + q.

Quadratics in turning point form A quadratic equation in turning point form is written as y = a(x − p)2 + q, where a, p, q are numbers. We can ﬁnd the key features and then use these to graph the parabola. The turning point is ( p, q). The axis of symmetry is x = p. 178

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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. Going the other way, we can also find the equation of a quadratic from the graph using these steps. 1. Identify the turning point from the graph and fill in the values of p and q in y = a(x − p)2 + q. 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 2 For the quadratic equation y = −2 (x − 3)2 + 8: a Is the parabola concave up or concave down?

Create a strategy

Apply the idea

Concavity depends on the sign of a: positive for concave up, negative for concave down.

Since the quadratic equation is y = −2 (x − 3)2 + 8 and a = −2 < 0, the parabola is concave down.

b Is the parabola more or less steep than y = x2?

Create a strategy

Apply the idea

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.

Since we have that a = −2, and −2 is further away from zero than 1 is, the parabola is more steep than y = x2.

c Find the coordinates of the turning point.

Create a strategy

Apply the idea 2

A quadratic of the form y = a(x − p) + q has a turning point at ( p, q).

The turning point of the quadratic equation 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 solve for y.

Apply the idea y = −2(x − 3)2 + 8 2

Write the equation

= −2(0 − 3) + 8

Substitute x = 0

= −2(9) + 8

Evaluate the power

= −10

Evaluate

The y-intercept is at (0, −10).

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179

e Sketch the graph of this quadratic. Include the axis of symmetry.

Create a strategy Plot the y-intercept and turning point and draw a curve through the points.

Apply the idea Since the axis of symmetry occurs at x = p, 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 − p)2 + q 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

Factorise the perfect square part

Simplify the constant

= (x + 2) − 4 – 1 = (x + 2) – 5

180

Adding and subtracting 4

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b Find the y-intercept of the curve.

Create a strategy Substitute x = 0 into the equation of the parabola.

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

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 value we found when using the standard form, which tells us we correctly completed the square.

c Find the turning point of the parabola.

Create a strategy We can find the turning point using the equation 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

Write the equation

y = (−2) + 4(−2) – 1

Substitute x = −2

y = −5

Evaluate

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181

d Is the parabola concave up or down? A

Concave up

Create a strategy

Concave down

Apply the idea 2

Look at the coefficient of x . 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, the parabola is concave up. So, the correct answer is option A.

e Plot the curve y = x2 + 4x − 1.

Create a strategy

Apply the idea

Plot the intercepts and turning point and draw a curve through the points.

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 is in the form y = a (x − p)2 + q, then: • The turning point form will be ( p, q). • The axis of symmetry will be x = p. • The y-intercept, can be found by setting x = 0 and evaluating to get (0, a( p)2 + q). 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 equation in factorised form is written as y = a (x − p) (x − q) where p and q are the x-values of the x-intercepts. y 7

−4 −3 −2 −1

2 1 −1

−1

x 1

−2

x 1

−1

−3

−4

y = (x − 4)2 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 Consider the graph shown: 12

10 8 6 4 2 −7 −6 −5 −4 −3 −2 −1 −2

x 1

−4 −6

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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).

b Find the equation in the form y = a(x − p)2 + q.

Create a strategy Use the coordinates of the turning point found in part (a) to fill in p and q in the quadratic equation of the form y = a(x − p)2 + q. 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).

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Example 5 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 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 solve for x using the null factor law.

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-values of the x-intercepts are x = 2, x = −4.

c Determine the coordinates of the turning point of the parabola.

Create a strategy The x-coordinate of the turning point is halfway 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).

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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 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

−1 −2 −3 −4

x 1

Idea summary If a quadratic equation 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.

Quadratics in standard form A quadratic equation in standard form is written as ax2 + bx + c = 0, where 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. 186

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Example 6 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 solve for x using the null factor law.

Apply the idea y = 4x + x2 Write the equation y = x(4 + x) Factorise 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-values of the x-intercepts are x = 0, x = −4.

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

= 4 × 0 + 02

Substitute x = 0

Evaluate

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d Find the coordinates of the turning point.

Create a strategy Use the equation

and substitute the value of x into the equation of the parabola.

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. y = 4x + x2 Write the equation = 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.

Apply the idea y 4 3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

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x 1

Example 7 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

Reﬂect and check The turning point lies on the axis of symmetry, which means x = 1 is the x-value of the turning point.

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).

c What is the y-intercept of this quadratic curve?

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 =8

Substitute 0 to x Evaluate

So, the y-intercept is (0, 8).

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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 equation in standard form is written as y = ax2 + bx + c, where a ≠ 0. To graph the parabola, we can: 1. Find the axis of symmetry using

2. Find the turning point by substituting 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.

5.04 Practice questions Understanding 1

190

Explorer 1–4

Adventurer 1–4

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?

Mathspace WA Curriculum – Year 10 mathspace.co

Trailblazer 1–4

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

Ex 2

Ex 3

What feature is at the point ( p, q) on the parabola deﬁned by y = a (x − p)2 + q?

What features are at the points ( p, 0) and (q, 0) on the parabola defined by y = a (x − p) (x − q)? 5ab, 6ac, 7a, 8a-c, 9, 10ad, 11ab, 12, 13ab, 14a, 15a-e, 16a-c

Fluency

Ex 1

5ac, 6ac, 7ab, 8ace, 9, 10b-d, 11bc, 12, 13bc, 14a, 15c-g, 16c-f

5ad, 6bd, 7cd, 8d-f, 9, 10d-f, 11cd, 12, 13cd, 14, 15h-l, 16d-h

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 = −(x + 3)2 − 6

y = 3 (x + 1)2 + 4

y = −0.5 (x − 6)2 + 1

For each quadratic: 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 quadratic. Include the axis of symmetry.

y = 2 (x − 1)2 − 3

y = 3 (x + 1)2 + 2

y = −3 (x + 5) − 4

For each quadratic equation: i

Write in the form y = a (x − p)2 + q.

Find the coordinates of the turning point.

iii

Sketch the graph of the parabola.

y = x2 + 6x

y = −x2 + 4x − 3

y = −x2 − 8x − 18

y = x2 + x + 3

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191

Ex 4

Each of the graphs has an equation of the form, y = a(x − p)2 + q. Find the values of a, p and q for each graph and write the equation. a

2 −3

−2

−1

x 1

−2

−1

y 12

8 6

x 1

−2

−6

−5 −4 −3 −2 −1 −2

−8

−4

−5 −4 −3 −2 −1

−1

−2

3 −3 −2 −1

−1

−3

−2

−4

−3

Mathspace WA Curriculum – Year 10 mathspace.co

x 1

192

−2

−4

x 1

2 −3

−1

−4

x 1

The parabola y = x2 has been translated 2 units to the right and 6 units down. a

Ex 5

Find the equation of the parabola.

Sketch the graph of the parabola.

y = x (4 − x)

For each quadratic: 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.

y = x (x + 4)

y = (x − 2) (x − 6)

y = (x − 3) (x + 1)

y = (5 − x) (x + 1)

y = (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)

Find the equation of the parabola in factorised form.

What is the turning point? y 14 12 10 8 6 4 2 −2 −1 −2

x 1

−4

5.04 Optional: Quadratic graphs mathspace.co

193

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 − p)2 + q? a

−4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

Ex 6

−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: i

Factorise to ﬁnd the x-intercepts.

Express the equation in the form y = a (x − p)2 + q by completing the square.

iii

Find the coordinates of the turning point.

Sketch the graph of the parabola.

y = x2 − 12x + 32

y = −2x2 − 8x + 10

Describe the concavity of the parabola.

Factorise, and ﬁnd the x-intercepts.

For each quadratic: i

194

7 6 5 4 3 2 1

−1 −2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1

−4 −3 −2 −1

7 6 5 4 3 2 1

iii

Find the y-intercept.

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

Find the axis of symmetry.

y = x + 6x + 5

y = x2 − 8x + 15

y = −x2 + 2x + 24

y = −x2 − 2x + 8

y = x − 2x − 8

y = 8 + 2x − x2

y = x2 + 2x − 3

y = 2x2 + 9x + 9

Mathspace WA Curriculum – Year 10 mathspace.co

Ex 7

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 parabola.

y = x2 − 4x + 1

y = x2 + 6x + 7

y = −4x2 − 4x + 1

y = −(x2 + 8x + 11)

y = x + 6x − 2

y = −x2 + 10x − 23

y = −x2 + 8x − 9

y = x2 + 2x − 11

Reasoning 17

17, 18, 23

17, 18, 20, 23

19, 21, 22, 24

Three different ways of expressing a quadratic equation are: y = a (x − p) (x − q), y = a (x − p)2 + q, 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.

Determine whether each statement is true or false. Justify your answer. a

The turning point of a quadratic graph is always 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 solutions of a quadratic equation set equal to zero can be determined by finding the x-intercepts of the corresponding quadratic graph.

Given a quadratic of the form y = a (x − p)2 + q, what are the conditions on a, p, and q 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.

For the equations: A = (x − 2) (x − 4) and B = 2 (x − 2) (x − 4) a

Which key features are the same for A and B?

Which key features are different for A and B?

5.04 Optional: Quadratic graphs mathspace.co

195

Marvin wants to understand how turning point form and standard form of a quadratic equation are related. Consider the turning point form: y = a(x − p)2 + q a

Expand a(x − p)2 + q and write the equivalent equation to y = a(x − p)2 + q 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 curves have the same intercepts.

Three different quadratic curves have the same turning point and share one common intercept.

Problem-solving 25–27 25

25–28

26–30

The height p, in metres, reached by a ball thrown in the air after t seconds is given by the equation: p = 10t − t2 a

Complete the table of values for p = 10t − t2: t

Graph the relationship p = 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: p = 163 + 38s − s2 where s is its horizontal distance from where it was thrown.

196

Graph the relationship p = 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 WA Curriculum – Year 10 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. y

(150, 10) x

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. 29

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 equation C = 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 equation in factorised form is y = a (x − m) (x − 12). If the equation has a turning point at the point (8, 15), show how we can ﬁnd the values of a and m.

5.04 Optional: Quadratic graphs mathspace.co

197

5.05 Optional: Quadratic equations After this lesson, you will be able to… • recognise that quadratic equations can have zero, one, or two solutions. • apply the null factor law to solve quadratic equations.

Quadratic equations A quadratic equation is an equation where the highest power of the variable is 2. It can be written in the form ax2 + bx + c = 0, where a ≠ 0.

ax2 + bx + c = 0 a b c

the coefficient of x2, not equal to 0 the coefficient of x the constant term

For example, in the equation 2x2 − 3x + 5 = 0, the values are a = 2, b = −3, and c = 5. Quadratic equations can have 0, 1 or 2 solutions. depending on the equation. Consider these examples: 1. x2 = −4 has 0 solutions because a squared number cannot be negative. 2. x2 = 0 has 1 solution: x = 0. 3. x2 = 9 has 2 solutions: x = 3 and x = −3, or x = ± 3. The symbol ± means “plus or minus,” representing both the positive and negative square roots.

Exploration Explore how many solutions these equations have by attempting to solve them: • x2 = 25 • x2 + 1 = 0 What do you notice about the number of solutions based on the value of the constant?

198

Mathspace WA Curriculum – Year 10 mathspace.co

Example 1 Solve for x: a x2 = 16

Create a strategy Reverse the square operation by taking the square root of both sides.

Apply the idea Start with the given equation

Take the square root of both sides

Simplify

The solutions are x = 4 and x = −4.

Reﬂect and check Check: 42 = 16 and (−4)2 = 16, so both solutions satisfy the equation.

b 81x2 − 64 = 0

Create a strategy Isolate the x2 term and take the square root.

Apply the idea Add 64 to both sides

Divide both sides by 81

Take the square root of both sides

Simplify

The solutions are

and

5.05 Optional: Quadratic equations mathspace.co

199

Create a strategy Isolate x2 using inverse operations, then take the square root.

Apply the idea Add 3 to both sides

Multiply both sides by 16

Take the square root of both sides

Simplify

The solutions are x = 12 and x = −12.

Example 2 Solve (7 − x)2 = 81.

Create a strategy Take the square root of both sides directly, then solve the resulting equations.

Apply the idea Take the square root of both sides

Simplify

Solve the two cases: 7 − x = 9 and 7 − x = −9. 7−x=9

First case

−x = 2

Subtract 7 from both sides

x = −2

Multiply both sides by −1

7 − x = −9

Second case

−x = −16

Subtract 7 from both sides

x = 16

Multiply both sides by −1

The solutions are x = −2 and x = 16.

Idea summary A quadratic equation has the form ax2 + bx + c = 0, a ≠ 0, and can have 0, 1, or 2 solutions. If rearranged to x2 = k, solutions are x = ± , where ± denotes both positive and negative roots.

200

Mathspace WA Curriculum – Year 10 mathspace.co

The null factor law The null factor law states that if the product of two or more factors equals 0, then at least one factor must be 0. For example, if

ab = 0 a b

first factor second factor

then either a = 0 or b = 0. We can use this to solve quadratic equations that are not in the form x2 = k. 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 3 Solve: a (x − 8) (x + 11) = 0

Create a strategy Apply the null factor law by setting each factor to 0 and solving.

Apply the idea x−8=0 x=8

Set the ﬁrst factor to 0 Add 8 to both sides

x + 11 = 0

Set the second factor to 0

x = −11

Subtract 11 from both sides

The solutions are x = 8 and x = −11.

b 6n(n + 12) = 0

Create a strategy Use the null factor law to set each factor to 0 and solve.

Apply the idea 6n = 0

Set the ﬁrst factor to 0

n=0

Divide both sides by 6

n + 12 = 0 n = −12

Set the second factor to 0 Subtract 12 from both sides

The solutions are n = 0 and n = −12.

5.05 Optional: Quadratic equations mathspace.co

201

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.

5.05 Practice questions Understanding

Trailblazer 2, 3, 5

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 = 16

−x2 = 4x − 3

A quadratic equation is expressed as a product of two factors set to zero, (x − a) (x − b) = 0. Determine the values of a and b for the following equations. (x − 2) (x − 7)

202

(x + 5) (x − 1)

x(x + 8)

The equation x2 − 144 = 0 has a positive integer solution of x = 12. Find its other solution.

Fluency 6

Ex 2

Adventurer 2, 3, 4bc, 5

Ex 1

Explorer 1–3, 4a, 5

6a–f, 7a–c, 8a-i

6a–g, 7c–f, 8a-i

6e–l, 7c–h, 8c–l

Solve: a

a2 = 2

k2 = 25

x2 = 121

t2 = 294

x2 − 121 = 0

4y2 = 100

15 = x2 − 10

36 = 25y2

−3k2 = −12

81k2 + 8 = 24

−16 = −81x2

Solve: a

(z + 3)2 = 49

(m − 7)2 = 81

(x − 6)2 = 2

(−3 + p)2 = 64

81 = (2 − n)2

(7 − x)2 = 81

(8l + 9)2 = 256

10 ( p2 − 7) = 930

Mathspace WA Curriculum – Year 10 mathspace.co

Ex 3

Solve: a

n (n + 9) = 0

−3p ( p + 6) = 0

(6x − 5)2 = 0

(2 − 5x)2 = 0

(r − 3) (r − 8) = 0

(t + 4) (t + 10) = 0

(7x − 4) (2x − 9) = 0

(4w + 9) (3w − 5) = 0

(5x − 11) (3x − k) = 0

Reasoning

9, 10, 12

y (3y − 10) = 0 (5k − 7)2 = 0

11–13

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.

Mark used the null factor law to solve the equation (x − 9) (x + 13) = 0 and found the solutions x = −9 and x = 13. Is Mark’s answer correct? Explain why.

Explain and justify why the equation x2 + 4 = 0 has no solutions.

Problem-solving 14–16 14

14, 16

16–18

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: R = −(t − 12)2 + 4 When will the revenue be zero?

The equation 4x2 + kx + 16 = 0 has one solution: x = 2. Find the value of the coeﬃcient k.

One solution to the equation ax2 − 32x − 80 = 0 is x = 4. Find the value of the coeﬃcient a.

The equation px2 + qx − 20 = 0 has solutions coeﬃcients p and q.

Given ax2 − c = 0:

and x = 4. Find the value of the

Solve for x, in terms of a and c.

For what values of a and c, would there be: i

No solutions

One solution

iii

Two solutions

5.05 Optional: Quadratic equations mathspace.co

203

5 Chapter review 1

For each graph: i

Identify the turning point.

State whether the turning point is a maximum or minimum.

iii

Identify the equation of the axis of symmetry.

Identify any intercepts that can be identiﬁed.

5 x

−1.5 −1 −0.5

0.5

−15

1.5

For each equation: i

Create a table of values from x = −2 to x = 2.

Plot the points and join with a parabolic shaped line.

y = 7x2 + 2

y = −4x2 + 5

y = −10x2 − 3

y = −6x2 + 9

y = 5x2 − 4

y = −13x2 − 2

y = 2x2 − 6

y = 11x2 + 4

This diagram shows only part of the graph of a quadratic relationship. a

Complete the table of values for the graph. x y

−3

−2

−1

Does the vertex represent a minimum or a maximum point?

If we knew the equation of the curve, what would be the sign of the coeﬃcient of x2?

16 14 12 10 8 6 4 2 −2 −4 −6

204

1.5

−10

−5

−1.5 −1 −0.5

0.5

For the equation y = −17x2 + 6: i

State whether the graph is concave up or down.

Find the turning point.

iii

Sketch the graph using at least ﬁve points.

Mathspace WA Curriculum – Year 10 mathspace.co

x 1 2 3 4 5 6 7 8 9 10

Identify the vertical translation amount and direction from y = x2, for these graphs: a

5 x

−6

−3

y 8

2 1

−4 −3 −2 −1 −2

−4

−6

−8

x 1

y = −x2 + 6

y = −x2 + 3

y = −x2 − 5

y = −x2 − 2

Find the equation of the quadratic in the form y = x2 + q that passes through the point: a

Identify the vertical translation amount and direction from y = −x2, for these equations: a

−10

−4 −3 −2 −1 −2

−5

5 −6

(2, 5)

(−5, 12)

(8, 15)

(3, −10)

y-intercept at y = 4, passes through (2, 12)

y-intercept at y = −2, passes through (4, 14)

y-intercept at y = −5, passes through (−7, −250)

y-intercept at y = 3, passes through (3, −24)

Chapter 5 review mathspace.co

205

Identify the horizontal translation amount and direction from y = x2 for these graphs: a

3 x

−2

x −6

−4

−2

x −4

−2

−10 −8 −6 −4 −2

−2

−2 −4

−5

Write the equation of the quadratic function that has the form y = (x − p)2 using the description of the horizontal translation from the base y = x2. a

Horizontally translated left 8 units

Horizontally translated right 10 units

Horizontally translated left 6.4 units

Horizontally translated right

Horizontally translated 4 units left and has a vertical reﬂection across the x-axis

Horizontally translated

units

units right and has a vertical reﬂection across the x-axis

For these equations, describe the transformations that have taken place from the base y = x2. a

y = (x + 8)2

y = −(x − 9)2

y = −(x − 5)2

y = (x − 11)2

y = (15 + x)2

A farmer is designing a parabolic irrigation arch where the water height is modelled by H = (D − 3)2 metres above the ground, with D being the horizontal distance from the arch’s centre. If the maximum height of the water stream (H) needs to reach 36 metres to irrigate a crop ﬁeld, ﬁnd the horizontal distance from the farmer to the centre of the arch.

206

Mathspace WA Curriculum – Year 10 mathspace.co

(Optional) Each of the graphs has an equation of the form, y = a(x − p)2 + q. Find the values of a, p and q for each graph and write the equation. a

16 14

4 2 1

y 20

x 1

−5 −4 −3 −2 −1 −2

−1 −2

−6 −5 −4 −3 −2 −1

y 10

−4

−6 −8

−10

−12

10 8 6 4 2

−2 −4 −6 −8 −10

−7 −6 −5 −4 −3 −2 −1

2 −8

−6

−4

−2

x 2

−4

−6 −8 −10 −12

(Optional) For each quadratic: 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.

y = x(x + 6)

y = x(5 − x)

y = (x − 4)(x − 8)

y = (x − 5)(x + 2)

y = (6 − x)(x + 3)

y = (2 − x)(7 − x)

Chapter 5 review mathspace.co

207

(Optional) A parabola has x-intercepts at x = −5 and x = 9. 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, −45)

The turning point is a maximum turning point and the y-intercept is at (0, 22.5)

The turning point is a minimum turning point and the y-intercept is at (0, −90)

The turning point is a maximum turning point and the y-intercept is at (0, 12.25)

(Optional) 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 parabola.

y = x2 − 6x + 5

208

y = x2 + 8x + 12

y = −2x − 8x − 3

y = −(x2 + 10x + 16)

y = x2 + 12x − 7

y = −x2 + 14x − 45

y = x2 + 10x − 15

z2 = 144

w2 = 169

64 = 16y2

−25 = −100x2

y = −x + 12x − 20

(Optional) Solve: a

b2 = 16

m2 = 36

x − 196 = 0

9y = 225

−5k2 = −45

49k2 + 15 = 64

20 = x − 16

(Optional) Solve: a

(z + 5)2 = 64

(m − 8)2 = 100

(x − 9)2 = 16

(−4 + p)2 = 81

144 = (3 − n)2

(10 − x)2 = 121

(6l + 11)2 = 169

12( p2 − 4) = 924

(Optional) Solve: a

m(m + 12) = 0

w(5w − 14) = 0

−4p( p + 8) = 0

(q + 10) = 0

(7x − 9)2 = 0

(6k − 11)2 = 0

(3 − 8x)2 = 0

(r − 5)(r − 10) = 0

(t + 6)(t + 15) = 0

(9x − 5)(3x − 8) = 0

(5w + 12)(4w − 7) = 0

(6x − 13)(2x − m) = 0

Mathspace WA Curriculum – Year 10 mathspace.co

“Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.” Emmy Noether

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. • 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 • Use a table of values to plot points and graph exponential functions of the form y = ax where a > 0. Identify and relate key graphical and algebraic features and use these to determine graphical solutions of related equations. Use digital tools to explore the shapes, features and related solutions to more complex exponential functions • Identify and distinguish between linear, quadratic and exponential functions represented by equations, tables of values and graphs • (Optional) Use algebraic techniques to solve exponential equations that involve terms with related bases • (Optional) Interpret and use base-ten logarithmic scales on graphs of real-life contexts • Analyse, represent and interpret problems in real-world situations involving real numbers, absolute and percentage error, linear inequalities, simultaneous equations, real-world formulae, quadratic or exponential functions, taxation, and/or compound interest

6 Exponentials and logarithms Chapter outline 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08

212 222 236 251 275 280 288 297

Carbon-14 dating uses exponentials to help archaeologists date ancient artefacts that are up to 50 000 years old!

6.01 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 is constant. 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 a 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

×3 3 =3 1

212

×3 ×3 9 27 =3 =3 3 9

Mathspace WA Curriculum – Year 10 mathspace.co

The table shows an example of exponential growth, an increase of one unit in x means that we triple the y-value.

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 a 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.

6.01 Exponential relationships mathspace.co

213

Example 1 For y = 2x: a Complete this table of values. x y

−3

−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

2x y

2−3 0.125

2−2 0.25

2−1 0.5

20 1

21 2

22 4

23 8

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 = 2x

−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 −3 −2 −1 −2

−4

−6

−8

C y

D 8

−4 −3 −2 −1 −2

x 1

2 −4 −3 −2 −1 −2

−4

−6

−8

x 1

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

Apply the idea

Look at the pattern in the y-values.

As x increases by 1, y is increasing by 3. This is a linear relationship, so it 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 relationship 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. To check:

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. Exponential relationships have an equation which has a variable in the exponent like y = 2x.

18 y 16 14 12 10 8 6 4 2

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

6.01 Practice questions Understanding 1

Explorer 1ab, 2, 3

Adventurer 1ac, 2, 3

Trailblazer 1cd, 2, 3

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

Describe the pattern in words.

Find the number of circles for step 4.

Find the next two terms. 1 10

2 15

Mathspace WA Curriculum – Year 10 mathspace.co

1 2 3 4000 400 40

Step 1

218

3 20

Step 2

Step 3

Describe the pattern in words.

Find the number of hexagons for step 4.

Step 1 Step 2

Fluency 4 Ex 1

4, 5, 6ab, 7, 8

Complete the table of values. 0 50 1

5 y b 5

4–8

For y = 5x: x

Ex 2

4, 5, 6bc, 7, 8

1 51 5

5 55 3125

Is y = 5x a linear or an exponential relationship?

Are these relationships 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

6 Ex 3

Step 3

−4 −3 −2 −1

x 1 2 3 y 0.8 1.6 2.4

4 3.2

x y

1 0.125

x 1

2 0.5

3 2.0

4 8.0

For these exponential relationships: Find y when x = 5.

Find the growth or decay factor.

x 1 2 3 y 10 35 122.5

x y

1 2

2 30

3 450

4 6750

x 1 2 3 y 0.123 0.246 0.492

x y

1 4096

2 1024

3 256

4 64

4 428.75 4 0.984

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Ex 4

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.

Reasoning 9

9, 10

The cumulative rainfall over a series of 4 days is represented on the graph. a

Time (minutes) 1 2 3 4 5 6

Describe the relationship between time and volume.

Determine if the points represent a linear or exponential relationship. Explain your reasoning.

Assuming the pattern continues over the next few days, ﬁnd the cumulative volume on day seven.

Step 3

Volume (mL) 3 9 27 81

10, 11

Volume (mL)

32 28 24 20 16 12 8 4

Time (days) 1

Consider the graphs: a

Find the rule for graph B.

Find the rule for graph A.

Determine which is an exponential relationship and which is a linear relationship. Explain your reasoning.

16 14

12 10 8 6 4 2

x 1

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Consider the table of values for x ≥ 1 where y1 = (1.05)x and y2 = 5x. Is it correct to say that y2 is always greater than y1 for all values of x ≥ 1?

x y1 y2

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 13, 15 12

12, 13, 15

12–15

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 relationship 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 relationship.

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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6.02 Characteristics of exponential graphs After this lesson, you will be able to… • recognise the standard form of exponential relationships as y = ax. • identify the y-intercept and horizontal asymptote of exponential curves. • determine the behaviour of exponential curves based on the value of a. • use tables of values, technology, or equations to sketch graphs of exponential relationships. • differentiate between exponential growth and exponential decay based on the value of a.

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 = ax where a is the growth or decay factor. 8

All equations of the form y = ax have these features in common: • The growth or decay factor is a. • The y-intercept is at (0, 1). • There is a horizontal asymptote at y = 0.

7 6 5 0<a<1

a>1

The value of a determines whether the curve increases or decreases. • If 0 < a < 1, the curve decreases. • If a > 1, the curve increases.

3 2 1 −4 −3 −2 −1

x 1

Notice the dashed line at y = 0. This line is called an asymptote. An exponential curve can get inﬁnitely close to an asymptote, but it can never cross it. This means that an exponential relationship of this form will not have an x-intercept.

Interactive exploration

Discover this concept in action online

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mathspace.co

As x increases in exponential growth relationships, the graph increases faster for higher values of a. As x increases in exponential decay relationships, the graph decreases faster for smaller values of a. To draw the graph of an exponential relationship, we can use a variety of strategies, including: • Completing a table of values and drawing the curve through the points • Using technology, such as a graphing calculator • Identifying key features from the equation

Example 1 Consider the equation y = 3x. a Complete the table of values. −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

, 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 y

−5

−4

Evaluate the exponent −3

−2

−1

243 59 049

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b Graph the exponential curve.

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 curve or a decreasing curve?

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 curve.

Example 2 Consider the equation

a Determine whether the exponential curve is increasing or decreasing.

Create a strategy

Apply the idea x

For equations in the form y = a , the curve is increasing if a > 1 and the curve is decreasing if 0 < a < 1.

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Since

, the curve is decreasing.

b Identify the equation of the horizontal asymptote.

Create a strategy For any relationship in the form y = ax, the horizontal asymptote is on the x-axis.

Apply the idea

Reﬂect and check

The equation of the horizontal asymptote is y = 0.

Because an asymptote is a line, it must be represented by an equation. Since exponential curves 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 Substitute x = 0 to ﬁnd the y-value of the y-intercept.

Apply the idea Write the equation

Substitute x = 0

Apply zero exponent law

The y-intercept is at (0, 1).

Reﬂect and check For any exponential equation of the form y = ax, the y-intercept will be at (0, 1).

d Find 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 equation

Substitute x = −1

Apply negative exponent law

Another point on the curve is (−1, 4).

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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 equation y = 8x. a Can the value of y ever be negative or zero?

Create a strategy

Apply the idea

Recall the location of the asymptote of the exponential relationship y = ax.

Since y = 8x is an exponential relationship of the form y = ax, 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.

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 ∞.

Create a strategy Substitute some positive values of x into y = 8x.

Apply the idea We can use x = 5, 10. y = 85 = 32 768 y=8

= 1 073 741 824

Substitute x = 5 Evaluate Substitute x = 10 Evaluate

We can see the values are increasing into inﬁnite values or ∞. The answer is option B.

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y tends towards 0.

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 curves, 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

Apply the idea x

Substitute x = 0 into y = 8 .

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.

Did you know?

Exponential growth explains how something small — like a video share — can quickly explode into millions of views. Each person passes it on to a few more, and the numbers grow faster and faster. This pattern shows up in real life too, from the spread of diseases to how your savings grow with interest. Once you recognise the steep curve of an exponential graph, you’ll start seeing it everywhere!

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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 increasing from left to right. • 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 graphs of

and y = 2x.

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 curves of the form y = ax is at (0, 1). For more points, substitute x = 1, 2 into y = 2x: y = 21

Substitute x = 1

=2 y=2

Evaluate the exponent 2

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

b Sketch

on the same Cartesian plane.

Create a strategy Plot the y-intercept and substitute x = −1, 1 into

for more points.

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Apply the idea Recall that the y-intercept for curves of the form y = ax 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

−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 curve.

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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x 1

Idea summary An exponential relationship can be modelled by y = ax

All equations of the form y = ax have these features in common:

• The y-intercept is at (0, 1) • There is a horizontal asymptote at y = 0

6 5 0<a<1

a>1

4 3 2 1

−4 −3 −2 −1

x 1

The value of a determines whether the curve increases or decreases. • If 0 < a < 1, the curve decreases • If a > 1, the curve increases As x increases in exponential growth relationships, the graph increases faster for higher values of a. As x increases in exponential decay relationships, the graph decreases faster for smaller values of a.

6.02 Practice questions Understanding

Explorer 1–4, 5a-d

Adventurer 1–4, 5b-f

Trailblazer 1–4, 5e-h

What part of the exponential equation y = ax determines whether it increases or decreases?

What is the name of the line that the graph of an exponential curve approaches but never crosses?

Is each statement true or false for y = ax? If it is false, correct the statement. a

If a > 1, then it is exponential growth.

If a < 0, then it is exponential decay.

If a is a fraction, then it is exponential decay.

If a > 0, then it is exponential growth.

The y-intercept is always (0, 1).

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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 curves: a

y = 3x

Ex 1

232

b f

Fluency 6

y = 0.97x

6a-d, 7abf, 8, 10, 11

y = 0.5x

y = 1.5−x

6b-e, 7c-f, 8, 9, 11

y = 1.05x

6c-f, 7d-h, 8, 11

For each exponential relationship: i

Complete the table of values.

iii

Determine whether the curve is increasing or decreasing.

x −1 0 1 y = 4x

x −1 0 1 y = 10x

−1

x −1 0

−1

Graph the exponential curve.

−1

y = 5x

Mathspace WA Curriculum – Year 10 mathspace.co

7 Ex 2

For each exponential relationship: i

Determine whether the curve 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.

y = 2x

e Ex 3

y = 4x

y = 7x

y = 8x

For the equation 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 curve 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

11 Ex 4

Consider the graphs of y = 5x and a

Sketch the graph of y = 5x.

Sketch the graph of

Compare the graphs of y = 5x and

−2

−1

. .

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Reasoning 12

12, 13

12–14

12–15

The graphs of two exponential relationships, R and S, are shown. a

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 curve will have smaller values of y? Explain your reasoning.

y R S

Do either of the exponential curves y = 9x or

have x-intercepts? Explain your reasoning.

For the exponential relationships

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 answer.

and

Problem-solving 16, 18, 19 16

16cd, 17–19

16cd, 17, 19–21

Find the missing coordinate in each ordered pair so that the pair is a solution to y = 5x: a

(3, ⬚)

(⬚, 5)

(−1, ⬚)

Find the missing coordinate in each ordered pair so that the pair is a solution of a

(0, ⬚)

(⬚, 9)

(−1, ⬚)

If the point (3, m) lies on the curve y = 2x, solve for m.

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The graph y = 2x is shown: a b

128 120 112 104 96 88 80 72 64 56 48 40 32 24 16 8

Find the value of y when x = 7. Find the value of x when y = 128.

−1

x 1

2 3 4 5 6 7 8

The graph of y = 3x is shown. Find the length of the interval PQ.

y P

−5 −4 −3 −2 −1

x Q (0, 1) 1 2 3 4 5

−x

For the exponential equation y = 3 : a

Rewrite 3−x with a positive index using the index laws.

Sketch the graph of y = 3−x.

Compare the graphs of y = 3x and y = 3−x.

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6.03 Exponential models After this lesson, you will be able to… • differentiate between exponential growth and exponential decay. • identify and define key features of exponential graphs. • express exponential relationships algebraically, highlighting either the growth or decay factor or rate. • apply knowledge of exponential models to solve real-world problems. • interpret the implications of the growth or decay rate in practical scenarios.

Exponential growth

Exponential growth happens when a quantity increases by a certain percentage over time. We can describe this kind of growth with an exponential formula:

y = abx

y 10

When we are thinking about growth or decay, this is with respect to time. So usually we are only interested in positive x-values. • a is the starting value (or initial value), when x = 0. • b is the growth factor, which shows how fast the value grows.

If b > 1, we have growth. If 0 < b < 1, it is decay.

7 6 5

y = 3(1.6x)

1 0

x 1

The compound interest formula: A = P(1 + r)t is an example of exponential growth where r is the growth rate (interest rate) of the investment. So if we connect the exponential growth formula, y = abx, with the compound interest formula we can see that b = 1 + r. This is greater than 1, also confirming that it is growing. We can use the key features of an exponential relationship to build its equation. First, we must find the y-intercept (or the initial value), then find the growth factor (b) or growth rate (r).

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Exploration Here are three exponential relationships represented in three different ways: Relationship 1: 8

Relationship 2: x y

0 1

2 1.3225

3 1.5209

4 1.7490

Relationship 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 0

1 1.15

x 1

1. Which of the relationships would you model using y = abx and why? 2. Which of the relationships would you model using y = a (1 + r)x and why? 3. 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 equation in the form y = abx. We usually refer to the growth rate as a percentage, just like in the compound interest formula. If you need to switch between forms, remember that b=1+r • b is the growth factor • r is the growth rate, usually a decimal or percentage

Example 1 Identify the y-intercept and the growth rate for each exponential equation: a y = 3 (1.725)x

Create a strategy This equation 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.

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Apply the idea

Reﬂect and check

The growth factor, b, is 1.725

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.

1.725 = 1 + r r = 0.725 The y-intercept is (0, 3), and the growth rate is 72.5%.

Create a strategy This equation 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

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 an equation, 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 equation, y = a (1 + r)x with the given variables: V = a (1 + r)t.

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Apply the idea

Reﬂect and check

The initial value of the memorabilia is $2900. This represents the value when t = 0, so this is the y-intercept.

Notice the similarities to the compound interest formula A = P (1 + r)t.

The growth rate is 9% which is 0.09 as a decimal, so the equation is V = 2900 (1 + 0.09)t.

b Evaluate the equation 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. equation and apply these units to interpret the meaning of the solution.

Idea summary An exponential growth relationship can be expressed as

y = abx a b

is the y-intercept, or initial value is the growth factor

In cases where we want to define the specific growth rate, like an interest rate, we use b = 1 + r and end up with a version of the compound interest formula: y = a(1 + r)x

Exponential decay We have seen that exponential relationships can be expressed in the form:

y 4

y = abx where: • a is the starting value (or initial value), when x = 0. • b tells us how fast it is decaying if 0 < b < 1.

3 2

y = 2(0.3)x

1 x 0

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For exponential growth, we saw that for a growth rate r: y = a (1 + r)x Exponential relationships 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 equation y = abx as y = 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.

Exploration Consider these exponential relationships: Relationship 1: A ball is dropped from a ledge. The height of the ball rebounds to 80% of its previous height after each bounce. 9

1 0

x 2

Relationship 2

Relationship 3

Relationship 4: Relationship 5: x y

0 10

1 9.6

2 9.216

3 8.847 36

1. Which ones can be classiﬁed as exponential decay? 2. Explain how you identiﬁed the relationships that decay.

An equation for exponential decay will have a decay factor that is less than 1, which will cause the y-values to decrease over time.

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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 = 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 = 400 (1 − 0.20)x = 400 (0.80)x We can use the key features of an exponential relationship to build its equation. 1. Find the y-intercept. 2. Find the decay factor or decay rate. 3. Choose the specific form of the equation depending on whether you want to highlight the decay factor or the decay rate.

Example 3 Identify the y-intercept and the decay rate for each of these exponential equations: a

Create a strategy

Apply the idea x

This equation is in the form y = ab . 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.

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

Apply the idea x

This equation is in the form y = ab . 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.

We see a = 1 and for the decay rate.

. Next, we need to solve

Equation for decay rate

Substitute

Evaluate the subtraction

The y-intercept is at (0, 1), and the decay rate as a percentage is 75%.

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Reﬂect and check Notice that the original equation tells us that we are keeping losing 75%, as shown with the decay rate.

. This means that we are

Example 4 A frog population, F, has been following the model F = 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 = 480(0.952)n F = 480(0.952)3 F ≈ 414 After 3 years, there would be 414 frogs remaining.

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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.

Apply the idea A good window setting is 0 < n < 120 and 0 < F < 480. 480 440 400 360 320 280 240 200 160 120 80 40

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 0

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 F 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 equation for exponential decay can be expressed as

y = abx a b

is the y-intercept, or initial value is the decay factor

In cases where we want to define the specific decay rate, like an interest rate, we use b = 1 − r and end up with a version of the compound interest formula: y = a(1 − r)x

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6.03 Practice questions Understanding 1

Explorer

of the previous week’s sales.

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 equation, 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?

y = 0.723 (1 − 0.05)x

y = 2500 (0.877)x

y = 5.9 (1.025)x

y = 10 000 (2)x

The local rat population is changing according to the exponential equation y = 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 relationship: x y

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?

Fluency 5

1–4

A store owner checked the sales report for the previous month and found that each week the sales were

Trailblazer

1–4

Are these situations exponential growth, exponential decay, or neither? a

Ex 1, 3

Adventurer

1–4

5–9, 11, 13, 14

6–9, 11, 13, 14

6–8, 10–13

The mass in kilograms of a baby orangutan at n months of age is given by the equation M = 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)

Sketch a graph of M. 6.03 Exponential models mathspace.co

245

6 Ex 2

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% per annum due to poor maintenance. Calculate its depreciated value. Round your answer 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

Ex 4

246

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)

Semi-Final

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?

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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, 16

15–17

15, 17, 18

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 relationship. a

The equation 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, rounded 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)

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 ).

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.

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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

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. 19, 20, 22, 23, 25, 26

19, 22–24, 26, 27

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. Copy and complete the table showing the number of newly infected people for that day. Day Number of people infected

248

Problem-solving 19–21, 23, 25, 26 19

of the

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 were 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?

Mathspace WA Curriculum – Year 10 mathspace.co

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: J = 179 (0.3)t • Pedro: P = 132 (0.6)t a

Find the highest blood sugar level measured for: i

Pedro

Find the rate, as a percentage, that the blood sugar level decreased each hour for: i

Juanita

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)

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?

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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 0

Days 4

Create equations to model 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 0

Do the given equations accurately model the percentage of the content Raymond and Sunyi recall, respectively: R = 100 (1 − 0.15)t and S = 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.

Investigation: Half-life

Investigate online

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6.04 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 a 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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Any exponential relationship can be described as a function, which is a special type of relation where each input corresponds to only one output. Functions can be written in function notation. This notation highlights the independent variable and names the function, which is helpful when working with multiple functions. When we are writing in function notation, instead of writing “y =”, we write “f (x) =”. For example, we can write y = 3x as f (x) = 3x, which means “f is a function of x”. Common letters to use for general functions are lower case f, g, and h.

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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

−4

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The curve has the same shape as

Example 1 This is a graph of y = 3x. a How do we shift the graph of y = 3x 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 to 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 1

y = 3x

−5 −4 −3 −2 −1 −1 −2

x 1

2 3 4 5 y = 3x − 4

−3 −4 −5

Example 2 The graph of

is shown.

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3

a Describe a transformation of the graph of that would obtain

−4

Create a strategy

Apply the idea

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.

Since the new equation is

k = 3 > 0, then it means that the transformation of the graph of g (x) is a translation up 3 units from

b Sketch the graph of

and

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

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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

3 2 1

curve

4 1 f ( x ) =    2

−5 −4 −3 −2 −1 −1

Then, shift each point up 3 units.

The horizontal asymptote shifts up 3 units too.

2 3 4 5

−2 −3 −4 −5

8 7

Connect the points to draw the curve

6 x 5  1 4 g ( x ) =  2  + 3

x 2  1 ( ) f x =   1  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.

The equation of the horizontal asymptote is y = 3.

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

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255

Vertical dilation of an exponential graph We will now look at exponential functions of the form y = a (bx).

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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 enlargement (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

y = 2x

4 x

y = 3 (2 )

The y-intercept for y = 2x is at (0, 1), and the y-intercept for y = 3 (2x) is at (0, 3).

−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 = 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 4 y = 0.5(2 x )

−6 −5 −4 −3 −2 −1 −2

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y = 2x x 1 2 3 4 5 6

Mathspace WA Curriculum – Year 10 mathspace.co

Both curves have y = 0 as the horizontal asymptote.

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

Both curves have y = 0 as the horizontal asymptote.

1 2 3 4 5 6 y = −3(2x )

−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.

Apply the idea For x = 1: y = 2x

= 21

Substitute 1 to x Evaluate

y = 3 × 2

= 3 × 21

Substitute 1 to x

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.

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Example 4 Consider the function

a Find the coordinates of the y-intercept.

Create a strategy Substitute x = 0 to ﬁnd the y-value of the y-intercept.

Apply the idea Write the function

Substitute x = 0

Use the zero power law

Evaluate

The y-intercept is at (0, 2).

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 on the x-axis.

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

258

, the function is decreasing.

Mathspace WA Curriculum – Year 10 mathspace.co

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

graph of

1 f ( x ) =   5

11 10 9 8 7 6 5 4 3 2 1

Then, multiply the y-values of the points by 2.

−4 −3 −2 −1

Substitute x = −1 and x = 0 to ﬁnd two points on the

x 3

Connect the points to draw the curve of

The horizontal asymptote stays at y = 0.

x 1

Idea summary If the function y = 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 across the x-axis. The key features become: • A y-intercept at (0, a) • Horizontal asymptote at y = 0

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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

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

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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

, the equation is given by:

Idea summary or any function of the form y = a (bx), we can ﬁnd the equation from a graph by F 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

Discover this concept in action online

mathspace.co

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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

−4

We 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.

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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. 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 to draw the graph.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

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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

x 1

−1

−2 −3 −4

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

−1

x 1

−2 −3 −4

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Translation upwards 1 unit.

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

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.

6.04 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1–4

For y = 2x and y = 2x + k: a

Are these key features the same or different? Rate of change

Asymptote

iii

y-intercept

Shape

Increasing or decreasing

What happens if k is positive?

What happens if k is negative?

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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?

Match each graph to one of the equations: i

y = 2x

iii b 8

x 1

−4

−6

−8

−4 −3 −2 −1 −2

266

x 1

−4 −3 −2 −1 −2

y = −2x

−4 −3 −2 −1 −2

−4

−6

−8

x 1

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 reflection across the x-axis

A vertical dilation by a factor of

Mathspace WA Curriculum – Year 10 mathspace.co

Fluency Ex 1

5–11, 13, 14c-f, 16, 18, 19bc

y = 2x + 4

The graph of a

y = 2x − 6

is shown.

Describe a transformation of the graph of that would obtain

. on the same

Sketch the graph of .

set of axes as

Write the equation of the horizontal asymptote of .

Ex 3

5–11, 14cf-h, 15, 16, 18, 19cd

Describe how the graph of y = 2x is transformed to get each graph: a

Ex 2

5–9, 12, 13, 14abg, 16, 17, 19ab

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 (2)x

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

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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

268

−4 −3 −2 −1

y 11 10 9 8 7 6 5 4 3 2 1

−4

x 1

Mathspace WA Curriculum – Year 10 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 = 4x + 3

d 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?

What is the equation?

(1, 20)

18 16 14 12 10 8 6 (0, 5) 4 2 −2

−1

x 1

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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?

−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

Reasoning

y = −(2x) + 4

21, 22, 24

y = 0.5 (2x) − 3

20, 22, 24

y = −3 (2x) − 1

20, 23, 25

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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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: x

−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. b 24

Graph y = −10x.

Find the values of x for which y = 0.

The function y = 3 (2x) is shown. 10 8 6

y y = 3(2x)

4 2 −10 −8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

Describe a transformation of the graph of y = 3 (2x) that would obtain y = −3 (2x).

Sketch the graph of y = −3 (2x) 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.

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271

Emir graphed y = 2 (3x) − 1 by transforming a few points. His work is shown. −1

y = 3x

y = 3x − 1

5 4 3 2 1

y = 2 (3 ) − 1

−4 −3 −2 −1

x 1

−1

−3

y-intercept: (0, 1)

y = 2 (3 ) − 1

y = 2 (1) − 1

y = 1

Horizontal asymptote: Translated downwards 1 unit to y = −1

1 −4 −3 −2 −1

One other point: (1, 5) a

Did Emir or Yalile draw the graph correctly?

Explain and correct the error.

−1

x 1

−2 −3

Problem-solving 26, 27

26, 28

27, 29–31

The graph of f (x) = 2x is shown. 5

f (x) = 2x

3 2 1 −5 −4 −3 −2 −1 −1

x 1

2 3 4 5

−2 −3 −4 −5

272

−2

Yalile graphed y = 2 (3x) − 1 by ﬁnding some key features. Her work is shown.

Describe a transformation of the graph of f (x) = 2x that would obtain g (x) = 2− x.

Sketch the graph of g (x) = 2− x on the same set of axes as f (x) = 2x.

Is g (x) = 2−x an increasing or decreasing function?

Mathspace WA Curriculum – Year 10 mathspace.co

For the function f (x) = 4x: a

Sketch the graph of f (x) = 4x.

Complete the table of values for −3

−2

−1

g (x)

Sketch the graph of

Compare the graphs of f (x) = 4x and

How can the graph of

be transformed to get the graph of

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, ⬚) (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? 1 −3 −2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

y x 1

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The equation of the graph is of the form y = a (bx) + k. y (2, 8)

8 6 4 (1, 2)

x −2

−1

1 (0, −1)

−2

−1

x y = a (bx) + k y = a (bx) y = bx

⬚ ⬚

−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?

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

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−2

−1

−2 −4 −6 (−1, −8) −8 −10 −12

x 1

6.05 Optional: 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.

Exponential equations An exponential expression is an expression of the form ax where the base, a, is a positive number and x is a variable. An exponential equation is an equation in which one, or both, sides are exponential expressions, for example: 3x + 2 = 81. The key property we need is that: If ax = ay, then x = y To solve an exponential equation, we can: 1. Write in the form ax = c. 2. Rewrite in the form ax = a⬚.

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.

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

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275

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.

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 3(2) − 4

= 125

Substitute x = 2

25 = 125

Simplify the exponents

15 625 = 15 625

Evaluate on a calculator

276

Write the equation

2+1

Mathspace WA Curriculum – Year 10 mathspace.co

Idea summary An exponential equation is an equation in which one or both sides are exponential expressions, for example: 3x + 2 = 81. The key property we need is that: If ax = ay, then x = y To solve an exponential equation, we can: 1. Write in the form ax = c. 2. Rewrite in the form ax = a⬚. 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.

6.05 Practice questions Understanding 1

Explorer 1, 3, 4

Adventurer

Trailblazer

1–4

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.

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

Complete each statement: a

89 = 8⬚

73 = (⬚)3

Fluency 5

with a base of 10

5–8, 9a-f, 10a-c, 11a-dg, 12ace

c g

4⬚ = 64

(4)2 = (22)⬚

6–8, 9d-i, 10b-d, 11d-i, 12b-f

d h

10⬚ = 1

(32)2 = 9⬚

6, 9h-l, 10d-f, 11e-j, 12d-h

Solve: a

10x = 100

10x = 0.01

10x = 0.0001

6.05 Optional: Exponential equations mathspace.co

277

Ex 1

Ex 2

Solve: a

3x = 36

7x = 1

3x = 27

2x = 64

5n = 625

Rewrite both sides of the equation with a base of 2.

Solve for x.

8x = 4

5x = 0.2

35x − 10 = 1

4x + 4 = 8

81x − 1 = 93x + 5

Solve:

Solve: 9y = 27 2 x+7

b 3

25y = 125 10x + 33

a x − 1 = a4 −3x − 1

(2 )

= 125

9x + 3 = 27x

8x + 3 = 322x − 1

36x − 1 = 2162x + 5

= 3125

Solve: a

5 × 10x = 500

10 × 2x = 80

30 × 2x − 6 = 15

54 × 3x − 6 = 6

40 × 3x + 1 = 3240

18 × 52x − 1 = 450

9x + 2 − 8 = 1

4x + 32 = 288

4−2x × 4x = 64

5x × 5x - 4 = 25

2x × 8x + 1 = 16

i 12

6x = 6–3

For each equation:

a Ex 3

9y = 81

Solve: a

2x × 2x + 2 = 16

Reasoning

13, 15

13–15

14, 16

Explain why 23x − 1 = 27x − 9 cannot be solved using index laws, while 23x − 1 = 8x − 9 can.

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Explain the reasoning for each step of this solution.

Given equation

1 2 3 4 5 6 7

Lorena was trying to ﬁnd the solution to the equation Her working is shown. a

Find and explain Lorena’s mistake.

Correct her solution.

Explain how to solve the equation roots.

Problem-solving 17, 19, 20acf 17

using your knowledge of squares and square

18, 19, 20b-f

18, 19, 20b-h, 21

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 = 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, in Hertz (Hz) of the nth key of an 88-key piano is given by a

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.

6.05 Optional: Exponential equations mathspace.co

279

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.

6.06

Optional: Logarithms

After this lesson, you will be able to… • represent numbers as powers of ten and find their base ten logarithms. • 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. So, 3 is the base 10 logarithm of the number 1000. log10 (1) = 0 because 1 = 100. log10 (0.01) = −2 because 0.01 = 10−2. In general:

log10 (10a) = a a

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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

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

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

6.06 Optional: Logarithms mathspace.co

281

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.

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

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 50 000

(1, 20)

(2, 200)

(3, 3631)

(4, 52 481) 4

x 6

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 log10 y

282

1 1.301

2 2.301

3 3.560

Mathspace WA Curriculum – Year 10 mathspace.co

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.

log10 y

(5, 5.4)

(4, 4.72)

4 (3, 3.56) 3 (2, 2.3)

(1, 1.3)

x 0

Example 2

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

4 Log of area

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 to find the log of each area. Then, find their corresponding value in the vertical axis.

Apply the idea 10 000 has 4 zeros while 100 000 has 5 zeros. This means the logs of the areas are 4 and 5, respectively. We can see from the histogram that the 4–5 column corresponds to 3 in the vertical axis. So there are 3 countries with an area between 10 000 km2 and 100 000 km2.

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Idea summary We can use logs to transform either (or both) axes depending on what is needed to better understand the data.

6.06 Practice questions Understanding 1

102

c d e f

104 −2

103

105

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 = ⬚ 3, 4, 5ab, 6ab, 7–9

3, 4, 5bc, 6bc, 7–9

Express the number as a power of 10.

Find the base ten logarithm of the number.

1 000 000

3, 4, 5cd, 6cd, 7–9

0.000 1

0.001

log10 1000 = 3 can be written in exponential form as 103 = 1000. Write these logarithms in exponential form: log10 100 = 2

log10 1 = 0

log10 10 000 = 4

log10 0.01 = –2

Use technology to ﬁnd the base ten logarithm, rounding your answer to four decimal places: 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: a

284

a 6

10−1

For each number:

a 5

The base ten logarithm of 100 is ⬚ because 10⬚ = 100

Fluency

Trailblazer 1, 2

The base ten logarithm of a number is the power of ten that makes the number. Copy and complete: b

Adventurer 1, 2

Evaluate:

Ex 1

Explorer 1, 2

log10 ⬚ = 1.57

log10 ⬚ = −1.2

Mathspace WA Curriculum – Year 10 mathspace.co

log10 ⬚ = 0.8

log10 ⬚ = 2.31

The points shown are plotted using a log scale. For each point, which coordinates represent the true value, (x, y)? 4

log10 y L

3 J

2 1 −1

x 1

3 K

−2

−3 −4

Point J: A

(1, 2)

(1, 20)

(2, −1)

(2, −10)

(4, 3981)

(4, 36)

(4, 3.6)

(7, 0.00316)

(7, 316.228)

Point M: A

Ex 2

Point L: A

(1, 100)

Point K: A

(7, −2.5)

The table shows the area (in km2) for 12 countries, using a log10 scale: Log of area

Number of countries

3–4 4–5 5–6 6–7

2 2 8 1

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?

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The histogram has been plotted on a log10 scale: Frequency 7 6 5 4 3 2 1 Log value −3

286

−1

What is the total frequency of scores between 1 and 100?

What is the total frequency of scores larger than

What is the total frequency of scores smaller than

Reasoning 10

−2

10, 11

? ?

10, 11

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.

10, 11

City Melbourne Adelaide Wollongong Geelong Hobart Toowoomba Ballarat Bunbury Warragul

In the graph shown, the horizontal scale is linear and the vertical scale is logarithmic. a

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?

Mathspace WA Curriculum – Year 10 mathspace.co

Population 4 929 200 1 396 783 310 479 278 708 231 707 146 955 105 924 80 106 44 625

B C

Problem-solving 12, 13 12

13, 14

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 lifespans of various animals and insects are shown in the table: Animal/Insect Ant African elephant Bowhead whale Bumblebee Chicken Dog Emperor penguin Fly Giraffe Hamster Ladybug Little penguin Monkey Shark

12–14

Lifespan (days) 20 25 500 71 400 21 2920 5110 8500 18 10 120 780 365 2190 7300 10 980

Lifespan in a log10 scale 4.41 1.32 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 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

Minor 2–3.9

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, rounded to one decimal place.

In which category does the quake fall?

6.06 Optional: Logarithms mathspace.co

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6.07 Optional: 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.

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

102

101

0.1

10−1

−1

0.01

10−2

−2

0.001

10−3

−3

0.000 1

10−4

−4

0.000 01

−5

0.000 001

10−6

−6

The order of magnitude forms a logarithmic scale of base 10.

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Increasing order of magnitude

Decreasing order of magnitude

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

Apply the idea

Multiply 450 by 8 and write the product in scientiﬁc notation.

Total = 450 × 8

Multiply

= 3600

Evaluate 3

= 3.6 × 10

Write in scientiﬁc notation

The order of magnitude is 3.

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, which 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.

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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, and every 1 unit down means decreasing by a factor of 10. The Richter scale for measuring earthquakes and the decibel scale for measuring sound are two common applications of logarithmic scales. 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 is recorded with 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

A sound with a reading of 70 on the decibel scale is 102 times more intense than a reading of size 50. A sound with a reading of 90 on the decibel scale is 103 times more intense than a reading of size 60.

290

Mathspace WA Curriculum – Year 10 mathspace.co

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 index law

Evaluate the exponent

Evaluate the expression and round

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

Subtract 50 dB from 80 dB

= 30 dB

Evaluate 3

On the decimal scale, a sound 1000 (10 ) times louder is assigned a decibel value of 30, so the bus terminal sound is 1000 louder than the normal speaking voice.

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291

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, and every 1 unit down means decreasing by a factor of b.

6.07 Practice questions Understanding 1

Trailblazer 1–3

100

1000

1 000 000

0.01

0.1

0.001

Write using scientiﬁc notation: a

600

7000

4 800 000

230

0.0076

0.012

0.00 009

0.1

Find the base ten logarithm of the number. 1 000

Fluency

Ex 2

Adventurer 1–3

Write as powers of 10:

Ex 1

Explorer 1–3

0.000 01

4–10, 12

4–13

5, 6, 8, 9, 12–14

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 combined total number of students enrolled at both universities?

The average distance of Mercury to the Sun is 5.791 × 107 km and the average distance from Uranus to the Sun is 2.871 × 109 km. By how many orders of magnitude do these distances differ?

292

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9 Ex 3

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: 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 9.1 than one of 6.3? Round your answer to the nearest whole number.

Earthquake Sumatra 2004 China 2008 Haiti 2010 Italy 2009

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 0

Richter scale 9.1 7.9 7.0 6.3

Time (hours) 1 2 3 4 5 6 7 8 9 10

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. For example, the concentration of hydrogen ions for a solution with a pH of 1 is 10 times greater than a solution with a pH of 2. 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?

6.07 Optional: Order of magnitude and logarithmic scales mathspace.co

293

12 Ex 4

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.

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 100 decibels, how many times louder is the bus terminal compared to a normal speaking voice?

Decibel value 150 120 110 100 90 70 60 50 30 20

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

lawn mower hair dryer normal conversation refrigerator library

80 70 60 50 40 30 20 10 0

−4

10 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12

294

Type of noise Jet plane Pneumatic drill Industrial Stereo music Inside a car Office Household Wind turbine Bedroom Falling leaves

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?

Mathspace WA Curriculum – Year 10 mathspace.co

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

15, 17

15–17

15–18

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.

A student notices that a logarithmic scale on a graph increases from 2 to 4 over a certain interval. If the scale is base 10, explain why this does not mean the measured quantity increased by 2 times. Then, determine how many times the quantity actually increased.

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.

A scientist measures the intensity of light from two sources on a base 10 logarithmic scale. Source A has a log intensity of 3.2, and Source B has a log intensity of 1.8. Explain how the difference in their log values relates to their actual intensity ratio. Then, calculate how many times brighter Source A is than Source B, rounding to the nearest whole number.

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Problem-solving 19, 22 19

0 50

1 500

2 5000

3 50 000

4 500 000

What is the order of magnitude of the algae population after 4 days?

Express the algae population after each day on a base 10 logarithmic scale, rounding to one decimal place.

If the pond’s capacity is 108 algae, how many days will it take for the population to reach this limit, assuming the same growth pattern continues? Explain your answer.

A telecommunications company measures signal strength in decibels-milliwatts (dBm), a logarithmic scale where 0 dBm corresponds to 1 milliwatt (mW), and every 10 dBm increase represents a 10-fold power increase. A cellphone signal is initially 20 dBm. a

What is the signal strength in milliwatts at 20 dBm?

If the signal drops to −10 dBm due to interference, how many times weaker is the new signal compared to the initial signal?

If the company boosts the signal from −10 dBm to 15 dBm, by how many orders of magnitude does the signal strength increase?

The table of values shows a log scale relating x and y. Form a logarithmic equation relating x and y. log scale measure ( y) 0 1 2 3 4

19–23

A biologist tracks the population of algae in a pond, which grows rapidly due to nutrient influx. The population is recorded over days as follows: Days Population

19, 20, 22

linear measure (x) 5 50 500 5000 50 000

= = = = =

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, rounded to one decimal place.

The population of shrimp is tracked over time and recorded as shown: Days Shrimp

296

1 4

2 16

3 64

4 256

5 1024

Plot the data on a Cartesian plane where x is the number of days and y is the population of shrimp.

Convert the data to a base 10 logarithmic scale.

Plot the data on a Cartesian plane where x is the number of days and y is the number of shrimp using a base 10 logarithmic scale.

Mathspace WA Curriculum – Year 10 mathspace.co

6 Chapter review 1

Are these relationships linear, exponential or quadratic? a

y = 5x + 7

y 1 2 3 x 0.8 1.6 2.4

b d

4 3.2

y = 2x2 + 7 y x

1 0.125

1 2 3 4 5

50 40 30

−6

−8

−10

−4 −3 −2 −1

For these exponential relationships: i

Find the growth or decay factor.

x 1 2 3 y 2000 500 125

4 31.25

Find y when x = 5. x y

1 8

2 20

3 50

4 125

Consider the pattern: a

Find the growth or decay factor.

Find the number of cubes for step 4. Step 1

−4

4 8.0

3 2.0

−5 −4 −3 −2 −1 −2

2 0.5

Step 2

Step 3

For each exponential relationship: i

Complete the table of values.

Graph the exponential curve.

iii

Is the curve increasing or decreasing?

x −1 0 1 x y=6

−1

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297

For the equation y = 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 curve have?

Is each y-value positive or negative?

9 8 7 6 5 4 3 2 1 −3 −2

−1

x 1

A piece of artwork was purchased for $7700, and it is expected to increase in value by 5.3% per year. a

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 now. 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.

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.

Describe how the graph of y = 10x is transformed to get each graph: a

298

y = 10x + 2

y = −10x

Mathspace WA Curriculum – Year 10 mathspace.co

y = 3 (10)x

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

Two exponential relationships, X and Y, are graphed.

1 2 3 4 5

One of the graphs is of y = 3 and the other graph is of

Which is the graph of

For the exponential function

Find the coordinates of the y-intercept.

What is the equation of the horizontal asymptote?

Is the function increasing or decreasing?

Graph the function.

(Optional) The graph of y = 4x is shown. a

Find the value of y when x = 4.

Find the value of x when

14 12 10 8 6 4 2 −5 −4 −3 −2 −1

x 1 2 3 4 5

(Optional) Solve: a

8x = 88

6x−4 = 36

3x = 81

c g

d 5−4x−1 = 1

105x+18 = 100x−3 Chapter 6 review mathspace.co

299

(Optional) Solve: a

3x × 9x+1 = 27

d all lie on the curve with equation y = 3x,

(Optional) Given the points (0, n), (k, 243) and ﬁnd the value of: a

2x × 2x+4 = 8

10 000

(Optional) For each number: i

Express the number as a power of 10.

Find the base ten logarithm of the number.

0.01

(Optional) 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?

10 9 8 7 6 5 4 3 2 1 0

Number of cities

1 2 3 4 5 6 Log of population

(Optional) 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. 21

300

(Optional) 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.

(Optional) For the number 0.000 04: a

Write the number using scientiﬁc notation.

Identify the order of magnitude.

Mathspace WA Curriculum – Year 10 mathspace.co

(Optional) 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?

(Optional) 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.

Type of noise Jet plane Rock concert Ambulance siren Stereo music Electric drill Hair dryer Refrigerator Bedroom Falling leaves

Decibel value 150 120 110 100 90 70 50 30 20

(Optional) 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. Earthquake Mexico 2017 Turkey 2023 Japan 2021 Botswana 2017

Richter scale 8.1 7.8 7.0 6.5

How many times stronger was the earthquake in Turkey compared to the earthquake in Botswana? Round your answer to the nearest whole number.

Chapter 6 review mathspace.co

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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 • Determine the solution to linear simultaneous equations in the forms y = mx + c or ax + by = c graphically and verify the solution by substitution • (Optional) Determine the solution to linear simultaneous equations in the forms y = mx + c or ax + by = c algebraically and verify the solution by substitution or using digital tools • (Optional) Identify the region on the Cartesian plane defined by linear inequalities

7 Simultaneous equations Chapter outline 7.01 7.02 7.03 7.04 7.05

Solve graphically Optional: Solve using substitution Optional: Solve using elimination Optional: Simultaneous inequalities Optional: Applications of simultaneous equations Chapter 7 review

304 312 320

328

Businesses use optimisation to improve efficiency. In 1987, a large airline removed a single olive from each of its in-flight salads, reducing costs by $55 000 a year!

7.01

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

Discover this concept in action online

mathspace.co

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.

304

Mathspace WA Curriculum – Year 10 mathspace.co

Apply the idea 8

y = −x + 5

−5 −4 −3 −2 −1 −2

x 1

−5 −4 −3 −2 −1 −2

−4

−6

−8

y = 5x − 7

x 1 2 3 4 5 y = 5x − 7

−8

For the line y = 5x − 7, m = 5 and c = −7. Plot the y-intercept of (0, −7).

For the line y = −x + 5, m = −1 and c = 5. Plot the y-intercept of (0, 5).

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).

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.

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 y

The point where the two lines meet is at (2, 3). This is the point of intersection.

8 y = −x + 5

The intersection point (2, 3) satisfies both equations, y = 5x – 7 and y = –x + 5.

4 2 −5 −4 −3 −2 −1 −2 −4 −6

x 1 2 3 4 5

When x = 2: • The first equation is y = 5 × 2 – 7 = 3. • The second equation is y = –2 + 5 = 3.

y = 5x − 7

−8

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.

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7.01 Practice questions Understanding

Explorer 1–3

Adventurer

Trailblazer

1–3

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

6 5 4 3 2 1

x 1 2 3 4 5 6

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

306

x 1 2 3 4 5 6

y 6 5 4 3 2 1

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

f 4

1 2 3 4 5

3 2 x 1 2 3 4 5 6

Mathspace WA Curriculum – Year 10 mathspace.co

1 −4 −3 −2 −1

−1

−2 −3 −4

x 1

4b-d, 5ab, 6a-d, 8, 9, 11, 12ab, 13ab

Fluency 4

4ace, 5bc, 6e-h, 7, 9, 11, 12ab, 13ab

4aef, 5bc, 6i-l, 7, 10, 11, 12cd, 13cd

Identify the solution to the simultaneous equations: a

y 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

5 4 3 2 1

4 3 2 1 −3 −2 −1−1

x 2 4 6 8 10

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7 −8 −9

2 4 6 8 10

−1−1 −2 −3 −4 −5 −6 −7

10 8 6 4 2

x 2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

f 4

3 2 x 1 2 3 4 5 6 7 8 9

1 −9−8−7−6−5−4−3−2−1 −1

x 1 2 3 4

−2 −3 −4

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Use the graph to solve the simultaneous equations: y

5 4 3 2 1 −2

−1

y=x−2 1 2

−1 −2 −3 −4 −5

x − 2y = 4 3

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 = 3x + 6

y=x+9

y=x+3

y = 4x − 3

y = 8 − 2x y = −x – 9 f

y = −x + 2 i

y = 1.5x

y = −x − 3 y = 2.5x − 4

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 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 this 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

308

y = 3x + 2

• Studio A: y = 50x + 250

y = 4 − 3x y = 7 − 2x

y = 4 − 0.5x y = 12 − 1.5x 7

2x − 4y = −8

y = −x + 3 −4x + 2y = −8

y = −1 j

• 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 this context?

Mathspace WA Curriculum – Year 10 mathspace.co

A company’s cost is represented by C = 0.4x + 2015, and its revenue is represented by R = 3x, where x represents the number of items produced.

y 2790

What coordinates satisfy both equations?

2325

What do the coordinates from part (a) mean in this context?

1860

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 (b) mean in this 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.

What does the point of intersection of the two lines mean in this context?

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

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

Reasoning

b 14, 15, 17, 18

y = 4x + 5

c 15–18

3y = 12x − 15

2y = 8x − 10

16–19

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?

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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.

2 4 6 8 10

y 5

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.

2 1 −2

−1

x 1

−2

Consider the simultaneous equations: y = −2x − 5 y = −2x + 4 Without graphing, is there a solution? Explain your answer.

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, 21 20

310

20, 21, 23

21–24

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.

Mathspace WA Curriculum – Year 10 mathspace.co

Rochelle and Mohamad are siblings. 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 this 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

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

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7.02 Optional: Solve using substitution After this lesson, you will be able to… • utilise the substitution method to solve for both variables in two linear simultaneous 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 these simultaneous equations using the substitution method. y = x + 11

y = 3x + 19

a First solve for x.

Create a strategy Substitute the expression for y from 1 , into the variable y in 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.

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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.

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.

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Example 2 Solve these simultaneous equations using the substitution method. p + 7q = –4

–6p + 2q = –20

Create a strategy 1 looks to be the easiest to rearrange as the coefficient of p is already 1. Use 1 to get an expression for p in terms of q. Then we will substitute that expression into 2 and solve.

Apply the idea p + 7q = –4

p = –4 – 7q Subtract 7q from both sides

We now have an equation for p, we will call this equation 3 . p = –4 – 7q

Substitute the value of p into 2 and solve for q. –6p + 2q = –20

–6(–4 – 7q) + 2q = –20

24 + 42q + 2q = – 20

24 + 44q = –20

2 Substitute p Expand the brackets Collect like terms

44q = –44

Subtract 24 from both sides

q = –1

Divide both sides by –44

Finally, we now find the value for p. p = –4 – 7q

p = –4 – 7 × –1

Substitute q = –1

p = –4 + 7

Evaluate the multiplication

p = 3

Evaluate

So the solution is p = 3 and q = –1.

Reﬂect and check If p = 3 and q = –1, then these values should satisfy both equations. We can do a quick check at 1 p + 7q = –4. If p = 3 and q = –1, then we have 3 – 7 = –4, so this is true. At 2 –6p + 2q = –20, if p = 3 and q = –1, then we have –18 – 2 = –20, which is also true.

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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

c Solve for y to ﬁnd the age of the son.

Create a strategy Substitute equation 1 into equation 2.

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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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7.02 Practice questions Understanding 1

Explorer

Adventurer

1, 2

1–3

Trailblazer 1, 3, 4

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

Is (5, 2) a solution of the simultaneous equations? x+y=7 x−y=3

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 (4, 17) a solution of the simultaneous equations? y = 6x – 7 4x + 2y = 67

Fluency Ex 1

5a-e, 6, 7a-d, 8, 10

5c-g, 6, 7c-f, 8, 10

5g-l, 6, 7e-h, 8, 9, 11

Solve the simultaneous linear equations using substitution: a

y = 6x + 56

y = 3x + 14

y = −3x − 23

y = −5x − 22

y = −3x + 22

y = 2x + 24

y = 5x + 37

y = −4x – 27

y = 7x + 38 5y = 35x + 265 2y = 10x + 54

x+y=2

y = −3x + 9

x + y = 10 6x − y = 0 j

y = 2x – 15

x + 4y = −37 3x − 4y = 20

5x − 2y = −10

y = 4x – 28 3x + 5y = 44

2x + 3y = 48

y = −5x + 3 4x − 5y = 8

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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

−7f + 2g = −0.4

−4a + 7b = 11.5

−2.1f + g = 0.36 −2a + 2.1b = 2.25 Ex 3

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 + 2 = 7(x + 2) 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 prices of apples and kiwi fruit, respectively. This can be modelled by the simultaneous equations: 2a + 3k = 10.08 5a + 7k = 24.08

Solve the simultaneous equations using the substitution method.

What is the cost of one apple and one kiwi fruit?

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 this 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.

318

Solve for the values of x and y.

What does the solution to the simultaneous equations mean in this context?

Mathspace WA Curriculum – Year 10 mathspace.co

Reasoning

12–14

12, 13, 15

12, 13, 15, 16

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 a

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.

Problem-solving 17, 18, 20a-d 17

17, 19, 20b-e

19, 20c-h, 21

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

y = x2

y = −5x d

(x + 3)2 + ( y − 3)2 = 4

(x + 1)2 + ( y + 4)2 = 25

y = 5x + 4 y = x2 + x + 8

y = 24

x = −5 y = −3x − 2

y = x2 − 5x y = (x − 5) (x − 2)

x2 + ( y + 7)2 = 7

y = −x + 1 y = −7 h

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The plan for a proposed tunnel through a mountain side is shown in the diagram:

y nnel

m the tu

Exit fro

Entrance to the tunnel

ed tunn

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.

7.03 Optional: Solve using elimination After this lesson, you will be able to… • apply the elimination method to solve equations simultaneously.

Solve using elimination As long as equivalence is maintained in 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 a similar way, so that we can reference them in our working later.

320

1 + 2

a+c=b+d

1 − 2

a−c=b−d

Mathspace WA Curriculum – Year 10 mathspace.co

We can also take multiples of equations before combining them with addition or subtraction, for example: 3× 1 −4× 2

3a − 4c = 3b − 4d

× 1 +3× 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.

Apply the idea 3x + y = −

(−4x + y = −11)

7x =

x =

Now that we have 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.

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Example 2 Use the elimination method to solve the pair of simultaneous equations: 2x + y = 4

x = 4y − 34 2

Create a strategy Rearrange 2 to have the same form as 1 , then identify a multiple to eliminate one of the variables. We can remove the variable x, by doing 1 − 2 × 2 .

Apply the idea Start with rearranging 2 by subtracting 4y from both sides, then call it 3 . x = 4y – 34 2

x − 4y = −34

To eliminate x, multiply 3 by 2.

x − 4y = −34

2x − 8y = −68

3 ×2

Then, subtract 3 × 2 from 1 . 2x + y = 4 −(2x – 8y = −68) 9y = 72 y=8 Now that we have 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.

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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.

7.03 Practice questions Understanding 1

Explorer 1, 2

Adventurer 1–3

Trailblazer 1–3

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

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

4, 5, 6ab, 7a-c, 8a-d, 9

4, 5, 6bc, 7b-d, 8b-f, 10

4, 6b-d, 7d-f, 8d-h, 11

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.

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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

Ex 1

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

Ex 2

Use the elimination method to solve each pair of linear equations: a

4x − y = 13

3x + 2y = 22

x = 4y − 16

2 2x = 3y + 9

2y = 2x − 4

6x = y + 15

4y − x = 1

3x − 2y = −15

7x + 13 = 3y − 11

2.5x + 3y = 11

x − 2y = −5

2 1.5x = 2.5y − 2

4.2x − 1.8y = 13.8 3.4x = 2.2y + 8.2

1 h 2

1 2 1 2

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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 this 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 this 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 this context.

Reasoning 12

12, 14

13, 14

13, 15

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

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.

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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

Original equations

y = −x – 2 y = 3x2 – 6 3y = −3x2 – 6 Multiply equation 2 by 3 4y = −12

Add the equations, solve for y

y = −3 −3 = 3x2 − 6

Substitute y, solve for x

3 = 3x 1 = x2 1=x

The solution is (1, −3).

Problem-solving 17, 20ab 16

16, 18, 20b-d

16, 19, 20d-f

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.

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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).

Mathspace WA Curriculum – Year 10 mathspace.co

Use the elimination method to solve the simultaneous equations. a

x2 − y = −2

y = 3x2 − 2

−x + y = 4

x2 + y = 5 −x + y = −1

x2 + y2 = 100

y = x2 2x2 + 10y = 68 e

x2 + y2 = 25

(x + 3)2 + ( y + 1)2 = 25

(x + 3)2 − y = 6

Did you know?

Simultaneous equations help solve real-world problems where multiple conditions must be met at once! For example, engineers use them to design traffic light systems by calculating the best timing for each lane at busy intersections. By analysing multiple variables at the same time, they ensure roads stay safe and efficient. Simultaneous equations are also used in economics to model supply and demand, and in robotics to determine precise movements based on multiple inputs.

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7 Chapter 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 = −x − 3

−8

−7

5 4 3 2 1

y = −3x −13

−6

−5

−4

−3

−2

−1

x + 2y = −6

y = −x − 3

y = −3x − 13 x + 2y = −6 3

x 1

−1 −2 −3 −4 −5

y = −3x − 13 x + 2y = −6

A boutique’s expenses are represented by E = x + 1220, and its sales revenue by S = 1.95x, where x is the number of dresses sold. y 3500 3000 2500 2000

1500

1000 500 x 0

328

256 512 768 1024 1280 1536

What coordinates satisfy both equations?

What do the coordinates from part (a) mean in this context?

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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. 6

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 Cartesian plane.

Find the length of one of the equal sides and the base of the triangle.

(Optional) Solve the simultaneous linear equations using substitution: a

y = 4x + 40

y = −2x – 11

y = 3x + 30 4x + 2y = −22 c

2a + b = 23

−3f + 2g = 6

b = −2a + 35 2f + 5g = 53 8

(Optional) 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 y = −x2 + x + 6

x = −1

(Optional) Solve the simultaneous linear equations using elimination: a

2x + 3y = 8

x − 3y = −14

3x − 6y = 48 x + 2y = 10

0.1x + 0.2y = 0.3 x − y = −22.5

(Optional) Use the elimination method to ﬁnd the solution(s) for each pair of simultaneous equations. a

x2 − y = 9

−x + y = 3

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Big ideas Ratios, and scale factors are tools for identifying similar figures and calculating their areas and volumes.

Curriculum Outcomes • Investigate, explore and determine the effect on the perimeter and area of shapes when they are enlarged or reduced by a scale factor • Investigate in order to determine the effect on surface area and volume when objects are enlarged or reduced by a scale factor

8 Scale factor Chapter outline 8.01 8.02 8.03

Scale and perimeter Scale and area Scale and volume Investigation: Volume versus surface area Chapter 8 review

332 339 344 351

Bonsai trees are miniaturised versions of full-sized trees, just like scale models in mathematics!

8.01 Scale and perimeter After this lesson, you will be able to… • explain how the scale factor impacts the perimeter of similar figures. • apply understanding of scale factors to calculate the perimeter of similar figures. • interpret and solve problems involving perimeters of similar figures and scale factors.

Scale and perimeter Since perimeter is the distance around the outside, the calculation of perimeter involves multiplying the same lengths based on the amount or adding different lengths together. For example: Square

P=4×s

s w

Rectangle

l a

Triangle

P=a+b+c

b a

Parallelogram

P = (2 × a) + (2 × b)

b d

Trapezium

P = 2(l + w)

P=a+b+c+d

b r

Circle

C = 2π r

When adding lengths with the same units, the unit remains unchanged. For example, cm + cm = cm. This means that if all side lengths of a shape are scaled by k, the perimeter is also scaled by k.

Plarge = k × Psmall Plarge the perimeter of the larger ﬁgure Psmall the perimeter of the smaller ﬁgure k the scale factor

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Exploration Consider two similar triangles. If the smaller triangle has side lengths 3 cm, 4 cm and 5 cm, and the scale factor is 2, calculate the perimeter of both triangles. What do you notice about the ratio of their perimeters?

Example 1 The two rectangles shown in the diagram are similar. Find the value of x, the perimeter of the larger rectangle.

x mm 28 mm 10 mm

30 mm

Create a strategy Calculate the scale factor using the corresponding side lengths, then multiply the perimeter of the smaller rectangle by the scale factor.

Apply the idea Scale factor = =3

Find the ratio of the corresponding lengths Evaluate

Perimeter of smaller rectangle = 28 mm

Use the given perimeter from the diagram

x = 28 × 3

Multiply the perimeter by the scale factor

= 84 mm

Evaluate

The perimeter of the larger rectangle is 84 mm.

Reﬂect and check Check: The smaller rectangle has a perimeter of 28 mm and length 10 mm, so its width is 4 mm (since 2 × (10 + 4) = 28). The larger rectangle has length 30 mm and width 4 × 3 = 12 mm. Its perimeter is 2 × (30 + 12) = 84 mm, which matches.

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Example 2 A triangle has side lengths of 8 cm, 13 cm and 15 cm. A second similar triangle has a perimeter that is 2 times larger than the ﬁrst.

8 cm 13 cm 15 cm

a Find the scale factor between the two triangles.

Create a strategy The scale factor is equal to the ratio of the perimeters of similar ﬁgures.

Apply the idea Scale factor = 2

Use the given perimeter ratio

The scale factor is 2.

b What is the relationship between the perimeters of the two triangles? A

The larger triangle’s perimeter is the same as the smaller triangle’s perimeter.

The larger triangle’s perimeter is 9 times the smaller triangle’s perimeter.

The larger triangle’s perimeter is 3 times the smaller triangle’s perimeter.

Create a strategy Use the scale factor from part (a).

Apply the idea Since the scale factor between the two triangles is 2, then each side length will be 2 times longer in the larger triangle. This means that the larger triangle’s perimeter is 2 times the smaller triangle’s perimeter, so the correct answer is option C.

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c Find the side lengths of the larger triangle.

Create a strategy Multiply each side length of the smaller triangle by the scale factor from part (a).

Apply the idea Side 1 = 8 × 2

Scale the ﬁrst side

= 16 cm

Evaluate

Side 2 = 13 × 2

Scale the second side

= 26 cm

Evaluate

Side 3 = 15 × 2

Scale the third side

= 30 cm

Evaluate

The side lengths of the larger triangle are 16 cm, 26 cm and 30 cm.

Reﬂect and check Check: Perimeter of the smaller triangle is 8 + 13 + 15 = 36 cm. Perimeter of the larger triangle is 16 + 26 + 30 = 72 cm. The ratio is

= 2, which matches the given condition.

Idea summary The ratio of the perimeters of two similar figures is equal to the scale factor, k. If the perimeter of the smaller figure is P, the perimeter of the larger figure is k × P.

8.01 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

True or false? a

Perimeter is calculated by adding the lengths of all sides of a shape.

When a shape’s side lengths are multiplied by a scale factor, its perimeter is multiplied by that factor.

The ratio of perimeters of similar ﬁgures is equal to the square of the scale factor.

A rectangle has a perimeter of 18 m. Give the perimeter of the new rectangle if: a

The rectangle was enlarged by a factor of 5.

A scale factor of 0.75 was applied to the rectangle.

A model fence has a perimeter of 45 cm. The actual fence has a perimeter of 15 m. Determine the scale factor used when creating the model based on the actual fence. 8.01 Scale and perimeter mathspace.co

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Fluency Ex 1

4–6, 7ab, 9, 10

4, 6, 7bd, 9–11

4, 6, 7cd, 8–11

Find the value of x for each pair of similar rectangles. a

72 m xm 4m

24 m

336 cm

13 cm

x cm 11 cm

The corresponding sides of two similar pentagons are 4 cm and 10 cm. The perimeter of the smaller pentagon is 25 cm. Find the perimeter of the larger pentagon.

A hexagon has a perimeter of 42 m. Another hexagon is created by scaling it up by a factor of 3.5. Find the perimeter of the larger hexagon.

Find the value of x for each pair of similar shapes. a

11 cm

164 cm

x cm

17 cm 13 cm

64 mm 16 mm 26 mm

26 mm

21 mm

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x mm

x mm 31 mm

580 mm

135 mm

34 m

106.81 m

Ex 2

Two similar trapeziums have corresponding sides in the ratio 3 : 8. The smaller trapezium is isosceles, with its two non-parallel sides (legs) equal, and its shorter base is 5 cm. The perimeter of the smaller trapezium is 33 cm. What is the length of the longer base of the larger trapezium?

A triangle has a perimeter of 35 cm with side lengths of 7 cm, 10 cm and 18 cm. A second similar triangle has a perimeter that is 4 times larger than the ﬁrst. a

Find the scale factor between the two triangles.

Find the perimeter of the second triangle.

A shape with a perimeter of 52 m is enlarged using the scale factor 2.5. Find the perimeter of the enlarged shape.

If a rectangle with a perimeter of 36 km is dilated by a factor of 0.25, what is the perimeter of the dilated rectangle?

Reasoning

12, 13

12–14

A circle has a circumference of approximately 25.12 m. Explain why a circle with a radius three times as large as a circumference of approximately 75.36 m.

Consider two similar garden plots: the ﬁrst with dimensions 6 m by 8 m, and the second with dimensions 15 m by 20 m. a

Find the length scale factor.

Find the perimeter of the smaller garden plot.

Find the perimeter of the larger garden plot.

Justify why the perimeter of the larger garden plot is 2.5 times that of the smaller garden plot.

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A pentagon has been enlarged with a scale factor of k. Describe the relationship between the perimeter of the two pentagons.

Problem-solving 15, 17, 19, 20 15

15–17, 19–22

15–18, 20–22

The ﬁgure shows a park trail used for a charity walk. To extend the event, the organisers dilate the trail boundaries by a factor of 2.2. What is the perimeter of the new trail?

12.2 m

A miniature racetrack has dimensions that are

of the actual racetrack. If the perimeter of

the actual racetrack is 750 m, ﬁnd the perimeter of the miniature racetrack. 17

A garden border has a perimeter of 12 m. In a blueprint, the corresponding perimeter is 60 cm. a

Find the scale factor from the blueprint to the garden border.

The longest side of the border in the blueprint is 25 cm. Determine the length of the longest side of the garden border in real life.

A triangular sculpture has side lengths of 3 m, 7 m and 9 m. a

Calculate the perimeter of the frame.

If the frame is scaled up by a factor of 4, give the side lengths of the enlarged frame.

Calculate the perimeter of the enlarged frame.

By what factor has the perimeter increased?

If a shape is enlarged with a scale factor of k, by what factor will its perimeter increase?

Leo designs custom picture frames. His standard frame has a perimeter of 80 cm. For a gallery exhibit, he creates a larger version with a perimeter of 240 cm. a

Find the scale factor between the two frames.

Leo uses 80 cm of wood to outline the standard frame. How much wood will he need for the larger frame?

A model of a historic castle wall is made in the ratio 1 : 1500. a

The perimeter of the actual castle wall is 900 m. Find the perimeter of the model in centimetres.

The model requires 24 cm of string to outline its perimeter. How much string, in metres, would be required to outline the actual castle wall?

A hexagon has a perimeter of 6y m. Determine the perimeter of the new hexagon if the original is dilated by a factor of: a

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The ﬁgures shown are similar. a

What is the scale factor k?

What is the value of l?

143 m

lm P = 442 m

P = 34 m

8.02 Scale and area After this lesson, you will be able to… • explain how the scale factor impacts the area of similar figures. • apply understanding of scale factors to calculate the area of similar figures. • interpret and solve problems involving areas of similar figures and scale factors.

Scale and area Since area is a two-dimensional measurement, the calculation of area involves two lengths multiplied together. For example: Square Square Rectangle

Rectangle

Triangle

h bh b h hb

Triangle

Parallelogram Parallelogram

Trapezium

Circle Circle

A = s2

s s

b a a h h b b r

A=l×w A= 1 ×b×h 2 A=b×h A = 1 × h × (a + b) 2 2

AA == πrπ2 r

When we multiply two lengths with the same units, the unit for the area is the length unit squared. For example, cm × cm = cm2. This means that if the side lengths of a shape are multiplied by a scale factor, the area of that shape will be multiplied by that scale factor twice (once for each dimension in the formula).

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Interactive exploration

Discover this concept in action online

mathspace.co

The ratio of areas in similar figures is equal to the square of the scale factor. For a scale factor of k, if the smaller figure has an area of A, then the area of the larger figure would be k2 × A.

Example 1 The two rectangles shown in the diagram are similar.

x cm2 15 cm2 12 cm

Find the value of x.

6 cm

Create a strategy Multiply the area of the smaller rectangle by the scale factor squared.

Apply the idea Find the ratio of the side lengths Evaluate

Multiply the area by the scale factor squared

Evaluate

Example 2 A triangle has side lengths of 6 cm, 12 cm and 16 cm. A second similar triangle has an area that is 9 times larger than the ﬁrst. a Find the scale factor between the two triangles.

Create a strategy Take the square root of the area scale factor.

Apply the idea Scale factor = =3

340

Take the square root Evaluate

Mathspace WA Curriculum – Year 10 mathspace.co

b What is the relationship between the perimeters of the two triangles? A

The larger triangle’s perimeter is the same as the smaller triangle’s perimeter.

The larger triangle’s perimeter is 9 times the smaller triangle’s perimeter.

The larger triangle’s perimeter is 3 times the smaller triangle’s perimeter.

Create a strategy Use the scale factor from part (a).

Apply the idea Since the scale factor between the two triangles is 3, then each side length will be 3 times longer in the larger triangle. This means that the larger triangle’s perimeter is 3 times the smaller triangle’s perimeter, so the correct answer is option C.

Idea summary The ratio of areas between two similar ﬁgures is equal to the square of the scale factor.

8.02 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

True or false? a

Area calculations involve the multiplication of two lengths.

If the side lengths of a shape are multiplied by a scale factor, the area of that shape will be multiplied by that scale factor three times.

The ratio of areas in similar ﬁgures is equal to the square of the scale factor.

A square has sides of 5 cm. Give the side length of the new square if: a

The square was enlarged by a factor of 12.

A scale factor of 0.2 was applied to the square.

A scale model of a building has a height of 32 cm. The actual building is 8 m tall. Determine the scale factor used when creating the model based on the actual building.

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Fluency Ex 1

4–6, 8, 9

4, 6, 8–10

4, 6–8, 10

Find the value of x for each pair of similar rectangles. a

320 cm2

x cm2

x cm2 5 cm

20 cm

7 cm 9 cm

27 cm

c 10 cm

x cm2 6 cm

12 cm

The corresponding sides of two similar triangles are 6 cm and 12 cm. The area of the smaller triangle is 18 cm2. Find the area of the larger triangle.

A square has a side length of 8 cm. Another square is created by scaling it up by a factor of 7. Find the area of the larger square.

Two rectangles are similar, with matching sides in the ratio 6 : 7. The smaller rectangle has an area of 576 cm2 and its shortest side has length 8 cm. Find the length of the longest side of the bigger rectangle.

A triangle has side lengths of 3 cm, 12 cm and 14 cm. A second similar triangle has an area that is 25 times larger than the ﬁrst.

Ex 2

Find the scale factor between the two triangles.

Describe the relationship between the perimeters of the two triangles.

A shape with an area of 36 cm2 is enlarged using the scale factor 3. Find the area of the enlarged shape.

If a square with an area of 25 m2 is dilated by a factor of 0.4, what is the side length of the dilated square?

Reasoning

12, 13

11–13

A rectangle has been enlarged with a scale factor of k. Describe the relationship between the area of the two rectangles.

A circle has an area of approximately 78.5 cm2. Explain why a circle with a radius twice as large as an area of approximately 314 cm2.

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Consider two similar ceilings: the ﬁrst with dimensions 5 m by 4 m, and the second with dimensions 20 m by 16 m. a

Find the length scale factor.

Find the area scale factor.

The smaller ceiling took 1.5 L of paint to cover it. How many litres of paint would be required to paint the second ceiling?

Justify why the amount of paint needed for the second ceiling is not simply four times the amount needed for the ﬁrst ceiling.

Problem-solving 15, 16, 18–20 14

14–16, 18–20, 22

All dimensions of the miniature stadium are

14, 16, 17, 19–22

of the actual stadium sizes. If the pitch area

of the actual stadium is 7200 m , ﬁnd the area of the pitch in the miniature version. 15

The figure shows the field on which a game of tag is being played during a physical education class. To make it more of a fitness challenge, the teacher dilates the boundaries of the field by a factor of 1.5 What is the area of the new field?

12.2 m

A rectangular sign has a side length of 1.5 m. In a photograph, the corresponding side has a length of 12.5 cm. a

Find the scale factor from the photograph to the sign.

The area of the sign in the photograph is 75 cm2. Determine its area of the sign in real life.

A rectangular prism has dimensions of 2 cm, 4 cm and 5 cm. a

Calculate the surface area of the prism.

If the prism is scaled up by a factor of 3, give the dimensions of the enlarged prism.

Calculate the surface area of the enlarged prism.

By what factor has the surface area increased by?

If a three-dimensional shape is enlarged with a scale factor of k, by what factor will the surface area increase by?

Maria runs a business where she makes and sells soft toys. Her most popular item is a teddy bear that is 14 cm long. For a special promotion, she decides to make a larger version of the bear that is 56 cm long. a

Find the scale factor between the two bears.

Maria uses 224 cm2 of felt to make the small bear. How much felt will she need to make the big bear?

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A model of the Eiffel tower is made in the ratio 1 : 12 960.

The height of the Eiffel tower is 324 m. Find the height of the model in centimetres.

The model requires 15 mL of paint to cover its entire surface area. How much paint, in litres, would be required to cover the entire surface area of the real Eiffel tower?

The Dominican Republic is shown on the map. According to the scale given, estimate the actual area of the Dominican Republic. Dominican Republic Santo Domingo 0 40 80 km

A square has sides of length x cm. Determine the area of the new square if the original is dilated by a factor of:

a 22

The ﬁgures shown are similar. What is the value of k?

k cm 6 cm Area = 300 cm2 Area = 75 cm2

8.03 Scale and volume After this lesson, you will be able to… • explain how the scale factor impacts the volume of similar objects. • identify and determine the scale factor based on the volumes of similar objects. • apply knowledge of scale factors and volume to find unknown sides in similar shapes. • solve problems involving volume of similar objects and scale factors.

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Scale and volume Since volume is a three-dimensional measurement, the calculation of volume involves three lengths multiplied together. For example: Vrectangular prism = length × width × height

V=l×w×h

Height Width

Length

Vcube = length3 V = s3

s s

Vtriangular prism = Area of triangle × prism height Height

This means that if the dimensions of an object are multiplied by a scale factor, the volume of that object will be multiplied by that scale factor three times (once for each dimension in the formula).

Exploration ×2

×2

Each side length of the cube in the image has been scaled by a factor of 2. 1.

If the original volume was 1 unit3, what is the volume of the new cube?

Did the volume also double? If not, how did it change?

If the original volume was 7 units3, what would the volume of the new cube be?

×2

This means that the ratio of volumes in similar objects is equal to the cube of the scale factor. For a scale factor of k, if the smaller ﬁgure has a volume of V, then the volume of the larger ﬁgure would be k3 × V.

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Example 1 The two rectangular prisms shown in the diagram are similar.

V = 11 cm3 2 cm V = x cm3 8 cm

Find the value of x.

Create a strategy Multiply the volume of the smaller rectangular prism by the scale factor cubed.

Apply the idea

Find the ratio of the side lengths

Evaluate

Multiply the volume by the scale factor cubed

704

Evaluate

Example 2 Spheres are always similar since they have only one deﬁning measurement (the radius). Two spheres have volumes of 40 cm3 and 625 cm3. Find the scale factor between the two spheres.

Create a strategy Find the volume scale factor, then take the cube root.

Apply the idea Find the ratio of the volumes

Evaluate

Take the cube root of the volume scale factor

Evaluate

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Idea summary The ratio of volumes in similar objects is equal to the cube of the scale factor.

8.03 Practice questions Understanding 1

Ex 1

Adventurer

1–3

Trailblazer

1–3

Is each statement true or false? a

Volume calculations involve the multiplication of four lengths.

If the side lengths of a shape are multiplied by a scale factor, the area of that shape will be multiplied by that scale factor three times.

The ratio of volumes in similar objects is equal to the cube root of the scale factor.

A cube has a side length of 2 cm. a

What is the volume of the cube?

What is the volume of the cube if we double the original side length to 4 cm?

What is the volume of the cube if we double the side length again to 8 cm?

What is the effect of doubling the side length on the volume?

Joe wanted to build a 5 m × 10 m pool in his backyard. He then decided he would increase the size of the pool by 50% but keep the depth the same. How much more water would he need to fill the pool with the new dimensions?

Fluency 4

Explorer

4–6, 8–10

4–6, 8, 9, 11

4, 5, 7, 8, 10, 11

Find the value of x for each pair of similar objects. a

V = 20 cm3

3 cm

V = x mm3 V = 36 mm3

3.5 m 1.4 m

V = x cm3 12 cm

V = 62.5 m3

V = x m3

4 cm

4 mm

Two similar cones have bases with radius 7 cm and 28 cm respectively. a

Find the scale factor of the height of the smaller cone to the height of the larger cone.

Find the scale factor of the volume of the smaller cone to the volume of the larger cone.

Calculate the volume of the larger cone, if the volume of the smaller cone is 852 cm3.

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For these two similar trapezoidal prisms: a

Find the length scale factor from the smaller prism to the larger prism.

Find the volume scale factor from the smaller prism to the larger prism.

20 cm

5 cm

The corresponding sides of two similar prisms are 5 cm and 15 cm. The volume of the smaller prism is 40 cm3. Find the volume of the larger prism.

Two similar prisms have volumes of 24 cm3 and 3000 cm3.

Ex 2

Find the scale factor between the two prisms. The two solids shown are similar. Find the value of x. x cm V = 193.7 cm3

41 cm

V = 24 212.5 cm3

The volume of two similar objects are 125 cm3 and 1331 cm3. Find the ratio of the larger side to the smaller side.

The volume of a soup can is reduced from 125 cm3 to 91.125 cm3. What is the scale factor?

SOUP

Reasoning 12

12, 13

13, 14

The length l, width w and thickness h, are the dimensions of a cement slab. If these dimensions are tripled, describe the effect on: a

The surface area of the cement slab

The volume of the cement slab

Basili’s old esky has a capacity of 20 L. Basili wants a new one with a capacity of 160 L, so he plans to buy one with dimensions 8 times the old one’s dimensions. Explain why Basili doesn’t need to buy such a large esky.

A game company makes six-sided dice in two sizes: a standard size with side lengths of s cm and a jumbo size with side lengths of j cm.

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A six-sided die is a cube. Explain why a cube is always similar to another cube.

Give an expression for the scale factor for the two dice in terms of s and j.

Give an expression for the volume scale factor for the two dice in terms of s and j.

Mathspace WA Curriculum – Year 10 mathspace.co

Problem-solving 15–19 15

16–20

18–22

For these two rectangular prisms: a

Find the length scale factor.

Find the surface area scale factor.

Find the volume scale factor.

Find the volume scale factor if the measurements of the smaller prism are doubled.

2.3 cm 1.9 cm

5.1 cm

3.22 cm 2.66 cm 7.14 cm

The surface areas of two similar triangular prisms are in the ratio 64 : 49. Give, as improper fractions: a

The scale factor.

The volume scale factor.

Consider the two spheres shown, where the smaller has radius 3 cm and the larger has a radius of 12 cm. a

Write the ratio (in the form m : n) of the radius of Sphere A to Sphere B.

Write the ratio of the volume of Sphere A to Sphere B.

If the matching dimensions of two similar ﬁgures are in the ratio m : n, what ratio are their volumes in?

Sphere B Sphere A 12 cm 3 cm

Two similar cylinders have volumes of 5760 cm3 and 90 cm3: a

What is the volume scale factor for the two cylinders?

How many times taller is the larger cylinder?

Find the value of h.

h cm 20 cm V = 5760 cm3 V = 90 cm3

The radii of two spherical balloons are 12 cm and 6 cm, respectively. a

What is the scale factor?

How many times greater is the surface area of the larger balloon?

How many times greater is the volume of the larger balloon?

If half the air is released from the smaller balloon, how many times more air does the larger now hold?

Valerie was making a trial birthday cake for her son. The trial cake dimensions are 10 cm for the length, 7 cm for the width and 3 cm for the height. The actual cake has dimensions 20 cm for the length, 14 cm for the width and 6 cm for the height. a

Find the ratio of the dimensions of the trial cake to those of the actual cake.

Deduce the ratio of the volume of the trial cake to that of the actual cake.

To make the actual cake, by what number must Valerie multiply each quantity of ingredients?

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A beehive consists of regular hexagonal cells with side length 3.2 mm and depth of 3.8 mm. A plastic container is built which is modelled on these hexagonal cells, with a side length of 6.4 cm. a

Find the scale factor between the cells of the beehive and the container.

What is the depth of the container?

A beehive with an estimated 70 000 cells holds approximately 45 000 cm3 of honey. What is the volume of a single cell? Round your answer to four decimal places.

What is the volume of the container? Round your answer to the nearest cm3.

A geologist found two pieces of pyrite, or fool’s gold. She found that the volume of Specimen B is 3.375 times greater than the volume of Specimen A.

Specimen A

Specimen B

What is the scale factor?

How many times greater is the area the base of Specimen B than Specimen A?

Investigation: Volume versus surface area

Investigate online

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8 Chapter review 1

A scale model of a building has a rectangular base with a perimeter of 80 cm. The actual building is 15 m tall and its model is 30 cm tall. a

Determine the scale factor used when creating the model.

What is the actual perimeter of the building’s base?

A model car is built using a scale of 1.20. The full-sized car has a rectangular base measuring 4 m long and 1.80 m wide. a

What are the dimensions of the model car?

What is the perimeter of the model car’s base in centimetres?

A square has sides of 6 cm. Give the side length of the new square if: a

The square was enlarged by a factor of 9.

A scale factor of 0.25 was applied to the square.

A rectangle has a length of 10 cm and a width of 4 cm. A new rectangle is created by applying a scale factor of 1.5 to the original. a

What is the perimeter of the original rectangle?

What is the perimeter of the scaled rectangle?

How does the perimeter of the scaled rectangle compare to the original?

Find the value of y for each pair of similar rectangles. a

450 m2 y cm2

32.5 cm

30 m y m2 10 m 13 cm 10 cm

The corresponding sides of two similar triangles are 8 cm and 16 cm. The area of the smaller triangle is 32 cm2. Determine the area of the larger triangle.

If a square has an area of 49 m2 and it is dilated by a factor of , how long is the side of the resulting square?

A triangle has been enlarged with a scale factor of p. How does the area of the dilated triangle relate to the original triangle’s area?

All dimensions of a toy house are

of the actual house sizes. If the living room area of the

actual house is 48 m , what is the area of the living room in the toy version? Chapter 8 review mathspace.co

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A square has sides of length y cm. Calculate the area of the new square if the original is enlarged by a factor of: a

Find the value of x for the pair of similar figures. a

6.5 cm V = 14.5 cm3 4.2 mm 2.8 mm 13 cm

V = 243 mm3

V = x mm3

V = x cm3

Two similar cones have a radius length of 5 cm and 20 cm, respectively. a

State the scale factor of the height of the smaller cone to the height of the larger cone.

State the scale factor of the volume of the smaller cone to the volume of the larger cone.

Calculate the volume of the larger cone, if the volume of the smaller cone is 250 cm3.

Two similar prisms have volumes of 27 cm3 and 729 cm3. If two, corresponding sides in the prisms have lengths of x cm and 9 cm, where x represents the length in the smaller prism, find the value of x.

An office fish tank has a capacity of 400 L. The manager wants to downgrade to a new fish tank with a capacity of 50 L. The manager estimates that the dimensions of the new fish tank should be half of the old tank’s dimensions. Is the manager correct? Justify your answer.

For these two rectangular prisms: 1.8 m 6.4 m

2.4 m

2.88 m 3.84 m 10.24 m

352

Are the two rectangular prisms similar? Justify your answer.

Find the scale factor of the surface area.

Find scale factor of the volume.

Find the volume scale factor if the measurements of the larger prism are halved.

Mathspace WA Curriculum – Year 10 mathspace.co

“The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.” Paul Dirac

Big ideas • Right-angled 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 • Use Pythagoras’ theorem and/or trigonometry to determine unknown sides and angles in right-angled triangles involving angles of elevation and depression • (Optional) Apply right-angled trigonometry to two dimensional situations involving navigational bearings • (Optional) Explore to establish and use the sine, cosine and area rule to determine unknown sides and angles for any triangle

9 Trigonometry Chapter outline 9.01 9.02 9.03 9.04 9.05

Pythagoras’ theorem Trigonometric ratios Angles of elevation and depression Optional: Bearings Optional: Sine and cosine rules Chapter 9 review

356 365 379

389

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!

9.01 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 shorter 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 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

Interactive exploration

Discover this concept in action online

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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 values of a and b

= 81 + 256

Evaluate the squares

= 337

Evaluate the sum

b Let c represent the length of the longest side. Find the value of c2.

Create a strategy

Apply the idea c2 = (18)2

Substitute the length of the longest side.

= 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.

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.

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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.

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

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Evaluate the square root

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Create a strategy Use Pythagoras’ theorem: a2 + b2 = c2.

Apply the idea We can substitute a = x, b = 10, and c = 20 into the formula: a2 + b2 = c2 2

Write the formula 2

x + 10 = 20

Substitute the values

x + 100 = 400

Evaluate the squares

x + 100 − 100 = 400 – 100

Subtract 100 from both sides

x = 300

Evaluate the difference

Take the square root of both sides

Reflect and check When finding a short side, the most important thing to remember is that the two given lengths need to go into different sides of the equation. 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.

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 Pythagoras’ theorem: c2 = a2 + b2.

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Apply the idea Notice that we have a right-angled triangle, and the length of the grappling hook’s rope will be the hypotenuse. One of the short sides 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 a = 10, b = (54 − 37), and c = l into the formula: c2 = a2 + b2

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

Idea summary Pythagoras’ theorem relates the three sides of a right-angled triangle. Sides a and b are the two shorter 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.

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:

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9.01 Practice questions Explorer

Understanding 1

1−3

Adventurer

Trailblazer

1−3

Write Pythagoras’ theorem in a form you could use to ﬁnd: a

The length of z.

The length of x.

Calculate the value of x, rounded to two decimal places. All values of x are positive. a

Ex 1

Ex 2

Ex 3

82 = 52 + x2

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

x2 = 22 + 32

4a-c, 5a-d, 7ab, 9

4d-f, 5c-f, 6, 7bc, 8a, 9

4f-h, 5e-i, 6, 7cd, 8, 9

Determine whether each of the following sets of values are Pythagorean triples: a

(12, 16, 26)

(4, 8, 12)

(8, 15, 17)

(9, 12, 15)

(12, 16, 20)

(7, 7, 10)

(8, 11, 14)

(7, 24, 25)

Find the length of the unknown side as an exact value: a

d c

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f 5

A right-angled triangle has 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.

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

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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

6 mm

9 Ex 4

18 mm Q

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 300 m joining the tops of the buildings is 200 m long, ﬁnd, to the nearest metre, the distance between the buildings.

200 m 450 m ?

Reasoning

10, 12a, 12b(i)(ii)

10, 12

11, 12

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.

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-angled triangle. a

If the triangle is enlarged by a factor of 3, does Pythagoras’ theorem still hold? Justify your answer.

Is a triangle with these sides a right-angled triangle? i

30, 40, 50

Problem-solving 13−16 13

35, 45, 55

iii

3k, 4k, 5k

13−17

Find the length of the diameter AB of the circle.

Find the exact length x.

21a, 28a, 36a

iv 14−18

24 cm O

A x

7 cm B 9 cm

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Tricia’s house has 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 metres, rounded to two decimal places.

5.6 m

3.6 m

10.2 m

Find the value of x, rounded to one decimal place. a

5 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?

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.

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?

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9.02 Trigonometric ratios After this lesson, you will be able to… • identify and define sine, cosine, and tangent, and describe the relationship between angles and sides in right-angled triangles. • use trigonometric ratios to find an unknown side or angle in right-angled triangles. • use trigonometric ratios with other geometric principles and formulas to find missing lengths, distances, and angles

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

Exploration We have three right-angled triangles with acute angles of 30° and 60°. 1. Is △ABC ∼ △DEF ∼ △GHI? Explain why or why not. G

2. Find the ratio for sin 30° for each of the triangles. 3. What do you notice about each ratio (use a calculator if you need to)? 4. Find the ratio for cos 30° for each of the triangles. 5. What do you notice about each ratio (use a calculator if you need to)?

30°

3 30° 2 60°

1 2

30°

2 3

2 60°

60° 1 F

6. Find the ratio for tan 30° for each of the triangles. 7. What do you notice about each ratio (use a calculator if you need to)? 8. Draw another similar triangle which would have the same ratios.

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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, then ﬁll in the side lengths.

Apply the idea The angle θ is also called ∠B.

Hypotenuse 13

Opposite

so we need the lengths of the adjacent and hypotenuse sides.

θ B

We are using the cosine ratio

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

Opposite

θ A

366

Adjacent

Mathspace WA Curriculum – Year 10 mathspace.co

The trigonometric ratios are:

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. 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 side length, rounded to two decimal places: a

f 47°

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

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11 g

42°

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 rightangled 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 measure. 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 measure of the angle.

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Example 3 Find the value of x to the nearest degree.

A 7

C 25

x B

Apply the idea With respect to the angle with a measure of x, the length of the adjacent side is 7, and the length of the hypotenuse 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 have substituted incorrectly.

Idea summary To find the measure of 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 measure of the angle.

Multi-step problems In some cases, the application of a single trigonometric ratio on a right-angled triangle will not be enough to solve the question. In this example, to find the length of h, we must first use our knowledge of trigonometry to find the length of x. Once we find the length of x, we add 70 to x to get the length of h.

xm 43° hm 70 m

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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 the 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 could calculate h. Let’s call the length of the vertical leg of the triangle x. First, calculate an acute angle in the right-angled 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 bottom left angle in the triangle must be 69°.

Substitute the values

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 x and y, rounded to the nearest whole number.

y cm

8 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

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Idea summary In multi-step problems, application of a single trigonometric ratio may not be enough to solve the problem. Extra addition, subtraction, or another application of trigonometry may also be needed.

9.02 Practice questions Understanding 1

Explorer 1, 2, 3a-c, 4

Adventurer 3b-d, 4

Trailblazer 3d-f, 4

Name the hypotenuse. a

A B

For the triangle ABC: a

Which side is opposite 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

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With reference to the angle θ, ﬁnd the ratio: i

iii b

34 5

θ 5

25 C

Write down the indicated ratios for each triangle using the side labels: a

tan θ

tan α

sin θ

A α

sin α

A α

θ B

cos θ

cos α

A α

θ B

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5a, 7, 8, 10−12, 14, 16, 17a-c

Fluency Ex 1

5b, 7, 8, 10−12, 14, 16, 17b-d

5b, 6, 7, 9−11, 13, 15, 16, 17c-e

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 side length, 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

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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

A string of lights of length 23.9 m is joined to the top of a tree, as shown in the diagram. 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

There is a 39° angle from the top of the NJ building, which is 300 m high, to the top of the LS building. The diagonal distance between the tops of the two buildings is 200 m. a

Find x, rounded to two decimal places.

Find the height of the LS building, rounded to two decimal places.

200 m 39°

300 m

376

Mathspace WA Curriculum – Year 10 mathspace.co

17 Ex 5

Find the side lengths, rounded to the nearest whole number: a

54°

16 cm 42°

26 mm

f cm h mm

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. b cm

θ°

21.32 cm 7.9 cm

9.02 Trigonometric ratios mathspace.co

377

Problem-solving 20−22

20−23

21−24

, 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.

Consider the octagon: 10 cm

12 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:

A sporting ground is in the shape of an irregular hexagon. 35 m 15 m

41 m

15 m 35 m

Find the perimeter of the ground.

Find the size of each of the four corner angles of the ground.

Calculate the size of angle DEC. D

6 x+7 A

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Mathspace WA Curriculum – Year 10 mathspace.co

12 − x C 4 B

9.03 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.

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.

1. What is the value of x? What property did you use to ﬁnd this? Is there another way to conﬁrm this?

x 50°

2. What is the angle of elevation from point C? 3. What is the angle of depression from point A?

4. What do you notice about the angle of elevation and depression?

50° C

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379

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. B

Dir

ect

Horizontal

dis

θ Dir

tan

ect

θ Horizontal

dis

Vertical

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.

tan

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

The angle of depression from point B to point D is 61.73°.

380

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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 a trigonometric ratio that relates the 13° and the opposite.

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

The distance AC is 17 326 m.

b Find the distance BC, to the nearest metre.

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

The distance BC is 13 950 m.

9.03 Angles of elevation and depression mathspace.co

381

c Now ﬁnd the distance covered by the jet in that one minute.

Apply the idea The distance covered by the jet in one minute is given by the length of AB. AB = 17 326 − 13 950

Subtract BC from AC

= 3376 m

Simplify the difference

The distance covered by the jet is 3376 m.

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 Pythagoras’ theorem, c2 = a2 + b2.

Apply the idea d2 = 52 + 42 2

d = 41 d=

Substitute the values

Take the positive square root of both sides

The distance from one corner of the floor to the opposite corner of m. the floor is

382

Evaluate the right-hand side

Mathspace WA Curriculum – Year 10 mathspace.co

c Find h, the height of the room. Give your answer to two decimal places.

Create a strategy Set up an equation involving the height, h, the diagonal on the ﬂoor, , and the angle of elevation, 57°.

h 57°

Apply the idea Use the tangent ratio h

Substitute the values

Multiply both sides by

57°

Evaluate using a calculator and round The height of the room is 9.86 m.

Idea summary We can solve problems involving angles of elevation and depression using these steps: 1. Draw a diagram if one was not provided. 2. Label all known information on the diagram, including the right angle. 3. Identify which angles or sides need to be found and in what order. 4. Set up an equation using the correct trigonometric ratio and solve for the missing side or angle. 5. Give your answer as a sentence.

Angle of elevation

Horizontal

Angle of depression

9.03 Angles of elevation and depression mathspace.co

383

9.03 Practice questions Understanding 1

Explorer 1−3

Adventurer 1−3

Trailblazer 1−3

Is angle θ an angle of elevation or depression? Explain how you know. 4 2

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

4−7, 9, 10

4−10

5−11

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.

D y

x 5

384

Mathspace WA Curriculum – Year 10 mathspace.co

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

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.

3° 12 m

A man stands at point A looking at the top of two poles. Pole 1 has a height of 8 m and an angle of elevation of 34° from point A. Pole 2 has a height of 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

Pole 2

25 m

Pole 1 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

9.03 Angles of elevation and depression mathspace.co

385

11 Ex 3

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:

60°

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.

Reasoning 12

12−13

13−14

An air traffic 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 from the tip of a tree’s shadow to the sun is 43°. The tree’s shadow is 20 m long. 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?

386

Find tan 90° on your calculator. How does this relate to your answer to part (b)?

Mathspace WA Curriculum – Year 10 mathspace.co

h 43° 20 m

Problem-solving 15, 17−21 15

15, 18−21, 23

16, 20, 22−23

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 to 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.

Buzz is standing 250 m from a building and measures the angle of elevation to 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.

9.03 Angles of elevation and depression mathspace.co

387

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 height

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?

Two hot air balloons are floating in different locations, and each is tied to the ground directly below. 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:

3.38°

1.22° 1600 m

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.

Roald is standing at point P and observes two poles, AB and CD, of different heights. P, B, and D are on horizontal ground: A C 7m B

29° 16 m

45°

18°

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.

388

Calculate the distance from Roald to the top of pole CD, rounded to two decimal places.

Calculate the distance from Roald to the top of pole AB, rounded to two decimal places.

Mathspace WA Curriculum – Year 10 mathspace.co

9 Chapter 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.

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.

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.

Consider the cone with slant height of 10 m and perpendicular height of 8 m:

10 m

Find the length of the radius, r.

Find the diameter of the cone’s base.

Chapter 9 review mathspace.co

389

Find the value of f, rounded to two decimal places: a

b 27°

10 cm

f cm

29° 5m

For △DEF, ﬁnd the exact value of : a

sin θ

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:

25.2 m

40° hm

1.2 m

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.

Find the value of the side length, rounded to the nearest whole number: a

20 cm

58° 23 mm

h mm

390

Mathspace WA Curriculum – Year 10 mathspace.co

67°

x cm

10 cm

The person in the picture sights a pigeon above him. Find the angle of elevation θ, rounded to two decimal places.

24 m

θ 17 m

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°.

14° B

16°

3000 m

Airport

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, rounded to the nearest metre.

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°. a

If the difference in height between the top of the building and Dave’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. Chapter 9 review mathspace.co

391

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 • Use absolute and percentage error to compare the result of using approximate rather than exact real numbers on final calculations • Use efficient strategies and apply formulas to determine the volume, capacity and surface area of composite solids, using appropriate units • Investigate, calculate and identify the impact of errors on the accuracy and outcome of results in measurement situations • (Optional) Explore, explain and apply efficient strategies and formulas to determine the surface area and volume of right pyramids, right cones, spheres and related composite solids • Analyse, represent and interpret problems in real-world situations involving Pythagoras’ theorem, trigonometry and angles of elevation and depression in right-angled triangles, volume, capacity and surface area of composite objects, the impact of errors in measurement and/or the effect of enlargement and reduction on perimeter, area and volume of similar figures and objects

10 Measurement Chapter outline 10.01 Surface area of prisms and cylinders 10.02 Volume of prisms and cylinders 10.03 Surface area and volume of composite solids Investigation: Rainfall collection Modelling investigation: Wind turbine design 10.04 Estimation and accuracy 10.05 Optional: Pyramids 10.06 Optional: Spheres 10.07 Optional: Cones 10.08 Optional: Composite solids Investigation: Instruments and methods Chapter 10 review

394 404 414 426 428

437

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?

10.01 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 a 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, top, and 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

394

Multiply the area by 2

Mathspace WA Curriculum – Year 10 mathspace.co

Add the areas Evaluate

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 in 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. 10.01 Surface area of prisms and cylinders mathspace.co

395

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 Substitute the values = 2π × 212 + 2π × 21 × 74 = 12 534.95 cm2 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

10.01 Practice questions Understanding 1

Explorer 1−3

Adventurer 1−3

Trailblazer 1−3

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

396

Mathspace WA Curriculum – Year 10 mathspace.co

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

4, 6acdf, 7a, 8ab, 10, 11ab, 15a, 17−21

Fluency

4, 6bd-f, 7b, 8cd, 10, 11cd, 12a, 13, 15b, 17, 21, 22

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

4, 5, 6c-f, 7b, 8cd, 9, 11d, 12b, 13, 14, 15b, 16, 21, 22

b 9.9 m

12 m 8.7 m

4m 7.5 m

5 cm

4 cm

23 cm 6 cm

4 cm

11 cm

9 cm

10 cm 5 cm

3 cm

5 cm

16 cm

4 cm

10.01 Surface area of prisms and cylinders mathspace.co

397

For each triangular prism: i

Find the value of x.

10 cm

x cm

4 cm

Find the surface area.

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 11 cm

5 cm

10 cm

3 cm

5 cm

13 cm

9 cm

4 cm

8 cm

12 m

16 cm

3 cm 17 cm 4m

For this triangular prism:

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, by how much would the surface area be overestimated? Round your answer to two decimal places.

10 m

For this rectangular prism: 5

5 5 5 5

5 8

b 398

Determine whether the net matches the rectangular prism.

iii

8 8 8

Find the surface area of the rectangular prism.

Mathspace WA Curriculum – Year 10 mathspace.co

7 5 7 5 7 8 7

7 5 7 7

11 Ex 3

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

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

10.01 Surface area of prisms and cylinders mathspace.co

399

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

25 m

The area of the circular face of 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, by how much would the surface area be overestimated? Give your answer rounded to two decimal places.

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 constructed from two rectangular and two triangular sheets of steel, forming a triangular prism with dimensions as shown in the diagram: Find the surface area of the roof of the shed (do not include the underneath rectangular face of the prism). Give your answer rounded to the nearest square metre.

400

Mathspace WA Curriculum – Year 10 mathspace.co

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.

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

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.

Reasoning 23

23−25

14 m

1.2 m

2.5 m

25−27

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? 24

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?

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.

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.

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

10.01 Surface area of prisms and cylinders mathspace.co

401

Write a simpliﬁed formula to ﬁnd the surface area of a cylinder that has its radius equal to its height.

Problem-solving 28, 30, 33, 35 28

28, 29, 31, 33−35

Find the surface area of the triangular prism, rounded to one decimal place.

29, 31, 32, 34−36

h cm

21 cm 46 cm

16 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.

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.

402

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 of bag A in centimetres.

Find the height 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?

Mathspace WA Curriculum – Year 10 mathspace.co

2.88 m

Bag A

7 cm

Bag B

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 she can 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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10.02 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. It’s common to use the word height to represent this dimension as we can always reorient the shape. 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.

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Example 1 Find the volume of this triangular prism.

5 cm 10 cm 5 cm

Create a strategy

Apply the idea

Use the volume formula V = Ah, where the area of the base is given by

We are given base length, b = 5, height of the triangular base, H = 5; and height of prism, h = 10. Write the formula

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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

Apply the idea

Use the volume of a prism formula V = Ah.

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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10.02 Volume of prisms and cylinders mathspace.co

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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

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is the radius of the cylinder is the height of the cylinder

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10.02 Practice questions Understanding

Explorer 1−3

Adventurer

Trailblazer

1−3

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

D 5 cm

5 cm 4 cm 3 cm

3 cm

4 cm

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Fluency 4

4, 5ace, 7, 8a-e, 9, 11, 12

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?

5ace, 7, 8b-f, 9, 11, 12

5bdf, 6, 7, 8e-h, 10, 11

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 408

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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

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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

13, 14

13−15

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. a

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?

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?

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.

410

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.

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Problem-solving 16, 17, 20, 23−26 16

16−18, 21, 22, 24, 26, 27

19, 25−30

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.

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?

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.

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.

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

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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

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Find the volume of the tent in cubic metres:

150 cm 290 cm 200 cm

The roof of a house is flat and rectangular, measuring 16 m by 11 m. 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 (b)?

Once the water tank is full, a further 5 mm of rain falls on the roof. How many litres of rainfall are not collected as a result of the tank overflowing?

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 is 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

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10.03 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.

If 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 - by multiplying the area of the base by the height of the 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.

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Apply the idea Large prism = 6 × 15 × 13 = 1170 cm

Small prism = 11 × 2 × 15 = 330 cm

V = 1170 − 330 = 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 21 cm metres. Round your answer to two decimal places. 21 cm 21 cm

r1 = 50 cm r2 = 54 cm r3 = 58 cm

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

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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.

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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.

10.03 Practice questions Understanding

Explorer 1, 2, 3a-c, 4ad

Adventurer 1, 2, 3a-c, 4ab

What is a composite solid? Give an example.

Explain what you would do to:

418

Calculate the surface area of a composite shape.

Calculate the volume of a composite shape.

Mathspace WA Curriculum – Year 10 mathspace.co

Trailblazer 1, 2, 3bc, 4bc

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

3 cm

2 cm 5 cm

4 cm

7 cm

8 cm 2 cm

5 cm

10 cm

2 cm

8 cm

c 5 mm 4 mm 3 mm 5 mm

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

9 mm 16 mm

11 cm

10 mm

6 cm 8 cm

4 cm

36 mm

53 mm

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Fluency 5 Ex 1

5, 6ad, 7, 8a-e, 9a-d

5, 6bd, 7, 8b-f, 8, 9c-f

5bc, 6bc, 8d-h, 9d-h, 10

Find the volume of these composite solids, rounded 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

Find the volume of these composite solids, rounded to two decimal places. a

11 cm

4 cm 7 cm 8 cm

8 cm 15 cm

5 cm

9 cm

4 cm 2 cm

12 cm 2 cm

420

2 cm

Mathspace WA Curriculum – Year 10 mathspace.co

9 cm

2 cm

7 Ex 2

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

8 Ex 3

Find the volume of these composite solids, rounded to two decimal places. a

b 15 m

10 m 8m

12 m

19.5 m

9m 11 cm

8m 6 cm

17 m

8 cm

4 cm

f 6 cm

3 cm

3 cm 12 cm

16 cm 12 m 6m

h 10 cm 14

5 cm 14 cm 5 cm

24 20

15 cm

10 cm

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9 Ex 4

Find the surface area of these composite solids, rounded to two decimal places. a

7 cm

16 cm

12 m 6m

1 cm

5 cm

13 cm

7 cm

4 cm 10 cm

9 mm

2 cm

16 mm 3 cm

10 mm

3 cm 53 mm

36 mm

10 cm

12 cm

1 cm

h 1 cm

7 cm

4 cm

1 cm

15 cm

5 cm

By considering the water bottle as a composite solid, estimate its surface area.

2 cm 1 cm

4 cm

20 cm

3 cm 422

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Reasoning 11

11, 12

11, 13

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.

8 cm

3 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 two identical holes carved out of it, each of which is a rectangular prism. All measurements are in metres. a

6.5 5

Explain how to find, and then calculate, the volume of the remaining solid. Round your answer to two decimal places.

14 6.5

Explain how to find, and then calculate, the surface area of the remaining solid.

Problem-solving 14ab, 16, 17 14

3 cm 6 cm

14ab, 15−17

14, 15, 18−20

A swimming pool is the shape of a trapezoidal prism.

51 m

Find the volume of the pool in m .

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 three quarters full, ﬁnd h.

1.4 m

10 m

3.3 m 50.7 m

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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.

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

The diagram shows the design for a marquee (tent). The material to make the marquee costs $44 per square metre. a

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?

3m 7m 10 m 8m

A company manufactures nuts shaped like regular hexagonal prisms, with cylindrical bolt holes cut out of the centre. a

424

59 cm

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. 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?

Mathspace WA Curriculum – Year 10 mathspace.co

0.5 cm

1.0 cm 1.2 cm

A swimming pool is composed of a trapezoidal prism joined to a half cylinder. 7m 3m 2m

0.8 m

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?

A steel shed is to be constructed with a rectangular cut-out at the front for the entrance. 2m

2.5 m 10 m 3m 4m

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?

If the steel sheets cost $18 per square metre, calculate the total cost of the steel required to build the shed.

Investigation: Rainfall collection

Investigate online

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MODELLING INVESTIGATION

Wind turbine design Explore how mathematics can help you model a wind turbine tower for a coastal WA school. Decide on the tower’s base and cap dimensions and material cost. Then determine its height, volume, surface area and cost while ensuring stability and efficiency.

Objectives • To analyse the wind turbine tower situation and decide what is needed to model it. • To use mathematics to model the tower’s dimensions, capacity and cost. • To explain your plan and how your choices affect the outcome.

Analyse the wind turbine tower design You are modelling a wind turbine tower with a cylindrical base and conical cap for a school. Pick the base radius, base height, cap height and material cost to plan its size and budget. Choose a base radius (for example 0.5 m to 1 m), base height (for example 5 m to 8 m), conical cap height (for example 1 m to 2 m) and cost per square metre (for example $20 to $30). Total height must be ≤ 10 m and surface area cost ≤ $1500. Sketch your tower manually to visualise it.

Conical cap

10 m 8 m (Base height)

Base radius

Investigate

426

Choose your base radius, base height, cap height and material cost per m2. Write down your numbers.

Will you use exact numbers? Why?

What might affect your plan? List one assumption and one thing that limits you.

Mathspace WA Curriculum – Year 10 mathspace.co

Represent with mathematics Use mathematics to model the tower’s total height, volume, surface area, cost and stability, including the effect of errors or enlargement.

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Calculate the total height, volume, surface area, cost and the angle of elevation of the conical cap. If there were errors in the measurements for the height or radius explore the effect of these on the cost.

Investigate 1.

Calculate the total height, volume and surface area using your dimensions.

Calculate the cost and the angle of elevation for the conical cap.

Determine the impact of a 5% radius error on volume.

Interpret your plan Look at your model and explain what it means for the wind turbine tower design. Verify if the plan meets the original restrictions; ≤ 10 m, cost is ≤ $1500 and angle is ≤ 60°, then adjust one of your chosen variables to assess it’s impact on the model.

Investigate 1.

Does your tower’s height, cost and angle of elevation meet the limits? Is the volume practical? Why or why not?

Change one thing. How does it affect volume, cost or stability?

Explain your turbine tower plan to the class. How do your assumptions affect it?

Discussion 1.

What was the trickiest part of modelling the tower with mathematics? Why?

How could you improve your design with more materials or a taller limit?

Talk to a friend about their plan. How was their tower size or cost different from yours?

Modelling Investigation: Wind turbine design mathspace.co

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10.04 Estimation and accuracy After this lesson, you will be able to… • explore rounding and how it influences the accuracy of calculations, especially in large data sets 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 The heights of five plants are measured. The measurements are as follows (in centimetres): Plant 1 2 3 4 5

Height 25.467 cm 30.234 cm 28.789 cm 32.512 cm 29.105 cm

a What is the total height of the plants without rounding or truncation?

Create a strategy Add the given heights of the plants.

Apply the idea 25.467 + 30.234 + 28.789 + 32.512 + 29.105 = 146.107 cm

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b What is the total height truncating to 2 decimal places?

Create a strategy

Apply the idea

Truncate each height to 2 decimal places, then add the truncated values.

Truncate each measurement: 25.467 → 25.46 30.234 → 30.23 28.789 → 28.78 32.512 → 32.51 29.105 → 29.10 Add the truncated heights: 25.46 + 30.23 + 28.78 + 32.51 + 29.10 = 146.08 cm

c What is the total height rounding to 2 decimal places?

Create a strategy

Apply the idea

Round each height to 2 decimal places, then add the rounded values.

Round each measurement: 25.467 → 25.47 30.234 → 30.23 28.789 → 28.79 32.512 → 32.51 29.105 → 29.11 Add the rounded heights: 25.47 + 30.23 + 28.79 + 32.51 + 29.11 = 146.11 cm

d How do the truncated and rounded values compare to the actual values?

Create a strategy Compare the values found from previous parts.

Apply the idea Exact total height: 146.107 cm Total height with truncation: 146.08 cm Total height with rounding: 146.11 cm The truncated value is further away from the actual total height than the rounded value is. Therefore, the rounded total height is a more accurate approximation of the exact total height compared to the truncated total height.

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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 actual 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) = 222.2 m

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Substitute the values Evaluate

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 cm

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 cm

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 = 367 061.696 cm3 The percentage error is given by:

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Substitute the values Evaluate

Idea summary Measuring 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

10.04 Practice questions Understanding 1

Explorer 1−3

Adventurer 1−4

Trailblazer 1−5

Are these statements true or false? a

Truncation and rounding errors can signiﬁcantly affect the ﬁnal results in a series of calculations.

In financial calculations, rounding or truncating repeatedly over time has no impact.

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 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.

What is the smallest number with three decimal places that would give an answer of 12.72 when rounded to two decimal places?

Fluency 6

6ab, 7, 8

6−8

Tony is calculating the surface area of the solid shown, where the height and diameter of the cylinder are both 5 cm. a

Find the smallest value that each of the measurements could be.

Find the largest value that each of the measurements could be.

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.

6−8 5 cm

8 cm

6 cm

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7 Ex 1

8 Ex 2

The weights of five boxes of fruits are measured (in kilograms). a

What is the total weight without rounding or truncation?

What is the total weight truncating to 2 decimal places?

What is the total weight rounding to 2 decimal places?

How do the truncated and rounded values compare to the actual values?

Box 1 2 3 4 5

Weight 12.783 15.482 11.926 13.643 14.217

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 halfway between two marks, in which case he rounds down, or more than halfway 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: a

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.

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, 10

9−11

10−12

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. a

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.

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?

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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 mv2, where m is the mass and v is the velocity. 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, where the ball rolls 5 metres, 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), 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), 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.

Problem-solving 13 13

13, 14

13−15

Keepsafe Family Bank pays a small amount of interest on their customers’ 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? 10.04 Estimation and accuracy mathspace.co

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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.

Find the latest date (the shortest time ago) that the artefacts can be from, measured in thousands of years.

Express 3.8 as a percentage of 36.8, rounded to one decimal place.

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.

Due to improvements in astronomical data, the current estimate of the age of the universe is 13.5 Byr, with an absolute error of just 200 million years. Find the percentage error in the current estimate, rounded to two decimal places.

Did you know?

Electricity meters at home often display rounded values—like 328 kWh—to make them easier to read. But energy providers record much more precise numbers, such as 328.462 kWh, to calculate your actual usage. This ensures you’re billed for exactly what you’ve consumed, even if the display shows a simpler figure.

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10 Chapter review 1

Find the surface area: a

16 m 6 cm

d 13 cm

7 cm

5 cm

12 cm

26 cm 8 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.

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

11 cm 20 cm 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?

For each ﬁgure, ﬁnd to two decimal places: i

The volume

6 cm

The surface area

9 cm 6 cm

6 cm

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 438

18 cm

6 cm

15 cm

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5 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.

15 cm

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.

Find the total surface area of the shed. Do not include the ﬂoor.

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.

4m 8m 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. a

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 fossilised 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.

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Big ideas Probability helps us analyse the chance that an event will occur and provides us with tools to make decisions about the future based on known information.

Curriculum Outcomes • Choose and construct appropriate sample spaces to show outcomes for twoand three-stage chance experiments both with and without replacement. Assign probabilities to events involving conditional statements, such as ‘if ... then’, ‘given’, ‘of’, ‘knowing that’ • (Optional) Use weighted tree diagrams and/or formulas to assign probabilities to two- and three-stage chance events including situations involving conditional probability • Represent secondary data in two-way tables or Venn diagrams and assign probabilities to outcomes involving conditional statements • Analyse, represent and interpret problems in real-world situations involving two- and three-stage chance experiments both with and without replacement, conditional probability or statements, boxplots, bivariate data and/or two-way tables • Conduct repeated chance experiments and simulations to model conditional probability and produce datasets using digital tools. Discuss, compare and analyse variation and estimated probabilities for conditional events

11 Probability Chapter outline 11.01 Review Venn diagrams and two-way tables 11.02 Conditional probability Modelling investigation: Animal tagging for environmental monitoring 11.03 Mixed probability concepts Investigation: Simulation 11.04 Optional: Geometric probability Chapter 11 review

442 455 466 468

485

Scientists use probability methods to determine the likelihood of space debris hitting land.

11.01 Review Venn diagrams and two-way tables After this lesson, you will be able to… • identify different probability notations and use the probability formula to calculate the likelihood of events. • apply the concepts of intersections and unions in solving probability problems. • interpret and construct Venn diagrams and two-way tables. • convert between two-way tables and Venn diagrams to represent information.

Probability notation Recall that probability is the likelihood of an event (set of outcomes) occurring. The sample space is the set of all the outcomes. To calculate the probability of an event occurring, divide the number of favourable outcomes satisfying the event by the total number of outcomes.

The probabilities of all the possible events in the sample space will add to 1. If we have an event, A, the complement of the event is given by A′. This is the set of outcomes that is not in set A. The probability of the complement is found by: P ( A′) = 1 − P ( A) The intersection of two sets is the outcomes both sets have in common. The probability of both A and B occurring is given by P ( A and B) or P ( A ∩ B). The union of two sets is the combination of outcomes in each set. The probability of either A or B occurring is given by P ( A or B) or P ( A ∪ B).

Example 1 In an experiment, a number is chosen randomly from the list of numbers shown: {2, 3, 5, 6, 7, 10, 12, 14, 15, 16, 19, 20} • Event A = an odd number is chosen • Event B = a multiple of 4 is chosen a Which of the following has the largest probability? A

B′

A∪B

A∩B

Create a strategy Since the sample space is the same for both events, compare the number of outcomes of each option and choose the one with the greatest number of outcomes.

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Apply the idea B′ = {2, 3, 5, 6, 7, 10, 14, 15, 19}

Numbers that are not multiples of 4

A ∪ B = {3, 5, 7, 12, 15, 16, 19, 20}

Numbers that are odd or multiples of 4

B = {12, 16, 20}

Numbers that are multiples of 4

A ∩ B = {}

No numbers are odd and a multiple of 4

A = {3, 5, 7, 15, 19}

Odd numbers

B′ has the greatest number of outcomes, so the correct answer is option A.

b Which of the following has a value of 0? A

P ( A ∩ B)

P ( A′ ∪ B)

P ( A ∩ B′)

P ( A′ ∩ B′)

Create a strategy Find the probability of each option. Use the formula:

Apply the idea There were 12 numbers in the original list, so this will be the denominator in our probabilities. No numbers satisfy A ∩ B

Evaluate

7 numbers are not odd or multiples of 4

5 numbers are odd and not multiples of 4

4 numbers are not odd and not multiples of 4

The correct answer is option A.

Idea summary The notation used for concepts in probability are: • • • • •

A — an event. A′ — the complement of event A. P ( A) — the probability of event A occurring. P ( A ∪ B) — the probability that either A or B occurs. P ( A ∩ B) — the probability that both A and B occurs.

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Two-way tables and Venn diagrams A two-way table and a Venn diagram are both ways to present information from two sets. We can easily convert a two-way table into a Venn diagram, or the other way around, by matching their corresponding elements.

A′

B′

Two-way table of events A and B

Venn diagram of events A and B

Did you know?

Wildlife researchers use two-way tables to track animal populations across different regions! These tables help identify where species are thriving or declining, and how factors like climate change or land use affect their habitats. By analysing the data, scientists can make informed decisions to support conservation efforts and protect biodiversity for future generations.

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Exploration Match each Venn diagram to the two-way table with the same highlighted region. 1. Which two models highlight event A? 2. Which two models highlight event A′? 3. Which two models highlight event A ∩ B? 4. Which two models highlight event A ∪ B? A

A′

B B′

Venn 1 A

Table 1 B

A′

B B′

Venn 2 A

Table 2 B

A′

B B′

Venn 3 A

Table 3 B

A′

B B′

Venn 4

Table 4

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For events A and B with sample space S, we can use our notation to show different areas of a Venn diagram and the related two-way table. S

The sample space is the list of possible outcomes that could result from the event or experiment. The notation for sample space as a list is S = {…}.

An event is shown as the entirety of a circle. A

The complement of an event is shown as the area outside of the event’s circle.

S A

The complement of event A can be written as A′, Ac,

The intersection of events is the overlap in the two circles.

S A

446

or ~ A.

The notation for the intersection of events A and B is A ∩ B.

The union of events A and B is shown by shading anything contained by the two or more event circles. The notation for the union of events A and B is A ∪ B.

Mathspace WA Curriculum – Year 10 mathspace.co

Example 2 Students in Irene’s class were asked if they owned a dog and asked if they owned a snake. The following two-way table shows that information. Owns a snake Doesn’t own a snake

Owns a dog 2 13

Doesn’t own a dog 3 11

Construct a Venn diagram that represents the information provided in the two-way table.

Create a strategy The Venn diagram should have two overlapping circles for each type of pet.

Apply the idea • The 2 that own a dog and a snake will go in the intersection. • The 3 that own a snake but not a dog will go in the snake circle outside the intersection. • The 13 that own a dog but not a snake will go in the dog circle outside the intersection. • The 11 that do not own a snake or a dog will go outside the circles. Dog owners

Snake owners

3 11

Example 3 Two events are deﬁned as: • Event A: Plays guitar • Event B: Plays piano People attending a concert were surveyed and the results are shown: B B′

A A′ 20 40 90 50

a Find the probability that a randomly selected person does not play guitar.

Create a strategy The probability that a randomly selected person does not play guitar is represented by P ( A′). First, we need to ﬁnd the number of people that do not play guitar. Then, we need to divide it by the total number of people that were surveyed.

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Apply the idea

Reﬂect and check

The number of people who do not play guitar is: A different way to ﬁnd the answer would be to ﬁnd the probability that a randomly selected 40 + 50 = 90 person does play guitar and subtract that The number of people surveyed is: number from 1. 20 + 90 + 40 + 50 = 200 The probability that a randomly selected person does not play guitar is:

b Find P ( A ∩ B).

Create a strategy P ( A ∩ B) is the probability that a randomly selected person plays both piano and guitar.

Apply the idea The number of people who play both piano and guitar is 20. The number of people surveyed is 200. The probability that a randomly selected person plays both piano and guitar is:

Example 4 The outcomes for events X and Y are shown in the Venn diagram. X 12

Y 82

a Find P (Y ).

Create a strategy The number of outcomes in Y are the outcomes in the intersection plus the outcomes that are only in Y. The total number of outcomes is the sum of all the outcomes in the sample space.

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Apply the idea The number of outcomes in Y is: 82 + 41 = 123 The total number of outcomes is: 53 + 12 + 82 + 41 = 188 The probability of event Y is:

b Find P (X ∪ Y )′

Create a strategy The number of outcomes in X ∪ Y are the outcomes in both circles. The probability of the complement is P (X ∪ Y )′ = 1 − P (X ∪ Y ).

Apply the idea The number of outcomes in event X or Y is: 12 + 82 + 41 = 135 .

The probability of the union of the events is

The probability of the complement of the union of the events is: Equation of complement

Substitute

Evaluate

Reﬂect and check Notice that the number of outcomes that are neither in X nor Y is 53. This means

Idea summary We can easily convert a two-way table into a Venn diagram, or the other way around, by matching their corresponding elements.

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11.01 Practice questions Understanding 1

Explorer 1, 3−5

Adventurer

Trailblazer

2−5

Two events are deﬁned as: • Event A: It will rain tomorrow. • Event B: There will be a storm tomorrow. Describe what the notation P (B) represents.

For these events: • Event A: Getting an odd number when a die is rolled. • Event B: Getting an even number when a die is rolled. Write the probability notation to describe the probability of event B in three different ways.

For these events: • Event A: A card randomly selected from a deck is red • Event B: A card randomly selected from a deck is a queen Write the probability notation to describe the probability of a randomly selected card being both red and a queen.

170 tennis players were asked whether they would support equal prize money for the women’s and men’s draw. Find the number of favourable outcomes for P (Males ∩ Support).

Support Males Females

A student creates this diagram of their favourite animals: • Event F: Selecting a four legged animal • Event S: Selecting an animal with stripes a

450

Draw a Venn diagram and shade the region that represents the favourable outcomes for the probability P (F ).

Draw a Venn diagram and shade the region that represents the favourable outcomes for the probability P (S′).

List the outcomes in F ∩ S

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Do not support 35

Turtle Hippo

Zebra

Lion

Tiger

Clown Fish

Crow

Kangaroo

Elephant

Fluency Ex 1

6, 9, 10, 12, 13, 15

6, 7, 9, 10, 12−15

6, 8−11, 14−16

In an experiment, a number is chosen randomly from the list of numbers shown: 1, 3, 5, 7, 9, 10, 15, 20, 25, 30, 35, 40 • Event A: An odd number is chosen. • Event B: A multiple of 5 is chosen. a

Which of these events has the largest probability? A

B′

A∪B

A∩B

P ( A′ ∩ B′)

Determine whether the probability of these events is zero. i

P ( A ∩ B)

P ( A′ ∪ B)

iii

P (( A ∩ B)′)

In an experiment, a card is chosen randomly from a standard deck of 52 cards. • Event A: A face card (King, Queen, or Jack) is chosen. • Event B: A diamond card is chosen. a

Which of these events has the largest probability? A

B′

A∪B

A∩B

P ( A′ ∩ B′)

Determine whether the probability of these events is zero. i

P ( A ∩ B)

P ( A′ ∪ B)

iii

P (( A ∩ B)′)

If a student from the library is randomly selected, we might select a student studying literature (L) or statistics (S). All of the students are studying one of these subjects, and no one is studying both. The probabilities are: • P (L) = 0.4 • P (S) = 0.6 a

Find the value of P (S ∩ L).

For this Venn diagram:

Find the value of P (S ∪ L). Rugby League Rugby Union

179

185

146

Complete the table: Play Rugby League

Don’t play Rugby League

Play Rugby Union Don’t play Rugby Union

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Ex 2

100 random people in Australia were surveyed, examining their carbon footprint and the city they lived in. The people were then categorised as living in either an urban or a regional location, and whether each person had a high carbon emission or a low carbon emission. High Emission Low Emission Total

Urban 37 24 61

Regional 13 26 39

Total 50 50 100

Construct a Venn diagram that represents the information provided in the two-way table. 11

A vet has 28 pets visit their practice in a day. The pets are categorised based on whether they have been vaccinated and whether they have been microchipped.

Vaccinated

Construct a two-way table based on the Venn diagram.

Microchipped

Ex 3

A healthy living initiative asked people to describe how often they go to the gym.

452

Male 33 18

How many people were surveyed?

If one person is chosen at random, what is the probability that they are a male who frequently goes to the gym? Give your answer as a fraction.

If one person is chosen at random, what is the probability that they rarely go to the gym? Give your answer as a fraction.

In a study, some people were asked whether they lie. A partially completed two-way table of the results is shown. Two events are deﬁned as: • Event L: The person lies. • Event C: The person is a child.

Female 34 35

Frequently Rarely

Complete the table.

Find P (C′).

A group of students were asked about their siblings. The 2 categories show if they have at least 1 brother (B), and if they have at least 1 sister (S): a

Use probability notation to describe the probability that a selected student has at least 1 sibling.

Calculate the probability that a selected student has at least 1 sibling.

Describe what the notation P (S′) represents.

Mathspace WA Curriculum – Year 10 mathspace.co

L C C′

L′

15 10

Total 25

Total

Have a sister Have a brother

9 8

Ex 4

A group of students were asked why they skipped breakfast. The two reasons given were that they were “not hungry” and they were “too busy”. Two events are defined as:

Not hungry

• Event H: Selecting a student that was not hungry • Event B: Selecting a student that was too busy

Find the value of P (H).

Find the value of P (H ∩ B).

Find the value of P (B′).

Ian makes a Venn diagram about the possible pets he could get from the local shelter. Ian is told that a pet will be selected at random.

Dogs

• Event D: The pet selected is a dog. • Event F: The pet selected is ﬂuffy. a

Find P (F′).

Find P (D ∩ F′).

Reasoning 17

Too busy

Fluffy

9 4

17, 18

17−19

For these two events: • A: Paul wins the badminton tournament. • B: Paul wins the golf tournament. Write the probability notation to describe the probability that Paul wins either the badminton tournament or the golf tournament, but not both. Explain your reasoning.

A group of tourists were asked whether they spoke Mandarin (M ) or Spanish (S). a

How many people speak both languages?

If one person is chosen at random, ﬁnd the probability that they speak neither language.

If one person is chosen at random, ﬁnd the probability that they speak only one of the languages. Explain your reasoning.

M M′

S 58 15

S′ 10 17

In an experiment, there are only two possible events A and B. If A occurs, B does not occur and vice versa. Are these probabilities true? Use a diagram to justify your answers. a

P ( A ∩ B) = 1

P ( A ∩ B) = 0

P ( A′) = P (B)

P ( A ∪ B) = P ( A)

P ( A ∪ B) = 1

P ( A) = P (B)

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Problem-solving 20, 21 20

20−22

This table describes the departures of trains out of a train station for the months of May and June:

21−23

May June

Departed on time 123 124

How many trains departed during May and June?

Find the probability that a train selected at random departed on time and in May. Give your answer as a percentage rounded to one decimal place.

If a train is randomly selected from the trains that departed in June, find the probability that it departed on time. Give your answer as a decimal rounded to three decimal places.

Find the probability that a train selected at random from the 2 months was delayed. Give your answer as a fraction.

In a survey, 150 individuals were asked whether they enjoyed drinking tea and coffee. A partially completed two-way table of the results is shown. The events are defined as: • Event T: The person likes tea. • Event C: The person likes coffee. If the probability that a randomly selected individual enjoys drinking both tea and coffee is

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Delayed 32 47

, what is the number of people in set T ∩ C?

T C C′

T′ 29 37

Verniece asked some people in her community whether they were pescatarian or not. 29 responders said they were pescatarian, of which 8 were children. 14 children said they were not pescatarian, and 11 adults said they were not pescatarian. a

Construct a model based on the results of Verniece’s survey.

Find the probability that a randomly selected responder is pescatarian.

Find the probability that a randomly selected adult is pescatarian.

In a factory, 45% of the employees only work on the assembly line, 30% only work in quality control, and 20% work both on the assembly line and in quality control. a

Construct a model to represent this data.

An employee is selected at random from the factory. i

ind the probability that the employee has neither worked on the assembly line nor F in quality control.

f the employees that work on the assembly line, ﬁnd the probability that the O selected worker also works in quality control.

Mathspace WA Curriculum – Year 10 mathspace.co

11.02 Conditional probability After this lesson, you will be able to… • calculate probabilities using the conditional probability formula. • interpret two-way tables, tree diagrams, 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? Conditions will reduce the sample space we are concerned with.

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

1. When randomly selecting a person from all those surveyed, how many people are we selecting from? 2. When randomly selecting a person from those who are not teachers, how many people are we selecting from? 3. Explain why your answers to the previous questions are different. 4. What is the probability of randomly selecting a person that is not a teacher and went on holiday? 5. Of those who are not teachers, what is the probability of randomly selecting a person that went on holiday? 6. 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.

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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)

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).

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

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B′

11 21 32

10 8 18

Total 21 29 50

Mathspace WA Curriculum – Year 10 mathspace.co

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

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 favourable outcomes in the 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 favourable outcomes in B′, so P ( A∣B′) =

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c Find P ( A′∣B′).

Create a strategy From the previous part, we know 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 favourable outcomes. So, Substitute known values

Simplify

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

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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. 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.

Apply the idea We can construct the following tree diagram for this situation:

Fine 0.4

0.9 0.1

0.7

0.6 Not Fine

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 (On time∣Fine) = 0.9 • P (Not fine) = 0.6 • 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 Not 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).

11.02 Practice questions Understanding 1

Adventurer

1−3

Trailblazer 1−4

Are these statements true or false? a

If P ( A) = 0.5, P (B) = 0.3 and P ( A ∩ B) = 0.15, then the events are dependent.

Using the two-way table shown, P (Maths∣Fail) = Pass Fail Total

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Explorer

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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)? A

P ( A)

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 positions 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) equal to, less than, or greater than P ( A ∩ B)?

Fluency

5−9, 11−13, 15, 18, 20

5−9, 11−14, 16, 18, 20

6−8, 10, 13, 14, 16, 17, 19−21

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.

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′)

Using the Venn diagram, ﬁnd: a

P ( A)

P (B′)

P ( A∣B)

P (B∣A)

P ( A∣B′)

A A′ Total

B′

5 7 12

6 2 8

Total 11 9 20

B 0.2

0.2

0.4 0.2

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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

List 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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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 the ﬁrst die is a two.

Mathspace WA Curriculum – Year 10 mathspace.co

Ex 4

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 a new software system that determines 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.

The probability of two independent events, A and B are P ( A) = 0.8 and P (B) = 0.5 Determine the probability of: Both A and B occurring

Neither A nor B occurring

A or B or both occurring

B occurring but not A

A occurring, given B occurs

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Reasoning 22

22, 23

23, 24

Given the conditional probability formula,

show that, P ( A∣B) × P (B) = P (B∣A) × P ( A) 23

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 B 5

A Venn diagram is shown in which the size of the regions are proportional to the number of elements in that region.

A B

Which conditional probability is greater, P ( A∣B) or P (B∣A)? Justify your answer.

Problem-solving 26−28 25

25, 26, 28

25, 26, 28, 29

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.

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.

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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 for the Greens 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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MODELLING INVESTIGATION

Animal tagging for environmental monitoring Explore how mathematics can help you model a shark tagging program in a WA marine park, like Ningaloo, using a safe classroom simulation. Simulate tagging sharks to estimate species proportions and lengths, then figure out what it means for conservation insights.

Objectives • To analyse the shark tagging simulation and decide how to investigate it. • To determine reliable datasets and use statistics to model shark populations. • To explain your findings and what they mean for the marine park.

Analyse the shark tagging situation You are modelling shark tagging in a marine park using a safe simulation of reef and tiger sharks. Pick your total shark population, species proportions, sample size and length ranges to estimate tagging success. Choose a total shark population (for example 50 to 100), reef shark proportion (for example 60% to 80%), sample size (for example 10 to 20 sets of 3 catches) and lengths (reef: 1-2m, tiger: 2-4m). Decide if you’ll simulate with replacement (sharks returned) or without (sharks removed). Sketch your plan manually to outline it.

Investigate 1.

Choose your total shark population, reef shark proportion, sample size and length ranges. Will you use replacement in your simulation? Write down your choice.

What question do you want to answer? Will your probabilities be exact or approximate? Why?

What might affect your simulation? List one assumption and one thing that limits you.

Determine your dataset and analysis Simulate shark tagging safely in class to create reliable datasets, then analyse species proportions and length distributions.

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Interactive exploration

Discover this concept in action online

mathspace.co

Simulate 15 sets of 3 catches with replacement (for example coin: heads = reef, tails = tiger heads = reef, tails = tiger) and without replacement (for example adjust probabilities after each catch), either in the applet or manually with a coin or die, recording species and lengths (reef: 1-2m, tiger: 2-4m).

Investigate 1.

Simulate your datasets with and without replacement. Record species outcomes and lengths. How did you ensure reliability?

Show a way to calculate the probability of one sequence and a conditional probability for both methods.

Make a two-way table for species outcomes and a boxplot for length distributions per species, comparing replacement scenarios.

Interpret your plan Look at your datasets and graphs to estimate the marine park’s shark population and explain what it means for conservation. Scale your results with and without replacement to 200 sharks, then adjust one factor (for example increase sample size by 5) either in the applet or by hand and compare estimates.

Investigate 1.

Predict the number of reef sharks in 200 for each method. Do the boxplots and tables look consistent? Why or why not?

Change one thing. How does it change your estimate?

Explain your shark tagging model to the class. How do chance and assumptions affect your conservation insights?

Discussion 1.

What was the trickiest part of modelling shark tagging with mathematics? Why?

How could you improve your simulation with more samples or tools?

Talk to a friend about their plan. How was their reef shark proportion different from yours?

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11.03 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.

ABC ACB BAC BCA CAB CBA

a Find the probability that A is drawn before B is drawn.

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) = . 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 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. Write the formula Substitute known values

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Reﬂect and check We can check our answer using the conditional probability formula and

, so Start with the conditional probability formula

Substitute known values

Evaluate the division

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.

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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

Total

E-reader Not e-reader 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

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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 who read on e-readers, what percentage read in both formats? Round your answer to the nearest percentage.

Create a strategy Divide the number of people who read on both formats by the number of people who read on an e-reader.

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 a calculator and round

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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.

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:

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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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11.03 Practice questions Understanding 1

Explorer 1−3

Adventurer

Trailblazer

1−3

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. A

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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6−8, 10, 11, 13, 15, 17, 18

Fluency 4

4−6, 8, 10−13, 15−18

4−6, 9, 11, 12, 14, 16, 19, 20

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.

A A′

B w x

B′ y z

0.3

0.15

0.25 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 b

3 2 4

List the sample space of two-digit numbers produced by this process.

2 3

Find the probability that:

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.

2 4 3

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

478

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 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′)

Mathspace WA Curriculum – Year 10 mathspace.co

L 43

47 7

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 watched any of the movies. • 39 had read and watched at least one of the stories. • 97 had read the book. • 4 had never read nor watched any of these. a

Complete the table for these results: Read the book Didn’t read the book Total

Watched the movie 39

Didn’t watch the movie 4 62

Total 97 166

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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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

Calamari

Lasagna

Risotto

Gelato Tiramisu Gelato Tiramisu Gelato Tiramisu

Spaghetti

Lasagna Soup Risotto

Gelato Tiramisu Gelato Tiramisu Gelato Tiramisu

480

If each combination of dishes is equally likely, ﬁnd the probability that Alessandra orders lasagna given that she ordered calamari.

Find the probability that Alessandra ordered spaghetti and tiramisu, given that she ordered soup.

Find the probability that Alessandra ordered risotto or spaghetti, given that she ordered soup.

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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 16

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 b

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.

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Reasoning 21

21−23

23, 24

For P ( A ∩ B): a

Write the formula of independent events.

Write the formula of dependent events.

Which formula, the one from part (a) or the one from part (b), can be used for both independent and dependent events? Explain.

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:

Student Hair Sick

A ✓ −

B ✓ ✓

C − −

D ✓ ✓

E − −

F ✓ −

G ✓ ✓

H −

Student Hair Sick

K −

L −

M −

✓

O − −

P ✓ −

Q −

✓

N ✓ ✓

J ✓ −

R ✓ ✓

S − −

T − ✓

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

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 is found, the two identical cards 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. a

Why does the second set of branches only have one fork?

Given that a candidate passes, calculate the probability that they passed on the ﬁrst attempt. Give the answer as a fraction in simplest form.

482

✓

I − −

Explain why it is inappropriate to assume that the probabilities apply to a randomly chosen candidate.

Mathspace WA Curriculum – Year 10 mathspace.co

P 0.6

0.7

0.4 F

0.3

Problem-solving 25, 27, 28 25

25, 27−29

26, 28−30

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 is 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 selecting student that submitted a project. Find the probability of selecting student that submitted a project given that they are a secondary student. 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.

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.

Comedy 10

Action 4

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.

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.

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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?

Investigation: Simulation

Investigate online

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Did you know?

Probability plays a huge role in baseball strategy! Coaches use it to decide when to take risks or play it safe during critical moments. For example, probability can help determine whether a player should attempt to steal a base or stay put, based on factors like the pitcher’s throw speed, the catcher’s reaction time, and the runner’s success rate in similar situations. By calculating these probabilities, teams make smarter, data-driven choices that can turn the game in their favour. It’s not just about instinct; probability helps teams predict which moves are most likely to score runs, making every decision a bit more strategic. In a close game, those small, calculated choices can make all the difference between a win and a loss!

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11 Chapter review 1

Construct a two-way table based on the information in the Venn diagram.

A spinner is divided into 6 equal 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 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: a

At least once

If three letters are randomly selected from the word CIRCUMFERENCE without replacement, ﬁnd the probability that: a

Exactly once

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 (A ∩ B)

P (B∣A′)

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.

N N

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.

B j l

A A′

EN EE

B′ k m

In a survey, 640 participants were asked whether they enjoy watching basketball or volleyball. Of the total respondents, 120 enjoy watching both sports and 70 enjoy neither. The total number who enjoy watching basketball is three times the number who enjoy watching both sports. a

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

486

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

Mathspace WA Curriculum – Year 10 mathspace.co

Total

100

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

Construct a tree diagram showing all possible combinations of food and drink Elijah can have for breakfast and their associated probabilities.

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 enjoy reading 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 enjoy reading 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: NSW YES NO

ACT YES NO

QLD YES NO

NSW

ACT

QLD

YES

NSW YES

YES

ACT

NSW YES

QLD NO

QLD

NSW

Find the probability that a randomly chosen person voted “YES”.

Of those that are not from NSW, ﬁnd the probability that a randomly chosen person voted “NO”.

Considering just the voters from NSW and ACT, ﬁnd the probability that a randomly chosen person was from NSW and voted “YES”.

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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 • Analyse, determine and interpret problems in real-world situations involving two- and three-stage chance experiments both with and without replacement, conditional probability or statements, boxplots, bivariate data and/or two-way tables • Represent and analyse boxplots. Explain differences between multiple boxplot datasets in terms of shape, spread and centre. Compare or match the shapes of boxplots to distributions depicting the same data • (Optional) Produce, organise and represent accurate and valid data in a cumulative frequency graph and use this to analyse quartiles and percentiles • (Optional) Determine the mean and standard deviation of a dataset. Investigate, analyse and interpret the effect of individual data values, including outliers, on the standard deviation • Critically analyse the claims, inferences and conclusions of statistical reports in the media and other real-life situations and identify potential sources of bias

12 Univariate statistics Chapter outline 12.01 Representations of data 12.02 Quartiles 12.03 Boxplots 12.04 Compare data sets Modelling investigation: Ferry schedules 12.05 Samples and populations 12.06 Optional: Cumulative frequency 12.07 Optional: Standard deviation 12.08 Optional: Comparisons using standard deviation Chapter 12 review

490 508 520 534 547 549

559

In medicine, scientists use standard deviation to assist in understanding drug effectiveness and side effects!

12.01 Representations of data After this lesson, you will be able to… • understand the purpose and usage of different types of data displays. • identify the best representation for a given data set. • create and interpret frequency tables, histograms, bar charts, column graphs, dot plots and stem and leaf plots. • apply knowledge of data displays to solve problems related to a data set.

Frequency tables Before displaying data with a graph, we ﬁrst need to organise it into meaningful groups. Frequency tables are helpful to organise data sets. Colour of cars in the school carpark Colour Frequency white 14 red 2 blue 3 black 1 yellow 1

Ungrouped frequency tables are the best choice to organise: • Categorical data Example: colour of cars • Discrete numerical data when there is a small number of possible values Example: number of siblings

Heights of year 9 students Height (cm) Frequency 145 ≤ Height < 150 3 150 ≤ Height < 155 10 155 ≤ Height < 160 8 160 ≤ Height < 165 13 165 ≤ Height < 170 1

Grouped frequency tables are best for: • Continuous numerical data Example: heights • Discrete numerical data when the data can take a large number of possible values Example: attendees at football match

For grouped frequency tables, all intervals must be of equal size, and there should not be gaps between intervals. We can see class intervals in a few different formats: • 1 –5, this class includes 1 and 5. Used for discrete data only. • 1 – < 5, this class includes 1, but not 5. Used for continuous data. • 1 ≤ x < 5, this class includes 1, but not 5. Used for continuous data. This is the most common representation. Assume this to be the case unless it is specifically indicated otherwise. • 1 < x ≤ 5, this class does not include 1, but does include 5. Used for continuous data. To perform calculations with the data set, we usually find the class centre. The class centre is the middle score of each class interval. For the interval 1–5, the class centre would be

Selecting the interval width is important. If the intervals are too narrow or too wide, the shape of the data distribution will not be apparent. As a guide, 6 to 12 intervals will typically be most useful for moderately sized data sets. 490

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Example 1 What would be the most appropriate way of representing data from: a A survey conducted of 1000 people, asking them how many languages they speak? A

Leaving the data ungrouped and constructing a frequency table

Grouping the responses and constructing a frequency table

Create a strategy Consider whether the data will have a wide range of responses.

Apply the idea

Reflect and check

Since we would expect that the responses for An example of a frequency table representing the number of languages people speak to be responses to this survey is shown: fairly limited, e.g., 1 to 6 languages, we can leave Languages spoken Frequency the data ungrouped and construct a frequency 1 462 table. 2 353 The correct answer is option A. 3 98 4 44 5 36 6 7 b A survey conducted of 1000 people, asking them how many different countries they know the names of? A

Grouping the responses and constructing a frequency table

Leaving the data ungrouped and constructing a frequency table

Create a strategy Consider whether the data will have a wide range of responses.

Apply the idea

Reflect and check

The number of countries a person can name An example of a grouped frequency table would be extremely varied. They might know representing responses to this survey is shown: just a few or be able to name over 190 countries. Countries Frequency This means the most appropriate way to represent this data is to group the responses and construct a frequency table. The correct answer is option A.

1 − 30 31 − 60 61 − 90 91 − 120 121 − 150 151 − 180 181 − 210

514 387 86 6 3 2 2

Notice that inequality notation was not used since the data is discrete.

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Idea summary Frequency tables are the best choice to organise categorical data and discrete numerical data when there is a small number of possible values. Grouped frequency tables are best for continuous numerical data and discrete numerical data when the data can take on a large number of possible values.

Histograms, bar charts and column graphs These data displays show the frequency of data values as the length of horizontal bars or vertical columns.

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Histograms are similar to column graphs, with vertical columns used to display numerical data, but are used in different contexts.

Data type Gaps

Number of students

Grouping Original data set

Bar chart (column graph) Categorical or discrete with small number of possibilities Gaps between bars to show separate categories Ungrouped data Original data preserved

No gaps between bars to show continuity Grouped data Original data grouped and not preserved

Each student in a class was surveyed and asked about the colour of their eyes.

12 10 8 6 4 2

The data is categorical and the results are displayed in a bar chart as shown.

Green Blue Hazel Brown Eye colour

492

Histogram Continuous or discrete with large number of possibilities

Mathspace WA Curriculum – Year 10 mathspace.co

Number of students

Each student in a class was surveyed and asked their heights.

12 10

The data is numerical and continuous, so the results are displayed in a histogram. The data is continuous data because it can take any value within a range. The height of each column represents the frequency of each data item within each interval.

8 6 4 2 135 140 145 150 155 160 165 Height (cm)

Unless noted otherwise, the left boundary is included in the class interval, while the right boundary is not. In this histogram, 135 is included in the first class interval, but 140 is in the second class interval.

A histogram may have a logarithmic scale when data is unevenly spread over a very wide range of values. We take the logarithm of each value and then group the data using the logarithmic values. The populations of the largest 100 cities in Australia is a great case for a logarithmic scale histogram. With a traditional scale, we can tell there are only a few large cities, but cannot easily read off how many fall in each interval.

100

Frequency

80 60 40 20 0

0 0 0 0 0 00 0 00 0 00 0 00 00 0 0 0 0 0 0 0 10 20 30 40 50 0

Population

When the data is converted to a logarithmic scale, we can see more detail and read off the number of cities with more than one million people more easily.

Frequency

40 30 20 10 0

4 5 6 7 Logarithmic population

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Example 2 In product testing, the time when faults are detected in producing a certain piece of machinery is recorded over the course of a day. The frequency table shows the results. Working hours 0≤t<4 4≤t<8 8 ≤ t < 12 12 ≤ t < 16

Number of faults 10 14 20 16

a Construct an appropriate data display.

Create a strategy The data is grouped continuous numerical data, so a histogram is most appropriate. Use the class intervals to draw the horizontal axis and the maximum frequency to scale the vertical axis, then draw the bars on the histogram.

Apply the idea We can draw a histogram using the class interval boundaries as the labels on the horizontal axis and the frequencies on the vertical axis. Number of faults

Faulty machinery 20 18 16 14 12 10 8 6 4 2 0

b How many faults were recorded throughout the day?

Create a strategy Add the frequency columns to find the total number of faults recorded.

Apply the idea Each value in the frequency column represents a fault that was recorded. Number of faults = 10 + 14 + 20 + 16 = 60

Add the frequencies Evaluate

There were 60 total faults recorded throughout the day.

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4 8 12 Working hours

c How many faults were recorded before 8 working hours had occurred?

Create a strategy

Apply the idea

Check the frequencies where the working hours If we look at the ﬁrst and second rows of the are less than 8. table, we see that there were 10 + 14, which equals 24 faults, so 24 faults happened before the 8 working hours ended.

d How many faults were recorded in the last 4 hours of the working day?

Create a strategy

Apply the idea

Look at the frequency for the last interval, which corresponds to the last 4 hours of the working day.

The last row of the table shows that 16 faults were recorded in the last 4 hours of the working day.

Example 3 The histogram shows the population of 30 different Australian cities plotted on a logarithmic scale.

Population of Australian cities 16 14 Frequency

12 10 8 6 4 2 0

5 6 Log Population

a One city has a population of 282 673 people. What is the log10 of 282 673 rounded to two decimal places?

Create a strategy Enter log10 282 673 into your calculator.

Apply the idea log10 282 673 = 5.451 284 328 08 ≈ 5.45

Evaluate using the calculator Round the answer

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b What population does the number 4 represent on the log population scale?

Create a strategy Since the values along the x-axis are logarithmic results and population is the input, we want to ﬁnd the value for x where log10 x = 4.

Apply the idea log10 (104) = 4 Definition of logarithms 104 = 10 000

Evaluate

The number represents a population of 10 000 on the log population scale.

c How many cities have a population of at least 1 000 000?

Create a strategy

Apply the idea

Identify where a population of 1 000 000 is represented on the log population scale, then count the relevant frequencies.

We have: log10 1 000 000 = 6 This means that any city in the 6 ≤ c < 7 class interval (denoting c as the log population of the cities) will have at least a population of 1 000 000. Therefore there are 5 cities with a population of at least 1 000 000.

d How many cities have a population of between 10 000 and 1 000 000?

Create a strategy

Apply the idea

Identify where a population of 10 000 and 1 000 000 is represented on the log population scale, then count the relevant frequencies.

We have: log10 10 000 = 4

log10 1 000 000 = 6

This means that any city in the 4 ≤ c < 5 and 5 ≤ c < 6 class intervals will have a population between 10 000 and 1 000 000. In total, there are 11 + 14 = 25 cities with populations within this range.

Idea summary Column graphs are used for categorical or numerical discrete data. They can have vertical or horizontal columns where the heights of the columns represent the frequencies. Histograms are used for numerical continuous data. They appear similar to column graphs but do not have spaces between the columns.

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Dot plots Dot plots are a graphical way of displaying the distribution of numerical or categorical data on a simple scale with dots representing the frequency of data values. They are best used for small to medium size sets of data and good for visually highlighting how the data is spread and whether there are any gaps in the data or outliers.

Days per week spent exercising

In a dot plot, each individual value is represented by a single dot, displayed above a horizontal line. The graph appears similar to a pictograph or column graph with the number of dots representing the total count. • To correctly display the distribution of the data, the dots must be evenly spaced in columns above the line. • The scale or categories on the horizontal line should be evenly spaced. • A dot plot does not have a vertical axis. • The dot plot should be appropriately labelled.

Example 4 Sally is a casual nurse. She used a dot plot to keep track of the number of shifts she did each week for a number of weeks.

Shifts per Week

a Over how many weeks did Sally record her shifts?

Create a strategy Each dot represents 1 week, so count all the dots in the graph.

Apply the idea Number of weeks = 5 + 3 + 1 + 8 + 2 + 5

Add the number of dots in each column

= 24 Evaluate

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b For how many weeks did she work 3 shifts?

Create a strategy

Apply the idea

Count the number of dots above 3 shifts on the dot plot.

The dot plot shows 5 dots at 3 shifts, indicating there were 5 weeks where she worked 3 shifts.

c How many weeks did she work less than 7 shifts?

Create a strategy

Apply the idea

Count the number of dots with less than 7 shifts on the dot plot.

Count all the dots at 3, 4, 5 and 6 shifts. By adding up the dots (5 + 3 + 1 + 8 = 17), we ﬁnd that she worked for a total of 17 weeks with less than 7 shifts.

d When Sally works at least 6 shifts a week, she buys a weekly train ticket. What proportion of the time did she buy a weekly train ticket?

Create a strategy Divide the number of weeks where she worked at least 6 shifts by the total number of weeks.

Apply the idea We count all the dots with a score of 6 or more. Examining the dot plot, we sum the dots, 8 + 2 + 5 = 15, indicating she worked for 15 weeks with at least 6 shifts. The total number of weeks, found in part (a), is 24. The proportion of time Sally bought a weekly train ticket is the time.

, which can be simplified as

Idea summary In a dot plot, each individual value is represented by a single dot, displayed above a horizontal line. The number of dots represents the total number of data values. We can read off the frequencies by counting the dots. They are best used for small to medium size sets of data and are good for visually highlighting how the data is spread and whether there are any gaps or outliers. 0

Days per week spent exercising

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Stem and leaf plots A stem and leaf plot, or stem plot, is used for organising and displaying numerical data. It is appropriate for small to moderately sized data sets. The graph is similar to a column graph on its side. An advantage of a stem and leaf plot over a column graph is the individual scores are retained and further calculations can be made accurately. In a stem and leaf plot, the right-most digit in each data value is split from the other digits, to become the ‘leaf’. The remaining digits become the ‘stem’. Stem

Leaf

0 3 6

5 5 6

0 3 6 8

The data values 10, 13, 16, 21, 26, 27, 28, 35, 35, 36, 41, 41, 45, 46, 49, 50, 53, 56, 58 are displayed in the stem and leaf plot shown.

6 7 8 1

5 6 9

Key: 2∣1 = 21 • The stems are arranged in ascending order, to form a column, with the lowest value at the top. • The leaf values are arranged in ascending order from the stem out, in rows, next to their corresponding stem. • A single vertical line separates the stem and leaf values. • There are no commas or other symbols between the leaves, only a space between them. • To correctly display the distribution of the data, the leaves must line up in imaginary columns, with each data value directly below the one above. • A stem and leaf plot includes a key that describes the way in which the stem and the leaf combine to form the data value.

Example 5 The stem and leaf plot shows the age of people entering through the gates of a concert in the first 5 seconds.

Stem

Leaf

2 6 8 9 9

0 2 3 4 5 7 7 8

2 2 4 7

4 5

9 Key: 1∣2 = 12

a How many people passed through the gates in the ﬁrst 5 seconds?

Create a strategy Each leaf represents the age of a person, so count the number of leaves.

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Apply the idea To ﬁnd the number of people surveyed, we can count the number of leaves in the stem and leaf plot, since each data entry will have one leaf. There are 6 people with an age between 10 and 19, 8 people in their 20s, 4 people in their 30s, and 1 person in their 50s, so we can add all these numbers to ﬁnd the total number of people. Total number of people = 6 + 8 + 4 + 1 = 19

b What was the age of the oldest person?

Create a strategy

Apply the idea

Find the largest number recorded.

The largest number will be in the largest stem which is 5, and have the largest leaf, which is 9. This stem and leaf make the number 59. The oldest person is 59 years old.

c What percentage of the concert-goers were under 24 years old? Round your answer to one decimal place.

Create a strategy

Apply the idea

Divide the number of people whose ages are less than 24 by the total number of people.

There are 9 people who are less than 24 years old, and the total number of people is 19.

Then, convert to a percentage by multiplying by 100%.

This gives us a fraction of

. Now, we need to

convert the fraction to a percentage. × 100% = 47.4% 47.4% of the concert-goers were under 24 years old.

Idea summary Stem

Leaf

0 3 6

5 5 6

0 3 6 8

6 7 8 1

5 6 9

Key: 2∣1 = 21

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A stem and leaf plot, or stem plot, is used for organising and displaying numerical data. It is appropriate for small to moderately sized data sets. An advantage of a stem and leaf plot is that individual scores can be seen. The right-most digit in each data value is split from the other digits, to become the ‘leaf’. The remaining digits become the ‘stem’.

Choosing an appropriate data display There are many considerations to be made when choosing an appropriate data display for a data set which has been collected. Histograms, dot plots, and stem and leaf plots have their own strengths and drawbacks.

Example 6 Determine the best type of data display(s) for each set of data. a The number of goals scored in each of the weekly school futsal games for the year.

Create a strategy First, decide the size and possible range of values of the data set.

Apply the idea

Reflect and check

The number of games in a season would be around 40 which is not too large. The number of goals per game would not vary greatly, likely between 0 and 5.

It could be difficult to find appropriate bin widths for a histogram without hiding the shape of the data.

A dot plot or bar chart would be appropriate.

b The height of all 196 Olympic gymnasts in the most recent summer games

Apply the idea

Reflect and check

Since there is a large number of data points and the data is continuous, a histogram would be the best graph to summarise the data. Since there might be significant height differences between male and female gymnasts, a histogram will show any gaps in the data.

Technology can ease some of the inefficiencies of graphing data. With technology, we can create and compare different displays to determine which visual best reflects the data.

Idea summary Strengths Histogram Bar chart Dot plot Stem and leaf plot

Drawbacks

Easily display large or spread out values of a data set; shape, centre and measures of spread are clearly visible. Easily display large data sets; shape clearly visible. Useful for individual data values. Shape is clearly visible. Useful for individual data values. Shape is clearly visible.

Cannot see individual data values Only used for categorical or discrete data Not suitable for large quantities of data Not suitable for large quantities of data

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12.01 Practice questions Understanding 1

Explorer

Adventurer

1−5

Trailblazer

2−6

Lucy counted the number of carrots she harvested from her garden over a few months and recorded the data in a table.

Month 1 2 3 4 5 6 7 8

In which months did Lucy harvest the same number of carrots?

Carrots 55 58 56 72 65 61 72 70

Mr. Hopkins recorded the hair colours of the students in his class in a column graph. Students’ hair colours 10

Number of students

9 8 7 6 5 4 3 2 1 0

Bonde

Brown

Black

Red

Colour

502

Which hair colour is most common?

Which hair colour is least common?

How many more students have red hair than blonde hair?

For the data provided in the table. a

Find the range.

Find the mode.

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Score 68 69 70 71 72 73

Frequency 16 41 30 31 49 29

The stem and leaf plot shows the ages of 20 employees in a company. a

Find the number of employees in their 30s.

What is the age of the oldest employee?

What is the age of the youngest employee?

Leaf 2

0 1

0 2 4 8 8

3 4 8

1 1

2 8 8 9 2 5

Key: 1∣2 = 12 years old 5

Find the class centre for the class interval 25–45.

How do we determine the frequency of each score or group of scores using: a

A dot plot

Fluency 7

Ex 2

A histogram

7, 9−11, 13, 16

A stem and leaf plot

7−12, 16

7−12, 14−16

For the surveys, should the data be grouped or ungrouped when constructing a frequency table? a

A survey of 2000 people, asking them how many pets they own

A survey of 2000 people, asking them how many different animals they can list

A frequency table shows the resting heart rate (in beats per minute) of some people taking part in a study.

Heart rate 45–54 55–64 65–74 75–84

Complete the table.

How many people took part in the study?

How many people had a resting heart rate between 55 and 74?

How many people had a resting heart rate below 65?

Class centre

Frequency 15 25 26 30

The ages of people (in years) when they bought their ﬁrst car from a particular neighbourhood are displayed on the histogram: Age Frequency 16 ≤ Age < 18 18 ≤ Age < 20 20 ≤ Age < 22 22 ≤ Age < 24 24 ≤ Age < 26 26 ≤ Age < 28

Age of first car purchase 20 16 Frequency

Ex 1

12 8 4 0

18 20 22 24 26 28 Age

Complete the frequency table based on the histogram.

How many people were surveyed?

How many people bought their ﬁrst car before they were 22 years old?

How many people bought their ﬁrst car when they were 24 years old or older?

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Ex 5

A group of recent college graduates were asked how old they were when they worked their first paid job. Their responses are shown in the dot plot. a

How many people were surveyed?

How many people worked their first paid job at 15 years old?

How many people were younger than 16 when they worked their first paid job?

If a person is at least 16 years old, they can drive themselves to work. What proportion of people surveyed could have driven themselves to their first paid job?

How many students took the test?

What was the highest score on the test?

What test score occurred the most frequently?

Age of first paid job

The scores for a recent spelling test are shown in a stem and leaf plot. The scores are out of 100.

12 13 14 15 16 17 18 19 20 Age

Spelling test results 6

2 3 3 6 6 7

0 1

2 3 4 5 6 9 9 9

4 5 7 9

How many marks separate the highest score and the most frequent score?

Minutes taken 1 2 3

How many people took part in the survey?

What proportion of people surveyed took 1 minute to make a decision?

Miss Yen recorded her students’ favourite pets in the column graph. Students’ favourite pets 15

Hamster

Dog

Cat

Fish

Pets

504

Which type of pet is most popular?

Which type of pet is least popular?

How many students picked a ﬁsh as their favourite pet?

How many students were surveyed in total?

Mathspace WA Curriculum – Year 10 mathspace.co

4 5 8

Key: 1∣2 = 12

In a survey, some people were asked approximately how many minutes they take to decide between brands of a particular product. The results are are shown in a frequency table.

Number of students

Ex 4

Bird

Frequency 13 17 12

An amusement park recorded the number of people that visited during a week of the school holidays and displayed the data in a column graph. Visitors to Adventure Land 290

Number of Visitors

270 250 230 210 190 170 150 130 Mon

Tues

Wed

Thurs

Fri

Sat

Sun

Day

Ex 3

Ex 6

On what day did Adventure Land have the most visitors?

How many more visitors were there on Thursday than Sunday?

On what day did Adventure Land have 220 visitors?

The histogram shows the income (in dollars) of 24 employees plotted on a logarithmic scale. a

A employee has an income of $46 974. What is the log10 of 46 974 rounded to two signiﬁcant ﬁgures?

What income does the number 5 represent on the log income scale?

How many people have an income of at least $10 000?

What percentage of people have an income less than $100 000?

How many people have an income of at least $10 000 but less than $10 000 000?

Frequency

12 11 10 9 8 7 6 5 4 3 2 1 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 log income

Determine whether a histogram, dot plot, or stem and leaf plot would be most appropriate for these data sets. Explain your answer. a

Heights of students in a cohort size of 300

Marks of students in a class size of 20

The results from a survey of 15 people asking how many pets they have

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Reasoning 17

17, 18

17−19

The frequency table shows the data distribution for the length of leaves collected from a species of tree in the botanical gardens.

Leaf length (mm) 0 ≤ x < 20 20 ≤ x < 40 40 ≤ x < 60 60 ≤ x < 80 80 ≤ x < 84

How many leaves less than 60 mm were collected?

What was the most common interval of leaf length collected?

Are leaves more likely to be greater than 60 mm in length or less than 60 mm in length? Explain your reasoning.

For three column graphs:

Tom

John Name

Kate

50 45 40 35 30 25 20 15 10 5 0

Tom

Is the tallest person the oldest? Explain your reasoning.

Is the heaviest person the oldest? Explain your reasoning.

John Name

Kate

Height 140

Is the heaviest person the tallest? Explain your reasoning.

Is the youngest person the lightest? Explain your reasoning.

Is it possible for a 10-year old to be lighter than an 8-year old based on the data displayed? Explain your reasoning.

Centimetres

120

100 80 60 40 20 0

Tom

John Name

Kate

The frequency table shows the average time spent travelling to work for 52 people. a

506

10 9 8 7 6 5 4 3 2 1 0

Mass

Kilograms

Years

Age

Frequency 5 11 19 49 43

Construct a histogram to display the data shown in the frequency table.

Mathspace WA Curriculum – Year 10 mathspace.co

Commute time (minutes) 0 ≤ x < 20 20 ≤ x < 40 40 ≤ x < 60 60 ≤ x < 80 80 ≤ x < 100 Total

Frequency 15 17 10 6 4 52

Determine whether each of the statements about the data is accurate and explain. i

he data suggests that living close to work is not a priority for people. Roughly equal T proportions of people live less than 40 minutes away as those that live more than 40 minutes away.

he data suggests that people prefer a shorter commute to work. The majority live T within 40 minutes travel, and in general the longer the commute the less people there are in that category.

iii

he data shows that everyone lives within an hour travel from their work, with the T most amount of people living between 20 and 40 minutes away.

he data shows that most people travel to work by car or by walking, since most T travel times are fairly short, and only a few people travel by bus or train.

Problem-solving 20

The histogram shows the length of time, in hours, that Daniel parked his car in a car park each day over a certain number of days. a

Estimate the total number of hours that Daniel’s car was parked in a car park.

Daniel received 152 hours of free parking through his job, and he is charged $4.30 per hour for any additional time. How much was he charged for parking?

20−22

The histogram shows the weights (in kilograms) of 26 zoo animals plotted on a logarithmic scale. a

A monkey has a weight of 45.8 kg. What is the log10 of 45.8 rounded to two signiﬁcant ﬁgures?

What weight (in kg) does the number −2 represent on the log weights scale?

How many animals have a weight of at least 1000 kg?

What percentage of animals have a weight less than 0.1 kg? Round your answer to two signiﬁcant ﬁgures.

How many animals have a weight of at least 0.1 kg but less than 100 kg?

Frequency

20, 22

Number of days

12 11 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 Length of parking time (in hours)

−2 −1 0 1 2 3 log weights

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The table lists the oil production in barrels per day for 10 countries. Construct a frequency histogram of oil production using a log scale. Country

Oil production (barrels per day)

Saudi Arabia

10 625 000

Norway

1 763 000

Ecuador

555 000

Brunei

113 000

France

61 000

Albania

21 000

Finland

13 700

Portugal

7100

Japan

4000

Costa Rica

300

log10 oil production

12.02 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. • 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:

The highest value in the data set

Minimum:

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

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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 the interquartile range (IQR). First quartile

Second quartile Q1

2, 3, Minimum

Third quartile Q2 10, 11, Median

5+5 2 Q1 = 5

13,

Fourth quartile

16,

Q3 18,

22, 39, Maximum

16 + 18 2 Q3 = 17

Q1 =

Q3 =

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 Lower quartile, Q1 Median, Q2 Upper quartile, Q3 Maximum

Lowest value At most 25% of the data is below this value 50% of the data lies on either side of this value At most 25% of the data is above this value 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

First locate the median, between the 4th and 5th scores:

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 25% of the data set. For a data set with n items, Q1 is the

value

Median is the

value

Q3 is the

value

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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 th where n is the total number of scores, to ﬁnd the position of the median.

Use the formula:

Apply the idea There are n = 13 scores in the list. Position of median =

Substitute n = 13

= 7th Median = 80

Evaluate Choose the 7th score

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 =

510

= 121

Mathspace WA Curriculum – Year 10 mathspace.co

Average the middle scores Evaluate

Idea summary • Q1 is the ﬁrst quartile (sometimes called the lower quartile). It is the middle score in the bottom half of the data set. • 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 usually not affected by outliers.

IQR = Q3 − Q1 Q1 Q3

Lower quartile Upper quartile

Interactive exploration

Discover this concept in action online

mathspace.co

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 + 5 + 5 = 18

Find the sum Evaluate the addition

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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 There are 18 scores in the list. Position of median = = 9.5

Substitute n = 18 Evaluate

Therefore the median is between the 9th and 10th score. Using the frequency table we can see that the 9th score is 38 and the 10th score is 38.

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 9 scores in the lower half excluding the median. The lower quartile will be the 5th 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 9 scores in the upper half excluding the median. The upper quartile will be the 5th in the upper half, which is the 9 + 5 = 14th 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

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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

Apply the idea

Find the median of the upper half of the scores.

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

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e Find the interquartile range.

Create a strategy

Apply the idea IQR = 16 − 9

Use the interquartile range formula: IQR = Q3 − Q1

Substitute the quartiles

Evaluate

Idea summary To calculate the interquartile range:

IQR = Q3 − Q1 IQR Q1 Q3

is the interquartile range is the ﬁrst quartile is the third quartile

12.02 Practice questions Understanding 1

Explorer

Adventurer

1−4

Trailblazer

1−4

Find the median for each data set of scores: a

2, 10, 28, 35, 50

8, 9, 11, 12, 13, 15

41, 42, 43, 44, 49, 50, 51, 53, 58

3, 4, 6, 7, 7, 8, 12, 14

For each data set, the median is shown. Find the upper and lower quartiles. a

0, 1, 4, 6 8, 9, 11 Median

22, 24, 25, 29, 32, 34 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 3

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

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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

5, 6, 7a-d, 8, 9, 11−14, 17

5, 7c-f, 8−11, 13, 15−17

5, 7e-h, 8, 10, 11, 15−18

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

8 Ex 2

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

19.5, 29.6, 39.1, 22.4, 15.8, 46.2, 35.4

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:

Median

Lower quartile

iii

Upper quartile

In which quartile does the mean lie?

Weight 16 17 18 19 20 21 22

Frequency 4 5 5 3 4 6 3

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Ex 3

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.

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

For this dot plot:

Matches 45 46 47 48 49 50 51 52

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.

The dot plot shows the ages of customers in a mobile phone store in one day:

Age of customers

516

Find the lower quartile.

Find the interquartile range.

Mathspace WA Curriculum – Year 10 mathspace.co

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:

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.

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

Dylan records the number of ﬁsh he catches for each ﬁshing trip over a period of time: a

How many ﬁshing trips did he go on?

Find the range in the number of ﬁsh caught.

Find the interquartile range.

Frequency

20 18 16 14 12 10 8 6 4 2 0

Number of fish caught

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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 Find the median value.

Find the lower quartile score.

Find the upper quartile score.

What percentage of games did the team score more than in the upper quartile?

What percentage of games did the team score less than in 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?

Reasoning 19

19, 22

20−22

20−23

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 20 16 12 8 4 0

518

Key: 6|1|2 = 16 and 12

Frequency

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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. a

Find the lower quartile.

Find the upper quartile.

Find the percentage of students who 0 have more pets than the upper quartile. Explain why it is not exactly 25%.

Number of pets

Is the percentage 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, 25a-f 24

24, 25

24−26

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? 12.02 Quartiles mathspace.co

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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.

12.03 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 is comprised of the following five statistics: Minimum Lower quartile Median Upper quartile Maximum

17 52 69 87 100

Minimum value

Median

Lower Quartile

Maximum value Upper Quartile

100

As you can see, the boxplot consists of a number line, a rectangle, and two horizontal lines (the whiskers). The box represents the middle 50% of the scores and its size tells us the interquartile range. 520

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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. Minimum and outlier

Upper quartile Q3

Median Q2

Lower quartile Q1

Maximum ∼25%

∼25%

Interquartile range

Outliers are data points that 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. Left skew

Symmetrical

10 15 20 25 30 35

A symmetrical data set is distributed around the centre with a similar frequency on the left and right.

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.

Right skew

Uniform

10 15 20 25 30 35

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

A uniform data set is evenly distributed across all values.

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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

Discover this concept in action online

mathspace.co

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

Example 2 For the boxplot shown, fill in the table with the corresponding values: Minimum Maximum Outliers Range IQR

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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:

Minimum Maximum Outliers Range IQR

1 43 1, 41, 43 42 7

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

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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.

Apply the idea The boxplot shows a slight right skew.

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

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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 Use the measures of spread and central tendency to compare the class marks.

Apply the idea 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.

Idea summary Boxplots are a graphical representation showing the ﬁve-number summary for a data set.

Minimum value

Median

Lower Quartile

Maximum value 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.

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12.03 Practice questions Understanding 1

526

1−4

Adventurer

Trailblazer

1−4

Fill in the missing numbers for each boxplot. a

Explorer

Minimum 20 Lower quartile 28 Median

Upper quartile

Maximum

Minimum 16 Lower quartile 33 Median

Upper quartile

Maximum

Minimum 26 Lower quartile 33 Median

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

Mathspace WA Curriculum – Year 10 mathspace.co

2 4 6 8 10 12 14 16 18 20

4 a

For the boxplot shown: 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? 5, 6ab, 7, 8, 9abgi, 11a, 12

Fluency 5

5, 6bc, 7, 8, 9cdhi, 10, 11b, 13, 14

6cd, 7, 8, 9efhj, 10−14

For the boxplot shown, ﬁnd the: a

Minimum

Maximum

Range

Median

Interquartile range 0

8 10 12 14 16 18 20

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

8 10 12 14 16 18 20

Scores

38 40 42 44 46 48 50 52 54 56 58 60

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Ex 1

Construct a boxplot to represent the information in the tables: a

Ex 2

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

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

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.

2 3 5 5 Key: 1|2 = 12

20 21 22 23 24 25 26 27 28 29 30 31

Leaf

10 11 12 13 14 15 16 17 18 19 20 21

Heart Rate

155 160 165 170 175 180 185 190 195 200

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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

Range 5 10 Q3 10 21

Maximum 13 22

IQR 6 3

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

Range 12 12

Class 10P

Complete the table: Class 10P

Class 10Q

Median Lower quartile Upper quartile Range Interquartile range Outliers b

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

Class 10Q

0 2 4 6 8 10 12 14 16 18 20 22

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

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Ex 4

The boxplots represent the daily sales made by Carl and Angelina over the course of one month. Angelina's sales

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

15, 17−19

15−17, 19, 20

16, 19−22

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.

530

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

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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

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.

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.

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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

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 treatment group.

Does the drug have a positive effect on patient recovery? Explain your reasoning.

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 where 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 a

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%?

Problem-solving 23, 25bd 23

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

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23, 24, 25a-c

23−26, 25a-c

Match the histograms on the left to the corresponding boxplots on the right: Histogram A

Boxplot 1

0 1

9 10

1 2 3 4 5 6 7 8 9 10

Histogram B

Boxplot 2

0 1

2 3 4 5 6 7 8 9 10

Histogram C

Boxplot 3

1 2 3 4 5 6 7 8 9 10

Histogram D

Boxplot 4

0 1

9 10

1 2 3 4 5 6 7 8 9 10

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Create a boxplot for the data shown in each graph. Monthly salaries of employees 30

Number of employees

20 15 10 5 20 25 30 35 40 45 50 Salaries (in thousands)

Debbie’s blog visitors

Number of days

Scores of students

10 9 8 7 6 5 4 3 2 1 0

100

Number of pages read

10 0

50 100 150 200 Number of visitors

For each of the conditions, use ﬁve of the digits from 1 to 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.

12.04 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.

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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 compare them.

Example 1 These 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

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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

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

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 bulbs with the best lifespan.

Reflect and check In fact, the best light bulb 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 light bulbs produced by Manufacturer B have a greater lifespan than all the light bulbs produced by Manufacturer A.

Example 2 This boxplots show the number of goals scored by two football players in each season. Sophie

Holly

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15 Goals

a Who scored the most goals in a season?

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.

b How many more goals did Holly score in her best season compared to Sophie in her best season?

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?

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?

Apply the idea Use the formula: IQR = Q3 − Q1 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 • Mode • Frequencies, grouped or individual • 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: 10 9 8 7 6 5 4 3 2 1 0

Histogram of approximately symmetric data

8 9 10 11 12 13 14 15 16 17 18 Score

Boxplot of symmetric data

Frequency

All representations of positively skewed data: 10 9 8 7 6 5 4 3 2 1 0

Histogram of positively skewed data

8 9 10 11 12 13 14 15 16 17 18 Score

Boxplot of positively skewed data

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All representations of negatively skewed data: Histogram of negatively skewed data

Frequency

10 9 8 7 6 5 4 3 2 1 0

8 9 10 11 12 13 14 15 16 17 18 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.

Example 3 Match the boxplot shown to the correct histogram.

1 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Create a strategy

Apply the idea

Look for characteristics of skewed and symmetrical distributions of data.

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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12.04 Practice questions Understanding 1

1−3

Trailblazer 1−3

0 10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

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 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90 100

10 9 8 7 6 5 4 3 2 1 0

540

1, 2

Adventurer

The histogram and the boxplot display the same data. 10 9 8 7 6 5 4 3 2 1 0

Explorer

10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90 100

Mathspace WA Curriculum – Year 10 mathspace.co

10 9 8 7 6 5 4 3 2 1 0

1 2 3 4 5 6 7 8 9 10

Histogram

Fluency 4

Which key features can be read off each display? a

Ex 1

Boxplot

4, 5, 8, 9

Dot plot

4−8

5−8

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

Complete the table using the two boxplots. English

Mathematics

Median Lower quartile Upper quartile Range Interquartile range b

In which test did the class tend to score better marks?

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Ex 2

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: a

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?

Repetitions

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

542

Is the pet ownership a little lower or higher in Block A than Block B?

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 number of pets?

Does either set of scores have an outlier?

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Block A

Block B

Ex 3

Match the histograms to the corresponding boxplots. Histogram A Boxplot 1 10 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

Histogram B Boxplot 2 10 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

Histogram C Boxplot 3 10 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

Histogram D Boxplot 4 10 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

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543

The parallel boxplots show the distances, in centimetres, jumped by two high jumpers.

Matt

Paul

140

150

160

170

180

Distance (cm)

Who had a higher median jump?

Who made the highest jump?

Who made the lowest jump?

Reasoning

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10−13

11−13

Two History classes, each with 20 students, were given a 10 question True/False test. The results for each class are shown. 10 9 8 7 6 5 4 3 2 1 0

Class 1 Number of Students

Number of Students

10, 11

5 6 7 Questions Correct

10 9 8 7 6 5 4 3 2 1 0

Class 2

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.

Besides studying, what is another factor that may have had an effect on the test results for each class?

Mathspace WA Curriculum – Year 10 mathspace.co

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 as the most expensive 2 book in Bookstore A.

25% of the books in Bookstore B are cheaper than the cheapest book in Bookstore A.

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. 12.04 Compare data sets mathspace.co

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Problem-solving 14 14

14, 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.

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

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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.

Mathspace WA Curriculum – Year 10 mathspace.co

MODELLING INVESTIGATION

Ferry schedules Design and conduct a statistical experiment to model tourist preferences for ferry schedules to Rottnest Island. Simulate departure time and activity choices, then propose an optimal timetable based on trip duration analysis.

Objectives • Design a statistics experiment to model ferry schedule preferences • Conduct the experiment and analyse data to propose an optimal schedule

Design the ferry preference experiment You are designing an experiment to model tourist preferences for Rottnest Island ferry schedules. Plan a survey to collect departure time and activity preferences, then simulate responses to analyse trip durations. Choose: A sample size (between 20 and 30 tourists) Departure preference proportions. For example, 60% morning, 40% afternoon. Activity proportions, for example, 50% cycling, 50% snorkelling. Assume morning trips last 2 to 4 hours and afternoon trips last 3 to 5 hours. Each ferry has 100 seats. Note your choices or sketch the survey plan.

Investigate 1.

Record your sample size and preference proportions. Why did you choose these values?

List one assumption (e.g., random sampling) and one constraint (e.g., ferry capacity).

Modelling Investigation: Ferry schedules mathspace.co

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Conduct and analyse the experiment Conduct your experiment by simulating tourist responses, analysing trip durations with boxplots, and proposing an optimal ferry schedule.

Interactive exploration

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Simulate responses for your sample (e.g., 25 tourists), assigning durations (morning: 2 − 4 hours, afternoon: 3 − 5 hours). Create a two-way table and boxplots for durations by activity. Propose a schedule fitting 100 seats per ferry.

Investigate 1.

Record your simulated dataset. How did you ensure reliability?

Compare boxplots for cycling and snorkelling durations. Describe differences in shape, spread, and centre.

Propose a ferry schedule for 500 tourists. How do your findings support it?

Discussion 1.

What was the most challenging part of your experiment? Why?

How could you improve your experiment with more data or a different method?

Compare your schedule with a classmate’s. How do your findings differ?

Did you know?

Video game developers use univariate statistics to measure things like average playtime or highest score in a level. By focusing on one variable at a time, they fine-tune gameplay to keep it fun and challenging. So the next time a game feels just right, stats might be behind it! 548

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12.05 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 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 behaviour, 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

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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.

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 of the retirement home must be of at least a retirement age (usually 65). While for Option C, the students attending a primary school will only range from ages 4 to 12.

The answers are options B and D.

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.

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Apply the idea

Therefore, there is estimated to be 4800 ﬁsh in the lake.

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

Reﬂect and check

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.

We sometimes call data which is all grouped together and summarised aggregate data and raw data disaggregated data.

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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.

Apply the idea

Reﬂect and check

No, Maryke cannot determine if there was a month where students, on average, spent more than $100 just from the average monthly spending data of Source A. The overall average does not provide details of spending in individual months.

Many times, it is best to ﬁnd secondary sources that report raw data values rather than data that has already been summarised. If you need to use the data for a different reason than the researcher who collected the data, the information in their report may not be useful for your statistical study.

Idea summary • I n 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 time-consuming, while a sample that’s too small may not represent the population and is not useful to make conclusions.

12.05 Practice questions Understanding 1

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Explorer 1−5

Adventurer 1−5

Trailblazer 1−4, 6

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 exactly the same.

Mathspace WA Curriculum – Year 10 mathspace.co

If the population has 42 000 members, would this sample size be large enough? a

4000

40 000

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 or high schools in Parramatta

100 children chosen from a Sydney junior rugby league team

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Fluency 7

7, 8, 9bc, 10, 11ab, 13, 15−17

7, 8, 9cd, 11cd, 13−17

Match each sample histogram to the population it was taken from. i

iii

10 9 8 7 6 5 4 3 2 1 0

9 8 7 6 5 4 3 2 1 0

0 10 20 30 40 50 60 70 80

Mathspace WA Curriculum – Year 10 mathspace.co

10 9 8 7 6 5 4 3 2 1 0

0 10 20 30 40 50 60 70 80

iv 10

0 10 20 30 40 50 60 70 80

9 8 7 6 5 4 3 2 1 0

0 10 20 30 40 50 60 70 80

9 8 7 6 5 4 3 2 1 0

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7, 8, 9ab, 10, 11ab, 12, 15−17

10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0

0 10 20 30 40 50 60 70 80

Ex 1

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?

All students who are in the school band.

Eight students in the hallway.

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?

Are the 400 chocolates that were tested a representative 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?

Are the 500 apples that were picked a representative 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.

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15 Ex 2

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 marks of every student from both classes. • Source B: This data set provides the average final mark of students from both classes. 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 mark for only one class?

Source B indicates that the average mark for one class is 85%. Can Yakov determine how many students scored exactly 85% using Source B?

Reasoning 18

18, 19, 21, 22

19, 21, 22

20, 21-23

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.

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Explain why these samples are biased: a

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.

Problem-solving 25, 26 24

24−26

24−27

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. a

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?

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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.

558

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 WA Curriculum – Year 10 mathspace.co

12 Chapter review 1

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?

0 3 3 3 3 5 8

0 2 2 4 5 7 7

0 2 5 7 9

0 Key: 10|0 = 100

The histogram displays amount of time that Aaliyah spent watching television on her streaming platform over different days.

Number of 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

Time spent watching television 15

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?

A musical has a large cast of students with various ages. The column graph shows the ages of all the students in the cast. a

How many students are in the cast?

What age is most common?

Find the range of the students’ ages.

Find the interquartile range.

Ages of students in a musical

20 Frequency

15 10 5 0

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These data sets each contain 10 movie ratings out of 10: Set A: 2, 3, 4, 5, 6, 7, 8, 9, 10, 10 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.

For the data set shown in the dot plot: a

Construct the ﬁve-number summary.

Draw a boxplot.

Describe the shape of the boxplot.

Stem

Leaf

5 8 8

0 2 5 5 6 8

0 1

2 3 3 5

Key: 1|2 = 12 8

The boxplot 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 boxplots summarise the results from a botanical study. The treatment group received an experimental fertiliser 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: Control group

560

12 15 18 21 24

Treatment group

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.

Mathspace WA Curriculum – Year 10 mathspace.co

Compare the interquartile ranges of the control group and the treatment group.

Does the fertiliser have a positive effect on plant growth speed? Explain your answer.

For the cumulative frequency histogram:

300

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.

Cumulative frequency

250 200 150 100 50 0

50 100 150 200 250 Scores

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 Number of attendees

Number of workshops (f )

Cumulative frequency (cf )

20 21 22 23 24 25

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?

Use the cumulative frequency histogram to estimate: a

The median

The lower quartile

The upper quartile

100 90 80 70 60 50 40 30 20 10

Score 44

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Match the boxplot shown to the correct histogram.

Score

7 8 9 10 11 12 13 14 15 16 17 18

562

D Frequency

9 8 7 6 5 4 3 2 1 0

8 9 10 11 12 13 14 15 16 17 18 Score

10 Frequency

9 8 7 6 5 4 3 2 1 0

Frequency

10 Frequency

8 9 10 11 12 13 14 15 16 17 18 Score

10 9 8 7 6 5 4 3 2 1 0

8 9 10 11 12 13 14 15 16 17 18 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?

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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

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

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.

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.

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.

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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 us to make 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 • Analyse, determine and interpret problems in real-world situations involving twoand three-stage chance experiments both with and without replacement, conditional probability or statements, boxplots, bivariate data and/or two-way tables • Analyse bivariate data represented in a two-way table, using proportions and comment on possible association between categorical variables • Represent the relationship between bivariate data in a scatter plot and draw a trend line by eye if appropriate. Use the graph and context to describe any association in terms of strength, direction, linearity and outliers. Make predictions and recognise and explain any limitations of the model • Critically analyse the claims, inferences and conclusions of statistical reports in the media and other real-life situations and identify potential sources of bias

13 Bivariate statistics Chapter outline 13.01 Two-way tables 13.02 Associations between categorical variables 13.03 Bivariate numerical data 13.04 Associations between numerical variables 13.05 Line of good fit 13.06 Interpolation and extrapolation 13.07 Misrepresentations of data Investigation: Misrepresentations in the media 13.08 Correlation coefficient Investigation: Conduct a statistical investigation Chapter 13 review

566 578 598 608 622 633 647 660 674

There’s a correlation between the number of honeybees and crop production!

13.01 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? 2. How many students prefer low-top sneakers and shopping online? 3. 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 or attributes. 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.

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 online classes and for traditional in-person classes. Round your answers to one decimal place.

Create a strategy Add 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 Write as a fraction of the total

Add the numerator

Convert fraction to percentage and round Write as a fraction of the total

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 online classes and for traditional in-person classes. Round your answers to one decimal place.

Create a strategy Add 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 Write as a fraction of the total

Add the numerator

Convert fraction to percentage and round Write as a fraction of the total

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.

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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.

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 favouring 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.

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13.01 Practice questions Understanding 1

Explorer

Adventurer

1−3

2−4

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

Trailblazer

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?

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. 1.5 hours or more Less than 1.5 hours Total

Year 8 students 7 18 25

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.

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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

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.

Fluency 5

Adult 50 71 11

5−9

5, 7, 9, 10

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

30 71

41 71 V

Total 84 85 169

Seventy-one 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 30

18 30

29 41

12 41

Sprinkles

Hot fudge

Total

Chocolate Vanilla Total

Laura asked some people in her community what their preference was between beef, chicken, and ﬁsh.

20 respondents said they preferred beef, 8 of those were children. 15 children said they preferred chicken, and 15 adults said they preferred ﬁsh. 35 children were surveryed, and there were 73 respondents overall. 572

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Beef Children Adults Total

Chicken

Fish

Total

Ex 1

Ex 2

Special diet 14 11 15 40

What does 28 mean in this context?

What does 11 mean in this context?

Insulin injections 14 9 16 39

Total 28 20 31 79

Sophia asked 38 children and 54 adults in her community about their dietary preferences. The options given were: vegetarian, vegan or neither. 40 respondents said they were vegetarian, 15 of those 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 respondents 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.

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This table shows the number of trains arriving either on time or late at a particular station:

Day

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?

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

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 Maths 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

13, 14

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 Fruit Insects Nuts Total

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Honey possum

Total

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.

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 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. 13.01 Two-way tables mathspace.co

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Problem-solving 15, 16 15

16, 17

16−18

Yuri surveyed a group of people about the type of employment they had. He recorded the data in the graph. 40 35

Number of People

30 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

How many people are unemployed?

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

If the proportion of all the animals recorded carrying the MC1R gene and having light fur is

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MC1R gene 36 28

What does the value of x mean in this context?

Mathspace WA Curriculum – Year 10 mathspace.co

, ﬁnd the value of x.

Agouti gene 37 x

Lucinda sets out to see how many of her classmates use the term “soccer” or “football” to describe her favourite sport. Lucinda plans to survey her classmates from her ﬁrst three periods, and creates the following table for organising the data. 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.

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

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? 13.01 Two-way tables mathspace.co

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13.02 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 left 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 proportion of survivors from each class, we need to calculate row percentages. To ﬁnd the row 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.

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We can also calculate the percentage in each class type that survived or died. To do this, we calculate column percentages. To ﬁnd the column 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 27.8%

Part-time 29.4%

Full-time 41.2%

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 table, 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 we have row percentage tables or column percentage tables, the values only sum to 100% in one of the margins. In this case, we would not get 100% when adding across the rows.

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 situation, 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 apparent association 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 cause the change.

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 582

Play netball 33 13 46

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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 percentages of heights, we should use a row frequency table. 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

Play netball 33 13

Do not play netball 72 30

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%

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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 with just 1% difference in values. Therefore, 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

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

If the values are similar, then we can say that 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. 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.

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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 that the proportion of death to survival in each column is different. This indicates that 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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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: • I f 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.

Did you know?

Just because someone enjoys maths and also loves strategic board games like chess or Codenames doesn’t mean one causes the other! In statistics, we call this an association—a pattern between two categorical variables, like “likes maths” and “plays board games.” But association isn’t the same as causation. There might be a third factor, like a shared love for logic or problemsolving, that connects the two. Understanding this helps us make better sense of survey data and avoid jumping to the wrong conclusions. 13.02 Associations between categorical variables mathspace.co

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13.02 Practice questions Understanding 1

Explorer

Adventurer

1−4

Trailblazer

1−4

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

Special diet 23 16

Insulin injections 15 12

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

588

Single

Married

Transportation

Food

Mathspace WA Curriculum – Year 10 mathspace.co

Married with kids Insurance

Single with kids

Entertainment

Other

Fluency 5

5−8, 11, 12

Adults 30 45 75

Children 40 55 95

Total 70 100 170

A teacher recorded the hair type and colour of their students. The results are shown in a column graph. 10 9 8 Number of students

Ex 1

6−14

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

5−9, 11, 12, 14

7 6

Straight

Curly

4 3 2 1 0

Brown

Blonde

Black

Complete the two-way table displaying the row percentages rounded to one decimal place. Straight Curly

Red

Red 5.9% 10%

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

50% 100%

Brown 66.7% 100%

Blonde 25% 75% 100%

Black

100%

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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

Comedy 30% 30%

Teens Children Total

Ex 2

Action

Drama 45%

Horror 5%

40%

Total 100% 100%

50% 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

590

Horror

Complete the two-way table displaying the column percentages rounded to one decimal place. Comedy

Action Drama Movie genre

Complete the two-way table displaying the row percentages rounded to one decimal place. Teens Children

Comedy

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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

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

Cardio

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.

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

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?

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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

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%

Personality type of smokers 100 80 Percentage

Ex 3

Smokers Non-smokers

40 20 0

592

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?

Mathspace WA Curriculum – Year 10 mathspace.co

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

Music preference

Music preferences by dietary restriction 100% 80%

Punk R&B Country & western Rap Pop Classical Rock & roll

60% 40% 20% 0%

Vegetarian Non-vegetarian Dietary restriction

Does this stacked column chart suggest that there is an association between music preference and dietary restriction? 13.02 Associations between categorical variables mathspace.co

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Reasoning

15, 16, 18

17, 19

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 your answer.

Mr. Tranor asked his class to pick their favourite subject. He displayed the results in a two- way table. Right-handed children Left-handed children

594

Regional 20 32 52

Maths 13 20

Music 9 12

Science 11 18

English 11 10

How many left-handed children did not pick maths as their favourite subject?

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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. of passengers Roll over No roll over Total

<5 335 1622 1957

5—9 18 42 60

> 15 2 5 7

10—15 17 42 59

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, 21 20

20, 21

20−22

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.

Aug Sep

In which month were children’s books the most popular?

Oct

Which variable is the response variable?

Nov

Do the divided bar graphs suggest that there is an association between the variables?

Fiction Design Environmental

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.

596

Jul

In which month were business books the most borrowed type of book?

Jun

Mathspace WA Curriculum – Year 10 mathspace.co

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

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?

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.

Did you know?

100% stacked column graphs make it easier to spot patterns in two-way tables—especially when the groups you’re comparing are different sizes! Instead of focusing on totals, these graphs show percentages, making it easier to compare categories fairly. For example, even if more boys were surveyed than girls, the graph can reveal that a higher percentage of girls prefer pop, while boys lean more towards rock or hip hop. It’s a clearer way to explore associations between groups—but remember, just because there’s a pattern doesn’t mean one thing causes the other!

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13.03 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

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Age (years)

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 part, 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 prices of ten houses are graphed against the house’s Price (thousands) land area on the following graph: 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 they 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)

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

600

The scatterplot of the data is shown.

Mathspace WA Curriculum – Year 10 mathspace.co

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 increase 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 increase, 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.

13.03 Practice questions Understanding 1

Explorer 1, 2ac, 3, 4

Adventurer 1, 2bc, 3, 4

Trailblazer 1, 2cd, 3, 4

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.

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.

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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

y 4

1 −4 −3 −2 −1 −1

x 1

−2

−3

−4

d 4

−4 −3 −2 −1 −1

602

−3

−4

6, 7, 10, 12, 14, 16

−2

Fluency

x 1

−4 −3 −2 −1 −1

6−8, 10−12, 14, 16

5, 7−11, 13−16

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

Mathspace WA Curriculum – Year 10 mathspace.co

What is the dependent variable?

Ex 1

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

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

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 0

Temperature 16 20

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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?

Annual salary 100 80 60 40 20 Height (cm) 0

Ex 3

120

160 200

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?

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

604

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

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

Global temperature anomaly (°C) 0.6 0.65 0.68 0.7 0.75 0.8 0.83 0.85 0.88 0.9

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?

Mathspace WA Curriculum – Year 10 mathspace.co

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

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 nine 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)

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.

Reasoning 17

3 14

Time (hours) Height (metres)

0 17

17, 18

17−19

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. 32 Number of trees in the garden 30 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

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.

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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

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.

606

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.

Mathspace WA Curriculum – Year 10 mathspace.co

Problem-solving 20, 21

20−22

Georgia has a high income but does not like her job. Which dot would represent her? Explain your answer.

3 2

Job Satisfaction

20−22

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)

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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13.04 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. • 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. If an association is linear, then it can be referred to as a correlation. To identify an association 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

Discover this concept in action online

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mathspace.co

Perfect positive correlation

Perfect negative correlation

Strong positive correlation

Strong negative correlation

Moderate positive correlation

Moderate negative correlation

Weak positive correlation

Weak negative correlation

No correlation

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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. Two variables may be strongly correlated, but not necessarily causal for a few reasons. 1. 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. 2. 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. 3. 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.

Apply the idea Examine the scatterplot and determine whether there is a discernible relationship between height and vision. 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 scatterplots correctly represents the 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

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 happens to the students’ marks in French as their marks in English 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

Apply the idea 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

None of them

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

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

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 into 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.

Idea summary To identify an association or a 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.

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13.04 Practice questions Understanding 1

Explorer

Adventurer

1−3

Trailblazer

1−3

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? a

b 100 Final exam mark

Final exam mark

100

x 0

10 20 Number of class attended

100

100 Final exam mark

Final exam mark

Number of classes attended

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:

Number of classes attended Running distance (km) 20

Using the scatterplot, are these statements true or false? a The number of weeks that Sumon trained for the half 15 marathon is the independent variable. 10 b The y-coordinates of the points represent the time spent by Sumon training. c There is evidence to suggest that the longer Sumon 5 trains, the further they can run. No. of weeks d Training for a half marathon causes further running 0 5 10 15 20 distance over time. e The correlation between the number of weeks training and the number of kilometres Sumon is able to run is positive. 614

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Complete each sentence with one of the words from this list. • contributing (third) variables • correlation

• association • causation • coincidence a b c d

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 ⬚.

Fluency 4

4−6, 8, 9, 12−14, 16

4−8, 10, 12, 14, 16

4−8, 10, 11, 15, 16

Does each scatterplot show a linear or non-linear relationship? a

5 x

x 0

5 x

x 0

5 x 0

x 0

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Describe the strength and direction for each linear association. a

5 x 0

5 x

x 0

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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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)

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%

618

20 85

GDP per capita $5 000 $10 000 $3 000 $15 000

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Ex 4

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

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 a novel and number of pages in the novel

Time spent training and athletic performance

Reasoning

17, 18

18, 20

18−20

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

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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.

Problem-solving 21, 23 21

21−24

A chemical company is testing the effect of different chemicals on slowing 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 remaining (%)

Snow remaining (%)

100

x 0

Temperature outside (°C)

100

100 Snow remaining (%)

Snow remaining (%)

x 0

Temperature outside (°C)

A snow sculpture company wants a chemical that will keep snow from melting.

620

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?

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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 of students who had answered Question 10 correctly against the percentage of students who had answered Question 7 correctly. 2

1 % that got Q10 correct

% that got Q7 correct

Determine whether the statements describe the data. i

There is no relationship between students’ performance in Q7 and Q10.

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.

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13.05 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? 2. 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

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. 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

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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 0 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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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 variable h is used instead of x. The value h = 0 represents a student that watched 0 hours of TV.

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

Apply the idea

In this case, the independent variable is “hours watching TV”.

It is possible to watch no TV and earn a mark of 97, so the answer is option A.

Is watching no TV possible in the context? Is earning a mark of 97 possible in this context?

c State the gradient of the line.

Create a strategy

Apply the idea

When an equation is of the form y = mx + c, the gradient of the line is m, the number in front of x. In this case, h is used instead of x, so the gradient will be the number in front of h.

The number in front of the independent variable (h) is −10, 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%

Multiply the decrease percentage by hours Evaluate

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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.

13.05 Practice questions Understanding 1

Explorer

Adventurer

1−4

Trailblazer

1−4

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

x 0 1

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2 3 4 5 6

8 9

Does there appear to be a linear relationship between the variables? a

5 x 0

Looking at the scatterplot:

x 0

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 0 1

2 3 4 5 6

8 9

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Fluency Ex 1

5, 6a-c, 7, 9ac, 10−12

5, 6c-e, 8, 9bd, 10−12

This scatterplot shows the data for two variables, x and y.

5, 6d-f, 8−12

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 0

x 0

5 x 0

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x 12

2 x

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

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

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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 0

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200 400 600 800 1000

Reasoning 13

13, 14

13−15

The heights (in centimetres) and the weights Weight (kg) (in kilograms) of 9 primary school children are shown on 60 the scatterplot: 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 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.

y 7

Frasier

6 5

4 Lorena said she noticed that there was a point that was far away, but decided to ignore it. Does Lorena’s 3 line meet the requirements for a line of good fit? 2 Explain.

Lorena

x 0 2 4 6 8 10 12 14 16 18

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Problem-solving 16−18 16

16−19

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?

17−19

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.

Mathspace WA Curriculum – Year 10 mathspace.co

11 10 9 8 7 6 5 4 3 2 1

x 0

632

2 3 4 5 6

8 9

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.

Year 2005 2006 2007 2008 2009 2010

Construct a scatterplot and find a line of good fit for this data.

More data has been collected, and the results are listed:

Y 0 1 2 3 4 5

Hives lost to CCD, (H)% 18 13 23 26 28 30

• 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?

13.06 Interpolation and extrapolation After this lesson, you will be able to… • understand and differentiate between interpolation and extrapolation. • 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.

Interactive exploration

Discover this concept in action online

mathspace.co

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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 y

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

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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. Temperature (°C) Number of chirps per minute

14 77

17 115

21 150

24 176

220

Chirps per minute

200 180 160 140 120 100 80 60 40 20

Temperature °C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

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 x-value for a given y-value. Start with y = 140 and find the x-value.

Apply the idea We can draw a line from our y-value, 140, horizontally to the line of the relationship, then another line vertically down to the x-axis.

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What is the x-value we end up at? 220

Chirps per minute

200 180 160 140 120 100 80 60 40 20

Temperature °C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

The answer is 20°C.

b According to the graph, how many chirps per minute will the crickets make if the temperature is 27°C?

Create a strategy When x = 27, find the value of y.

Apply the idea We can draw a line from our x-value, 27, 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? 220

Chirps per minute

200 180 160 140 120 100 80 60 40 20

Temperature °C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

The answer is 210 chirps per minute.

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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 x = 19, find the value of y.

Apply the idea We can draw a line from our x-value, 19, 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? 220

Chirps per minute

200 180 160 140 120 100 80 60 40 20

Temperature °C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

The answer is 130 chirps per minute.

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 number of chirps per minute is 77 and the maximum number of chirps per minute is 176. 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 redictions can be made either using a graph of the line of good ﬁt or substituting P 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

13.06 Practice questions Understanding 1

Explorer 1, 2, 3a

Trailblazer 1, 2, 3c

Is each statement true or false? a

A line of good ﬁt can only be used for interpolation, not extrapolation.

Extrapolation using a least-squares regression line 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.

Does each graph show interpolation or extrapolation? a

y 90 80 70 60 50 40 30 20 10

b 90

80 70 60 50 40 30 20 x 0 10 20 30 40 50 60 70 80 90

638

Adventurer 1, 2, 3b

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x 0 10 20 30 40 50 60 70 80 90

y = 0.45x + 7.1, at x = 3

y = 1.75x - 3.2, at x = -2.4 4a, 5, 6, 8, 9, 11, 13

y = -2x - 7, at x = 5

4b, 5, 6, 8, 9, 11, 13

4c, 5−7, 10, 12

A prediction for the y-value when x = 5 is made from the data set: x y

4 0

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

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:

Ex 1

Are these predictions an extrapolation or an interpolation? a

x 0 10 20 30 40 50 60 70 80 90

Find the y-value for the given least-squares regression line and x-value.

Fluency 4

94.4

96.4

94.4

97.8

94.9

95.9

94.4

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

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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)

1.1

1.5

2.1

2.5

3.5

Reaction time in seconds ( y)

4.66

4.1

4.66

3.7

3.6

3.4

The data has been graphed along with a line of good fit found by eye.

Reaction time 5

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.

4 3 2 1 0

Number of hours of sleep 2 3 4 5

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

2 1995

The data has been graphed along with a line of good fit found by eye. a

Predict the number of years until there are no ﬁsh left in the river.

Predict the number of ﬁsh remaining in the river after 7 years.

According to the line of good ﬁt, how many years are there until there are 900 ﬁsh left in the river?

3 1602 2000 1800 1600 1400 1200 1000 800 600 400 200

4 1695

5 1311

t 2 4 6 8 10 12 14 16 18 20

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

Approximately, what is the temperature when there are 200 people at the beach?

25 230 225

People

200 175 150 125

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

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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?

Is it reasonable to use the line of good fit to predict pressure within each of these range of temperatures? i

300 ≤ Temp ≤ 320

300 ≤ Temp ≤ 600

iii

0 ≤ Temp ≤ 320

280 ≤ Temp ≤ 340

2580 2560 2540 2520 2500 2480 2460 2440 2420 2400

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

35 70

29 45

34 71

33 67

25 30

25 28

34 65

34 75

The data has been graphed along with a line of good ﬁt: a

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

20 ≤ Temp ≤ 40

iii

20 ≤ Temp ≤ 40

0 ≤ Temp ≤ 60

32 60

100 90 80 70 60 50 40 30 20 10

35 74

33 63

28 35

33 62

32 65

Sales

Temp (°C) 22 24 26 28 30 32 34 36 38 40

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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 drawn by eye. a

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?

A 9000

Predict the altitude of the plane 100 seconds into the descent.

How many seconds did the plane take to descend to the ground?

1700 593

8000 7000 6000 5000 4000 3000 2000 1000 t 0 200 400 600 800 1000 1200 1400 1600

A car company looked at the relationship between how much it had spent on advertising (A) and the amount of sales (S) each month over several months. The data has been plotted on the 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 sales in thousands of dollars, and A is the amount spent advertising costs.

S 800 700 600 500 400 300 200 100 A 0

1000 2000 3000 4000 5000 6000 7000 8000

Find the value of c, the vertical intercept of the line. c

Use the line of good ﬁt to estimate the number of sales next month if $4800 is to be spent on advertising.

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 1 2 3 4 5 6 7 8 9 10

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Reasoning 14

15−17

15, 17

14, 15, 17

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. a b

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 (a) reliable? Explain your answer.

x 0 1 2 3 4 5 6 7 8 9 10 11 12

Are these predictions examples of interpolations or extrapolations? Justify your answer. a

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.

40 50 60 80 90 100

Fuel efficiency (L/100 km) 6.5 5.8 5.2 4.9 5.4 6.1

Year 2015 2016 2017 2018 2019

Revenue ($ million) 10 12 15 18 22

Speed (km/h)

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 different hours of study, a prediction for the test score at 15 hours of study is made from the data set.

Test score

80 70 60 50 40 30 20 10

Hours of study 0 2 4 6 8 10 12 14 16 18

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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)

14 12 10 8 6 4 2

00 00 00 00 00 00 00 10 20 30 40 50 60 70 Size (square metres)

Discuss the limitations of using interpolation and extrapolation for making predictions in the context of climate change.

Problem-solving 19−22 18

18, 20, 22−24

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 0

19, 21−23

1000

2000

3000

4000

5000

6000

7000

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.

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.

A 800 700 600 500 400 300 200 100

T 0

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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)

1 3.9

2 3.7

3 3.5

4 3.1

5 3.1

6 2.9

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.

7 2.7

The ﬂeet manager for the Australian Automotive Association wants to estimate how car maintenance costs, C (in hundreds of dollars), are related to the distance, K (in thousands of kilometres), driven each year. The data collected and the least-squares regression line are shown on a scatterplot. C 550 500 450 400 350 300 250 200 150 100 50 K 0

The equation of the least-squares regression line for the graph is C = 9.9K + 146.2

Interpret what the coeﬃcient of K represents in this context. b

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.

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.

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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. y 100 80 60 40 20 t 1

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? A

Extrapolation is less reliable since we should use a 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 good fit, are either interpolation or extrapolation reliable? Explain. 9

8 7 6 5 4 3 2 1

x 0 1

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2 3

4 5 6

8 9

13.07 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 realworld examples. • develop skills to critically evaluate data displays and graphs for potential misrepresentation.

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 5 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

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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

Apply the idea 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. 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.

Reﬂect and check 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.

Idea summary Common ways that data may be manipulated during reporting: • • • • • • • • •

648

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

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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

1. Are they showing the same or different data? Are they conveying the same message? Explain. 2. What do you notice that is different about the graphs? Explain. 3. When would each graph be useful for a media source to show to viewers? Why or why not?

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.

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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 Party 5/9 Sydney Ave, Barton ACT 2600

The Australian Labour Party released this graph after Tony Abbott 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.

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. The cabinet sizes of each country are not the same, so the amount of women in Tony Abbott’s cabinet may be proportionally more comparable to the amount of women in other countries’ cabinets. Option C is correct. 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 on the vertical axis, a proportion would be more appropriate. If a country has 10 members in the parliament and one is female, then that is more representative than a country that has 60 members in the parliament and only three are female.

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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

Apply the idea

Many of the issues relate to the scale and type The y-axis does not use a linear scale. Instead, of graph. Analyse the axes to see what might be a logarithmic scale was used but it was not misleading. specified.

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

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Idea summary Some of the ways in which the axes and scale 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

13.07 Practice questions Understanding 1

Explorer 1−3

Adventurer

300 people were asked if they would support any of the three candidates for an election. a

What should the percentages in a conventional pie chart add up to?

What do the percentages in this pie chart add up to? Supports Huckabee Is this an appropriate data display?

Trailblazer

1−4

1−4 Presidential run 2012

Supports Palin 70% 63% 60% Supports Romney

652

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.

Mathspace WA Curriculum – Year 10 mathspace.co

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 zero.

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

Fluency 5

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

Ex 1

Cases on log10 scale 1.5 2.3 2.24 2.53 2.4 2.6

5, 6, 9−11, 13, 14, 16

5, 6, 8, 10, 11, 13, 14, 16

5, 7, 8, 11, 12, 14−16

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 and comparing that to the response of 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? a

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.

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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 included the data since 2011 accurately 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 priced at: $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 the building says “Most units cost $250 000.”

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?

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. Percentage

Brand A 65%

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 company using?

Which statement describes why the claim is misleading? A

The statement is inaccurate.

The survey was conducted by the company that owns 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?

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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

Turnover million

50 000 40 000 30 000 20 000 10 000 0 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Year

The ﬁgure shows the number of tourists from each country to the United States:

IN CO IL

DE FR

Brazil: 613 347

France: 930 265

Germany: 1 263 344

Japan: 2 169 716

Mexico: 4 295 124 United Kingdom: 2 905 909

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?

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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? 15

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. Medication in blood over time Medication concentration in blood (ng/mL)

Ex 3

100 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?

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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

17, 18, 20

17, 19, 20

19−21

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.

Consider the line graphs of the same set of sales ﬁgures: 9000

Sales ($)

8000 7000 6000 5000 4000 3000 2000 1000

Month 0

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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:

Month 5

Quarterly operating deficit 55

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.

50 $ (million)

45 40 35 0

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

658

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?

Mathspace WA Curriculum – Year 10 mathspace.co

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-ageing 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?

Problem-solving 22, 23 22

22, 23

23, 24

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

2018

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

Norway

Number of COVID-19 tests per million people

Source: Reddit May 2020 a

Explain why the graph and comment are misleading.

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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

2000

1996

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.

Investigation: Misrepresentations in the media

Investigate online

mathspace.co

13.08 Correlation coefficient After this lesson, you will be able to… • calculate and interpret the value of the correlation coefficient r and coefficient of determination r2 for a given data set. • apply the correlation coefficient and coefficient of determination to interpret relationships between variables in real-life scenarios.

Correlation coefficient We have describe the form, strength, and direction of a scatterplot in words. For linear data sets, we will how quantify the association with a number. The correlation coefficient, r, is a value that describes the strength and direction of a linear relationship between two variables. Sometimes it is called the Pearson’s correlation coefficient. 660

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Interactive exploration

Discover this concept in action online

mathspace.co

The value of the correlation coefficient varies from −1 to +1, where -1 describes perfect negative correlation and +1 describes perfect positive correlation. If the points are trending upwards from left to right, the correlation coefficient is positive. If the points are trending downwards from left to right, the correlation coefficient is negative. The cutoff points for interpreting r-values can vary depending on the field, industry, or researcher as there is no universal standard. We will use the following here as a general guide: Correlation Very weak Weak Moderate Strong Very strong

r-values 0 ≤ r < 0.2 0.2 ≤ r < 0.4 0.4 ≤ r < 0.6 0.6 ≤ r < 0.8 0.8 ≤ r < 1

In actual statistical analysis, the interpretation of correlation would always take the context and nature of the data into account, as the strength of a relationship may have different significance in different settings. The closer the points are to being in a straight line, the closer r is to +1 or -1. 9 8 7 6 5 4 3 2 1

9 8 7 6 5 4 3 2 1

y 4 3 2 x

1 2 3 4 5 6 7 8 9

No correlation r = 0.072

Moderate negative correlation r = −0.684

x 20

40 60 80

Strong negative correlation r = −0.974

The calculation of r can be done by hand, but is typically done using technology. It is important to be able to distinguish between causal relationships (when changes in one variable cause changes in the other variable) and correlation where the two variables are related, but one variable does not necessarily influence the other. Even when two variables have a strong relationship and is close to 1 or -1, we cannot say that one variable causes change in the other variable. Causation can only be determined from an appropriately designed statistical experiment.

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Example 1 For the graph depicted, choose the correlation coeﬃcient that best represents it. A 1

y 20 15

B 0 C −1

D −0.64

5 x 5

−5

Create a strategy Check if you could ﬁt a straight line to the points plotted.

Apply the idea Based on the graph, a straight line cannot be ﬁtted on the plotted points. This means that the two variables in data set are not linearly correlated. So, we can say that the correlation coeﬃcient is 0, and the correct answer is option B.

Example 2 Given the following data: x y

1 4

4 4.25

7 4.55

10 4.4

13 4.45

16 4.75

19 4.2

a Calculate the correlation coefficient and round your answer to two decimal places.

Create a strategy Use technology to ﬁnd the correlation coeﬃcient.

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Apply the idea Using the Statistics mode in your calculator, enter each x-value along with its y-value into a data table on your calculator.

Use the Two-Variable Regression Analysis:

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Find the summary statistics:

Look for the correlation coeﬃcient (r): r = 0.47

b Choose the best description of this correlation. A

Moderate negative

Strong positive

Weak negative

Moderate positive

Strong negative

Weak positive

Apply the idea The value of r = 0.47 and is between 0.4 and 0.6. So there is a moderate positive correlation between the variables. The correct answer is option D.

Idea summary ry Ve

r Ve

Mathspace WA Curriculum – Year 10 mathspace.co

k ea W

Correlation does not imply causation.

e at er

ro St

−0.8 −0.6 −0.4 −0.2

ro st

ry Ve −1

Perfect negative linear correlation

664

0.2

No correlation

a er

ea W

0.4

0.6

ro St

0.8

r Ve

ro st

Perfect positive linear correlation

Coefficient of determination The coeﬃcient of determination, r2, is related to the correlation coefficient, r. Since it is the square of a number between −1 and 1, the coefficient of determination is between 0 and 1, and is often shown as a percentage. If we have already identiﬁed the value of r, this can simply be squared to get the value of r2. For instance, if r = 0.8 then r2 = 0.64, and if r = −0.9 then r2 = 0.81. r2 tells us the proportion of the response variable ( y) that can be explained by the variation in the explanatory variable (x). For example, if r2 = 0.92 then we can say that 92% of the variation in the response variable is explained by the variation in the explanatory variable. The closer the value of r2 is to 1, the more that the variation in the response variable is explained by the variation in the explanatory variable. Note that this does not mean that the closer r2 is to 1, the more the x-variable is causing the y-variable to happen. Correlation does not imply causation.

Example 3 A scientist investigated the link between the number of cancer cells killed by a certain drug and the strength of the drug used. The results were recorded and the coeﬃcient of determination r2 was found to be 0.92 Which of the following is true? A There is a strong relationship between the strength of the drug used and the cancer cells killed. B The number of cancer cells killed causes the strength of the drug used. C We cannot infer a causal relationship between the strength of the drug used and the cancer cells killed. D The strength of the drug used causes the cancer cells to be killed. E There is a weak relationship between the strength of the drug used and the cancer cells killed.

Create a strategy The coeﬃcient of determination with a value close to 1 is considered strong, and a value close to 0 is considered weak.

Apply the idea Option A is correct because it describe the strong relationship as the recorded coeﬃcient of determination, 0.92, is close to 1. Option B is incorrect because the statement does not make any sense as we see the number of cancer cells killed depends on the strength of the drug used. Option C is correct because a high coefficient of determination value does not necessarily imply a causation (nor does a low one). Option D is incorrect because again correlation does not infer causation. Option E is incorrect because the relationship between the two variables is described as strong. The correct answers are options A and C.

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Example 4 The heights (in centimetres) and the weights (in kilograms) of 8 primary school children are shown on the scattergraph below.

Weight 60 55 50 45 Height 110 115 120 125 130 135 140 145

a Calculate the value of the coeﬃcient of determination. Round your answer to two decimal places.

Create a strategy Use the linear regression function on your calculator. The data set as a table of values is: Height Weight

106 45

112 46

113 46

122 52

127 50

132 53

137 57

142 60

Apply the idea Using the Statistics mode, enter each x-coordinate along with its y-coordinate into a data table on your calculator then ﬁnd the linear regression. Look for the coeﬃcient of determination (r2): r2 = 0.93

b Calculate the value of the correlation coeﬃcient. Round your answer to two decimal places.

Create a strategy Take the square root of the coeﬃcient of determination.

Apply the idea Take the positive square root of 0.93

Evaluate and round

Reﬂect and check When we take the square root, we need to consider the positive and negative square root. Since this association is positive, we will take the positive square root.

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c What percentage of the variation in weight is accounted for by the height of the child? Round your answer to the nearest whole percent.

Create a strategy Convert the coeﬃcient of determination into a percentage.

Apply the idea We have already found a value for the coeﬃcient of determination (as a decimal). 0.93 = 0.93 × 100%

Multiply by 100%

= 93%

Evaluate

d Consider these two comments on the claim “The weight of a child is primarily inﬂuenced by their height.” Which do you think is most correct? A

This claim is valid and is supported by the strong relationship between the two variables.

hile this claim is supported by a strong relationship between the two variables, we cannot W state causality as there may be other factors inﬂuencing the outcome.

Create a strategy Consider your answers to previous parts of this question and choose the statement that you think is most correct.

Apply the idea The claim describes a strong relationship as the coeﬃcient of determination, 0.93, is close to 1. But a high coeﬃcient of determination (or correlation) value does not necessarily imply causation. So the correct answer is B.

Idea summary The coefficient of determination, r2, tells us the proportion of the dependent (response) variable ( y) that can be explained by the variation in the independent (explanatory) variable (x).

13.08 Practice questions Understanding 1

Explorer 1ab, 2–3, 5

Adventurer 1bc, 2–5

Trailblazer 1cd, 2–5

For each correlation coefficient, find the coefficient of determination. a

r = 0.94

r = -0.5

r=1

r = -0.85

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Data set A has a correlation coeﬃcient of r = 0.1 while data set B has a correlation coeﬃcient of r = 0.6. Which data set has the stronger correlation?

Is each statement true or false?

The coefficient of determination can be negative.

We can tell the direction of the association from the coefficient of determination.

We can tell the strength of the association from the coefficient of determination.

10 is a possible correlation coefficient.

Let r(x, y) represents the correlation coeﬃcient for the relationship between variable x and y, and r(x, z) represents the correlation coeﬃcient for the relationship between variables x and z. For each pair: i

Determine whether the two linear relationships have the same direction.

Which variables have a stronger correlation.

r(x, y) = 0.3, r(x, z) = 0.9

r(x, y) = −0.9, r(x, z) = −0.5

r(x, y) = 0.8, r(x, z) = −0.5

r(x, y) = 0.3, r(x, z) = −0.9

For each situation, describe the correlation in context. a

A researcher plotted the life expectancy of a group of men against the number of cigarettes they smoke a day. The results were recorded and the correlation coeﬃcient r was found to be −0.88

A researcher was evaluating the relationship between the number of years in education a person completes and the number of pets they own. The results were recorded and correlation coeﬃcient r was found to be −0.3 6, 7, 8abc, 9–14

Fluency 6

Ex 1

6, 7, 8bcd, 9, 11–15

6, 7, 8cde, 9, 11–13, 15, 16

Describe the linear relationship between the variables with these correlation coeﬃcients: a

0.96

0.66

0.36

−0.06

−0.34

−0.56

−1

For each scatterplot: i

Describe the type of linear relationship between the variables.

Estimate the value of the correlation coeﬃcient.

5 x

x 5 668

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5 x

x 5

y 15

15 10 10 5

5 x 5

8 Ex 2

For each set of bivariate data: i

Calculate the correlation coeﬃcient to two decimal places.

Describe the strength of the relationship between the two variables.

x y

9 19

4 12

35 75

34 63

32 61

18 37

28 57

10 22

20 38

3 7

x y

2 88

4 83

7 60

0 95

3 84

2 88

0 104

6 72

4 85

3 84

x y

3 7

4 7.4

5 7.88

6 7.64

7 7.72

8 8.2

9 7.32

x y

65 25.5

72 24.2

48 16.8

84 10.4

78 24.8

56 9.6

63 6.3

64 21.4

77 13.7

93 16.3

x y

13 3.6

13 12.5

9 12.1

12 12.8

9 11.1

15 9.7

10 11.4

12 9.7

12 10.6

14 8.9

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Noah is a coffee vendor. He records the maximum temperature of the day and the number of coffees sold. The results are recorded in the table: Maximum temperature (°C) Number of coffees

28 17

32 37

31 25

33 39

31 23

26 7

25 19

29 34

35 42

Construct a scatterplot for the data.

Use technology to calculate the correlation coeﬃcient. Round your answer to two decimal places.

Describe what happens to the sales of coffee as the temperature increases.

Calculate the value of r2 for the following data sets whose correlation coeﬃcient is given. Round your answer to two decimal places. a

Correlation coeﬃcient is −0.61

Correlation coeﬃcient is −0.44

A linear association between two data sets is such that the correlation coeﬃcient is −0.72 What proportion of the variation can be explained by the linear relationship? Round your answer to the nearest percent.

Ex 3

A linear association between two data sets is such that the coeﬃcient of determination, r2 is 0.66 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.

Mae has a small herb garden with a thyme plant. She suspects that the growth of the thyme plant is linearly dependent on the amount of tea she drinks. For a month, she keeps a daily log of the amount of tea she drinks and the millimetres that the thyme plant has grown. The resultant coeﬃcient of determination r2 was found to be 0.13

14 Ex 4

Is there a linear relationship between the growth of the thyme plant and the amount of tea that Mae drinks?

Can we say that there is a causal relationship between the growth of the thyme plant and the amount of tea that Mae drinks?

During an alcohol education programme, 10 young adults were offered up to 12 drinks and were then given a simulated driving test where they scored a result out of a possible 100. The results are displayed in the table: Number of drinks Driving score a

670

2 61

3 65

5 44

3 62

4 36

2 73

7 32

6 41

9 18

2 54

Calculate and round your answers to two decimal places. i

The coeﬃcient of determination

The correlation coeﬃcient

What percentage of the variation in the driving score is accounted for by the number of drinks consumed? Round your answer to the nearest percent.

Is this claim valid: “Anyone driving poorly on the road has been drinking”?

Mathspace WA Curriculum – Year 10 mathspace.co

Which Bank is researching the number of applications they receive for home loans per day based on their advertised interest rate. The results are displayed in a table. a

Calculate, rounding your answers to two decimal places: i

The coeﬃcient of determination

The correlation coeﬃcient

What percentage of the variation in the number of applicants is accounted for by the interest rate? Round your answer to the nearest whole number.

Is this claim valid: “If you want people to take out a home loan, then you need the interest rates to be low”?

Interest rate 3.2 4.2 4.4 0.3 6.5 7.3 4.8 0.7 10.3 2.4 8.9 10.7

Advertisers are researching the optimal length of an advertisement so that viewers don’t click ‘Skip’.

Minutes 5.1 5.9 0.4 1.4 0.8 4 4.1 0.6 2.7 5.8 1.4 0.7 1.1 0.8 4.8

The results in the table alongside show the length of the advertisement in minutes, and the number of proportion of viewers who clicked ‘Skip’: a

Calculate and round your answers to two decimal places. i

Explain the meaning of the value of r2 in context.

Is this claim valid: “If an ad is over 2 minutes in length it means everyone presses ‘Skip’ ”?

Reasoning 17

17, 18

Number of applicants 170 158 155 196 134 125 154 191 95 176 110 95

17, 18

Proportion 1 1 0.6 0.4 0.7 0.8 0.8 0.3 0.9 0.8 0.7 0 0.4 0.8 0.8

18, 19

The table shows the number of fans sold at a store during days of various temperatures: Temperature (°C) Number of fans sold

6 12

8 13

10 14

12 17

14 18

16 19

18 21

20 23

Is the value of the correlation coeﬃcient, r, positive or negative?

Is there a causal relationship between the variables? Explain your answer.

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Explain why it would not be appropriate to calculate the correlation coeﬃcient for the data set shown.

y 800 700 600 500 400 300 200 100

x 1

2 3 4 5 6

8 9

−100 −200

Use the correlation coeﬃcient to interpret the relationship between the variables in these studies: a

A study found that the correlation coeﬃcient between heights of women and probability of being turned down for a promotion was found to be −0.90

A study found that the correlation coeﬃcient between population of a city and number of speeding ﬁnes recorded was found to be 0.83

A study found that the correlation coeﬃcient between length of hair and length of ﬁngernails was found to be 0.07

A study found that the correlation coeﬃcient between number of bylaws a council has about dog breeding and number of dogs available for adoption at the local shelter was found to be 0.55

Problem-solving 20, 21 20

21–23

A cafe manager collected data about sales of hot cocoa during a winter weekend. The data included the outside temperature and the number of hot cocoa sold every two hours. Outside temperature in °C, t No. hot cocoa sold, n

672

20–22

−17.8 20

−16.7 16

−15 3

−16.7 16

−17.2 18

−16.1 17

−14.4 8

−15.6 12

Using technology, determine whether a linear or exponential model good ﬁts this data.

Calculate the regression model for this data set. Round all values to the nearest tenth.

Using your answer from part (a), determine whether the correlation coeﬃcient is a relevant statistic in this example. If so, calculate and interpret the correlation coeﬃcient within the context of this problem.

Mathspace WA Curriculum – Year 10 mathspace.co

The following table shows the years of teaching experience for various teachers across Australia and their current salary: a

Calculate the value of the correlation coeﬃcient. Round your answer to two decimal places.

Calculate the value of the coeﬃcient of determination. Round your answer to four decimal places.

What percentage of the variation in teacher salaries can be explained by their years of teaching experience? Round your answer to two decimal places.

Is it reasonable to suggest that teaching experience is the sole inﬂuence on teacher salaries?

Years of experience 5 6 6 6 5 3 10 11 1 11 11 6

Salary ($) 72 400 67 200 65 700 67 400 73 600 64 500 87 400 82 000 57 300 86 900 84 700 68 800

If not, give an example of a different factor that affects teaching salaries. 22

Research on the number of cigarettes smoked during pregnancy and the birth weights of the newborn babies was and results shown in a table.

Average number of cigarettes per day (x) 46.3 13.0 21.4 25.0 8.6 36.5 1.0 17.9 10.6 13.4 37.3 18.5

Birth weight in kilograms ( y) 3.9 5.8 5.0 4.8 5.5 4.5 7.0 5.1 5.5 5.1 3.8 5.7

Using technology, calculate the correlation coeﬃcient between the average number of cigarettes per day and birth weight. Round your answer to three decimal places.

Describe the correlation between these two variables.

Use technology to ﬁnd the line of good ﬁt. Give all values to two decimal places.

Use your regression line to predict the birth weight of a newborn whose mother smoked on average 5 cigarettes per day. Round your answer to two decimal places.

Describe the validity of the prediction in part (d).

Create a data set which has a correlation coefficient between −0.9 and −0.8

Investigation: Conduct a statistical investigation

Investigate online

mathspace.co

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13 Chapter review 1

Emrik conducted a survey in his neighbourhood about the type of car people preferred. The options given were: Truck, Sedan, or SUV. 36 respondents said they preferred trucks, of which 20 were under 30 years old. Among 22 respondents who preferred sedans, 14 people were older than 30 years old while among 28 respondents who preferred SUVs, 18 people were under 30 years old.

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 team or club they are currently a member of. The results are shown in the table. Physical therapy Education Software engineering Graphic design Total

Sports team 90 58 60 10 218

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 are members of a team or club other than the dance club?

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

674

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 WA Curriculum – Year 10 mathspace.co

The 100% stacked column graph shows the results of a survey on beverage preferences: a

Which is the explanatory variable, beverage preference or preference on time of day?

Does the 100% stacked column graph indicate an association between the variables?

Beverage preferences 100% 75%

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%

High school or less 25 20 7 0 52

Some university 20 20 12 3 55

Bachelor’s degree 3 10 25 8 46

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 brass 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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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

Number of sick days 7 5 4 3 1 0

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.

Describe the strength and direction for each linear association. a

5 x

676

x 0

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.

Mathspace WA Curriculum – Year 10 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.

Write the equation of the line of good ﬁt.

iii

Describe the correlation of the data set as positive or negative.

ii b

16 14

10 8

6 4

x 0

x 2 4 6 8 10 12 14 16 18

The height above the ground (in centimetres) of a radish sprout over time is shown on the scatterplot: Height (cm) 25 20 15 10 5 Age (days) 5

20 25

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 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?

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677

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.

M (in dollars)

900 800 700 600 500 400 300 200 100

C (in days) 0

12 16 20 24 28

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. Prefers BetaPhone 70%

Prefers GammaDevice

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Mathspace WA Curriculum – Year 10 mathspace.co

70%

67%

Prefers AlphaModel

(Optional) For each scatterplot: i

Describe the type of linear relationship between the variables.

Estimate the value of the correlation coefficient.

21 20

18 15

12 10

9 6

0 1 2 3 4 5 6 7 8 9 10

(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.

(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 (mAh) 2500 3000 1500 2800 2000 3500 2600 2100 3200 2400 2900 1700 2300 2700 3400

Satisfaction proportion 0.9 0.8 0.6 0.7 0.6 0.9 0.7 0.65 0.85 0.7 0.8 0.55 0.7 0.8 0.85

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Answers

g 4p9

h −13

10 4

i 16u v

1.01 Index laws Understanding

k m

l 22

m x60

n −9

o p17

1 Subtraction

p 64y12 12

q 16x 3

3 a i Yes

ii Yes

iii Yes

iv No

v No b i False

ii True

c 6 4 a 10

b 9

e 11

f m4

c 2

c b5 7

e 6y

f 12y

i u

j 2w

b 0 d

c 56m n

l 72y17

c 3m7

d k13

e a5

f m3

g j7

k 9j2k3

8 a −32y15

b −96y11

5 5 5

c −30s t r

d 504y24

e −10

f 3y4

9 a j24

b w22

c y12

d 81a8

e 25h14

f x5y10

j x36

k −125x12 l

b k31

c 24r58

10 a w36v24 39 4

45 60

e 160r s f a b 11 a 24 e 11 12 a x28y8

680

h am+n

k 15z

8 5

6 a 40g h

d y10 11

g −2a

7 6

7 a u11

b 1 f 1

b 10x2y11z15

c 2p14

d −5x2y14z11

f 9v5y11

g −6u11

h 4x2

Reasoning

b a10 7

13 a n20r20

d 0

Fluency 5 a a8

r 1125y16 t c86

s x

j 81x9

g p

c 1 g

14 Yes. The expression in the numerator allows the fraction to be separated into two parts, each with a denominator of a2. The division rule can be applied to both terms. 15 Indira and Ernestino are both correct. Chad is incorrect. Answers may vary. The expression in the numerator is a product, so the division rule does not get applied to both parts of the numerator. We can check by substituting, for example, a = 2. By simplifying each expression, we see that Indira and Ernestino’s first lines of work are equivalent to the original expression, while Chad’s is not. 16 a The coefficients of each term must multiply to 9, and the indices should add together when multiplying the same base. The solution to this problem does not multiply the coefficients and instead multiplies the indices. The solution is 9s8.

d x67y15

b When raising a product to a power, each part of the product must be raised to the power. Therefore, 2 must also be raised to the power of 5, leading to the solution 32x15.

h x60

c When dividing terms with the same base,

d 1 h 6

b p2

c 0

d 81a8b20c4

f y7

Mathspace WA Curriculum – Year 10 mathspace.co

subtract the indices, so

not t . The solution is 6. d When raising a power to a power, we multiply the indices rather than add them which is done here. Then, we will have a different answer when we divide like bases. The solution is x2y22.

e The entire expression 4a must be raised to the power of zero, leading the expression to simplify to 1.

b an

7 a u−4

b 2r−6

c xy−2

d 4p−6q−7

8 a

9 a 20y6

b 28a2

h 25m3

10 a 6

c 23

d 11

g 4

h 4

11 a 32y8

b 60y5

e 6y12

f 4y11

12 a

o 3x8

13 a

6 a

Problem-solving 17 There will be 26 = 64 bacteria after 6 hours. 18 We know that 3x is not the same as x3, in general, but the three values x = 0, x = , and will make 3x = x3 be a true statement. x= 19 a = 3, b = 6 20 a True

b False

c False

21 16p = 24p 22 a 3a+6

b 343a−21

e 8h7

1.02 Negative indices Understanding 1 a True

b False

c False

d True

2 Less than 1. can be expressed as a−1.

Fluency 4 a 2−4

2−3

2−2

2−1

10−3 10−2 10−1 100 101 102

103

104

b 10−4

10 100 1000 10 000

c 3−4

3−3

3−2

3−1

Reasoning 14 Both Oreste and Donovan are correct. Using

5 a

written as

is undefined, as there is no number that when multiplied by 0 will equal to 1.

l −

the index law

, we can show that their

answers are equivalent. 15 Applying the negative index law, 0−3 can be , which equals

. This expression

Answers mathspace.co

681

16 The −1 index only applies to the variable directly in front of it, since there are no brackets. The .

correct working is

17 As a becomes a very large positive number, the value of x−a approaches 0 while remaining positive. This is because the negative index law rewrites the expression as

. Since x > 1,

the denominator x becomes an extremely large number as a increases. Dividing 1 by an increasingly large denominator results in a value that gets progressively closer to 0. For example, 10−10 = 0.000 000 000 1.

22 The negative index law requires us to write the reciprocal of the base value and apply the positive index to the expressions.

, which we

base 2. However,

know is greater than the base . There are many examples of this same pattern, such as 3−1 < 3 and

23 4.1 minutes 24 a 3 × 10−6 Farads

b 5 × 10−6 Farads

Volts or 0.03 Volts

18 a • 2p : −4, 0, 8, 20

, which is less than the

Therefore,

• 4p : −8, 0, 16, 40 • 2−p + 4p : −4, 1,

1.03 Optional: Simplify surds

• 2−p × 4p : −32, 0, 1,

Understanding

b • For 2p and 4p: Since these expressions are directly proportional to p, as p increases, the values of 2p and 4p will also increase linearly. Thus, they will become very large as p becomes significantly large.

• For 2−p + 4p: As p increases, 2−p = approaches zero because the denominator becomes very large. Therefore, 2−p + 4p will be dominated by the 4p term and will increase linearly with p. • For 2−p × 4p: Although 4p increases linearly, the term 2−p decreases exponentially towards zero. The product of a linearly increasing term and an exponentially decreasing term will approach zero. Therefore, 2−p × 4p will get closer and closer to zero as p becomes very large.

Problem-solving 19 There are many different representations, such as: •

•

• 22

• 4

b Yes

c Yes

d No

2 a 4

b 25

c 9

d 1

e 8

f 64

3 a True

b False

c False

d True

e False Fluency 4 a Rational b Rational c Neither d Irrational e Irrational f Irrational g Rational h Neither i Rational j Rational k Irrational l Neither m Rational n Rational o Rational p Irrational 5 a

6 a

7 a 1, rational c

21 a n = −3

b n = −4

c n = −3

Mathspace WA Curriculum – Year 10 mathspace.co

, irrational

e 9, rational

, irrational

h 11, rational

, irrational

m 3, rational

n 2, rational

, irrational

8 a

g 6

b 4

c 1

d 6

e 60

682

1 a No

9 a

18 a 8x

Reasoning

e 6h

10 a Yes b No, it is an approximate value and not exact is an irrational number that because cannot be expressed as a terminating decimal. 11 Jordan’s error was that he raised 27 to the power of 3 using the yx button instead of taking button; the correct its cube root using the calculation should result in an answer of 3.

b 9z5

c 4v6

f 2x

g 3a

d 12b3 h 5a7b3

1.04 Optional: Add and subtract surds Understanding 1 a like surds b a power that matches the degree of the radical. For example, no square powers if a square root, no cube powers if it is a cube root and so on

12 a Yes, because 17 is not a perfect square and there are no further simpliﬁcations or common factors in the square root expression.

2 a Not like surds

b No, because 50 can be simpliﬁed further by factoring the perfect square 25.

c Like surds

d Not like surds

e Not like surds

f Like surds

g Like surds

h Not like surds

c No, because 125 can be simpliﬁed by factoring out the perfect square 25.

3 a False

b True

c False

d False

e True

f True

g False

h True

4 a

d Yes, because 21 is not a perfect square and there are no further simpliﬁcations or common factors in the square root expression.

Fluency

can be simpliﬁed further.

13 a No.

Using 20 = 4 × 5, we can simplify , then = becomes

b We could use 16 × 5 and that would mean that we can fully simplify in one step. 14 a False, the square root of a negative number is not a real number. b True, the square root of a square is the absolute value of the original number, so this would be ∣p∣, which would also be rational. c True, the square root of a square is the absolute value of the original number, so this would be ∣p∣, which would be positive. d False, the square root of a negative number is not a real number, so we can’t square an undeﬁned number. This is also not a real number.

b Like surds

5 a

6 a

7 a

Problem-solving Reasoning 15 a c 16 17

8 a 5x + 3y b We can simplify both by collecting like terms. They have the same coeﬃcients, but one has variables and the other has surds. c

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683

9 a For example, k = 16, m = 9, gives on the left hand side, and on the right hand side. b Yes, k = m = 0 c The equation is not generally true, but is true if k = m. are not like surds, so they cannot 10 a and be added or subtracted. b These are not like surds, so no simplification is possible. We cannot add the radicands; they must be like surds in order to add their coefficients. 11 We simplify surds to make them easier to understand, compare and work with. Moreover, to add or subtract surds, they must have the same radicand and the same index. Working with surds in their simplest form makes it easier to see how different terms combine. Problem-solving units

b 218 m

13 a

Fluency 3 a 75

b 288

c 108

d 80

e 50

f 35

g 6

h 5

4 a

5 a

6 a

7 a

8 a

14 a 34 cm is a bit less b It would be an overestimate. , and is a bit less than than . c d 2.51%

9 a 2

b 4

i 2

10 a 6

15 a b c 1830

1.05 Optional: Multiply and divide surds

b 10

g 16

h 10

11 a 5

f 2

Reasoning

Understanding 1 a True

b True

c False

d False

e True

f True

g True

h True

i True

j False

2 a b

is not

12 a Sometimes a surd. For example, a surd, but

is a surd.

b Always a surd. For example, is a surd.

, which

c Never a surd. This is the deﬁnition of a rational number, so it can’t be a surd.

For example, , which is not a surd. and are 13 She is incorrect. For example, is not a surd. surds, but

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Mathspace WA Curriculum – Year 10 mathspace.co

14 Yes. For example,

and are surds, but is not a surd.

15 a For example, a = 2, b = 3, gives

1 a 6x5

b −3a9

b For example,

, gives c = 5

c For example,

, gives

16 a Prime numbers cannot be square, so is a surd as it is the square root of a number that is not square.

Chapter 1 review

, which is an integer.

b Since a and b are different prime numbers, would be a ab cannot be square. So surd.

d 54m9 n8

c 8y 2 a u8

b 25y8

c 16x12

3 a = 5, b = 4 4 a 53a−2

b 4v13 y

5 a

17 All papers in this series will have an area of the , where n is an integer. If n is form (an integer multiple of even, then it will be three), which would be a surd. If n is odd, then it (an integer multiple of will be of the form three), which would also be a surd. Problem-solving

units2

b 1.69 m2

c 20 736x4

21 b 46 m2

22 a c 40 m2

d The difference is 6 m2, approximately 13% of the area which is a signiﬁcant underestimate.

, irrational

10 a

c Answers will vary. Answers should consider what makes a reasonable height and weight for an adult in conjunction with the requirements to meet the 2 m2 surface area. For example, a height of 165 cm and weight of 90 kg supports the formula and is around 2 m2 surface area, however a height of 1 cm and weight of 14 400 kg does solve the question mathematically but is quite unreasonable contextually.

d 25m3

7 a

9 a m2n6

20 a

c 9, rational 11 a 54

c , irrational

d -5, rational

b 4

d 6

12 13 a 14 a

d units

15 16 a

17 a

2.01 Factorise algebraic expressions

23 24 a 0.14

8 a

b 100 cm2

18 a

6 a 20y6

b 0.04

c 0.63

25 21 m, rounded to the nearest metre.

d 1.44

Understanding 1 a nm

b 5x

c 4ab

d 2bn

e 4bx

f 3ab

g (x + 1)

h (4 − x)

Answers mathspace.co

685

2 a 2x (x2 + 2x + 1)

b 5y ( y2 + 3y + 2)

d 4k (x2 + 3x + 4)

4 a 3a (x2 + 2x − 2)

b 2x (x2 − 8x + 1)

c a (x − 3x + 6) 3 3pr Fluency 2

c 2ax (x + x + 2)

d 5x ( y + 2y + 3)

e 4b (b2 − 4b + 5)

f 6q (r2 − 6r − 2)

g 3xz ( yz + 7y + 3)

h 4ac (−bc + 2b + 1)

i −6ehg ( fh + 5f + 2)

j 10rtu (su + 7s − 4)

5 a (2y − 3) (x + 2) c 2 (n − 3) (2m2 + 3) 3

b (3m2 + 5) (n + 2) d (5y3 + 7) (z + 5) 2

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

f 2b (c − 2) (3a − 2)

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

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

i −(7 − x )(2y + 1)

j (8 − c2)(7 − 5d)

k (4m + 7)(m − 5)

l 2(z + 2)(10z − 3)

6 2x − 1 7 a 5j (4 − 3k − 11jk) b p2 q2 (3 + 4p2 q2 + 5p4 q4) c 2y (z − 5x + 6xyz)

multiplying the factors back together 4x(4x2 + 3x + 2) = 4x(4x2) + 4x(3x) + 4x(2) = 16x3 + 12x2 + 8x. The result matches the original expression, confirming that our factorisation is correct. Problem-solving 12 a 3 (x2 + 2x + 1) b The entire expression 3(x2 + 2x + 1) is the length of the rectangle. It is 3 times the width. This means that the term x2 + 2x + 1 represents the width of the rectangle. 13 a Let the width of the smallest field be x so that its area is x2. The largest field has an area of 9x2 and the middle field has an area of x (x + 5). b The area of the entire field is the sum of each smaller field: x2 + 9x2 + x (x + 5). The factored form of this expression is x (11x + 5) so the width of the field is x and the length is 11x + 5.

d pqr (1 + pq + p2 q2)

14 It is a square when x = . For all other values of x, the rectangle is just a rectangle.

e h (−f − j + g)

15 Example answer: 8x + 30 = 2(4x + 15)

f 5m (5 − 3n − 2mn)

2.02 Expand binomial products Reasoning 8 The expression has not been fully factorised, as the terms of 2x3 + 3x2 − 4x contain a common factor of x. 9 Xander’s error lies in factoring the highest common factor. He hasn’t considered the last term −5xy in the original expression. Consequently, the correct factored form of the expression should be 5xy(7x + 2y − 1). 10 The highest common factor will have the lowest possible index of the terms because if it were any higher, not all the terms could be divided by that factor. 11 When factorising an algebraic expression, you can check your answer by expanding the factors and verifying that they multiply back to the original expression. Consider the algebraic expression 16x3 + 12x2 + 8x. Its factorised form is 16x3 + 12x2 + 8x = 4x(4x2 + 3x + 2). We can check if our factorisation is correct by

686

Mathspace WA Curriculum – Year 10 mathspace.co

Understanding 1 a 6x2

b 20y2

d −10p2

c −21z

2 a (x + 2) (x + 1) = x2 + 3x + 2 b (x − 3) (x + 2) = x2 − x – 6 c (x + 2) (x − 5) = x2 − 3x – 10 d (x − 3) (x − 4) = x2 − 7x + 12 3 a (x + 3)2 = x2 + 6x + 9 b (x − 3)2 = x2 − 6x + 9 c (x + 6)2 = x2 + 12x + 36 d (3x − 8)2 = 9x2 − 48x + 64 4 a (x + 5) (x − 5) = x2 – 25 b (x + 2) (x − 2) = x2 − 4 c (x − 8) (x + 8) = x2 − 64 d (x + 14) (x − 14) = x2 − 196

Fluency

Reasoning 2

5 a x −9x + 8

b x + 5x + 4

c x2 − 9x + 18

d x2 − x − 12

e x2 − 12x + 35

f x2 − 6x + 5

g x − x − 30

h x2 − 7x − 18

6 a 6x2 + x − 12

b 12x2 + 7x − 10

c 4x2 − 7x − 2

d 12x2 − x − 20

e 2x + 2x − 12

f 6x2 + 9x − 60

g 10x2 − 15x − 10

h −18x2 − 3x + 6

7 a

8 a x2 + 14x + 49

b c2 + 12c + 36

c e + 10e + 25

d y2 − 10y + 25

e 49 − 14z + z2

f 36 + 12a + a2

g a + 2ab + b

h a2 − 2ab + b2

9 a 9x2 + 42x + 49

b 4y2 − 20y + 25

c 4y − 12y + 9

d x4 + 2x2 + 1

10 a 5m2 − 50m + 125 b 6n2 + 72n + 216 c 8q2 + 48q + 72 d 147x3 − 210x2 y + 75xy2 11 a x2 − 49

b 64 − m2

c 49 − 36p2 2

d 81y2 − 1

f 0.04x2 − 9

e 81x − 4y 12 a

14 No. Explanations may vary. One possible answer: To expand perfect squares, we square the first term, square the last term and double the product of the terms. 15 36 × 24 can be written as (30 + 6)(30 − 6). We can apply the formula of difference of two squares which states that for any two numbers a and b, (a + b)(a − b) = a2 − b2. In our case, a = 30 and b = 6. So, (30 + 6)(30 − 6) simplifies to 302 − 62. This gives us 900 − 36 = 864. 16 (x + 3)(x + 3) = x2 + 6x + 9 and 13 × 13 = 169. As we can see, the structure of the results in both cases follows a similar pattern, reflecting the general formula for squaring a binomial. The pattern emerges when a binomial expression (in the first case, a mathematical expression and in the second case, a number) is squared. The first term in the result is the square of the first term of the binomial, the last term is the square of the second term of the binomial and the middle term is twice the product of the terms of the binomial. 17 The student’s expression is incorrect because they didn’t apply the correct formula for squaring a binomial. The correct formula for the square of a binomial is (a + b)2 = a2 + 2ab + b2, not a2 + b2. Therefore, when you square a binomial, you must remember to include the 2ab term which represents the twice the product of the two terms. 18 a We can expand the expression using the square of a binomial formula, which is (A − B)2 = A2 − 2AB + B2. Our expression is in this form, with A being

13 a 75x − 108xy 2

b 243x − 192y c 71

d 25t2 + 90t − 36r2 + 81 2

e 384x − 486y

and B being 3b. Let’s start by squaring A, which is

Then, we calculate 2AB, which is :

f 20x3 − 405xy2 Lastly, B2 is (3b)2 B2 = (3b)2 = 9b2

Answers mathspace.co

687

Now, substituting A2, 2AB, and B2 back into the formula, we get:

v No. This creates one of the side lengths to be negative, which creates a negative volume. c Maximum occurs at length = 7.145 and Volume = 525.365, so any answer around length = 7 would be acceptable.

Simplifying, we get:

2.03 Factorise binomial products

Understanding 1 a 2

b The resulting expression is not a trinomial because it contains terms with variables in the denominator and the expression has more than 3 terms.

c (c + 5)

b 5x

2 a 81 = 92

d 2d2

b 100 = 102

d 25 = 52

c 16 = 4 3 x+4 4 n+6

Problem-solving

5 a Yes

19 a P = 4x + 16 metres

b No

b A = x2 + 8x − 20 square metres.

Fluency

c The length of the ﬁeld would be (6 + 10) which equals 16 metres.

6 a (m − 6) (x + y)

The width of the ﬁeld would be (6 − 2) which equals 4 metres. The perimeter of the ﬁeld would be 24 + 16 which equals 40 metres.

c Yes

d No

b ( y − z) (x − w)

c (4w + 3x) (5y − z)

d ( y − 4) (8y − 3)

e (g + h) (3f + g + h)

f (2x2 + 3z) (2y − 1)

g (2c − d) (c + 5d + 3)

h (3x2 − 5y) (x + 4y)

7 a (x + 2) (x + 5)

b (x − 3) (x + 8)

c (a + 8) (a2 + 1)

d (x − 2y) (8 + z)

The area of the ﬁeld would be 16 × 4 which equals 64 square metres.

e (3 + x) (8 + y)

f (x + z) (7y + w)

20 a 6x2 + 36x + 54 units2

g (p + 2) (2m + 3)

h (4x + 3z) (1 + 8y)

i (2x + 3z) (1 + 6y)

j (df + rz) (3f − 11y)

b x3 + 9x2 + 27x + 27 units3

8 a (x + 3)2

21 (4x2 − 8x + 4) cm2 22 a 24x2 − 49 3

b 62x2 − 25y2

23 n + 3n + 2n 3

24 a −y + 4y + 96y b i N o, because we cannot have a negative length and y = −10 would create 2 negative lengths, even though the volume calculation would be viable. ii No, because you cannot have a zero length, and zero length yields zero volume. iii Yes, all the side lengths are viable and the volume is 200 cubic units. iv No, because the side length 12 − y becomes zero and zero length yields zero volume.

b (8 + e)2

d (u − q)2

c (7 − p)

e 6 (c − 2)

f 5 (q + t)2

g (4w − 5)2

h (8r + 3)2

j (b − r)2

9 a (x + y) (x − y)

b (n − 5) (n + 5)

c (v − 1) (v + 1)

d (11 − v) (11 + v)

g (4 + 3y) (4 − 3y)

h (xy − 7) (xy + 7)

i (5m + 7) (5m − 7)

j (9x + 4y) (9x − 4y)

k 7 (x − 3) (x + 3) 2

l 5 (3t + 2) (3t − 2) 2

10 a 3y (x − 2y + 3x y ) b (4xy + 3z)2 c (11c − 7d) (11c + 7d)

688

Mathspace WA Curriculum – Year 10 mathspace.co

d (x2 + y) (3y + z)

20 a

e 4b (2a2 b3 + 3c2) f 7 (2b + 1) 2

x 2

g 7hp (2h r + 3h + 7p )

h 2 (8w + 3t) (w − 4t2)

i (4q2 + 3r)2 j 3d (4f − h) (4f + h)

b (x + 6) (x + 4)

Reasoning 11 No, if we multiply (a + b) (a + b), the result is a2 + 2ab + b2. The sum of two squares is not equal to (a + b) (a + b). 12 a The error is in the constant term of the factors. The correct factorisation should be x2 + 4x + 3 = (x + 1)(x + 3).

2.04 Factorise monic quadratic trinomials Understanding 1 a 9 and 10 c −6 and −4 2 a • (1 × 18)

b −9 and 7 d −8 and 11 b •

• (−1 × −18) •

b The error is in the signs of the factors. The correct factorisation should be x2 − 4x + 4 = (x − 2)(x − 2). 13 a i Perfect square trinomial ii (x + 1)2 b i Common factor then difference of squares ii n (n + 11) (n − 11) c i Factoring by grouping ii (2x + 5)(x − 3)

• (2 × 9) •

(2 × 32)

• (−2 × −9) •

(−2 × −32)

• (3 × 6) •

(4 × 16)

• (−3 × −6) •

(−4 × −16)

•

(8 × 8)

• (−8 × −8) c •

(−1 × 30)

•

(1 × −30)

d i Perfect square trinomial ii (3x − 2y)2

•

(−2 × 15)

•

(2 × −15)

e i Difference of squares

•

(−3 × 10)

•

(3 × −10)

•

(−5 × 6)

•

(5 × −6)

ii (x + 16 + y) (x + 16 − y) f i F actor by grouping then difference of squares ii (x + 4) (2x + 1) (2x − 1) 14 The trinomial cannot be factorised using perfect squares because the middle term, −72a, is not equal to twice the product of the square roots of the ﬁrst and last terms.

(1 × 64) (−1 × −64)

3 a m+6

b m + 10

c y+2

d x−4

4 a = 12, b = 7 Fluency

Problem-solving 15 a 256

b 392

16 a t + 7 units

c 261

d 660

b 4 (t + 7) units

17 x + 12x + 36 2

18 a (x + 4)

b (a − 3) (a + 3) 2

19 (x − 1) (x + 1) (x + 1)

5 a (x + 2)(x + 3)

b (x + 4)(x + 5)

c (t + 7)(t + 8)

d ( y + 6)( y + 12)

6 a (x + 8) (x + 3)

b (x + 4)2

c (x + 9) (x + 8)

d (x + 8)2

e (t + 5) (t + 2)

f (t + 7)(t + 1)

g ( y + 4)( y + 5)

h (x + 3)(x + 5)

i (x + 3)(x + 4)

j ( y + 7)( y + 2)

k (x + 5) (x + 8)

l (x + 6)2

Answers mathspace.co

689

13 a Exactly one of p or q must be negative.

7 a (x − 3) (x − 1)

b (x − 6) (x − 10)

c (x − 7) (x − 12)

d (x − 5) (x − 7)

b Both p and q must be positive.

e (x − 7) (x − 1)

f ( y − 2) ( y − 8)

c Both p and q must be negative.

g (t − 6) (t − 7)

h (x − 3) (x − 17)

i ( y − 7) ( y − 11)

j (t − 12) (t − 8)

k (x − 11) (x − 1)

l (x − 15) (x − 3)

8 a (x + 2) (x − 3)

b (x + 5) (x − 6)

c (x + 5) (x − 4)

d (x + 6) (x − 9)

e (x + 7) (x − 10)

f (x + 13) (x − 9)

g ( y − 16) ( y + 3)

h (t + 14) (t − 6)

i ( y − 15) ( y + 3)

j (x + 14) (x − 4)

k (x + 12) (x − 2)

l (x + 7) (x − 4)

9 a 2 (x + 2) (x + 5)

b 4 (x + 2) (x + 4)

c 4 (x − 9) (x + 8)

d 3 (x − 5) (x − 2)

e 5 (x − 2) (x + 3)

f −2 (x − 2) (x + 11)

g −4 (x − 5) (x + 2)

h −3 (x − 2)2

i −(x + 4) (x + 2)

j −(x − 3) (x − 4)

k x (x + 3) (x + 4)

l 2x (x + 5) (x + 3)

Reasoning 10 Octavia is correct, since the terms share the highest common factor of −3. The trinomial factorises to −3 (x2 + 5x + 6) = −3 (x + 3) (x + 2) first. 11 Omar used the fact that 144 is a perfect square and checked to see if it was a perfect square trinomial. Since it was, he efficiently factored the expression without needing to find a pair of numbers. Dima found two numbers that had a sum of 24 and a product of 144. She then used those to fill in the factors and then simplified. Both strategies work and lead to the same answer. Omar needed to recognise the pattern, while Dima needed to recognise that she could simplify since the factors were the same. 12 a The error is in the signs of the factors. The correct factorisation should be x2 + 7x − 18 = (x − 2)(x + 9). b The error is in the signs of the factors. The correct factorisation should be x2 − 11x + 30 = (x − 5)(x − 6).

690

Mathspace WA Curriculum – Year 10 mathspace.co

14 a No, this would be the area of the whole square and would be equivalent to x2. b Yes. x2− Area II is the same as Area I. Area I has length of x and width of x − 4, so it would have an area of x (x − 4). c Yes. Area I has length of x and width of x − 4, so it would have an area of x (x − 4). d No, this would be Area II which would be equivalent to 4x. 15 a (x + 9)(x + 8) = x2 + 17x + 72 b (x + 2)(x + 1) = x2 + 3x + 2 c (x + 1)(x − 9) = x2 − 8x − 9 16 a a and b must be a factor pair of 36. To maximise the length of the entire ﬁeld you would increase the length by 36 and the width by 1. b a and b must be a factor pair of 36. We want to minimise the difference of the factor pair. The differences are: • 1 and 36 = 35 • 2 and 18 = 16 • 3 and 12 = 9 • 4 and 9 = 5 • 6 and 6 = 0 If both portions are extended by 6 then the areas in the bottom left and top right will be the same, which would minimise the difference in area. Problem-solving 17 (m + 5) (m + 9) 18 a (x + 6) (x + 3)

b (m + 5) (m + 10)

19 a (u − 8)(u − 3) b (x + 4)(x − 2)(x + 3)(x − 1) 20 a (x + 3) (x + 5)

b 2

c 9x2 + 72x + 135 21 a (x − 2) (x + 2) (x2 + 3) b ( y − 2) ( y + 2) ( y − 1) ( y + 1) c (a + 1) (a + 8) d (d − 7)2

22 To determine the values of c and d, use factoring to get the area into the form of ( y + c) ( y + d). The expression y2 + 8y + 15 can be factorised into ( y + 3)( y + 5). This means that the values are c = 3 m and d = 5 m. 23 x = 7

3.01 Simple interest Understanding 1 a $5400

b $1080

2 a Divide by 12

b Multiply by 12

3 a True

b False

c True

Chapter 2 review 1 a x2 − 10x + 16

b x2 + 4x + 3

c x2 − x − 30

d x2 − 5x − 36

Fluency 4 a $342

b $7942

5 $10 080

2 a Yes b No, ( y − 3)2 = y2 − 6y + 9 2

b 64 − 25p

3 a m – 16 c 9x2 − 0.49

c 4p + 12pq + 9q

b 2

5 a m + 10m + 25

b n − 8n + 16 2

6 9x2 − 72x + 144 cm2 7 a 5b( y2 + 2y − 3)

b −3ce(d2 − 3d + 2)

8 a (a + 3)(a + 4)

b ( p − 2)( p + 6)

c (x + 3)( y + 5)

d (m − 3)(k − 4)

9 a (k + 8) (k2 + 1)

b 5 (r + s)2 d

e (a + b) (5 + x)

f 5 (x + 4y) (x2 − y) b (x + 8)

c (1 − x)

d (x + 9) (x + 8)

e (x + 13) (x − 9)

f ( y − 16) ( y + 3)

g (x + 12) (x − 2)

h (x + 5) (x − 4)

11 a ( y + 5)

b (a − 6)

c (b + 2) (b + 7)

d (c − 2) (c − 9)

e (d + 5) (d − 2)

f (e − 8) (e + 3)

12 a p = 5 and q = 7

b 44 units

b 10.95%

c 7.84%

10 a 1.5%

b 0.5%

c 0.12%

11 a $693.00

b $692.75

c $2164.80

d $657.99

12 a $4996.53

b $10 329.30

c $1470.75

d $8835.44

e $10 509.60

f $4980.36

13 Option 2 14 Option 1

b 9%

b 4x − 28 cm

15 a 3x2 + 19x + 20 square metres 2

b x − 2x + 1 square metres c 2x2 + 21x + 19 square metres

c 4%

17 $80 18 $78 585 19 Fortnightly Reasoning 20 Yarel has multiplied an annual interest rate by the number of months, so the units of time in his calculation are not the same. He must either convert 22 months to interest rate of 8% p.a. to

13 4y − 3 metres 14 a x − 7 cm

b 6 quarters

9 a 7.56%

16 a 8%

7 1.82 years

15 $1634

c 7 (5m2 − 7)

10 a (x − 5)

b 5 years

8 a 1.5 years

4 a −24x2 + 14x + 24

6 a $2500 2

years or convert the per month.

21 The interest from Option 1 will be 0.5 × 2 × 12 = 12% of the principal, while the interest from Option 2 will be 2.5 ×

= 8.75% of the

principal, so Option 1 will earn Ariela the most interest.

Answers mathspace.co

691

22 When calculating interest we multiply by the loan term, so a shorter loan term will result in less interest paid. When we calculate repayments, we divide by the number of repayments. If making monthly repayments, for example, the number of repayments is 12 × T. A smaller T value means a smaller number of monthly repayments, which means that the repayment amount may be greater.

7 a

Balance at beginning of year

Interest earned

First year

$3200

$160

Second year

$3360

$168

Third year

$3528

Fourth year

$3704.40

$176.40 −

Balance at beginning of year

Interest earned

b $504.40 8 a

Problem-solving First year

$3500

$133

Second year

$3633

$138.05

b 6%

Third year

$3771.05

25 a $2784

b $5490

Fourth year

$3914.35

$143.30 −

c $10 764

d $2490

23 a $960

b $290

24 a $2400

26 a $1000

b $200

b $414.35

c 2%

9 a

27 5 years 28 25.36% p.a. 29 $2857.14 30 Option 1

3.02 Compound interest Understanding 1 a Always true

Year

Balance + interest

Value

Interest earned

Start of investment, 0

−

$2090

$87.78

First year, 1

$2090 + $87.78

$2177.78

$91.47

Second year, 2

$2177.78 + $91.47

$2269.25

$95.31

Third year, 3

$2269.25 + $95.31

$2364.56

$99.31

Fourth year, 4

$2364.56 + $99.31

$2463.87

$103.48

Fifth year, 5

$2463.87 + $103.48

$2567.35

−

Year

Balance + interest

Value

Interest earned

Start of investment, 0

−

$2090

$88

First year, 1

$2090 + $88

$2178

$91

Second year, 2

$2178 + $91

$2269

$95

Third year, 3

$2269 + $95

$2364

$99

Fourth year, 4

$2364 + $99

$2463

$103

Fifth year, 5

$2463 + $103

$2566

−

b Never true c Always true

d Sometimes true, this is only for the ﬁrst period. e Always true f S ometimes true, it depends on the interest rates and principal. For the same principal and interest rate, this is true after the ﬁrst period. 2 Account B

c $1.35

3 Graph 1 4 a No

b No

5 a $9000 c Quarterly

c No

d Yes

b 7% d 3 years

Fluency 6 a Investment A

b Investment A

c 11 years

692

Mathspace WA Curriculum – Year 10 mathspace.co

10 a 500 000 × (1 + 0.012)24 b $165 736.40 11 a $7706.73

b $6617.25

12 a $5705.83

b $705.83

13 a $4477.00

b $1477.00

14 a $25 631.28

b $631.28

15 a $2529.86

b $1761.93

Problem-solving

c $837.74

20 $53 136.44 21 $53 516.75

Reasoning 16 a $1794.85

b $1797.60

22 a 0.02

c $2.75

d The more frequently interest is compounded, the greater the total interest earned will be if all other conditions are the same. This is because more frequent compounding results in more periods where interest is earning “interest on interest”. 17 a Jadria should choose Plan 2, because she would have to pay back less interest. For Plan 1, the amount of interest she would have to pay is $9000(0.029) (3) = $783. For Plan 2, the amount of interest she would have to pay .

b No, Plan 2 will not always be the better choice. As Plan 2 involves compound interest, it is in the form of an exponential relationship, and Plan 1 is in the form of a linear relationship. If we were to draw the graphs of both plans, then the graph of Plan 2 will eventually exceed the graph of Plan 1. In fact, after only 4 years, the interest needing to be repaid on Plan 2,

will exceed that of Plan 1, $9000(0.029) (4) = $1044.

18 4.75% compounded annually. Although the other rates are lower, they are compounded more often which works out to accruing slightly more interest than 4.75% compounded annually. • 4.75% compounded annually: He would owe $26 187.50 after one year • 4.7% compounded quarterly: He would owe $26 195.87 after one year • 4.65% compounded monthly: He would owe $26 187.60 after one year 19 a $6367.25 b The value in the table has been rounded to the nearest whole dollar, so the amount of interest will be slightly different.

23 a

b $3750 Value ($)

3000 2500 2000 1500 1000 500 0

No. of years 20 25

b 24 years 24 24 years 25 $60 803.96 26 5.116% 27 18.5 years

3.03 PAYG and tax Understanding 1 The Medicare levy is calculated as 2% of an individual’s taxable income for the financial year. 2 Personal income tax is a tax that must be paid based on the income of individuals, households, or other legal entities. 3 a Pay As You Go b Pay As You Go is an amount of tax that is deducted from each pay weekly, fortnightly, monthly, or occasionally quarterly. c Pay As You Go ensures that there isn’t a huge sum due at the end of the financial year. 4 Deductions are specific and approved expenses that can be subtracted from an individual’s gross income to reduce the amount of income subject to tax. They lower an individual’s taxable income, which, in turn, reduces their income tax owed.

Answers mathspace.co

693

Fluency

= $4288 + $6000 Multiply 0.30 by $20 000 to ﬁnd the additional tax amount

5 $10 140

6 a $3172

= $10 288 Add the base tax amount and the additional tax amount

b He will receive a refund. 7 $124.54

24 a No. Low paid workers will pay less tax, but 18c for each $1 does not raise enough tax for those earning over $45 001 to pay the same tax.

8 $24 507 9 $84 737 10 a $2750

b 3.7%

11 $25

c $31 500 plus 37c for each $1 over $135 000 instead of $31 288 plus 37c for each $1 over $135 000.

12 $36 13 $108.78 14 a $18.22

b $29.81

15 $1268.27 16 $1200

Problem-solving 25 a $261.85

b $312.25

c $574.10

d $37 440

e $36 865.90

17 x = $89 057

26 a $20 064.70

18 $7530

b $1951.78

c $7104.22

19 a $0

b $2977.60

c $18 806.50

d $35 603.68

e $53 612.60

f $67 752.50

27 −$3508; Since the result is negative, Jasmine will owe $3508 in taxes rather than receiving a refund.

b $3959.10

28 Michael will receive a tax refund of $11 031.

20 a $55 217.75 c $63 176.85 21 a $50

b New taxable income: $48 700, Tax payable: $5398 c $202 Reasoning

23 a Stewie made an error in Step 3. $7400 is incorrect. He must have overlooked and multiplied it by 0.37, which is in the formula of the next row. b Tax = $4288 + 0.30 × ($65 000 − $45 000) Apply the corresponding tax formula

29 a $148 122

b $36 143.14

c $1394.50 per fortnight 30 a $52 354 c $1047.08

b $6494.20 d $10 036

e $2494.72

Chapter 3 review

22 Yes, she is correct as she pays no income tax on $17 000, therefore, her only payment is the Medicare levy of 2%.

= $4288 + 0.30 × $20 000

Evaluate the brackets

694

b $25 000 × $0.18 = $4500, so they would pay $4500 − $4288 = $212 more.

Mathspace WA Curriculum – Year 10 mathspace.co

1 a $787.57

b $610.61

c $1623.24 2 a $1080

b $10 080

3 $63.92 4 a Simple interest. The amount of interest is the same every period, so this is simple interest. b Compound interest. The amount of interest changes every period, so this cannot be simple interest. We can see that the interest is 5% of the opening balance.

c Compound interest. This equation shows that every year, we multiply by 1.03. So add a percentage of the previous balance, which is how compound interest works. d Simple interest. This equation shows that $30 is added every year. Since this is a constant amount, this is simple interest.

Fluency 9 a i

6 a $3637.78

b $2342.37

−4

−3

−2

4 3 2 1

−4 −3 −2 −1 −1

f C ompound interest. This graph is increasing at an increasing rate, compound interest has more interest added each period. b $1045.45

−1

y ii

e Simple interest. This graph shows constant growth. Simple interest has the same amount added each period, so the graph is linear.

5 a $7045.45

−3 −4

c $1243.66

−1

8 a $20

−2

b i

7 $71 422.66

y 4

b $30

9 x = $92 000

10 $8250

11 a $0

b $8386.60

c $49 971.15

−2

d $94 822.25

−1

4.01 Linear graphs c i

Understanding 1

−1

−9

−7

−5

Gradient

y − intercept

y = 3x − 4

−4

y = −5x + 2

−5

y=4

x=3

undefined

none

−3−2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

2 B 3 x=6 4 y = −1 5 y = 0, x = 0 6 a x = 6, y = −2

b x = −7, y = 7

7 a m = 2, c = 1

c m = 0, c = −3 8 a Yes

b No

y x 1 2 3 4 5 6 7 8 9

d c Yes

d No

Answers mathspace.co

695

d i

−1

y 4

4 3

−4 −3 −2 −1 −1

−1

−2 −3

h i

−4

e i

−1

7 6

2 1

1 x −2

f i

−1

10 a

−1

−3

−1

−2

y b

y 2 1

y 3

x −2

4x −2 3

−1

1 −2

−1

−2

−1 −2

11 a

−3

g i

696

−1

Mathspace WA Curriculum – Year 10 mathspace.co

−1

9 y 8 7 6 5 4 3 2 1 −4 −3 −2 −1−1 −2 −3 −4

x 1

c (0, 3)

d (−1.5, 0) f x = −3.5

12 a

−1

−6−5−4−3−2−1 −1 −2 −3 −4

e y=9

7 6 5 4 3 2 1

x 1 2 3 4 5 6 7

y 2 1

−1

−2

x 1

−1

−2

−3 −4

−5

2 1

−4 −3 −2 −1 −1

−9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18

−2 −3 −4

c (0, −2)

d (3, 0)

13 a

3 y 2 1 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13

9 8 7 6 5 4 3 2 1 −1

−1 −2

x 1 2 3 4 5 6

14 a

y x 1

y 6 5 4 3 2 1 x

−1

x 1

Answers mathspace.co

697

b i −1

iii

3 2 1 −3

−2

−1

x 1

−1

−2 −3

7 6 5 4 3 2 1 −3 −2 −1 −1 −2 −3

−4

ii (0, 5)

x 1 2 3 4 5 6 7

ii (0, −2)

c i 5

iii

2 1

1 x −4 −3 −2 −1

−4 −3 −2 −1 −1

−2

−3

−4

6 4 2 −4 −3 −2 −1 −2

iii

−4

−6

−8

15 a i 2 7

−4 −3 −2 −1 −1

ii (0, 4)

iii

ii (0, 1)

d i

−2

−3 −4

6 5 4

iii

1 −2

−1

ii (0, 3)

e i

3 2 x 1

3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

698

Mathspace WA Curriculum – Year 10 mathspace.co

x 1

f i 4

ii (0, 4)

iii

d i x = 2.5

iii

7 6 5 4 3 2 1

3 2 1

−4 −3 −2 −1 −1

−2 −1

−3 −4

16 a i x = 1.5 iii

ii y = −3 4

ii y = 5

−1 −2

x 1

−3

ii y = 1

f i iii

y 2

x 1

1 2 3 4 5

x −2

−1

c i x = 1.5

−2

ii y = 3 y

ii y = −4

g i

3 2

iii

−4

−3

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−3

−4

ii y = 3

−2

x 1

−2

b i x = −1.5

−4 −3 −2 −1 −1

−4

−2

5 4 3 2 1

ii y = −2

iii

−4 −3 −2 −1 −1

iii

x 1

e i

iii

x 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Answers mathspace.co

699

ii y = 3

h i iii

1 −4 −3 −2 −1

x 1

−1

−4

18 a

x 1

y = 2x + 7

−6 −5 −4 −3 −2 −1 −1 −2

b 7

y −1

−1

4 3

−3

−4

x 1 2 3 4

−2

x 1 2 3 4

x 1

−5

−3 −4

−5

x 3

y = 4x − 3

−6 −5 −4 −3 −2 −1 −1 y = −2x − 4 −2

−2 −3 −4

700

−3

−2

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

7 6 5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

−2

−3

2 1

−4 −3 −2 −1 −1

−2

17 a

Mathspace WA Curriculum – Year 10 mathspace.co

−6

x 1

6 5 4 3 2 1 −3 −2 −1

−1 −2 −3 −4

d i x = −9

ii y = −10

iii

y 2

y = −3x + 6

−10 −8 −6 −4 −2 −2

x 1

−4 −6 −8 −10

19 x = 7 20 a i x = 2 iii

21 a

ii y = 3 4

3 2

2 1 1

−4 −3 −2 −1 −1

−2 −3

−2

−4

−3 −4

b i x=7 iii

ii y = −4

7 y 6 5 4 3 2 1 −7−6−5−4−3−2 −1 −1

2 1

−4 −3 −2 −1 −1 −2

−3

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

−4

3 2 1

c i x = −6

ii y = 8

iii

−4 −3 −2 −1 −1

−2

−3

−4

4 2 −8 −6 −4 −2

x 2

−2 −4

Reasoning 22 The gradient or slope of a line determines its steepness. A positive gradient slants upwards from left to right because for every step we move to the right (positive x-direction), we also move upwards (positive y-direction). On the

Answers mathspace.co

701

other hand, a negative gradient slants downwards from left to right because for every step we move to the right, we move downwards. 23 A line is uniquely determined by two points. The x-intercept and y-intercept of a linear equation provide two distinct points on the line (unless the line is vertical or horizontal). By connecting these two points, we can sketch the entire line. 24 y = kx + d

Gradient k

y − intercept d

y = −px + r

−p

y=k

25 Changing m to 2m effectively doubles the gradient or slope of the line. This will make the line twice as steep, but it will still cross the y-axis at the same point, c. This is because the slope of a line determines its steepness and direction, while the y-intercept locates it on the y-axis.

550 500 450 400 350 300 250 200 150 100 50 0

Understanding 1 A 2 B 3 a Horizontal

b Vertical

c Horizontal

d Vertical

4 a 250

b 12.5

Fluency 5 a 115

b B

6 a The straight line graph starting at the origin is the cost function. The curved graph with y-intercept of 50 is the revenue function. b The intersection is the break-even point where cost is equal to revenue. 7 a Linear

b Non-linear

c Linear

d Non-linear b No

120 110 100 90 80 70 60 50 40 30 20 10

Sales

Price 5

d −5

c 8

8 a V

Problem-solving 26

4.02 Linear models

10 15 20 25 30 35

9 a

Based on the graph, it is estimated that 350 toys would be sold if the price was $25. 27 The cost equation is C = 60 + 0.60x, where x is the number of kilometres driven. Substituting x = 250, it would cost $210 to rent a car for a day and drive 250 kilometres. 28 29 y = x + 2 30 C = 5q + 150 31 y = 1.25x + 0.5

Number of minutes 0 passed (x) Heart rate ( y)

93 99 105 108

111 y 108 105 102 99 96 93 90 87 84 81 78 75 72

x 1 2 3 4 5 6 7 8 9 10 11

c 3 beats per minute

702

Mathspace WA Curriculum – Year 10 mathspace.co

10 a

Number of minutes passed (x)

Depth of diver in metres ( y)

0 0.8 1.6 2.4 3.2

b 3.6

13 a

T 25 20 15

3.2 2.8

2.4

1.6 1.2

b 5 °C

0.8

d T = 5x + 5

0.4

c 0.8 m/min 11 a

d 19.2 m

48 C 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 0

5 10 15 20 25 30 35 40 45

b 21 °C c 190 chirps per minute

15 a 18

12 a 5° C

20 40 60 80 100 120 140 160 180

d 120 chirps per minute

c $1.20

d $90.00 b 3° C

16 14

c y = 5x + 3

28 y 26 24 22 20 18 16 14 12 10 8 6 4 2 0

28 T °C 26 24 22 20 18 16 14 12 10 8 6 4 2 0

b C = 1.20x

14 a

c 5 °C

10 8 6 4 2 0

x 1

d 1

2 3 4 5 6 7 8 9

b 2 cm

c 26 cm

d 11

e h = 2d + 2

Answers mathspace.co

703

16 a

19 a The variables are income and number of hours worked.

8500 8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500

b 800 700 600 500 400 300 200 100 t

x 5

10 15 20 25 30 35

c For example, y = 25x d $625

c 1700 seconds

e 5 hours

d A = −5t + 8500 1.8

200 400 600 800 1000 1200 1400 1600 1800

b 6000 m

17 a

20 a The variables are operation cost and number of hours operated.

1.6

b For example, y = 150x

1.4

c $3300 d 36

1.2 1

Reasoning

0.8

21 a Litres of petrol used per kilometre

0.6

b Dollars earned per hour

0.4 0.2

1 2 3 4 5 6 7 8 9 10 11

b F = 0.10d + 0.30

b 4 of A

d 3 km

c Example answer: Exchange rates between different countries’ currencies change on a daily basis. So this graph may give accurate conversions one day, but not the next.

C 6 5 4

23 a

3 2 1 t 0

8 10 12 14 16

b $2.80 c 4 minutes d C = 0.30t + 0.40

704

d Ascent or descent in metres per second 22 a 4 of B

c $1.70

18 a

c Heartbeats per minute

Mathspace WA Curriculum – Year 10 mathspace.co

120

−8

b The baseball is moving downwards instead of upwards at T = 4. c Example answer: The model is useful to track the velocity of the ball while it is in the air, but once it hits the ground or if it hits an object, the model will not be accurate. Also, the velocity may change due to weather conditions.

24 a

Number of minutes passed, x

Problem-solving 0

20 50

Amount of fuel 250 225 200 175 150 left in tank, y

b y = 250 − 5x c The amount of fuel in the car is decreasing at a constant rate of 5 litres per minute. d Example answer: The model is useful to track the fuel consumption of the car while it is moving at a constant speed. But as the car changes speed the fuel consumption will change, and the model will not be accurate. 25 a is the acceleration - that is, the rate of increase in the velocity. b 18 is the velocity at which the ball is initially rolled. c d 46.9 m/s e Example answer: The model is useful to track the velocity of the ball as it rolls down the hill, but once the hill ﬂattens the velocity will decrease as time increases. Therefore, the model will no longer be accurate. 26 a is the amount by which the temperature is rising every minute. b y = 3 is the temperature of the room before the heater has been turned on. c d 13 °C e Example answer: The model is useful to track the temperature for a couple of hours, but eventually we can assume the heater will be turned off, or a window or door will be opened, which will affect the temperature and the model will no longer by accurate. For instance, after two hours the temperature according to the model will be 43 °C, so the heater will most likely be turned off.

27 a 16 km 28 a 104 °F

b 1.6 km c −29 °C

b 10 °C

29 $77.08 30 a 50 °F c Below

b 1.8 °F d F = 1.8C + 32

e 95 °F 31 a −300 is the amount the ﬁsh population decreases by each year. b F = 6000 c F = −300t + 6000 d 3300 ﬁshes e 14 years 32 a 2 is the increase in earnings for each additional shirt ironed. b y = 4 is the earnings independent of how many shirts are ironed. c $32 d 12 shirts 33 a 20 mi/hr 34 a

b 80 mi/hr

is the amount of water that is ﬂowing out

of the hole every minute. b 30 is the capacity of the bucket. c d 3L 35 a 10 is the variable cost of manufacturing per chair. b y = 85 is the ﬁxed cost of manufacturing. c For example, y = 10x + 85 d $335

4.03 Linear inequalities Understanding 1 a i 25 26 27 28 29 30 31 32 33 34 35 ii –33–32 –31 –30–29–28–27–26–25

Answers mathspace.co

705

h i a < 0.4

b For x ≥ 29: i No

ii No

iii Yes

iv No

iii No

iv Yes

For x < −29: i No

ii No

2 The inequality symbol must be reversed. 3 a False

b True

c False

d True

−1

−0.5

0.5

i i b ≤ 2.8 ii 0

j i s < 7.5 ii 5

Fluency 4 a x ≥ 14

b v≥0

c c>3

d x > −8

e b≤0

f g≥9

g s < 19.8

h y < −1.52

5 a y>1

c e < −3

d u ≥ 0.74

g h > −4

h m>4

i m < −34.6

j x ≥ −1 b

6 a c m ≥ −24

d y < −14

e a>4

f k ≤ 45

g y > 40

h x≤3

−10

−5

b i y<0 ii

−10

−5

−10

−5

−10

−5

−15

706

0.5

1.5

−5

mi ii n i q>1 ii

−10

o i u≥8 ii 0

p i x≤6 −10

−5

8 x≥3 b There is no greatest value. 10 p + 15.25 ≤ 32 11 A − 33.40 ≥ 32 12 x + (63 + 73 + 98) ≥ 300 x ≥ 66 b h ≥ 61.4

c 62

−10

−5

b x ≤ 1630.10 15 a B ≤ 3.4 16 W ≤ 4

−5

14 a x + 488.30 ≤ 2118.40

g i m>8 ii

−5

13 a 5h + 72 ≥ 379

f i b ≥ −12 ii

l i x>5

A ≥ 65.40

e i v>2 ii

p ≤ 16.75

d i a ≥ −1 ii

9 a q > 16

c i j<4 ii

−10 −8 −6 −4 −2 0 2 4 6 8 10

7 a i x>1 ii

k i y≤5 ii

i x ≤ 8.2

Mathspace WA Curriculum – Year 10 mathspace.co

c $1630.10 b 3

17 a No

b Yes

c Yes

d Yes

18 a Yes

b Yes

c Yes

d No

19 a No

b No

c No

d Yes

20 a No

b Yes

c No

d No

Reasoning

29 150 grams ≤ 0.18w + 0.03(1500 − w) ≤ 225 grams Least value is 700 grams and greatest value is 1200 grams

Chapter 4 review 1 a

21 x < 4 means x is any number less than 4. It doesn’t include the endpoint (4). 0 ≤ x ≤ 4 means x is any number between 0 and 4. It does include the endpoints, so x can also be 0 or 4.

−3

−2

−1

−2

y 15

22 a Step 1 is incorrect. Isabelle has subtracted 25 from both sides but the left-hand side should now be −2x not 2x.

10 5

b 25 − 2x < 31 Step 1: −2x < 6 Step 2: x>−3

x −4 −3 −2 −1

23 The minimum should be at −9 instead of −8 and the dot at that point should be ﬁlled as −9 is included in the solution.

c 4

b 14 rockmelons

26 Lachlan, James 27 a x + 650 ≥ 950

3 2 1 −4 −3 −2 −1 −1

x 3

−2 −3

c No, he gets paid for washing the whole window, not parts.

25 a

b (0, −2)

b No, washing 8 windows would only generate $16 which would give him a total of $38.10, which is not enough money to afford the equipment.

Problem-solving

2 3 4

2 a

24 a No, washing exactly 9 windows would generate $18, which with the $22.10 he already has enough money to afford the equipment.

d Yes, if Ryan washed 9 windows he would earn enough to buy the equipment. If he washed more than 9 windows, he would have more than enough money.

−4

y −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7

b $960

c Yes 28 a 5031 kg

b No

Answers mathspace.co

707

4 a x-intercept: (6, 0) c

b y-intercept: (0, 4)

y 4 3 2

ii Maximum point

b i Concave up

ii Minimum point

c i Concave down

ii Maximum point

d i Concave up

ii Minimum point

ii Minimum

x 1

500 490 480 470 460 450 440 430 420 410

4 a i Concave down

5 a i The turning point is (0, 5).

5 a

Fluency

iii The equation of the axis of symmetry is x = 0.

iv There are no x-intercepts and the y-intercept is at (0, 5).

b i The turning point is (0, 3). ii Maximum iii The equation of the axis of symmetry is x = 0. iv The x-intercepts are at (−0.5, 0) and (0.5, 0), while the y-intercept is at (0, 3). t 2

6 a i

8 10 12 14

b V = −5t + 500

−2

−1

c The rate of water loss, −5 mL/min

b d = 15t

6 a 15 km/h

c 67.5 km

7 a $0.50

b $10

c C = 0.5m + 10

d $20

8 a 38 L

b 11.84 gal

9 a Yes

b No

10 a 12t + 8b ≥ 100

b No

11 a 25c + 2k ≥ 150

b No

12 8 4 −2

b i

c 6

5.01 Graphs and features of quadratics Understanding b Concave down

2 a True

c False

b False

d True

e False 3 The turning point lies right on the axis of symmetry. Knowing the axis of symmetry makes it possible to ﬁnd the turning point, and vice versa. Both elements help in understanding the shape and position of the graph.

Mathspace WA Curriculum – Year 10 mathspace.co

−1

−2

−1

−16

−1

−16

4 2 −2

1 a Concave up

708

−1 −2

−4 −6 −8 −10 −12 −14 −16 −18

y x 1

c i

−2

−1

−50

−14

−2

−14

−50

d i

−1

−2

−1

−18

−1 −2 −4 −6 −8 −10 −12 −14 −16 −18

−2

−1

−3

−6

−7

−6

−3

y −2

−1

x 1

−1

−3 −5

x 1

−6

−7 −8

h i

−2

−1

−5

20 y 18 16 14 12 10 8 6 4 2 −2

−4

−1 −5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −55 −60

−2

10 y 8 6 4 2 −2

f i

g i

e i

−5 −10 −15 −20 −25 −30 −35 −40 −45 −50

y −2

−1 −2 −4 −6

7 a

−2

−1

−61

−16

−1

−16

−61

−2

−1

55 50 45 40 35 30 25 20 15 10 5

−2

−1

−3

−2

−1

−2

−5

−2

x 1

x y

b Minimum c Positive

Answers mathspace.co

709

equal to c. When a < 0, the parabola opens downward, and the y-value of the turning point, also c at x = 0, is the maximum value. Thus, the range is all y-values less than or equal to c.

8 i Concave down ii (0, 4) iii

5 −2

−1 −5

12 a i D

−10 −15 −20 −25 −30 −35 −40 −45 −50 −55 −60 −65 −70 −75 −80

Reasoning 9 Answers may vary. An explanation may be: When a > 0, the parabola opens upward, and the square of any number (positive or negative) multiplied by a is always greater than or equal to 0. Thus, the minimum value of y occurs at the vertex, where x = 0, and y = c. 10 Always true. For y = ax2 + c, the parabola has a turning point determined by the sign of a. If a > 0, it opens upward and has a minimum value of c at x = 0. If a < 0, it opens downward and has a maximum value of c at x = 0:

4 3

y maximum

2 1 minimum −4 −3 −2 −1 −1

11 The y-value of the turning point is signiﬁcant in determining the range of the quadratic function y = ax2 + c because it represents the minimum or maximum value of the function, depending on the sign of a. When a > 0, the parabola opens upward, and the y-value of the turning point, which is c at x = 0, is the minimum value. Thus, the range is all y-values greater than or

Mathspace WA Curriculum – Year 10 mathspace.co

iv B

b D

c C

d A

14 a The parabola opens upward because a = 20 is positive. It has a minimum value. b (0, −80) c The minimum value is −80 at x = 0, representing a $80 loss when no cupcakes are sold (for example, ﬁxed costs). d 20x2 = 500, so x2 = 25 and x = ±5. Only x = 5 is practical, meaning selling 5 dozen (60 cupcakes) yields a $420 proﬁt. Hours

Fee

2 3 4

−4

iii C

Problem-solving

15 a

−3

710

13 a C

−2

ii A

b For y = ax2 + c, we solve ax2 + c = 20 by rewriting it as ax2 = 20 − c. If a is positive, the parabola opens up, and there are solutions only if c is less than 20. A bigger a makes the parabola steeper, so the solutions (where the graph hits y = 20) are closer to the y-axis. For example, with y = 16x2 − 36, the solutions are farther apart than with y = 49x2 − 36. If a is negative, the parabola opens down, and solutions only happen if c is more than 20, because the graph needs to reach down to 20. For example, y = −4x2 + 15 doesn’t hit 20 since its highest point is 15 and it goes down. The c moves the parabola up or down, deciding if 20 is even possible to reach.

Fee 800 600 400 200 0

$25 $100 $225 $400 $625 $900

c y = 25x2 e 10 hours

Hours 6

d $1600

16 a

160

241

D 250

5 The connection between the y-intercept and the value of q in y = x2 + q is that the y-intercept is equal to q, because when x = 0, the equation becomes y = q. Fluency

200

6 a 6 units upwards

150

c 1 unit upwards

100

7 a 4 units upwards

b 3 units downwards

b 2 units upwards

c 3 units downwards

50 x 1

8 a i

b 9x2 = 144, so x2 = 16 and x = ±4. Only x = 4 makes sense, meaning the rocket reaches 160 metres after 4 seconds.

5.02 Vertical translations Understanding

4 3 2 1

2. Turning point: The turning point of both parabolas is still on the y-axis. For y = x2, the turning point is at (0, 0), while for y = x2 + 7, the turning point is at (0, 7).

5 4 3 2 1

y = x2 + 3

y = x2 1

ii 3

x 4

iii (0, 3)

c i

yy 3 −4 −3 −2 −1

y = x2 2 3

xx 4

−3 −6 −9

3. Axis of symmetry: Both parabolas have the same axis of symmetry, which is the vertical line x = 0. 4 The entire parabola shifts below the x-axis so there are no longer x-intercepts.

x 4

y = x2 − 5

−4 −3 −2 −1 −1 −2 −3 −4 −5

3 Change:

1. Shape: Both graphs have the same parabolic shape, meaning they are both “U” shaped.

y = x2 1 2 3

iii (0, −5)

b i

2 Example answers: y = x2 − 4, y = x2 − 8

Stays the same:

ii −5

1 Examples answers: y = x2 + 5, y = x2 + 10

Vertical translation: The graph of y = x2 + 7 is a vertical translation of the graph of y = x2. The whole graph has been shifted 7 units upwards.

d 1 unit downward

−4 −3 −2 −1 −1 −2 −3 −4 −5

c The vertex at (0, 16) represents the starting height of 16 metres when the rocket is launched at x = 0.

unit upwards

−12

ii −12

y = x2 − 12

iii (0, −12)

Answers mathspace.co

711

d i

Reasoning

y 20

13 a y = −x2 + 3

iii (0, 10)

b y = x2 − 11

c y = x2 − 49

d y = x2 − 8

9 a y=x

10 a y = 2x2 + 3, The graph has been narrowed (vertically enlarged) by a factor of 2 and vertically translated up 3 units. 2

b y = x − 1, The graph has been vertically translated down by 1 unit. c y = −5x2 − 4, The graph has been vertically reﬂected across the x-axis, narrowed (vertically enlarged) by factor of 5 and vertically translated down 4 units. , The graph has been vertically

reﬂected across the x-axis, widened (vertically compressed) by a factor of vertically translated up

and

unit.

16 a The vertical translation of 10 metres means that the initial height of the toy rocket is 10 metres above the ground. To include this in the equation, we need to add 10 to the original height equation: h = −4t2 + 30 + 10. The new equation becomes h = −4t2 + 40. b 40 metres c 3.16 seconds d

h (metres)

35 30 20 15 10

5 0

t 1

c The initial height of the rocket is given by the value of h when t = 0: h = 4(0)2 + 10 = 0 + 10 = 10 metres. This value corresponds to the y-intercept of the graph.

712

Problem-solving

12 a 46 metres 50 45 40 35 30 25 20 15 10 5

d y = x2 − 12

11 hmax = 20 metres and t = 2 seconds b

b y = 3x2 − 4

15 Yes, vertical translations always result in a new turning point. When a function is vertically translated, every point on the graph, including the turning points, is shifted up or down by a constant value. This means that the turning points will have the same relative position to the rest of the graph, but their coordinates will be different due to the vertical shift.

−5

ii 10

14 a y = x2 c y=x −3

y = x2 1

d y = −x2

c y = −x − 1

y = 2x2 + 10

−4 −3 −2 −1

b y = −2x2 + 8

Mathspace WA Curriculum – Year 10 mathspace.co

t (seconds) 1

17 a The coefficient −2 in the equation h = −2t2 + 8 represents the effect of gravity on the water’s height, as it causes the parabolic arc to open downwards. The constant term +8 represents the initial height of the water above the fountain when t = 0. b 8 metres c 2 seconds d The two fountains have similar parabolic arcs as both equations have the same coefficient of −2, indicating the same effect of gravity on the water’s height. However, the constant term differs, with +8 for the first fountain and +10 for the second fountain. This means that

the second fountain has an initial height of 10 metres, which is 2 metres higher than the ﬁrst fountain. Consequently, the maximum height of the water in the second fountain is also 2 metres higher than the ﬁrst fountain.

5 The relationship between the x-value of the turning point and the value of p in the equation y = (x − p)2, is that the x-value of the turning point is equal to p. Fluency

18 a b As we move further and further away from the hill (increasing x), the elevation (h) will continue to increase indeﬁnitely according to the equation , which is a parabola opening upwards. This indicates a problem with the model, as it does not realistically represent the elevation of a hill, which should have a maximum height and then decrease as we move further away.

6 a Right

b 1 unit left d 5 units left

8 a 1 unit right

d 5 units left

e 3 units left

g 6.5 units right

h 5 units right

9 a i

In this case, the constant term 1500 represents the initial water level in the tank.

b i

1 Example answers:

x 1 2 3 4 5

iii (−1, 0) 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2

• y = (x + 2)2 • y = (x + 5)2 ii 1 c i

3 When comparing the graph of y = x2 and y = (x − 2)2, the shape and width of the parabolas remain the same, but the vertex shifts from the origin (0, 0) to the point (2, 0) due to a horizontal translation of 2 units to the right.

1 2 3 4 5

iii (1, 0) 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

ii 9

unit left

ii 1

Understanding

4 When the graph of y = −x2 is shifted left, the y-intercept changes. For example, if we shift y = −x2 two units to the left, the new y-intercept will be (0, −4).

7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

5.03 Horizontal translations

• y = (x − 8)2

b 3 units left

c 10 units right

c i T he amount of water in the tank is affected by the vertical translation in the equation.

2 Example answers:

d Left

c 1 unit right

b It takes approximately 22.36 minutes for the tank to be completely drained.

• y = (x − 3)2

c Right

7 a 3 units right

19 a 1500 cm

ii The rate the water drains is affected by the dilation in the equation. In this case, the coeﬃcient of the x2 term, −3, represents the rate at which the water drains.

b Left

x 1 2 3 4 5

iii (−3, 0)

Answers mathspace.co

713

d i 7 6 5 4 3 2 1

d i

−5 −4 −3 −2 −1 −1 −2

7 y 6 5 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5

ii x = −6 ii

11 a y = (x + 7)

iii

10 a i

7 y 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

ii x = −1 b i

d y = (x − 4.6)2

e y = −(x + 3)2

f y = −(x − 2)2

12 a The graph has been horizontally translated 6 units to the left.

d The graph has been horizontally translated 13 units to the left. e The graph has been horizontally translated unit to the left. x

2 3 4 5 6 7

f T he graph has been horizontally translated 0.15 units to the right. 13 y = − (x − 6)2 or y = −(x − 4)2

ii x = 5

14 There are two values for D that solve the equation approximately 7.32 metres and −5.32 metres, however negative distance doesn’t make sense in this context, so the answer is 7.32 m.

iii (5, 0) 7 y 6 5 4 3 2 1

−4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

ii x = −2

b y = (x − 13)2

c The graph has been vertically reflected across the x-axis and horizontally translated 3 units to the right.

7 y 6 5 4 3 2 1

c i

iii (−6, 0) 2

b The graph has been horizontally translated 9 units to the right.

iii (−1, 0)

−2 −1−1 −2 −3 −4 −5 −6 −7

714

x 1 2 3

Reasoning x 1

iii (−2, 0)

Mathspace WA Curriculum – Year 10 mathspace.co

15 A quadratic of the form y = (x − p)2 always yields non-negative y-values because squaring any real number, whether positive, negative or zero, results in a positive value of zero. 16 The maximum y-value is 5. This is always the case because, for any value of p, (x − p)2 will always be zero or positive since it’s a square number.

17 Sample Answer:

For a given value of x to get the same y-value as y = x2 for the function y = (x − 3)2 we need to add 3 to the x-value. For example: For the quadratic equation y = x2 when x = 2, y = 4, to get a y-value of 4 for the quadratic y = (x − 3)2 we need x − 3 to be 2. So we add 3 in order to get the point (5, 4). Problem-solving

72 64 56 48 40 32 24 16 8

y (in dollars)

x (month) 1 2 3 4 5 6 7 8 9 10 11 12

5.04 Optional: Quadratic graphs

18 a There are two possible horizontal positions: 7 metres to the right of the shooter (positive direction) and 1 metre to the left of the shooter (negative direction). b The target should be placed at a height of 484 metres above the ground if the marksman wants to hit a target 25 metres away. 19 a When t = 0, s = 4 km/hr. b When s = 9, t = −1, 5. We take positive values only so t = 5 hours. c s = 0 when t = 2 hours d Fastest speed occurs at the end of the journey at 6 hours s = 16 km/hr

Understanding 1 a A parabola is a ∪-shaped curve that opens either upward or downward. b i I f the coeﬃcient a is positive, the curve is ∪-shaped opening upward. If a < 0, then the curve is ∩-shaped, opening downwards. ii Changing the constant term c in a quadratic equation shifts the entire graph vertically. If c is positive, the graph moves upward, and if c is negative, the graph moves downward. The magnitude of the shift depends on the value of c. 2 The graph is concave upward.

16 s(km/hr) 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

axis of symmetry

10 8

y-intercept

6 4

x-intercepts

2 t(hours) 1

20 a The maximum revenue occurs at the end of the range, so at month 12. b R = (12 − 4)2 = 64, which is $64 000 c Sales reach 6000, set R = 6. So, 6 = (m − 4)2. Solve for m we get two values as: ± + 4 = m the ﬁrst time the sales reach 6000 is in month 1.55, which we would say is month 2. d At month 4, the revenue is 0.

−2

x 1

turning point

3 Turning point 4 The x-intercepts Fluency 5 a i H orizontal shift 3 units to the right and vertical shift 4 units down ii (3, −4)

Answers mathspace.co

715

iii

7 6 5 4 3 2 1

iii

−1 −2 −3 −4 −5

x 1

ii (−3, −6)

x 1

−1 −2 −3 −4 −5

b i R eﬂection about the x-axis, horizontal shift 3 units to the left and vertical shift 6 units down

4 3 2 1

2 3 4 5 6 7 8 9

6 a i Concave up

ii More steep

iii (1, −3)

iv (0, −1)

y 3

iii

y 1

−5 −4 −3 −2 −1

−5 −6

−1

−7

−2

−8

−3

−9

b i Concave down

c i H orizontal shift 1 unit to the left, vertical shift 4 units up and dilated (compressed) by a factor of 3.

ii Less steep

iii (2, 2)

y 3

ii (−1, 4) iii

−3 −2 −1

2 11 10 9 8 7 6 5 4 3 2

−5 −4 −3 −2 −1

1 −1

x 1

−1 −2 −3

c i Concave down x 1 2 3 4

d i H orizontal shift 6 units to the right, vertical shift 1 unit up and dilated (compressed) by a factor of 0.5. ii (6, 1)

iii (−5, −4) v

ii More steep iv (0, −79)

x y −9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12

716

Mathspace WA Curriculum – Year 10 mathspace.co

d i Concave up

ii More steep

iii (−1, 2)

d i

iv (0, 5)

18 16 14 12 10 8 6 4 2 −4 −3 −2

ii iii

y 5 4 3

−1

2 1

7 a i y = (x + 3)2 − 9

ii (−3, −9)

iii

x −3 −2

−6 −5 −4 −3 −2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9

b y = (x − 1)2

c y = −x + 6

d y = 3(x + 2)2

e y = 0.5(x + 2)2 − 3

f y = −2(x − 3)2 + 4

9 a y = (x − 2)2 − 6 b

6 5 4 3 2 1 −1

ii (2, 1) y 1 x −6 −5 −4 −3 −2 −1

8 a y = x2 + 4

b i y = −(x − 2)2 + 1 iii

−1

−1 −2 −3 −4 −5 −6

x 1

1 2 3 4 5 6 7

−1 −2

10 a i y = 0

ii x = 0, −4

iii x = −2

iv (−2, −4)

−3

−4

3 1 −4 −3 −2 −1 −1

c i y = − (x + 4)2 – 2

−3

y x −9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18

−2

ii (−4, −2) iii

x 1

−4

b i y=0 iii x = 2

ii x = 0, 4 iv (2, 4)

Answers mathspace.co

717

4 3

1 −2 −1

−1

1 −1

−3 −4

−1

x 1

c i y = 12

ii x = 2, 6 iv (4, −4)

y 2

8 6 4 2

x 1

−4

d i y = −3

ii x = 3, −1

iii x = 1

iv (1, −4)

28 24

16 12

x 1

−1

1 2

b y = −2(x − 1) (x + 7)

1 −1

−8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16

−2

11 a y = (x − 1) (x + 7)

−2

iii x = 4 12

−2

8 4

−2

−8 −7 −6 −5 −4 −3 −2 −1

x 1 2

−3

c y = 3 (x − 1) (x + 7)

−4

e i y=5

ii x = 5, −1

iii x = 2 v

−2 −1

x 1

f i y=3

ii x = 1, 3

iii x = 2

iv (2, −1)

718

Mathspace WA Curriculum – Year 10 mathspace.co

y −8 −7 −6 −5 −4 −3 −2 −1−4

−8 −12 −16 −20 −24 −28 −32 −36 −40 −44 −48 −52 −56

iv (2, 9) 9 8 7 6 5 4 3 2 1

15 a i Concave up

d y = −0.5(x − 1) (x + 7)

ii (x − 1) (x − 3), (1, 0), (3, 0)

iii y = 3

iv x = 2

24 20

y 4

−8 −7 −6 −5 −4 −3 −2 −1

b (2.5, 12.25)

13 a

and y = x2 − 3 and y = −x2 + 3

and y = 2x2 − 6

iii y = −36

iii (6, −4) y

v (0, −36)

y −6 −4 −2 −4 −8 −12 −16 −20 −24 −28 −32 −36

ii y = (x − 6)2 − 4

iv x = 0

14 a i y = (x − 4) (x − 8), (4, 0) and (8, 0)

ii (x + 6) (x − 6), (−6, 0), (6, 0)

and y = −3x + 9

−2

−1 −2 −3 −4 −5

b i Concave up

5 4 3 2 1

1 −1

12 a y = −(x − 6) (x + 1)

−1

1 2

v (2, −1)

x 2

c i Concave down ii (2 + x) (2 − x), (−2, 0), (2, 0) iii y = 4

iv x = 0

v (0, 4)

y 4 3

b i y = −2(x − 1) (x + 5), (1, 0) and (−5, 0)

ii y = −2(x + 2)2 + 18

iii (−2, 18) iv

18 16 14 12 10 8 6 4 2 −6 −5 −4 −3 −2 −1

−2

−1

x 1

−1 −2

d i Concave up ii (x + 3) (x + 1), (−3, 0), (−1, 0) iii y = 3

iv x = −2

v (−2, −1)

x 1

2 3

Answers mathspace.co

719

h i Concave down

y 4

ii −(x + 4) (x − 2), (−4, 0), (2, 0)

iii y = 8

iv x = −1

2 x −4

−3

−2

−1 −1

e i Concave up ii (x + 5) (x + 1), (−5, 0), (−1, 0) iii y = 5

iv x = −3

−6 −5 −4 −3 −2 −1

iii y = −8

iv x = 1

vi −2

−1

−1 −2 −3 −4

16 14 12 10 8 6 4 2

iv x = 4

v (4, −1)

iii y = 8

9 8 7 6 5 4 3 2 1

x 1

iv x = 1 24 21 18 15 12 9 6 3

−4 −3 −2 −1 −3

720

iv x = 1

g i Concave down

−1 −2 −3 −4 −5 −6 −7 −8 −9

ii −(x + 2) (x − 4), (−2, 0), (4, 0)

ii −(x + 4) (x − 6), (−4, 0), (6, 0) iii y = 24

x 1

j i Concave down

−2

v (1, −9)

ii (x − 3) (x − 5), (3, 0), (5, 0) vi

ii (x + 2) (x − 4), (−2, 0), (4, 0)

f i Concave up iii y = 15

i i Concave up

7 6 5 4 3 2 1

−4 −3 −2 −1

v (−3, −4)

9 8 7 6 5 4 3 2 1

v (−1, 9)

v (1, 25)

−2

−1

v (1, 9)

x 1

k i Concave up

ii (x + 3) (x − 1), (−3, 0), (1, 0) iii y = −3

x 1 2 3 4 5 6

Mathspace WA Curriculum – Year 10 mathspace.co

iv x = −1

v (−1, −4)

c i

2 1

4 3 2 1

−4 −3 −2 −1 −1

iii y = 1

−4 −3 −2 −1−1

−2

x 1

−2 −3 −4 −5 −6 −7

−3 −4

l i Concave up ii (2x + 3) (x + 3), (−3, 0), d i x = −4 iii y = 9

10 9 8 7 6 5 4 3 2 1 −4

16 a i x = 2 iv

−3

−2

−1

−9 −8 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12

iii y = 1

e i x = −3

−7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14

x 1

−2 −3 −4

b i x = −3

2 1

ii (−3, −11)

−4 −3 −2 −1 −1

5 y 4 3 2 1

−1

ii (2, −3)

ii (−3, −2)

iii y = 7 9 8 7 6 5 4 3 2 1

−9−8 −7 −6−5−4 −3 −2 −1 −1 −2 −3 −4

iii y = −11

ii (−4, 5)

f i x=5 iv

x 1 2 3 4

4 2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 −22

ii (5, 2)

x 1 2

iii y = −2 y x 1 2 3 4

iii y = −23

y x 1 2 3 4 5 6 7 8 9 10 11

Answers mathspace.co

721

g i x=4 iv

8 6 4 2 −1 −2 −4 −6 −8 −10

h i x = −1

iii y = −9

ii (4, 7) y

x 1 2 3 4 5 6 7 8 9

ii (−1, −12)

iii y = −11

y 2 −4 −3 −2 −1 −2

x 1

−4 −6 −8 −10 −12

Reasoning 17 For all forms, to find the y-intercept, we can substitute x = 0 and evaluate fully using the order of operations. It’s easiest to find the y-intercept using the standard form y = ax2 + bx + c since when x = 0, ax2 and bx become 0. This leaves just y = c and so the y-intercept is (0, c). 18 a False. The turning point of a quadratic graph can be located at any point in the Cartesian plane, not necessarily at the origin. It depends on the values of the coefficients in the quadratic equation. b False. If the coefficient of the x2 term is negative, the graph will open downwards. The sign of the coefficient determines the direction of the graph. c True. All quadratic graphs have a line of symmetry that passes through the turning point. The line of symmetry divides the graph into two symmetric halves. d True. The constant term in the quadratic equation represents the y-intercept of the graph. It is the value of y when x equals zero.

722

Mathspace WA Curriculum – Year 10 mathspace.co

e True. The roots of a quadratic equation are the values of x where the graph intersects the x-axis. They can be found by locating the x-intercepts of the quadratic graph. 19 a a > 0 means the curve will be concave up and so has a minimum turning point, q < 0 means the turning point will be below the x-axis, so the curve will cut the x-axis at two points. p can take any value. b a < 0 means the curve will be concave down and so has a maximum turning point, q = 0 means the turning point will be on the x-axis, so the curve will touch the x-axis at one point. p can take any value. c a > 0 means the curve will be concave up and so has a minimum turning point, q > 0 means the turning point will be above the x-axis, so the curve will not cut the x-axis at all. p can take any value. 20 a (x − 2)2 + 5

b (x − 4)2 − 2

c The equation would become (x − (2 + a))2 + (5 + b). The turning point would move from (2, 5) to (2 + a, 5 + b) 21 a They have the same x-intercepts, both are concave up, and they share an axis of symmetry. b The y-intercept and turning points are different. 22 a y = ax2 − 2apx + ah2 + q b y = ax2 + (−2ap)x + (ap2 + q) so b = −2ap c y = ax2 + (−2ap)x + (ap2 + q) so c = ap2 + q d Vertex form and standard form are equivalent quadratic equations in different forms. The equations give us different information more efficiently. We can find the vertex, (p, q), efficiently using vertex form. We can find the y-intercept, c, efficiently by plugging in 0 for x and evaluating the corresponding output using standard form. We can also find the axis of symmetry,

, in standard form

and use it to ﬁnd the y-value of the vertex. In both forms, the leading coeﬃcient, a, is the vertical stretch/compression factor and tells us if the parabola will open up or down.

23 The turning point of a quadratic graph is the point where the graph reaches its minimum or maximum value. The x-coordinate of the turning

point can be found using the formula

where a and b are the coeﬃcients of the x2 and x terms, respectively. 24 a If two quadratic curves have the same intercepts, it means they both pass through the same points on the x-axis and y-axis. If these are three different points, then the two quadratics are the same. If they have less than three intercepts, however, there can still be differences in their shapes, such as the steepness or direction of their parabolas. For example, the curves represented by y = x2 and y = 2x2 both have a single intercept at (0, 0), but the latter is a steeper parabola. b If three quadratic curves have the same turning point, it means they all have the same vertex (the highest or lowest point of the parabola). Additionally, if they share one common intercept, it means they all pass through the same point on either the x-axis or y-axis. If the shared intercept is different to the turning point, then the three quadratics are the same. If the shared intercept is also the turning point, however, then there can still be differences in their shapes. For example, the curves represented by y = (x − 1)2, y = 2(x − 1)2 and y = −1(x − 1)2 share a single x-intercept of (1, 0) which is also their turning point, but they have different steepness and directions of their parabolas. They also have different y-intercepts.

h 25

10 5 t 1

2 3 4 5 6 7 8 9

c 24.75 m 26 a

d 25 m

500 450 400 350 300 250 200 150 100 50

5 10 15 20 25 30 35 40

b s = 19

c 524

27 a 6 m

c x = −4

b 3m

y 5 4 3 2 1 x −5

−4

−3

−2

−1

28 29 a Elise should expect to pay $900

Problem-solving 25 a

b The price would decrease by $393.75

9 16 21 24 25 24 21 16 9

9 10 0

1000 900 800 700 600 500 400 300 200 100 0

cost ( y)

metre (x) 2

Answers mathspace.co

723

d The decrease in price per square metre makes the parabola wider. The x and y-intercept, domain and range, turning point, axis of symmetry, end behaviour, and intervals of the domain for where the curve is positive/ negative and increasing/decreasing stay the same. 30 Since the turning point lies on the axis of symmetry, we know that 8 is the average of 12 and m. So then, we must have m = 4. If we substitute the coordinates of the turning point into the equation, we get: 15 = a(8 − 4)(8 − 12) .

If we solve this equation we get

5.05 Optional: Quadratic equations Understanding 1 5 2 A quadratic equation can have 0, 1, or 2 solutions. 3 a a = 2, b = −7, c = 5

b a = −3, b = 8, c = −7

c a = 1, b = 0, c = −16

d a = −1, b = −4, c = 3

4 a a = 2, b = 7

b a = −5, b = 1

c a = 0, b = −8

Reasoning 9 We can ﬁnd the solutions for x by taking the positive and negative square roots of k, that is, . 10 Answer may look like: The equation can be rewritten as x2 = 9. As both 32 and (−3)2 are equal to 9, there are two solutions. 11 Answer may look like: To write the equation in the form (x − a) (x − b) = 0, we can use the given solutions to ﬁnd the factors. Since x = 4 is a solution, one factor is x − 4. Similarly, since x = −3 is a solution, the other factor is x + 3. Therefore, the equation can be written as (x − 4)(x + 3) = 0. 12 Mark’s answer is incorrect. When using the null factor law, the correct equations should be x − 9 = 0 and x + 13 = 0. Solving these equations, we get x = 9 and x = −13 as the correct solutions, not x = −9 and x = 13. 13 Rearranging gives x2 = −4. There are no real numbers that will give −4 when squared. Problem-solving

5 x = −12

14 t = 14, t = 10

Fluency b k = ±5

c x = ±11

e x = ±11

f y = ±5

g x = ±5

i k = ±2

k x = ±15

6 a

7 a z = 4, −10

b m = 16, −2

15 k = −16 16 a = 13 17 p = 2, q = −3 18 a b i a = 0, c = All real numbers and a > 0, c < 0 or a < 0, c > 0

d p = 11, −5

e n = −7, 11

f x = −2, 16

h p = ±10

iii a > 0, c > 0 or a < 0, c < 0

Chapter 5 review

8 a n = 0, −9 c p = 0, −6

d q = 0, −8

ii a ≠ 0, c = 0

1 a i The turning point is (0, 22). ii Maximum

724

h r = 3, 8

i t = −4, −10

Mathspace WA Curriculum – Year 10 mathspace.co

iii The equation of the axis of symmetry is x = 0. iv The x-intercepts are at (−1.1, 0) and (1.1, 0), while the y-intercept is at (0, 22).

b i The turning point is (0, −15).

d i

ii Minimum iii The equation of the axis of symmetry is x = 0.

−2

−1

−15

iv The x-intercepts are at (−1.5, 0) and (1.5, 0), while the y-intercept is at (0, −15). 2 a i

−2

−1

−2

y 25

e i

15 10

−2

−1

−4

y 15

−1

y ii

5 −2

−1 −2 −4 −6 −8 −10 −12 −14 −16 −18

8 6 4 2

2 10

b i

−2

−1

−11

x −2

y 4 2 −2

c i

−1

x 1

−2 −4 −6 −8 −10 −12

f i

−1

−43

−13

−3

−13

−43

y −1

−5 −10 −15 −20 −25 −30 −35 −40 −45

−2

−1

−54

−15

−2

−15

−54

−1

−5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −55

g i

−2

−1

−2

−1

−4

−6

−4

Answers mathspace.co

725

4 y 3 2 1 −2

h i

−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

b 7 units downwards

c 5 units upwards

d 4 units downwards

6 a 6 units upwards

b 3 units upwards

c 5 units downwards

b y = x2 − 13

d y = x2 − 19

c y = x − 49

−2

−1

8 a y = 2x2 + 4. The graph has been narrowed (vertically enlarged) by a factor of 2 and vertically translated up 4 units. b y = x2 − 2. The graph has been vertically translated down by 2 units. c y = −5x2 − 5. The graph has been vertically reﬂected across the x-axis, narrowed (vertically enlarged) by a factor of 5 and vertically translated down 5 units.

45 40 35 30 25 20 15 10 5

d 2 units downwards

7 a y=x +1

y ii

3 a

x 1

5 a 8 units upwards

d y = −3x2 + 3. The graph has been vertically reﬂected across the x-axis, narrowed (vertically enlarged) by a factor of 3 and vertically translated up 3 units. x

−2

−1

−3

−2

−1

−4

−6

−4

9 a 5 units right

b 4 units left

c 2 units right

d 6 units left

b y = (x − 10)2

10 a y = (x + 8)

c y = (x + 6.4)2

e y = −(x + 4)2

b Minimum c Positive 4 i Concave down

11 a The graph has been horizontally translated 8 units to the left.

ii (0, 6) iii i

−2

−1

−62

−11

−62

5 −2

726

−1

−5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −55 −60 −65

y x 1

Mathspace WA Curriculum – Year 10 mathspace.co

b The graph has been vertically reﬂected across the x-axis and horizontally translated 5 units to the right. c The graph has been horizontally translated 11 units to the right. d The graph has been horizontally translated 15 units to the left. e The graph has been vertically reﬂected across the x-axis and horizontally translated 9 units to the right. f The graph has been horizontally translated units to the right. 12 There are two values for D that solve the equation: 9 metres and −3 metres. However, since negative distance doesn’t make sense in this context, the answer is 9 m.

13 a y = (x + 3)2 + 5

b y = (x − 4)2 − 2

c y = −2(x + 2) + 3

d y = 3(x + 5)2 − 4

e y = −(x − 6)2 + 8

f y = 2(x + 4)2 − 10

14 a i y = 0

ii x = 0, −6

iii x = −3

iv (−3, −9)

4 y 3 2 1 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

y 2

−4 −8 −10 −12

x 1

e i y = 18

ii x = 6, −3

iii x = 1.5

iv (1.5, 20.25)

y 25 20

b i y=0

ii x = 0, 5

iv (2.5, 6.25)

7 6 5 4 3 2 1

5 x −4−3−2 −1

c i y = 32

ii x = 4, 8

iii x = 6

iv (6, −4)

1 2 3 4 5 6 7

f i y = 14 x

−1−1 −2 −3 −4

1 2 3 4 5 6

−6

iii x = 2.5 v

−3 −2 −1 −2

ii x = 2, 7

iii x = 4.5 v

18 y 16 14 12 10 8 6 4 2 −2 −1 −2 −4 −6 −8

30 25

iv (4.5, −6.25)

1 2 3 4 5 6 7 8 9

20 15

15 a y = (x + 5)(x − 9)

5 −2 −5

d i y = −10 iii x = 1.5

x 2

8 10 12

ii x = 5, −2 iv (1.5, −12.25)

y −4 −2 −5 −10 −15 −20 −25 −30 −35 −40 −45

x 2

Answers mathspace.co

727

b y = −0.5(x + 5)(x − 9)

b i x = −4

ii (−4, −4)

−8 −7 −6 −5 −4 −3 −2 −1 −3

−4 −2

c y = 2(x + 5)(x − 9)

c i x = −2

iii y = −3 6

−5 −4 −3 −2 −1

−2 −6 −8

d i x = −5

iii y = −16

ii (−5, 9)

10 y 8 6 4 2 x

20 15 −10 −8 10 5 x

ii (3, −4)

iii y = 5

y 8 6 4 2

−2

x 1

−4

728

x 1

−4

16 a i x = 3

d y = −0.25(x + 5)(x − 9)

−4 −2

1 2

ii (−2, 5)

y −4 −2 −10 −20 −30 −40 −50 −60 −70 −80 −90 −100

iii y = 12

Mathspace WA Curriculum – Year 10 mathspace.co

e i x = −6

−6

−4

−2 −2 −4 −6 −8 −10 −12 −14 −16 −18

ii (−6, −43)

iii y = −7 yx

−14 −12 −10 −8 −6 −4 −2 −5 −10 −15 −20 −25 −30 −35 −40 −45

f i x=7 iv

ii (7, 4)

−5 −10 −15 −20 −25 −30 −35 −40 −45

h r = 5, 10

i t = −6, −15

10 12

6.01 Exponential relationships Understanding

g i x=6 15

iii y = −45

iii y = −20

ii (6, 16) y

10 5 −5

x 2

−10

ii 80, 160

b i Adding 5

ii 25, 30

c i Multiplying by 3

ii 202.5, 607.5

d i Dividing by 10

ii 4, 0.4

2 a Multiplying by 3

b 108 circles

3 a Multiplying by 4

b 128 hexagons

Fluency

−15

4 a

−20

h i x = −5

1 a i Multiplying by 2

ii (−5, −40)

5 5 y 1

iii y = −15 y x

5 25 125

625 3125

−12 −10 −8 −6 −4 −2 −5

b Exponential

−10

5 a Exponential

−15

c Quadratic

d Exponential

−20

e Exponential

f Exponential

g Linear

h Exponential

−25 −30

b Linear

−35

6 a i 3.5

−40

b i 15

ii 101 250

c i 2

ii 1.968

d i 0.25

ii 16

17 a b = ±4

b m = ±6

c z = ±12

d w = ±13

e x = ±14

f y = ±5

g x = ±6

h y = ±2

i k = ±3

j k = ±1

k x = ±16

ii 1500.625

7 a 5

b 125 triangles

8 a The volume increases by a factor of 3 each minute. b

Time (minutes)

Volume (mL)

9 27

18 a z = 3, −13

b m = 18, −2

c x = 13, 5

d p = 13, −5

e n = −9, 15

f x = −1, 21

h p = ±9

243

729

g 19 a m = 0, −12 c p = 0, −8

c 6561 mL

d q = 0, −10

Answers mathspace.co

729

Reasoning 9 a The volume increases by a factor of 2 each day. b The points represent an exponential relationship as they increase by a constant ratio of 2.

The relationship between time and mass is exponential because the ratio between consecutive values of the mass is constant 0.8 15

Number of pallets Number of bricks 0

They do not lie on a straight line. c 256 mL 10 a graph B = b graph A = 2x c Graph A is an exponential relationship as it is increasing at an increasing rate and has a constant base of 2. Graph B is linear as it increases at a constant rate of . 11 No, y2 > y1 for some values of x ≥ 1. However, as y1 is increasing at an increasing rate while y2 increases at a constant rate, at some point y1 will become greater in value.

12 a Points A, C, F

b 6 units

c Points B, D, E

d 2

Time

Population

160

320

The relationship between time and population is exponential because the ratio between consecutive values of the population is constant 2. 14

730

19 500

19 000

18 500

18 000

17 500

The relationship between number of pallets and number of bricks is linear because there is a constant difference of −500 between the consecutive values of the pallets, which is equal to the gradient of the line.

6.02 Characteristics of exponential graphs Understanding

Problem-solving

20 000

Time

Mass

100

51.2

40.96

32.768

26.2144

Mathspace WA Curriculum – Year 10 mathspace.co

1 The base, a. 2 Asymptote 3 a True b False, if 0 < a < 1, then it is exponential decay. c False, if 0 < a < 1, then it is exponential decay. d False, if a > 1, then it is exponential growth. e True 4 a As x increases, y increases at an increasing rate. b As x increases, y decreases at a decreasing rate. 5 a Increasing

b Decreasing

c Decreasing

d Increasing

e Increasing

f Decreasing

g Decreasing

h Increasing

Fluency 6 a i

x y = 4x

−1

4 16 64

y 2 3 1 4 1 1 2 1 1 4 1 3 4 1 2 1 4

30 25 20 15 10 5 x −2 −1

iii Increasing b i

−1

x y = 5x

−2

5 25 125

3 iii Decreasing e i

−1

−1 0

150

100

4 1 3 2 3 1 2 2 2 1 1 2 1 1 2

50 x −2 −1

100

1000

iii Increasing c i

−1

x y = 10x

−2

−1

iii Decreasing

1000

f i

800

−1 0

600 400

200

5 2

x −2 −1

iii Increasing d i

−1 0 2

3 2

1 2

x −2

−1

iii Decreasing

Answers mathspace.co

731

ii y = 0

7 a i Increasing

iii (0, 1)

iv Answers may vary, for example: (2, 4) v

5 4

−4 −3 −2 −1

3 2

−4 −3 −2 −1

iii (0, 1) iv Answers may vary, for example:

ii y = 0

b i Increasing

ii y = 0

e i Increasing

iii (0, 1)

iv Answers may vary, for example:

7 6

5 4

4 2 x

−4 −3 −2 −1

7 6

5 4

−4 −3 −2 −1

3 2

g i Decreasing

d i Increasing

ii y = 0

iv Answers may vary, for example: (1, 7)

−4 −3 −2 −1

iv Answers may vary, for example: (−2, 9)

ii y = 0

c i Increasing

iii (0, 1)

f i Decreasing

−4 −3 −2 −1

x 1

ii y = 0

iii (0, 1) iv Answers may vary, for example: (2, 64)

732

Mathspace WA Curriculum – Year 10 mathspace.co

x 1

ii y = 0

iii (0, 1) iv Answers may vary, for example: (−1, 5)

11 a

y 7

−4 −3 −2 −1

ii y = 0

h i Decreasing

b 7

iii (0, 1)

5 4

3 2

−5 −4 −3 −2 −1 −1

5 4 3 1 −4 −3 −2 −1

x 1

8 a No. For a > 0, the expression ax is greater than zero for all real x.

Reasoning 12 a Graph R b For x < 0, the graph of y = (5)x is below the graph of y = (3)x. This is because negative values of x result in fractional y-values, and for any negative value of x, (5)x will result in a smaller value than (3)x.

b y tends towards ∞. d 1

f 7

e 0

13 No, both functions are always greater than zero, because a positive base to any power is always greater than zero.

5 4 3 2 1 −5 −4 −3 −2 −1 −1

b 1

x 1 2 3 4 5

c The two graphs are reﬂections across the y-axis.

9 a

1 2 3 4 5

iv Answers may vary, for example:

c 0

−5 −4 −3 −2 −1 −1

14 a i False

ii False

iii True

iv True

1 2 3 4 5

b Shared y-intercept of (0, 1) c As x gets larger, the y-values approach 0. c 16

d None. For all values of x, it will have a positive value.

15 Example justiﬁcation: When graphed on the interval 0 < x < 2, we can see that the two

e It gets larger, approaching inﬁnity.

curves start at the same point, but

f It gets smaller, approaching zero.

ends below the graph of

10 a Positive b y = 0 is a horizontal asymptote of the curve.

that

. This tells us

is approaching y = 0 faster,

Answers mathspace.co

733

which means it is decreasing more rapidly. The pattern continues even for larger values of x. y y1

d i Exponential growth ii 10 000

iii 100%

3 a 840

b 4%

4 a 1.4

b Increase by 40%

Fluency x 0

5 a 2.4 kg b

Months

Mass (kg)

1.8000

1.9800

Problem-solving 16 a 125

b 1 b −2

17 a 1

d −2

c c 3

d 3

18 m = 8 19 a y = 128

b x=7 c

20 80 units

2.1780

2.3958

2.6354

2.8989

3.1888

Mass (kg) 4

21 a b

27 24 21 18 15 12 9 6 −x 3 y=3 −4 −3 −2 −1

2 1 1

6 a y = 1300(1 + 0.08)v x

c The two graphs are reﬂections across the y-axis.

6.03 Exponential models

Months 6

b $2806.60

7 41 199 people 8 6591 bees 9 $503 579 10 $5.20 11 $12 072

Understanding 1 a Exponential decay c Neither

b Exponential growth d Exponential decay

e Neither iii 5%

d 256 players 13 a 5000

iii 12.3%

c i Exponential growth ii 5.9

1 2

c p = 2r

b i Exponential decay ii 2500

Number of rounds (r) Number of players (p)

b Any power of 2 such as 2, 4, 8, 16, 32, 64 …

2 a i Exponential decay ii 0.723

12 a

iii 2.5%

b The constant percent rate of change is 2.3%. This means that the population will increase by 2.3% every minute. c 299 629

734

Mathspace WA Curriculum – Year 10 mathspace.co

17 a

N (’000s)

(N)

(C)

480

2880

17 280

103 680

622 080

3 732 480

22 394 880

t 30

120 150

b C = 480 × 6N−1

e Approximately 100 minutes 14 a

c 29 023 764 480 sales

Week number

Weight of treats (g)

2000

1000

500

250

125

62.5

31.25

It will take 6 weeks to get the dog down to less than 50 g of treats per week. b

d No, this is not a reasonable goal. It is impossible for them to have approximately 29 billion sales in one quarter. To achieve this, they would need to sell more than three phones to every single person in the world. This level of growth is unsustainable. They need to abandon the assumption that growth will change in exactly the same way in the coming quarters and try a more sophisticated model. 18 a

, where w is the number of

weeks on the plan and T (w) is the weight of treats in grams.

Seconds passed (t)

Undissolved sugar in grams ( y)

Reasoning 15 The decay rate is the percentage decrease from one term to the next, while the decay factor is the number being multiplied to the previous term. For example, if 20% is the decay rate, 0.8 is the decay factor. 16 a a represents the initial temperature of the substance, and r represents the rate at which it cools down.

c It is decreasing at a decreasing rate. d No, the model shows the amount of undissolved sugar is decreasing at a decreasing rate and has an asymptote of y = 0, so theoretically, some sugar will always remain undissolved. e If the sugar dissolves twice as quickly, it can ,

be modelled by the equation

b T = 145(1 − 0.4) c 0.88 °C

d It will never reach 0 because there is no time that could make (0.6)t equal to zero.

which is equivalent to

The amount of sugar that dissolves between the second and third second is

Exponential relationships have a horizontal asymptote at y = 0.

, which is less than .

e Substance P

Problem-solving 19 a 256

b The 8th stage

Answers mathspace.co

735

20 a

Day

Number of people infected

5 25 125 625

b 5n

c Both equations do accurately model the amount of content remembered.

c 8 days 10

d 2 = 1024 people 21 a

Example reasoning: From the graph, we see that Raymond remembers about 85% of the previous day’s content, which means he forgets 15%. That is consistent with the given equation. Sunyi remembers about 87%, according to the graph, which means she forgets 13%. This is also consistent with the given equation.

Value ($) 3500 3000 2500 2000 1500 1000 500 0

No. of years 2 4 6 8 10 12 14 16 18 20

b 13 years b 3 bounces

22 a 23 2020 24 24 hours 25 a i 179

ii 132

b i 70%

ii 40%

c Juanita’s blood sugar level peaked at a higher amount and decreased more rapidly. 26 a 0.01 or 1% b A = 38 200(1 − 0.01)n c $35 964.54 d The Motorbike 27 a The graph represents the content Maximilian and Lucy remember. After 1 day, it appears that Maximilian remembers 52%, which means he forgets 48%. After 1 day, Lucy remembers 62%, which means she forgets 38%. An equation that models the percentage Maximilian recalls is P = 100(1 − 0.48)t. An equation that models the percentage Lucy recalls is Q = 100(1 − 0.38)t. b Example explanations: No. As the models are equations for exponential decay, there is a horizontal asymptote of y = 0 that the curve will get closer and closer to but never reach.

736

In reality, the amount they remember will probably not continue to decrease at a percent rate. They may always remember a small percentage or they may forget all of it by a certain point.

Mathspace WA Curriculum – Year 10 mathspace.co

d Maximilian and Lucy forget things a lot quicker than Raymond and Sunyi. Possible reasons may include some of the following ideas: • One factor may be the type of information used in the studies. Maximilian and Lucy were given information to memorise. Raymond and Sunyi are given content to learn in school, which means they are most likely reviewing that content regularly. Raymond and Sunyi know that the information given to them needs to be retained as they will likely be assessed on it in school. • Maximilian and Lucy may not have known about the design of the study, so they may not have tried to actively retain the information they were given. • Finally, if the information given to Maximilian and Lucy was not very interesting, it would be easier to forget.

6.04 Transformations of exponentials Understanding 1 a i The same

ii Different

iii Different

iv The same

v The same b If k is positive, the graph of y = 2x + k moves upward from y = 2x. c If k is negative, the graph of y = 2x + k moves downward from y = 2x.

2 a i Different

ii The same

8 y = 4x

iii Different

iv Different

9 a A vertical dilation by a factor of 0.5, which is a compression.

v The same b If a > 1, the graph will be stretched and have a higher y-intercept. c If 0 < a < 1, the graph will be compressed and have a lower y-intercept. d If a is negative, the graph will be reﬂected across the x-axis.

b 8 7 6 5 4 3

b y = −2

3 a x

c y=2

b Yes

c No

d No

−3 −2

b The graph is translated down 6 units. c The graph is dilated (compressed) by a factor of . d The graph is dilated (stretched) by a factor of 5. 6 a A translation down 5 units

−1

b Vertical dilation by a factor of 2, which is a stretch. 11 a Translated down 2 units b Reﬂection across the y-axis c Translation up 3 units d Dilation by a factor of 2 12 Graph R

13 a A dilation by a factor of 3 b x

−5 −4 −3 −2 −1 1 2 3 4 5 −1 x −2 g( x ) = 1 − 5 3 −3 −4 −5

−2 −1

f (x) = 4x

g(x) = 3(4)x

12 48

c 16

14 12

c y = −5

7 a A reﬂection about the x-axis b

g(x) = 0.5(8x)

10 a Vertical dilation by a factor of , which is a compression.

5 a The graph is translated up 4 units.

5 4 3 x 2 f ( x) = 1 3 1

f (x) = 8x

Fluency

4 a No

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

8 6

y g(x) = 3(4)x

f (x) = 5x

−3 −2 −1

4 2

f (x) = 4x x 1

x 1 2 3 4 5

g(x) = −5x

c As x increases, the function values get increasingly large in the negative direction.

Answers mathspace.co

737

14 a i (0, −4) iv

ii y = −5 6 5 4 3 2 1

b i (0, 3)

18 16 14 12 10 8 6 4 2

1 2 3 4 5

ii y = 2 8 7 6 5 4 3 2 1

ii y = 0

e i

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

iii Increasing

−4 −3 −2 −1

iii Decreasing

f i (0, 2)

iii Increasing

ii y = 0

iv 4

iii Decreasing

3 2 1 −4 −3 −2 −1

−4 −3 −2 −1 −1 −2

2 3

x 1

−1

−2 −3 −4

c i (0, −3) iv

ii y = −4 3

iii Decreasing g i (0, 2)

2 1

18 16 14 12 10 8 6 4 2

−4 −3 −2 −1 −1

2 3 4

−2 −3 −4 −5

d i (0, 4) iv

ii y = 3 8 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2

h i (0, −3) iv

−3 −4 −5 −6 −7

738

Mathspace WA Curriculum – Year 10 mathspace.co

3 4

ii y = 0

−2 x

x 1

−4 −3 −2 −1 −1

1 2 3 4 5

iii Increasing

−4 −3 −2 −1

iii Increasing

ii y = 0

iii Decreasing

y x 1

15 a y = 5, which tells us a = 5.

b The growth factor is 4, which tells us b = 4.

c y = 5(4x)

2 1

16 a y = −4, which tells us a = −4. , which tells us

b The decay factor is

−4 −3 −2 −1 −1

−2

−3 −4

b y=3

17 a

d i (0, −4)

ii y = −1

iii Vertical dilation by a factor of 3, a stretch. A reﬂection across the x-axis.

Translated 1 unit downwards. ii y = 1

19 a i (0, 4)

Translated 1 unit upwards. iv

y −4 −3 −2 −1 −1

iii Vertical dilation by a factor of 3, a stretch.

x 1

−2

−3 −4

−5

−6

−7

−8

3 2 1

−4 −3 −2 −1

3 4

20 The coordinates of the y-intercept are (0, a). When a dilation is applied to f (x) = bx, the equation becomes f (x) = a (b)x. If we substitute x = 0 into the transformed equation, we will get y = a.

ii y = 4

b i (0, 3)

iii Reﬂected across the x-axis. Translated 4 units upwards. iv

21 By reﬂecting the graph across the y-axis

3 2

22 For x < 0, the graph of

1 −4 −3 −2 −1 −1

x 1

−2 −3

is below the

graph of y = (3) . This is because the y-values will be half of the y-values of 3x, which are smaller in value than (3)x. 23 a Yes. For any value of x, 10x is always positive. Therefore, −10x is always negative.

−4

c i (0, −2.5)

Reasoning

ii y = −3

iii Vertical dilation by a factor of 0.5, a compression. Translated 3 units downwards.

Answers mathspace.co

739

Problem-solving

y 50 −3 −2

−1 −50

26 a A reﬂection about the y-axis b g(x) = 2−x

−100 −150 −200

c There is no x-value for which y = 0.

27 a

b Emir should have gone in the order of operations, vertical dilation, reflection, and then translation, but he did translation ﬁrst and then vertical dilation. 0

y = 3x

y = 2(3x)

y = 2(3x) − 1

−3 −2 −1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

e The graph of

16 64

x 1 2 3 4 5

can be reﬂected

across the y-axis to get the graph of

2 1

x 1

−2 −3

740

x 1 2 3 4 5

d The two graphs are the same.

−4 −3 −2 −1 −1

g(x)

25 a Yalile’s graph is correct, Emir’s is incorrect.

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

c y = 3(2x). The number of bacteria can’t be negative and all values of y = −3(2x) are negative, so it must be y = 3(2x).

−1

5 4 3 2 1

−5 −4 −3 −2 −1 1 2 3 4 5 −1 −2 x −3 y = −3(2 ) −4 −5

f (x) = 2x

c Decreasing

5 4 x 3 y = 3(2 ) 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−250

24 a A reﬂection across the x-axis

5 4 3 2 1

Mathspace WA Curriculum – Year 10 mathspace.co

28 a

2 If the bases on both sides of the equation are the same, then the exponents will be equal. By setting the exponents equal to each other, we can solve for the variable.

Point on (−1, 2) (0, 1) original graph Point on new graph

(−1, 6) (0, 3)

3 a 53 e 6

b c 7

−2

b 25 f 10 b 7

c 3

d 0

e −3

f −2

g 2

h 2

5 a x=2

b x = −1

c x = −2

d x = −4

6 a x=6

c x = −3

d x=3

f y=2

g x=6

h n=4

Fluency

4 y=

1  2

3 2

y=3

1  2

1 −4 −3 −2 −1

−1

x 1

d Above

e x=0

e Above

29 30 a y = −4. Tells us k = −4. b

−1

−1 2

⬚

⬚ ⬚

⬚

−1

y = a(bx) + k

−1

y = a(bx)

y = bx

y = a(bx) + k y = a(bx) y = bx c a=3 d

e y = 3(2x) − 4 31 a y = 5(2x) − 10

d 34

4 a 9

6 5

c 43

−3

6.05 Optional: Exponential equations Understanding

7 a i 23x = 22

b i 24x = 2−1

c i 2−3x = 25

d i 2−x = 2−2

ii x = 2

8 a x = −2

b y = −3

c x = −5

d x = −1

9 a

c x=5

d x=2

f x = −3

g x = −2

i x=6

j x=2

l x = −7

10 a x = −3

b x = −2

c x=9

d x = −7

e x=1

11 a x = 2

b x=3

c x=5

d x=4

e x=3

g x = −1

h x=4

j x=1

12 a x = 1

b x = −3

c x=3

1 a False

b True

e x=0

c True

d False

Answers mathspace.co

741

Reasoning

b x=3

20 a

13 In 23x − 1 = 27x − 9, we cannot solve using exponent laws because the bases 2 and 27 are not powers of the same number, so we cannot write them in the form ax = ay. 3x − 1

x−9

=8 can be solved because the Solving 2 bases 2 and 8 can both be rewritten as powers of 2. The equation can then be solved for x by comparing the exponents directly.

e x = 4, −1

g y=4

21 Answers may vary. For example, (29−5)1 = 48−6.

6.06 Optional: Logarithms

Given equation

Understanding 2

Rewrite 16 with a base of 4

Distributive property

Quotient exponent law

Since 24 = (22)2, we can rewrite 24 with a base of 4

ax = ay, so x = y

Solve the linear equation

15 a When rewriting the right-hand side of the equation with a negative exponent in step 2, Lorena did not apply the negative to both terms. Step 2 should have been 42p = 4−(2p+1). b 16 When is squared, it equals 7. So the equation can be solved as shown: y = 4

Given equation Square both sides

b 0.1

c 10 000

d 1000

e 1

f 10

g 0.01

h 100 000

2 a The base ten logarithm of 100 is 2 because 102 = 100 b The base ten logarithm of 1000 is 3 because 103 = 1000 c The base ten logarithm of 0.01 is -2 because 10-2 = 0.01 d The base ten logarithm of 1 is 0 because 100 = 1 e The base ten logarithm of 65 is 1.8129 because 101.8129 = 65 f T he base ten logarithm of 224 is 2.3502 because 102.3502 = 224 Fluency 3 a i 106

ii 6

b i 10−4

ii −4

c i 100

ii 0

−3

ii −3

d i 10 2

Evaluate

b 100 = 1

4 a 10 = 100

Rewrite 49 with base 7 Power of a power law

Problem-solving 17 a 5 hours

b 10 hours

c 12 hours

18 a n = 8

b k=4

c m = −2

19 a 440 Hz

b 262 Hz

c n = 73

742

1 a 100

Mathspace WA Curriculum – Year 10 mathspace.co

c 104 = 10 000

d 10−2 = 0.01

5 a 1.6532

b 3.8860

c 0.7126

d −2.2373

6 a 37.154

b 0.063

c 6.310

d 204.174

7 a B

b A

c C

d B

8 a 2 countries

b 10 countries

9 a 13

c 6

b 25

Reasoning

City

Population

Log scale

Melbourne

4 929 200

6.693

Adelaide

1 396 783

6.145

Wollongong

310 479

5.492

Geelong

278 708

5.445

Hobart

231 707

5.365

Toowoomba

146 955

5.167

Ballarat

105 924

5.025

Bunbury

80 106

4.904

Warragul

44 625

4.650

Log scale

Frequency

4–5

5–6

6–7

11 a B

b C

Frequency

10 a The data is very large and very spread out. Trying to ﬁt the information on the same scale would be diﬃcult and would not show us the trends in the data. b

3 2 1 1 2 3 4 5 Log of lifespan

14 a R = 3.8

b Minor

6.07 Optional: Order of magnitude and logarithmic scales Understanding 1 a 102

b 103

c 106

d 101

e 100

f 10−2

g 10−1

h 10−3

2 a 6 × 102

b 7 × 103 6

c 4.8 × 10

d 3.5 × 101

e 2.3 × 102

f 7.6 × 10−3

−2

c A

h 9 × 10−5

g 1.2 × 10 3 a 3

b −5

c 0

d −1

Problem-solving Fluency

12 a $79 432.82

4 −7

b $100 000 to $1 000 000 13 a

5 a i 8 × 103

Lifespan (days)

Lifespan in a log10 scale

1.30

African elephant

25 500

4.41

Bowhead whale

71 400

4.85

7 4

Bumblebee

1.32

Chicken

2920

3.47

8 2

Dog

5110

3.71

Emperor penguin

8500

3.93

1.26

Giraffe

10 120

4.01

Hamster

780

2.89

Ladybug

365

2.56

Little penguin

2190

3.34

Monkey

7300

3.86

Shark

10 980

4.04

Animal/Insect Ant

Fly

ii 3

b i 9.644 × 105

ii 5

c i 8 × 10−4

ii −4

d i 2.574 × 10−1

ii −1

6 3

9 a 16

b 631

10 a 5.2 b 100.1 or approximately 1.26 times larger 11 a The concentration of hydrogen ions in the pH 8 solution is 100 times greater than in the pH 10 solution. b The concentration of hydrogen ions in the pH 5.4 solution is 104.6 times greater than in the pH 10 solution.

Answers mathspace.co

743

c The concentration of hydrogen ions in the pH 5.4 solution is 102.6 times greater than in the pH 8 solution.

c The population increases by a factor of 10 each day (e.g., 50 ×10n). Starting at 50 = 5 × 101, solve 50 × 10n = 108 : 10n = 2 × 106, so n = log10(2 × 106) ≈ 6.3. Thus, it takes 6.3 days, or 7 days when rounded up.

12 a 1 000 000 b 100 000 times louder 13 a 104 times

b 103 times

c 106 times

d 107 times

14 a 102.8 times

b 103.6 times

20 a 100 mW b 103 or 1000 times weaker c 2.5 21

c 106.4 times

22 R = 8.7

Reasoning 15 The order of magnitude for DNA is −9, and the order of magnitude for a hydrogen nucleus is −15. These magnitudes have a difference of −9 − (−15) = 6, so they differ by 6 orders of magnitude.

23 a

16 On a base 10 logarithmic scale, the values 2 and 4 represent 102 = 100 and 104 = 10 000. The difference is not 2 times but a ratio of 4−2

= 10

1000 900 800 700 600 500 400 300 200 100

x 1

= 10 = 100 times, so the quantity b

increased by 100 times.

Base 10 log scale 0.602

1.204

1.806

256

2.408

1024

3.010

Therefore, source A is 25 times brighter than Source B.

1 x

Problem-solving

19 a 5

Chapter 6 review Population

Base 10 log scale

1 a Linear

b Quadratic

1.7

c Linear

d Exponential

500

2.7

5000

3.7

e Exponential

f Quadratic

50 000

4.7

500 000

5.7

2 a i 0.25

ii 7.8125

b i 2.5

ii 312.5

3 a

744

18 On a base 10 logarithmic scale, the difference 3.2 − 1.8 = 1.4 means the intensity ratio is 101.4. Calculating 101.4 ≈ 25.12 or 25.

b 101.5 times

Population of shrimp

17 a 102 times c Yes, if houses collapse for earthquakes of magnitude 6, then it would take an earthquake of smaller magnitude to cause only a crack. A magnitude of 6.2 would cause more than just a crack in a building.

Mathspace WA Curriculum – Year 10 mathspace.co

b 2 cubes

4 a i

−1 0

x y = 6x

6 36

216

7 a y = 7700 (1 + 0.053)n b $16 707.69 8 $12 068

y 200 160

9 a 10 300

b Decreasing

c 9.8%

d 6818 bees

10 a The graph is translated up 2 units.

120

b The graph is reﬂected across the x-axis.

c The graph is dilated (enlarged) by a factor of 3.

40 x −1

11 a A translation down 4 units

5 4 3 2 1

iii Increasing b i

−1

f (x) = 7x −5 −4 −3 −2 −1 −1 −2 −3 g (x) = 7x − 4 −4 −5

y 4 1 3 2 3 1 2 2 2 1 1 2 1 1 2

x 1 2 3 4 5

c y = −4 12 Graph Y b y=0

13 a d

−1

c Increasing

12 10

iii Decreasing 5 a y=0

b (0, 1)

6 4

c One point is

−8 −6 −4 −2

15 a x = 8

e x=6

5 x

6 a None

−1

14 a y = 256

−2

x 2

b Positive

16 a 17 a n = 1

b x = −2 b x=4

c x = −2

d y = −3

h x = −8

c x=0

b k=5

c m = −2

18 a i 10−2

ii −2

b i 104

ii 4

19 5 cities

Answers mathspace.co

745

Animal

Weight (in kg)

Weight in log10 scale

Ferret

0.8

−0.10

Cat

3.2

0.51

Whale

32 840

4.52 3.44

Rhinoceros

2754.23

Arabian wolf

25.12

1.40

Hamster

0.15

−0.82

Shetland pony

127

2.10

21 a 16 to 32 years old

b 9 years old

22 a 4 × 10−5

b −5

b i

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

c i y = −x + 3

b 10

24 a 1000

5.1

times

25 20

Understanding 1 The x- and y-coordinates at the point of intersection satisfy both equations. 2 The lines would be parallel and never intersect. b 0

e 0

f 1

c 1

b (−6, 4)

e (6, −3)

f (−4, −2)

c (2, −4)

5 a x = 6, y = 4

d i

10 8 6 4 2

d Inﬁnite

x 2 4 6 8 10

x −10−8 −6 −4 −2 2 4 6 8 10 −2 −4x + 2y = −8 2x − 4y = −8 −4 −6 −8 −10

d (3, −6)

b x = 0, y = −2

c x = 4, y = 0 6 a i

y = −x − 9

ii (0, 3)

Fluency 4 a (5, −1)

x 2 4 6 8 10

−10−8 −6 −4 −2 −2 −4 −6 y=x+3 −8 −10

7.01 Solve graphically

3 a 1

y y=x+9

ii (−9, 0)

23 3

ii (4, 4) 10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

e i

y y = 4x − 4 y = 8 − 2x

x 2 4 6 8 10

ii (2, 4)

746

10 8 6 4 2

Mathspace WA Curriculum – Year 10 mathspace.co

y 10 8 y = 3x + 6 6 4 2 y = −x + 2 x −10−8 −6 −4 −2 2 4 6 8 10 −2 −4 −6 −8 −10

ii (−1, 3)

f i

j i

10 y = −x − 3 8 6 4 2

x 2 4 6 8 10

−10−8 −6 −4 −2 −2 y = −1 −4 −6 −8 −10

y 6

y = 4x − 3

−4 −2

2 −2

k i

y = 4 − 3x

y y=

3 x−1 2

−4

−6

−6 −4 −2 2 4 6 −2 y = 1 x − 2 2 −4

ii (1, 1) h i

−6

y 8

−8

y = 3x + 2

ii (-1, -2.5)

y = 7 − 2x

2 −4 −2

l i

x 2

−2

−4

y = 4 − 0.5x 10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10 y = 1.5x

−10−8 −6 −4 −2 −2

ii (1, 5)

ii (2, 3)

2 4 6 8 10

ii (4, 6)

i i

−10−8 −6 −4 −2 −2 −4 −6 −8 y = 2.5x − 4 −10

ii (−2, −1) g i

y 16 14 12 10 8 6 4 2

y = 12 − 1.5x

x 2 4 6 8 10

−4

−6 −8 x 2 4 6 8 10

ii (1, 2) 7 a

500 400 300 y = 50x + 250 200 100 −5 −4 −3 −2 −1 −100 −200 −300 −400 −500

y = 100x + 50 x 1 2 3 4 5

Answers mathspace.co

747

b (4, 450) c Renting either studio A or B for 4 hours costs the same amount, $450. 8 a

24 22 20 18 16 14 12 10 8 6 4 2

y y = 2x + 10

y = 4x

x 1 2 3 4 5 6 7 8 9 10 11

c No

d Yes

13 a No

b No

c Yes

d Yes

15 These equations actually represent the same line, just written in different forms. This means there are inﬁnite solutions. Every point on one line satisﬁes both equations.

c Non-members and members pay the same amount for 5 classes, $20. 9 a (775, 2325)

16 No, because there is an inﬁnite number of lines that go through that point. The gradient could be any real number.

b After making 775 items, the company’s revenue will equal their costs. This means they will not make a proﬁt and this is the break-even point. 1100 1000 900 800 700 600 500 400 300 200 100

17 a To intersect at (0, 3), the line must have a y-intercept of 3. Therefore, c = 3.

Dollars

b The value of m ≠ 5. If m = 5 and c = 3, then the second line is the same as the ﬁrst and there is an inﬁnite number of solutions. C

18 No, the lines have the same gradient, so they are parallel and will never intersect.

19 Non-integer solutions represent fractional quantities or parts of a product, which may not be practical in real-life scenarios. In such cases, the company might need to round to the nearest whole numbers. This may not result in the ideal way of distributing the goods, but it ensures that the goods are distributed in whole units, which is more practical.

100 150 200 250

b (200, 800) c This is the break-even point. For 200 toys, both the cost and revenue are $800. 30000 27000 24000 21000 18000 15000 12000 9000 6000 3000 0

Problem-solving 20 a y = x + 3, 2x + 2y = 18

y = 10 000 + 772.60x

y = 1200x x 2 4 6 8 10 12 14 16 18 20 22 24 26 28

b (23.4, 28 076.7)

748

b No

14 No, it’s not possible. Simultaneous linear equations represent two lines. Two lines can either intersect at a single point (one solution), coincide resulting in inﬁnite solutions or be parallel (no solution). They can’t intersect at exactly two points.

b (5, 20)

11 a

12 a Yes

Reasoning

10 a

c After about 23 weeks, Laura will be spending more money than she has in her bank account.

Mathspace WA Curriculum – Year 10 mathspace.co

2x + 2y = 18 10 8 6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

y y=x+3

x 2 4 6 8 10

c 6 m and 3 m 21 a Let x = Mohamad’s age and y = Rochelle’s age Equation 1: y + x = 21

Since the two lines do not intersect in the first quadrant, they will never reach the same height on the same day.

Equation 2: y = 4x + 11 b

22 20 18 16 14 12 10 8 6 4 2 0

y = 4x + 11

7.02 Optional: Solve using substitution y + x = 21

Understanding 1 a Answer varies:

y 2 4 6 8 10 12 14 16 18 20 22

c Rochelle is 19 years old and Mohamad is 2 years old. 22 a Let x = the number of parts Elvia completes and y = the number of parts Kang completes. Equation 1: x + y = 9 Equation 2: y = 2x − 6 b

24 The point of intersection on the graph has negative x and y-values. In terms of the height of a plant over time, only positive values make sense.

10 y 9 8 7 6 5 4 3 2 1 −2 −1 −1 −2 −3 −4

Tomi will get y = 2(6 − y) − 3, which simplifies to y = 12 − 2y − 3, then finally 3y = 9. This approach is efficient as it results in a simple equation where we can solve for y directly. b Answer varies: Carla will get x + 2x − 3 = 6, which simplifies to 3x − 3 = 6, then finally 3x = 9. This approach is also efficient because it results in a simple equation where we can solve for x directly. c Answer varies: Regie will get 6 − x = 2x − 3, which simplifies to 3x = 9. This approach is efficient because it results in a simple equation where we can solve for x directly.

x+y=9 y = 2x − 6

d Answer varies: x

1 2 3 4 5 6 7 8 9 10

c Kang did 4 parts of the assignment and Elvia did 5 parts. 23 Answers will vary. For example: Let x be the number of days and y be the weight of the rice in kilograms. Maria has 15 kgs of rice and cooks 5 kgs of rice every 7 days as represented by the purple line in the graph. Arjun has 12 kgs of rice and cooks 1 kg of rice every 2 days as represented by the green line in the graph. The solution (14, 5) means that after 14 days, Maria and Arjun have the same amount of rice remaining.

Junard will get

, which simpliﬁes

to y + 3 + 2y = 12, then finally 3y = 9. This approach is slightly less efficient because the equation involves a fractional coefficient, which can be a bit harder to work with. Once the fraction is handled, the resulting equation is also simple. 2 Yes, it is a solution. 3 a True

b True

c False

d True

e False 4 No, it is not a solution. Fluency 5 a x = −8, y = 8

b x = −5, y = 3

c x = −8, y = −3

d x = −6, y = −3

e x = −3, y = 5

f x = 6, y = 4

g x = 1, y = 6

h x = 8, y = 4

Answers mathspace.co

749

i x = −5, y = −8

j x = 8, y = 1

l x = 12, y = 8

15 No. If we substitute one equation into the other, we do not end up with a true statement: 3(x + 5) = 3x + 5 3x + 15 = 3x + 5 15 ≠ 5

6 a b Inﬁnitely many solutions b x = −6, y = 5

Since this is always false, the two equations would not have any solutions.

c x = 6, y = −5

d x = 2, y = 3

16 a x = 4, y = 2

e f = 0.4, g = 1.2

f a = 1.5, b = 2.5

7 a x = 7, y = 4

b Yes c Yes, this is the only solution to the simultaneous equations. The solution (x = 4, y = 2) is the point of intersection for all three lines represented by these equations. As these are linear equations, and each pair of them intersects at this point, there can’t be another solution for all three equations simultaneously. This means the simultaneous equations has exactly one solution.

8 The son’s current age is 4 and the mother’s current age is 40. 9 a a = $1.68 and k = $2.24 b The cost of one apple is $1.68 and the cost of one kiwi fruit is $2.24. 10 a x = 250 000 and y = 810 000 b In 2008, there were 250 000 jobs created and in 2013, there were 810 000 jobs created.

d 4

11 a x = 8, y = 14

2 1

b In the group, there are 8 adults and 14 children.

−2 −1 −1

Reasoning

−2

12 If the process of substitution leads to a false statement, it indicates that the simultaneous equations have no solutions. In other words, the lines represented by the equations do not intersect at any point.

−4

x 1 2 3 4 5 6 7 8 9

−3

Problem-solving 17 a The fixed charge is $5 and the charge per km is $10.

13 If you get a true statement, it indicates that the simultaneous equations have inﬁnitely many solutions. This could mean that the equations represent the same line.

18 Each racket is $50 and each ball is $5

14 Example explanation:

19 35 sausages and 52 soft drinks.

In this case, it may be easier to make y the subject in the ﬁrst equation and then substitute it into the second equation. This would leave us with an equation in terms of x, so we could then solve for x. We would then substitute the value of x and solve for y using either equation. Finally, we could substitute the values of x and y into both original equations to check our answer. This would avoid fractions and unnecessary rearranging.

750

Mathspace WA Curriculum – Year 10 mathspace.co

b $255

20 a (0, 0), (−5, 25)

b (8, 24), (−3, 24)

c (2, 14)

d (−5, 3)

e (3, −2)

g (−1, 1), (2, −8)

h (1, 0),

7.03 Optional: Solve using elimination Understanding 1 C 2 Multiply the ﬁrst equation by 7 and the second by 5. 3 a True

b True

c False

b x = −3

Fluency 4 a 7

5 a Multiply the ﬁrst equation by 5 and the second equation by 7. b x = −1, y = −3

and y = −4

b i 2x + 3y = 5 5x + 4y = 2 ii x = −2 and y = 3 c i 4x + 3y = 20 and 8x − 3y = 4 ii x = 2, y = 4 d i 6x + 5y = 240 and x + 3y = 66 ii x = 30, y = 12 7 a

12 You would add the two equations. Because the y terms have opposite coeﬃcients, they would negate each other out, leaving you with an equation in x to solve. Once you’ve found the x-value, you can substitute it into either equation to find y. 13 If you add the two equations, both variables would be eliminated, resulting in a true statement 0 = 0. This means there are infinitely many solutions. Each original equation represents a line, and these lines coincide (i.e., are the same line), hence there are infinitely many solutions. 14 Answer may look like:

6 a i 3x + 11y = −48 and −6x + 21y = −76 ii

Reasoning

b x = 2, y = 8

If the coefficients of one variable are the same or opposite in both equations, it’s often easier to use the elimination method because you can add or subtract the equations to eliminate that variable right away. With substitution, you would first have to solve one equation for one variable, which could be more work. 15 a When Elias substituted y to solve for x, he only found one value for x because he forgot that −1 could also be a solution to x2 = 1. b (−1, −3)

c x = 2, y = −9

d x = 8, y = 3

Problem-solving

e x = 40, y = −2

f x = −8, y = −2

16 a

x + y = 30 6x − 3y = 90

8 a x = 4, y = 5

b x = 3, y = −1

c x = 3, y = 1

d x = 5, y = 15

e x = −3, y = 1

f x = 2, y = 2

17 x = 10, y = 7

g x = 5, y = 4

h x = 6, y = 4

18 x = 5, y = 5

9 a There are 17 four-passenger cabs and −2 six-passenger cabs. b The solution is not valid because it is impossible to have a negative value of a passenger cab. 10 a x = $2, y = $4 b One pen costs $2 and one ruler costs $4. 11 a R = 5, C = 4

b x = 20, y = 10

19 a 2016

b 68%

20 a (−1, 3) and (2, 6) b (−3, −4) and (2, 1) c (−1, 1) and (1, 1) d (−8, −6) and (8, −6) e (−4, 3) and (4, 3) f (−6, 3), (0, 3) and (−3, −6)

b Marjan purchased 5 rhododendrons and 4 chrysanthemums.

Answers mathspace.co

751

c Base is 6 m and the other side lengths are 8 m each.

Chapter 7 review 1 a i

7 a x = −10, y = 0

b Infinitely many solutions

c No solution

−6 −5 −4 −3 −2 −1 −1

−2 −3 −4

ii (−4, 3) b i

6 5 4 3 2 1

d f = 4, g = 9

8 a (−1, 7) and (−1, 1)

b (1, 6) and (−4, −14)

9 a x = −2, y = 4

c x = 6, y = −15

d x = −14, y = 8.5

10 a (4, 7) and (−3, 0)

b (−4, 8) and (4, 8)

8.01 Scale and perimeter Understanding x

−6 −5−4−3−2 −1 −1 −2 −3 −4 −5 −6

1 2 3 4 5 6

1 a True 2 a 90 m

4 a x = 12 m b x = 0, y = −3

c x = −4, y = −1

b After selling 1280 dresses, the boutique’s revenue from sales is equal to their expenses. This is their break-even point, implying any dresses sold beyond this quantity will lead to a proﬁt.

5 62.5 cm

7 a 68 cm

9 a 4

5 No, the lines have the same gradient, so they are parallel and will never intersect.

Reasoning

y x + 2y = 22

13 a 2.5 y=x+2

4 2 x

752

b 140 cm

12 The circle has been dilated with a scale factor of 3. This means that its perimeter (circumference) will increase by a factor of 3, and 3 × 25.12 = 75.36 m.

6 a y = x + 2, x + 2y = 22

d 17 m

10 130 m 11 9 km

b 356 mm c 116 mm

4 D

b x = 91 cm

6 147 m

3 a (1280, 2500)

b 13.5 m

Fluency

2 a x = −5, y = 2

c False

3 0.03

ii (2, −1)

b True

Mathspace WA Curriculum – Year 10 mathspace.co

b 28 m

c 70 m

d The perimeter scales linearly with the scale factor. Since the length scale factor is 2.5, the perimeter of the larger garden plot is 2.5 times the perimeter of the smaller garden plot.

14 The enlarged pentagon has a perimeter that is k times the perimeter of the original pentagon.

13 a 4

Problem-solving 15 83.6 m 16 30 m 17 a 20

b 5m

18 a 19 m

b 12 m, 28 m, and 36 m

c 76 m

d 4

e k 19 a 3

b 240 cm

20 a 60 cm

b 360 m

21 a 18y m

b 6y2 m

22 a k = 13

15 334.89 m2

b False

b 10 800 cm2

16 a 12 17 a 76 cm2

b 6 cm, 12 cm and 15 cm c 684 cm2 d 9 e k2

c True b 1 cm

3 0.04

18 a 4

b 3584 cm2

19 a 2.5 cm

b 2 519 424 L

20 48 400 km2 21 a 4x2 cm2 b x4 cm2

c 1 cm2

22 k = 12

Fluency 4 a x = 20

14 18 m2

b l = 11 m

Understanding 1 a True

c 24 L

Problem-solving

c 6m

8.02 Scale and area

2 a 60 cm

b 16

d The amount of paint needed is not simply four times the amount for the ﬁrst ceiling because the paint covers area, not length. The area scale factor is the square of the length scale factor, so the amount of paint needed is sixteen times the amount for the ﬁrst ceiling.

8.03 Scale and volume b x = 567

c x = 30 5 72 cm2 6 3136 cm2

Understanding 1 a False

b The larger triangle’s perimeter is 5 times the smaller triangle’s perimeter. 9 324 cm2 10 2 m

c 512 cm

c False 3

d Multiplies it by 8

3 1.5 = 2.25 more times Fluency 4 a x = 1280

b x=4

c x = 36 000

5 a 4

b 64

c 54 528 cm3

6 a 4

b 64 3

Reasoning

7 1080 cm

11 The enlarged rectangle has an area that is k2 times the area of the original rectangle.

8 5

12 The circle has been dilated with a scale factor of 2. This means that its area will increase by a factor of 22 = 4, and 4 × 78.5 = 314 cm2.

b 64 cm

2 a 8 cm

7 84 cm 8 a 5

b False

9 x = 8.2 10 11

Answers mathspace.co

753

Reasoning 12 a The surface area will increase by factor of 32 = 9. b The volume will increase by factor of 33 = 27. 13 Capacity is the amount that an object can hold, given in litres in this case. The required volume (which is the space the esky takes up) of the new esky is eight times the old esky. This means that the required dimensions of the times the old dimensions. new esky are If Basili buys an esky with dimensions 8 times his old one, he will end up with an esky with a capacity of 20 × 83 = 10 240 L, which is much larger than he needs. 14 a Since all sides of a cube are the same length, the side lengths of two different cubes are always in the same ratio. All the faces are also squares, and so all adjacent sides are perpendicular to each other. This means any cube is similar to another cube. b

c The perimeter increased by the same factor as the scale factor 1.5, because perimeter is a linear measurement. 5 a y = 50 6 128 cm 7 3.5 m 8 The area of the dilated triangle is p2 times the area of the original triangle. 9 0.75 m2 10 a 9y2 cm2

b y4 cm2

11 a x = 116 12 a 4

c 1 cm2

b x = 72 c 16 000 cm3

b 64

13 x = 3 cm 14 Capacity refers to the volume an object can hold (expressed in litres or similar units). The desired volume of the new ﬁsh tank is 8 times smaller than that of the old tank. This implies that the required dimensions of the new tank are

= 0.5, which is half the dimensions of

the old one. This means the manager’s estimation is correct.

Problem-solving 15 a 1.4

b 1.96

16 a

c 2.744

d 0.343

15 a Yes because all corresponding sides are in the same ratio:

17 a 1 : 4

b 1 : 64

c m3 : n3

18 a 64

b 4

c h = 80

19 a 2

b 4

c 8

20 a 1 : 2

b 1:8

c 8

21 a 20 c 0.6429 cm3 22 a 1.5

d 16

b 7.6 cm d 5143 cm3 b 2.25 times greater

Chapter 8 review 1 a 0.02

b 40 m

2 a 20 cm long and 9 cm wide b 58 cm 3 a 54 cm

b 1.5 cm

4 a 28 cm

b 42 cm

754

b y = 812.5

Mathspace WA Curriculum – Year 10 mathspace.co

b 2.56

c 4.096

d 0.512

9.01 Pythagoras’ theorem Understanding 1 a z2 = x2 + y2

b x2 = z2 − y2

2 a x = 8.49

b x = 3.61

c x = 6.24

d x = 8.06

3 a 45

b 49

c No, not a right-angled triangle, 45 ≠ 49.

9.02 Trigonometric ratios

Fluency 4 a No

b No

c Yes

d Yes

e Yes

f No

g No

h Yes

5 a c=5

1 a AB

d c = 17

Understanding

2 a AC

b BC b BC

c BC

3 a i

iii

b i

iii

c i

iii

d i

iii

e i

iii

f i

iii

d AC

e ∠ACB

g Hypotenuse: h Side: i Side: 6 10 m 7 a 14.1 cm

b 11.2 mm

c 29 m

d 13.6 cm

8 a 10.8 cm

b 19.2 mm

9 132 m Reasoning 10 The triangle can be a right-angled triangle since 32 + 52 = 72 (9 + 25 = 34 ≠ 49). Hence, by the Pythagoras’ theorem, it’s not a right-angled triangle. 11

4 a i

b i

c i

102 = 82 + BD2

∴ BD = 6 cm 102 = 82 + AC2

Fluency

∴ AC = 6 cm ∴ BD = DC = 6 cm, so D is the midpoint of BC. 12 a 152 = 225 2

5 a i

iii

b i

iii

9 + 12 = 225 ∴ 152 = 92 + 122 and Pythagoras’ theorem holds. b i Yes

ii No

iii Yes

iv No

Problem-solving 13 a 25 cm

6 b x=

14 h = 5.61 m

7 a f = 4.08

b f = 6.12

c g = 9.46

d a = 11.59

15 a x = 4.9 m

b x = 1.8 cm

8 b=8

16 a $210

b $420

9 d = 3084

17 205 cm

10 29.0°

18 Beth, by 69 m

11 a i x = 10

ii 53.1°

b i x=4

ii 53.1°

12 d = 1.48

Answers mathspace.co

755

2 a σ or ∠CBD

13 1.96 m

b θ or ∠ACB

c They are the same size.

14 h = 4.4 m

3 a α or ∠BCA

15 θ = 9.88° 16 a x = 155.43 m

b 425.86 m

17 a h = 21

b f = 22

b Draw a horizontal line from A to the left of A and the angle of depression is from the horizontal line to the line AC.

d x = 34

c α

c h = 154, x = 72 e x = 12

Fluency

Reasoning

4 a x = 37.23°

b y = 52.77°

18 An isosceles triangle can be split into two right-angled triangles. Therefore, the adjacent side can be seen as half the length of the base.

5 a y = 36.87°

b x = 53.13°

Using the formula:

7 x = 182

6 θ = 52.52°

8 d = 229 Simplify the numerator and make A the subject of the identity. Take the inverse cos of both sides.

9 a AB = 11.86 m 10 a AC = 9409 m

b AC = 16.24 m

c BC = 4.4 m b BC = 5808 m

c 3601 m

11 a d = 7.81 m

19 Using the formula:

c 66.08°

Multiply both sides of the equation by b, and make b the subject of the equation by dividing both sides by 0.3.

b h = 13.53 m d 69.72°

Reasoning 12 Viggo is incorrect as the angle of depression must be measured from the horizontal. The correct angle of depression is 23.9°. 13 Fiona is not correct as an angle of elevation is measured from the horizontal.

14 a The shadow gets longer as the angle of elevation gets smaller.

Problem-solving 20

b The length of the shadow is zero.

21 22 a 135°

b I = 135°

23 a 131.2 m

b 101.3°

24 31.8°

c tan 90° is undeﬁned. The sun directly overhead means the angle of elevation is 90° and it’s not possible to draw a triangle, therefore the shadow has no length. Problem-solving 15 a x = 455.84 m

b 425.84 m

9.03 Angles of elevation and depression

16 a h = 127.38

b 129.1 m

Understanding

17 d = 1022.58

1 It is an angle of elevation as it is the angle formed from the horizontal looking upwards.

756

Mathspace WA Curriculum – Year 10 mathspace.co

18 h = 17 m 19 d = 10.86

20 θ = 45.97 21 a 29.26°

5 8 cm b 18.76°

c 4.61°

d 100 cm

22 a 60 m

b 1601 m

23 a PC = 22.65 m

b PA = 18.29 m

c 96 cm

Chapter 9 review 1 a f = 2.77

b f = 8.91

2 a

b θ = 30°

3 a 50°

b x = 19.30 m

6 a 288 m2

b 451.26 m2

d 392 cm2

e 380 cm2

f 204 cm2

7 a i 5 cm

ii 228 cm2

b i 8 cm

ii 608 cm2

8 a 216.49 cm2 2

c 174.49 m 9 a

c h = 20.50 m

b 380 cm2 d 534 cm2 b

c 11.32 m

4 a h = 20

b x = 26

10 a i No

5 θ = 54.69°

ii No

iii Yes

iv No

b 262 square units 11 a 23 184.95 cm2

b 452.39 cm2

c 1570 m

c 207.35 m2

d 439.82 m2

7 θ = 28.01°

12 a 75.40 m2

b 75.40 m2 b 50.27 m2

6 a AC = 12 032 m

b BC = 10 462 m

8 a h = 92.97 m

b 94.6 m

13 a 12.5664 m

9 a x = 13.13

b a = 6.85

14 h = 69

10 a B = 32°

b ∠B = 74° or ∠B = 106°

15 a i x = 11.00 cm

ii 376.95 cm2

b i z = 4.90 m

ii 858.76 m2

11 ∠ABC

16 a 91 m

b 50 m

12 a = 19.71

17 a 312π cm2

b 12.98 cm2

13 ∠C = 115°

18 596 cm2

14 a 5.94 m2

b 24.30 m2

19 103 m2

15 a x = 51°

b A = 33.2 m2

20 11 998 cm2

16 1389.32 m2

21 109.78 m2

10.01 Surface area of prisms and cylinders

22 158.4 m2 Reasoning

Understanding 1 a 6 faces

23 Example answer: 2

b x cm

c SA = 6x

2 7 faces 3 a 6 squares b 2 circles and a rectangle c 4 triangles and a square d 6 rectangles and two hexagons Fluency

The formula depends on how many circular faces the cylinder has as each circular face has an area of π r2. A closed cylinder has two circular faces so the formula is SA = 2π r2 + 2π rh. A cylinder open at one end has one circular face and the formula is SA = π r2 + 2π rh. A cylinder open at both ends has no circular faces and the formula is SA = 2π rh.

4 150 cm2

Answers mathspace.co

757

24 They are both correct as the formulas are the same. Scott’s formula is the factorised version of Lachlan’s formula.

35 Number of cans = 9

25 a Option 2 has a smaller surface area (219.5 cm2) than Option 1 (about 245 cm2) so Quicker Oats should choose Option 2 to minimise the amount of materials used.

10.02 Volume of prisms and cylinders

b Maximising profits means selling the smallest volume of chocolate chip oats per package. Option 2 has a smaller volume (185.25 cm3) than Option 1 (about 282.74 cm3) so to maximise profits, Quicker Oats should choose Option 2 to minimise the amount of product used. 26 a The surface area is made up of: • Two rectangles for the curved walls, both with heights of 3 cm and lengths equal to the circumference of circles of radii 2 cm and 10 cm respectively. • A small circular base with a radius of 2 cm that fits into the large circular base, making a complete circle with a radius of 10 cm. • A large circular base with a radius of 10 cm. b Surface Area = 854.51 cm2

36 212.4 cm2

Understanding 1 a True

b False

2 The volume of both a rectangular prism and a cylinder involves ﬁnding the base area and multiplying it by the height. However the methods of ﬁnding the base area, the interpretation of the height, and the visual representation of the volume differ between the two shapes due to their distinct geometric properties. 3 Options B and D Fluency 4 a i Triangle iii 90 cm

b i Trapezium 5 a 520 cm3

b 114 cm3

c 64 cm

d 231 cm3

e 840 cm3

f 90 cm3

28 2978.7 cm2

7 Height = 5 cm

, and

b Any dimensions which result in a total surface area between 325 cm2 and 335 cm2. For example, 11.65 cm × 9.71 cm × 2.43 cm. 30 20.00 m

31 a 11.13 m2 32 a h = 24 d 836 cm2 33 495.1 cm

b h = 21

c 886 cm

e Bag B

34 186.32 cm2

758

8 a 785.4 cm3 3

Mathspace WA Curriculum – Year 10 mathspace.co

b 1335.2 cm3

c 678.6 cm

d 226.2 m3

e 45.9 cm3

f 70.7 cm3 3

g 307.9 cm

h 955.0 cm3

9 a 1696.5 cm3

b 2309.1 cm3

c 59.7 cm

10 883.57 cm3 11 16 286.02 L

b 4 L costs $98

ii 99 cm2

iii 495 cm3

6 Base Area = 80 cm2

For example,

ii 9 cm2

Problem-solving

29 a Any scale factor s which satisﬁes the equation s2(350) = a, where a is the new surface area of the box and 325 ≤ a ≤ 335.

d True

e False

27 Surface Area = 4π r2

c True

12 864.19 cm3 Reasoning 13 a 1.782 m3 b 2 m3

c Oliver’s father’s rounded measurements give a volume that is 0.218 m3 larger. 14 Doubling the length and width of the right prism while keeping the height the same results in a four-fold increase in the volume. 15 a 37.31 m3 b 37.29 m3 c The exact value is larger by 0.02 m3. d In these real-world situations, using rounded or approximated values can lead to small discrepancies in calculations. In the case of the cylindrical container, it could mean not having enough water for the experiment if a large number of cylinders needed to be filled. It is essential to understand the potential impact of approximations and make decisions accordingly. e 37 310 000 mL Problem-solving 16 1 134 000 L 17 a Vidar should get the flat base type. b Yolanda should get the flat base type. 18 The second can with a diameter of 18 cm and a height of 16 cm. 19 a Morcom Aquatic Center b The pool at Morcom is 1.8 times the volume of the pool at Sanford Seminole. c 20 a 1335.18 cm3

b 4005.54 cm3

c 28 L

30 14 kL

10.03 Surface area and volume of composite solids Understanding 1 A composite solid is a 3D ﬁgure that is composed of simpler geometric ﬁgures. Examples could include a house shape (a rectangular prism with a triangular prism for the roof) or a cone with a hemisphere (like an ice cream cone) on top. 2 a To calculate the surface area of a composite solid, we need to calculate the area of all of the faces that are on the outside of the composite solid. b To calculate the volume of a composite solid, we need to calculate the volume of each of the solids that make up the composite solid and combine them appropriately. Sometimes this requires addition, sometimes subtraction. 3 a Two rectangular prisms b A trapezoidal prism and a triangular prism c A cylindrical hole in a rectangular prism 4 a A rectangular prism and a triangular prism. b Three cylinders, two of which are identical. c Two trapezoidal prisms. d A rectangular prism and a cylinder. Fluency 5 a 672 cm3

b 171 cm3 3

d 3136 cm3

c 1920 cm

6 a 1608.50 cm3

b 883.57 cm3

d 609.47 cm3

c 263.89 cm

21 37.5 L 22 a 35 343 L

b 35.343 kL

23 a 140 m2

b 8820 L

7 0.57 m3 8 a 1140 m3

b 424 m3 3

d 272 cm3

24 90 m

e 982.35 cm

f 601.65 m3

25 67.5 L

g 7164 units3

h 1800 cm3

c 1203.75 m

9 a 1512 cm2

26 4.35 m3 27 a 176 m2 28 a 2540 m 29 a 7771 m

c 228 cm2

b 3.52 m3

c 3520 L

b 406.4 m

d 365.45 cm2 2

e 6755.25 mm

d 880 L 2

b 429.37 m2

b 1631.91 m3

g 120 cm

f 867.08 cm2 h 621.24 cm2

10 157π cm2

Answers mathspace.co

759

Reasoning 11 The simple solids and their required dimensions to ﬁnd the volume are: • A rectangular prism for the base. We need to know the length, width, and height. • A cylinder for the pole. We need to know the radius or diameter of the pole, and its height.

b Calculate the surface area of the original cube, subtract the four rectangles which are the ends of the holes and add in the rectangular surfaces which are parts of the prism going through the shape. 6 × 142 − 4 × 5 × 6.5 + 4 × 14 × 6.5 + 4 × 14 × 5 = 1690 m2

• A rectangular prism for the backboard. We need to know the length, width, and height. One of these dimensions can be thought of as the thickness.

Problem-solving

• A cylinder that has had a cylinder cut out of it for the hoop. We need to know the inner and outer radii or diameters, and the vertical thickness of the hoop.

15 a 1658.76 cm2

14 a 1198.5 m3

b 299.625 m3

c h = 0.5875 m b 131.95 cm2

d 3298.67 cm2

c 1507.96 cm 16 13 815.88 cm2 17 a 68 m2

12 a i

b 70 m2

Volume = 300 cm

d 376 m2

c 100 m

e $16 544 18 a 16.3 cm2

19 a 36.47 m

Volume = 300 cm3

b 614

b 36 469 L

c 34 016 L

b Calculate the total surface area of the rectangular prism:

d 1717 L

e 21

2 × 7 × 6 + 2 × 6 × 8 + 2 × 7 × 8 = 292 cm2 Subtract the triangle at each end of the triangular hole:

20 a 127.1 m2

b 140 m2

c $2520

10.04 Estimation and accuracy Understanding

Add in the three rectangular faces that are part of the triangular prism going through the shape. Pythagoras’ theorem is needed to work out that the width of two of these rectangles is

1 a True

b False

c True

d False

2 6.35, 6.36, 6.37, 6.38, 6.39, 6.40, 6.41, 6.42, 6.43, 6.44 3 5.44 4 0.15

cm 2

5 12.715

= 361 cm (nearest whole number) 13 a To ﬁnd the volume of prism, use the formula: Volume of Prism = A × h, where A is the base area and h is the height. To ﬁnd the base area, subtract the area of the two rectangles from the square. Area of Rectangle = l × w Area of Square = side × side Volume = (14 × 14 − 2 × 6.5 × 5) × 14 m3 Volume = 1834.00 m3

760

Mathspace WA Curriculum – Year 10 mathspace.co

Fluency 6 a • Length of the rectangular prism: 7.5 cm • Height of the rectangular prism: 7.5 cm • Width of the rectangular prism: 5.5 cm • Diameter of the cylinder: 4.5 cm b • Length of the rectangular prism: 8.5 cm • Height of the rectangular prism: 8.5 cm • Width of the rectangular prism: 6.5 cm • Diameter of the cylinder: 5.5 cm

c Total surface area = 341 cm2

momentum is directly proportional to velocity, so errors in the velocity measurement have a less drastic impact on the momentum calculation.

d Total surface area = 461 cm2 7 a Total weight = 68.051 kg b Total truncated weight = 68.03 kg c Total rounded weight = 68.05 kg

Problem-solving

d The truncated value is slightly below the actual total, while the rounded value is very close to the actual weight.

13 $9 125 000

8 a 7.2%

b 14.6%

c 4.2%

d 5.3%

14 a 40.6 thousand years ago b 33.0 thousand years ago c 10.3%

Reasoning

15 a 48.15%

9 The company would overestimate the weight of the materials, which could lead to overpayment if the materials are bought or sold based on weight. They could reduce this error by recalibrating or replacing the weighing scale.

Chapter 10 review

10 a Different rounding and level of accuracies being used. b Caleb rounded his to the nearest metre, Ida used to the nearest tenth of a metre and Lavinia to the nearest hundredth of a metre. c Yes, they have all used different levels of accuracy. 11 Truncating the interest rate could lead to slightly lower total interest payments. Over a large number of customers, this could result in signiﬁcant savings for the bank. 12 a Alex will use: Mass: 0.37 kg (the closest value to Liam’s measurement that Alex’s scale can measure) Velocity: Using Liam’s measurements, the time taken is

, which Alex

will measure as 2 seconds, giving a velocity of

ii 0.92 Joules

b 216 cm2

d 608 cm2

c 612 cm 2 h = 71

3 Design B because it has a smaller surface area (494.12 cm2) than Design A (531.93 cm2) . 4 a x = 12.10 cm

b 537.21 cm2

5 a 490 cm3

b 200 cm3 3

ii 0.82 kg m/s

e 13.41% f T he percentage error in Alex’s kinetic energy calculation is higher than the percentage error in his momentum calculation. This is because kinetic energy is proportional to the square of the velocity, so errors in the velocity measurement are ampliﬁed in the kinetic energy calculation. In contrast,

d 140 cm3

c 2040 cm

6 Base Area = 110 cm2 7 a 552.92 cm3

b 2261.95 cm3

8 a 3.42 m3

b 0.684 m3

9 a i 864 cm3 b i 1526.81 cm

ii 648 cm2 3

ii 894.58 cm2

10 a 180 cm3

b 824.67 cm3

11 a 58 904.86 cm3

b 2120.58 cm2

c 2434.73 cm2

d 3063.05 cm2 b $1802

13 a Trina’s c 0.025 cm

c 26.09% d i 0.93 kg m/s

1 a 544 m2

12 a 488 m2

. b i 1.16 Joules

b 1.48%

b Kareem’s 3

14 By rounding down, the total calculated viewing time will be slightly less than the actual time spent by users. Over a vast number of users, this could result in a signiﬁcant discrepancy between the tracked and the actual viewing times. 15 The products may be overcooked if the timer is consistently slow, which can affect the texture and taste.

Answers mathspace.co

761

16 a 525 Myr

b 475 Myr

c 5%

d 2%

High emission

Urban

11.01 Review Venn diagrams and two-way tables

Understanding

Not microchipped

Microchipped

1 Probability of a storm occurring tomorrow. 2 • P (B)

Vaccinated

• P (A′)

Not vaccinated

• 1 − P (A)

12 a 120

3 P (A ∩ B)

13 a

4 49

Support

Do not support

Males

Females

5 a

L′

Total

C′

Total

b F

14 a P (S ∪ B)

c The probability of selecting a student with no sisters. 15 a b

16 a

Reasoning 17 We want either A or B to occur, but not both. This could happen in two ways:

c {Zebra, Tiger}

• Event A occurs and event B does not occur Fluency

• Event B occurs and event A does not occur

6 a B b i No

ii No

iii No

iv Yes

• Events A and B′ occur

7 a B b i No

ii No

8 a 0 9

762

We can rewrite an event not occurring as its complement.

iii No

iv No

• Events B and A′ occur A ∩ B represents event A and B occurring.

b 1

• A ∩ B′ occurs • B ∩ A′ occurs

Play Rugby League

Don’t play Rugby League

Play Rugby Union

185

P ((A ∩ B′) ∪ (B ∩ A′))

Don’t play Rugby Union

179

146

If we were doing the calculation, we could also do P (A ∪ B) − P (A ∩ B).

Mathspace WA Curriculum – Year 10 mathspace.co

Now, A ∪ B represents event A or B occurring.

18 a 58

23 a A Venn diagram or two-way table may be used. A Venn diagram is shown:

c The “number of favourable outcomes” is the number of tourists who speak only one of the languages.

Assembly line

45%

This is made up of tourists who only speak Mandarin or who only speak Spanish.

Quality control

20%

30% 5%

The “total number of outcomes” is the total number of tourists.

b i

The probability that a person speaks only

11.02 Conditional probability

one of the languages is

19 Since there are no outcomes where both A and B can occur at the same time, the events must be mutually exclusive. A Venn diagram that represents this situation is shown:

Understanding 1 a False

b True

c True

d True

2 3 A 4 a Patricia is chosen second, after Aaron has been chosen ﬁrst.

a No, there is no intersection between sets A and B. b Yes, there is no intersection between sets A and B. c Yes, the outcome is either in set A or set B, and there are no other possible outcomes. d No, the sets are not necessarily the same size. e Yes, if the outcome is not in set A then it is in set B. f No, set B is not an empty set.

b 37.7%

c 0.725

6 0.18 7 a

b 0.4

9 a 10 a i 260

22 a A Venn diagram or two-way table may be used. A two-way table is shown:

Children

5 0.8

21 The number of people in set T ∩ C is 36.

Fluency

8 a 0.4

Problem-solving 20 a 326

b Greater than

Pescatarian

Not pescatarian

Total

Adults

Total

b ii 237

iii 111

iv 208

b 11 a {0, 1, 2, 3, 4, 5}

c 12 a

13 0.6875 b

c 14 a

Answers mathspace.co

763

24 P (A∣B) is greater because:

15 a b Given that the ﬁrst die is a two, the possibilities are six: (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6).

and

But according to the size of the circles on the Venn diagram:

Out of them, there are only two possibilities with sum of 7 or more: (2, 5) and (2, 6). The answer is 16 a

17 a

c Problem-solving

18 a (0.4 × 0.8) + (0.6 × 0.7) = 0.74

25 a

Comedy News

Comedy

Horror Animation

19 a 0.75 × 0.05 = 0.0375

Comedy Comedy

b 0.25 × 0.95 = 0.2375 c 0.0375 + 0.2375 = 0.275

Horror Animation

Current Affairs

Comedy

Horror

Animation Comedy

20 a 177

Horror Animation

Animation

21 a 0.8 × 0.5 = 0.4 b (1 − 0.8) × (1 − 0.5) = 0.1 c 1 − 0.1 = 0.9

b 26 a

d (1 − 0.8) × 0.5 = 0.1

0.2

e P (A∣B) = P (A) = 0.8 (A and B are independent events)

0.7

0.8

Reasoning

0.85

0.3

0.15

Since P (B ∩ A) = P (A ∩ B), therefore P (A∣B) × P (B) = P (B∣A) × P (A) 23 a

e Since B is inside of A, if the event B has occurred, the event A must have occurred. So, P (A∣B) = 1.

764

Mathspace WA Curriculum – Year 10 mathspace.co

C T

b 27 a

28 a

9 a

Labor (next election)

Liberal (next election)

Greens (next election)

Total

Labor (last election)

Liberal (last election)

Greens (last election)

Total

140

200

29 a

Believe in Tooth Fairy

Do not believe in Tooth Fairy

Total

100

b 38.75%

10 a 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 b 1, 2, 3, 4, 6, 8, 12, 16 c 4 numbers from set A: 10, 14, 18, 20 and 2 numbers from set B: 1, 3. In total, 6 numbers are unique.

e H and S

Do not Believe believe in Total in Santa Santa

d 11 a

Watched Didn’t watch Total the movie the movie

11.03 Mixed probability concepts Understanding 1 a Diagram iv

b Diagram i

c Diagram ii

d Diagram iii

2 a P (A ∩ B) = P (A∣B) × P (B)

Read the book

Didn’t read the book

Total

104

166

b i

b P (A) + P (B) = 1 c P (A ∪ B) = P (A) + P (B) − P (A ∩ B) d P (A ∩ B) = P (A) × P (B)

12 a i x = 40

iii

ii y = 400

iii z = 200

b 360

3 B

Fluency

13 a

4 w = 0.3, x = 0.3, y = 0.2, z = 0.2

b 10

14 a

5 6 a {23, 24, 32, 34, 42, 43} b i

iii

c 15 a

7 a

8 a 3

B Y B

R Y B R

Answers mathspace.co

BY RB RR RY YB YR YY

765

b i

iii

v 0

vii

16 a

c Two events A and B are considered independent if P (A∣B) = P (A).

Using the fact that

, we

can rewrite this condition for independence as P (A ∩ B) = P (A) × P (B).

Serve in 0.55 Serve in 0.81

We calculated P (S ∩ H) and P (H) × P (S), and with these results we can conclude that these events are independent.

0.45 Fault

23 a 0.19 Fault

b 0.45

b The probability of picking an identical pair depends on how many cards are left. Since the number of cards decreases as pairs are picked, the probability of picking a pair is affected by how many cards have been removed.

c 0.0855

17 a Star Jet

Star Jet

Air Tiger

In this case, it is dependent. Star Jet Air Tiger

24 a No second attempt is needed if the exam is passed on the ﬁrst attempt.

Air Tiger

b c Candidates will have a variety of abilities and preparedness for the examination.

b i ii

Problem-solving

18 a

19 a

20 0.856

25 a

26 a 0.18

Reasoning

27 a 27

21 a P (A ∩ B) = P (A) × P (B)

c 0.38

b 43

d 0.64

b P (A ∩ B) = P (A∣B) × P (B)

28 a (0.55 × 0.02) + (0.45 × 0.08) = 0.047

c The formula P (A ∩ B) = P (A∣B) × P (B) can be used for both independent and dependent events.

The answer is 4.7%.

For dependent events, we have to use the formula from part (b). For independent events, P (A∣B) = P (A) since it doesn’t matter whether B occurred or not. So, we can use the formula from part (b) and still get the same answer as the formula from part (a). 22 a i

b 0.23 29 22.5% 30 10.58%

Chapter 11 review 1

iii

766

b 0.45

e 0.9

2 a

Mathspace WA Curriculum – Year 10 mathspace.co

Y′

82 12

X′

41 53 b Independent

3 a

4 a

12.01 Representations of data Understanding 1 4 and 7

5 0.2

2 a Black

b Blonde b 72

6 a

3 a 5

4 a 5 employees b 58 years

c 6 students

c 20 years

5 35 7 a 0.7

b 0.65

8 a

6 a By counting the number of dots above a particular outcome.

9 j = 0.4, k = 0.4, l = 0.1, m = 0.1

b By ﬁnding the heights of the bars using the frequency axis.

10 a

c By counting the number of leaves for a particular stem.

240

120

210

Fluency 7 a Ungrouped

8 a

Heart rate

Class centre

Frequency

45–54

49.5

b 11 a

55–64

59.5

65–74

69.5

75–84

79.5

Went hiking

Did not hike

Total

Went swimming

Did not swim

b 96 people

Total

100

c 51 people

b i

12 a

Egg sandwich 6 10

4 10

Muffin

b 0.3

iii 5 10

Orange juice

b Grouped

9 a

d 40 people

Score

Frequency

16 ≤ Age < 18

18 ≤ Age < 20

20 ≤ Age < 22

22 ≤ Age < 24

5 10 5 10

Iced coffee

24 ≤ Age < 26

Orange juice

26 ≤ Age < 28

5 10

Iced coffee

c 0.8

b 82 people

c 44 people

d 26 people 10 a 20

b 3

c 12

b 99

c 71

d 28

13 a

11 a 28

14 a

12 a 42 people

13 a Bird

b Cat

15 a

c 8 students

d 36 students

Answers mathspace.co

767

14 a Wednesday

b 60

c Tuesday 15 a 4.7

b $100 000

c 16

b i N o, the number of people who live less than 40 minutes away from work is greater than those who live more than 40 minutes away. ii Yes, since a higher proportion of people live within 40 minutes of their workplace.

d 75%

e 16

iii No, there are 10 people who travel an hour or more to work, even though it is true that most people live between 20 and 40 minutes away.

16 a Histogram, as this is continuous data with a large spread of values. b Stem and leaf plot, as we want to know individual data values across a medium range and small data set. c Dot plot, as there would be concentrated values in a very small range.

iv No, there is no indication about what mode of transportation the people surveyed use to travel to work. Problem-solving

Reasoning

20 a 180.5 hours

b $122.55

17 a 35

21 a 1.7

c 1

d 3.8%

Country

Oil Production (barrels per day)

log10 Oil Production

Saudi Arabia

10 625 000

7.0

Norway

1 763 000

6.2

Ecuador

555 000

5.7

Brunei

113 000

5.1

18 a Yes, the tallest person is Kate from the height graph and she is also the oldest person from the age graph.

France

61 000

4.8

Albania

21 000

4.3

Finland

13 700

4.1

b No, the heaviest person is John and he is not the oldest. The oldest is Kate.

Portugal

7100

3.9

Japan

4000

3.6

Costa Rica

300

2.5

b Between 60 mm and 80 mm

c No, the heaviest person is John and he is not the tallest. The tallest is Kate. d No, the youngest person is Tom and the lightest person is Kate. e Yes, based on the data, Kate who is 10 years old, is lighter than John who is 8 years old. John weighs 45 kg while Kate weighs 20 kg. 19 a

4 Frequency

c Greater than 60 mm in length because ‘more likely’ refers to how frequent a group of lengths is compared to the others. In the frequency table, we can see that there are 92 leaves greater than 60 mm in length that were collected compared to 35 leaves less than 60 mm in length that were collected.

b 0.01 kg

e 19

3 2 1

Frequency

0 15 10

3 4 5 6 7 8 Log Oil Production

12.02 Quartiles

Understanding 0

0 20 40 60 80 100 Commute time (minutes)

1 a 28

b 11.5

2 a Q1 = 1, Q3 = 9 c Q1 = 15, Q3 = 26.5

768

Mathspace WA Curriculum – Year 10 mathspace.co

c 49

d 7

b Q1 = 24, Q3 = 32 d Q1 = 16, Q3 = 24.5

3 10.5

h i 15.8, 19.5, 22.4, 29.6, 35.4, 39.1, 46.2

4 a iv: Minimum: Lowest value b v: Lower quartile, Q1: At most 25% of the data is below this value c ii: Median: 50% of the data lies on either side of this value d iii: Upper quartile, Q3: At most 25% of the data is above this value e i: Maximum: Highest value Fluency 5 a 74

b 61

c 114

d 53

6 a 72

b 51.5

c 85.5

d 34

7 a i 6, 10, 15, 17, 24, 39, 40 ii 7 iv 10

iii 17

ii 7

iii 29.6

iv 19.5

v 39.1

vi 19.6 8 a 18.93 kg b i 19 kg

ii 17 kg

iii 21 kg

9 a 51

b 7

c 1

10 a 47

b 7

c 4

c 2

11 a 5

b 16 b 26.5

c 7

13 a 72 hours

b 85 hours

c 13

14 a

Score (x)

Freq ( f )

v 39

b i −6, −4, −1, 7, 7, 9, 9 iv −4

v 9

Totals

vi 13 c i 4, 8, 10, 14, 15, 19, 20 iii 14

iv 8

v 19

vi 11 d i 20, 22, 28, 32, 40, 42, 43, 54 ii 8

iii 36

iv 25

v 42.5

vi 17.5 e i 69, 71, 78, 79, 82, 84, 85, 88 ii 8

iii 80.5

iv 74.5

v 84.5

vi 10 f i 100, 102, 110, 113, 115 ii 5

iii 110

iv 101

v 114

vi 13 g i 197, 198, 202, 205, 207, 228 ii 6

iii 203.5

iv 198

v 207

vi 9

6 iii 7

ii 7

d 17

12 a 19.5

vi 29 ii 7

c 11

b 7

c 6

d 8

e 2

15 a 1

b 3

c 2

16 a 75

b 8

c 3

17 a 79

b 64

c 84

d 20%

18 a 74

b 79.5

c 24

d 35

e 16.5

f 19.75

e 20%

g The Emerald Eagles’ scores are approximately symmetrically distributed around the median, while the Sapphire Sharks’ scores show a slight right skew, indicating the majority of scores on the lower side and some higher scores pulling the mean upwards. Reasoning 19 a 8 marks

b 9 marks

c 1

d In this data set, the scores less than 7, while constituting the minimum values, don’t signiﬁcantly impact the mean and median due to their low frequency. However, they do increase the range and could potentially affect the ﬁrst quartile by shifting it lower.

Answers mathspace.co

769

20 a Set 2 has the largest interquartile range. This indicates that the middle 50% of data in Set 2 is spread over a larger range of values compared to the other two data sets. b Set 1 appears to be symmetric as the differences between corresponding quartiles are nearly equal, indicating a symmetrical distribution. Set 2 is skewed right because the difference between the third quartile and the maximum is much larger than the difference between the minimum and the ﬁrst quartile, indicating a few higher values causing a right skew. Set 3 is skewed left as the difference between the minimum and the ﬁrst quartile is much larger than the difference between the third quartile and the maximum, indicating a few lower values causing a left skew. 21 a For Group 1, the range is 39, the interquartile range is 23 (116.5 − 93.5), the median is 99 and the mean is approximately 104.24. For Group 2, the range is 30, the interquartile range is 18 (105 − 87), the median is 100 and the mean is approximately 97.48.

consider the dataset: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The lower quartile is 3 (25th percentile) and the upper quartile is 8 (75th percentile). The data values between Q1 and Q3 are 4, 5, 6 and 7, which represent 50% of the total dataset. Problem-solving 24 65 minutes. This means that about half of

25 a

the ﬂights had a delay less than

minutes minutes.

and half had a delay of more than b 10 minutes

c 45 minutes

d 35

e 83

f 83%

g 22

h 22% 26 a 142 minutes

b 27

c 137 minutes

12.03 Boxplots Understanding 1 a

b The mean of Group 1 is higher than that of Group 2, which suggests that Group 1, on average, scored higher on the EQ test. c The spread of Group 1 is larger than that of Group 2, indicating that there is a greater variability in EQ scores within Group 1 than within Group 2. 22 a 2.5

b 8

c There are only 2 students who have more than 8 pets (the upper quartile), which is equal to 8% of the total number of students (2 out of 25). It is not exactly 25% because the quartiles are determined by dividing the data into 4 equal parts, but this does not guarantee that the percentage of values above or below a quartile will always be 25%. 23 Yes, the percentage of data values between the upper and lower quartiles is always exactly 50%. This is because the lower quartile (Q1) represents the 25th percentile of the data and the upper quartile (Q3) represents the 75th percentile of the data. The difference between the 75th and 25th percentiles is 50% of the data. For example,

770

Mathspace WA Curriculum – Year 10 mathspace.co

2 a i 19

ii 11

b i 34

ii 15

3 a True

b False

e True

f True

4 a i 50%

ii 25%

c True

d False

iii 50%

iv 75%

c 15

d 10

Data

v 75% b Second c First and third Fluency 5 a 3

b 18

e 7 6 a

Minimum

Lower quartile

Median

20 30 40 50 60 70 80 90 100

8 a 2

b 40

d 38

e 6

9 a i

c 2, 3, 40

Minimum

Lower quartile

Upper quartile

Median

Maximum

Upper quartile

Minimum

Maximum

Lower quartile

Median

Upper quartile

Maximum

Minimum

Lower quartile

Median

Upper quartile

Maximum

Minimum

Lower quartile

Data

20 22 24 26 28 30

iii Left skewed b i

Minimum

Lower quartile

Median

51.5

Median

31.5

Upper quartile

36.5

Maximum

7 a

Data

20 30 40 50 60 70 80 90 100

Data

20 22 24 26 28 30 32 34 36 38 40

iii Symmetrical

Answers mathspace.co

771

c i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

40 42 44 46 48 50 52 54 56 58 60

iii Somewhat symmetrical, with slight left skew f i

iii Somewhat symmetrical, with slight right skew d i

Minimum

Median

Upper quartile

Maximum

60 67

Median

Upper quartile

Maximum

Lower quartile

Minimum Lower quartile

Data

iii Symmetrical g i

iii Somewhat symmetrical, with slight right skew e i

Minimum

Lower quartile

Median

50.5

Upper quartile

Maximum

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

iii Somewhat symmetrical, with slight right skew

772

Mathspace WA Curriculum – Year 10 mathspace.co

h i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

iii Symmetrical 10 a The cyclist’s heart rates varied by 36 beats per minutes (bpm). 20

iii Somewhat symmetrical, with slight right skew i i

Minimum

Lower quartile

Median

Upper quartile

Maximum

b The middle half of the cyclist’s heart rates varied by 19 beats per minute. 11 a i D ata set A - skewed left, data set B - symmetrical ii Data set B

b i D ata set A - skewed right, data set B - skewed left ii Data set B

iii Data set A

12 a

Class 10P

Class 10Q

Median

Lower quartile

Upper quartile

Data

Range

Interquartile range

Outliers

None

b Class 10P 18

iii Right skew j i

iii Data set B

c Interquartile range

13 a 31 years c 25%

Minimum

14 a 49

Lower quartile

e Carl

b 17 years d 7

e 1st

f 5

b 50

c 12

d Carl

Median

15.5

Upper quartile

Reasoning

Maximum

15 The typical pulse rates increases from about 68 to about 108 and are spread out over a larger range after exercise. 16 Answers may vary. One possible explanation: A boxplot visually displays the median, providing a measure of the central value of the data set. The interquartile range, represented by the box, indicates the variability within the central half of the data, offering insight into the

Answers mathspace.co

773

spread of data. Lastly, the ‘whiskers’ of the boxplot highlight the overall range of data, contributing to our understanding of the data’s distribution. 17 a

• We can expect the Delta Dragons to win since in about 25% of games they scored more points than the highest number of points scored by the Gamma Geckos. 19 a Negatively skewed

Scores

b Positively skewed c Yes. The group that received treatement had a lower median and most people had symptoms for less than 9 days, whereas the control group had half the people having symptoms for more than 9 days. 50

100

Scores

20 a Yes, because correcting the error will not affect the positions of the Q1, Q2, and Q3 points. As the data point 23 is less than the maximum but greater than the Q3 division then the boxplot will remain unchanged. b This changes the range from 21 to 14, a decrease by 7. c Because the positions of Q1 and Q3 were unchanged, there is no impact on the IQR.

100

c Science. The highest score is the maximum value of the data. Science has a highest score of 91, while music has a highest score of 88. Science has a minimum score of 52, while music has a minimum score of 43. Science has a median on 74 and Music has a median score of 66. 18 a The median score of the Gamma Geckos is 5 points higher than the median of the Delta Dragons. b The range of the Delta Dragons is 12 points wider than the Gamma Geckos.

21 Answers should specify the expected minimums, maximums, or differences to justify the context. a For example, the price of items at the discount store. Most items will be $1, but some discount stores have items that also cost up to $5 giving them a range of $4. b For example, the ages of every person in a kindergarten classroom. Most of the people are the kindergarteners whose ages are close together giving an interquartile range of 1–2 years old, but the teacher is also in the room so the range could be anywhere from 15–70 depending on how old the teacher is. 22 a

c The interquartile range of the Delta Dragons is 19 points wider than the interquartile range of the Gamma Geckos. d Answers can support either team, but should use summary statistics to justify the response. For example: • We can expect the Gamma Geckos to win since the median number of points scored is 5 points higher than the median number of points scored by the Delta Dragons.

b 20 c 25% d 52 and 85

774

Mathspace WA Curriculum – Year 10 mathspace.co

95 100

e There are now 17 scores, so there are 13

26 a Any ﬁve-number summary with an interquartile range of 2 and a range of 8. For example:

= 76.5%. At least

values below Q3 and

75% of the data lies below Q3. Problem-solving

• Minimum: 1

• Lower quartile: 2

• Median: 3

• Upper quartile: 4

• Maximum: 9

This has the largest possible range of 8 and the smallest possible interquartile range of 2. b Any ﬁve-number summary where the range is three times the interquartile range. For example:

• Lower quartile: 2 • Upper quartile: 4

• Maximum: 7

This has a range of 6 and an interquartile range of 2.

24 • Boxplot 1/Histogram A • Boxplot 2/Histogram B

12.04 Compare data sets

• Boxplot 3/Histogram D • Boxplot 4/Histogram C 25 a

• Minimum: 1 • Median: 3

Understanding

Monthly salaries (in thousand dollars)

1 a The histogram

b The boxplot

c The boxplot

d The histogram

2 a Negatively skewed

b Bimodal

c Symmetrical 15

3 a • Frequencies for each group • Mode

Scores of students

• Shape • Outliers b • Quartiles 60

100

• Median • Shape

• Outliers

Debbie’s blog visitors

• Range and interquartile range c • Frequencies for each group • Mode 0

100

150

200

• Shape • Outliers

Number of pages read

Fluency 4 a 5

English Mathematics Median

Lower quartile

Upper quartile

Range

Interquartile range

Answers mathspace.co

775

b Mathematics

13 a

5 a Upmarket restaurant

b $22

c Upmarket restaurant

d $36

e 26 items 6 a Weightlifter A b The range and interquartile range c The median

65 000 70 000 75 000 80 000 85 000 90 000 95 000 100 000 105 000

d Weightlifter B

b 20 000

7 a Lower

b 0

c 50%

c 2

d Positively skewed

e 4

f Block B

d From the boxplot it appears the highest 25% all earn $100 000. However, this is using class centres, from the data provided we can only conclude that the top 25% earn in the range $95 000 to $105 000.

g No 8 • Histogram A/Boxplot 2 • Histogram B/Boxplot 4 • Histogram C/Boxplot 1

Problem-solving

• Histogram D/Boxplot 3 9 a Matt

b Matt

14 a

Cooper

c Matt

Reasoning 10 a 5.2

b 8.9

c 3

d 3

e No. The results are mostly 4 or 5 out of 10 and the mean is only 5.2 f Y es. The results are mostly 9 and 10 out of 10, and the mean is 8.9

b $10 c i True

ii True

12 a The boxplot has a long tail to the right which indicates positive skew, while the histogram does not appear to be skewed. The median for the histogram is roughly in the middle, while the median of the boxplot is located further to the left. b The boxplot indicates that the data is widely spread, while the histogram indicates that most of the data occurs near the median.

776

Mathspace WA Curriculum – Year 10 mathspace.co

44.8

47.1

49.1

54.7

Median

52.5

60.45

55.75

65.25

Maximum

66.6

68.5

b Yes, Cooper has an outlier of 66.6. c

g Answers may vary. One possible answer is that one class may have covered more material than the other class. 11 a Bookstore A, it has a lower range and interquartile range.

Marion

Minimum

Cooper Marion

55 60 50 Time (seconds)

d Cooper e No, a high outlier does not change what the minimum score is, hence would not alter the selection for pole position. 15 a Minimum

Tim

Odi

29.8

26.0

33.2

34.8

Median

35.0

38.4

37.4

40.0

Maximum

39.6

42.8

Mean (1 d.p.)

35.1

37.2

Range

9.8

16.8

Interquartile range

4.2

5.2

b Tim throws more consistently, his range, interquartile range and standard deviation are all signiﬁcantly smaller. c All of Tim’s scores are consistent, so no outliers for his data. Odi has an outlier of 26, which is far from the rest of his data set. d Odi appears to be the better hammer thrower. His mean is higher and other than the outlier the other values of the 5 number summary are all larger for Odi. In particular, his top 25% of throws are larger than any throw from Tim. Odi’s median is also signiﬁcantly higher (3.4 m) than Tim’s. e As the next lowest throw is more than 6 m larger than his outlier it appears this throw may have been a misadventure and it would be reasonable to remove it when looking at his mean. Including the outlier would distort how high he usually throws and what he would likely achieve in a competition.

1 a False

b False

e False

f False

c True

d True

2 a No

b No

c Yes

d Yes

3 a No

b Yes

c Yes

d Yes

e Yes

f No

4 a The population is all the residents of the town, and the sample is the 500 residents surveyed. b The population is all sales at the bookstore, and the sample is the 20 receipts she looks at. c The population is all students at Antony’s school. The sample is the 85 students he surveyed. b Yes

c Yes

d No

b No

c No

d Yes

e No e Yes

d Yes

9 a Example answers: • All the seniors • The ﬁrst 30 students to arrive at school that day • All the students that were in the library at lunch time b Example answers: • All students of a primary school in Sydney • All students in Year 7 in a particular suburb • 100 children chosen from the Sydney under 15s baseball tournament c Example answers: • The members of 3 teams chosen from the local hockey tournament • All students from a local school who compete in a school sports competition • All the active members of a local football club

• All the students from a local school who own a dog 10 Fair 11 a Biased

b Biased

c Not biased

d Not biased

12 a All products in the factory

b 30%

c 61%

d No

13 a All apples in the orchard

b 62%

c Yes 14 a Yes

b No

c No

d No

e Yes 15 Approximately 610 bees 16 1 089 600 17 a Source A

b Source A

c No

Reasoning 18 Jenny’s claim is more valid since she has surveyed more people.

Fluency 7 a ii

c No

• 30 people walking their dog at the local park

Understanding

6 a No

b Yes

d Example answers:

12.05 Samples and populations

5 a Yes

8 a Yes

b iii

c i

d iv

Answers mathspace.co

777

19 This is a very convenient sample so would be very inexpensive and not take much time. However, a disadvantage is that there’s a chance that someone at the dog park doesn’t have a pet dog such as a professional dog walker or a friend of a dog owner. There is also the possibility that someone with accessiblity needs would not be able to access the park so would be excluded. 20 a The sample is not representative of the target population. b The sample is from self-selecting participants, i.e. only those that made an effort to respond. c The sample is not representative of the target population. 21 a Example answer: Assign a number to everyone on the electoral role and use a random number generator to decide who to call. b The sample will not include anyone who was out for the evening for work or social reasons, and the results are based on less than half of the people they tried to contact, so the sample is not representative of the populations. 22 a The proportion from the second day, 12% is twice the proportion from the third day, 6%. b The differences in sample proportions are due to the varying sample sizes (750 dingoes in the first sample and 100 in the second) and natural random variation. A larger sample size generally yields more representative results, and the uneven distribution of tagged dingoes across the area can lead to different proportions in each sample due to chance and spatial distribution factors. c The sample statistic from the second day is more accurate because sample size is larger, and thus more likely to be representative of the population of dingoes. 23 a The sample proportions are different due to sampling variability. It is impossible for every sample to have the exact proportions as the population. b Both samples are valid, so they could use either one. Often we will take the average of many sample proportions to get a more accurate estimate of the population parameter.

778

Mathspace WA Curriculum – Year 10 mathspace.co

Problem-solving 24 a Example answers: • Check additional reputable secondary sources to see which information is corroborated by other authors or creators. • Look at the sources or references cited by each source. A well-cited claim with primary sources to back it up is generally more reliable. • Check the date of publication. Newer sources, like the 2015 documentary, might have had access to more recent research or previously unavailable information. However, it’s essential to note that newer doesn’t always mean more accurate. • Evaluate the intent and potential bias of each source. For instance, a textbook might aim for neutrality and comprehensive coverage, whereas a documentary might focus on specific stories or narratives. b Example answers: • Different sources may offer various perspectives or focus on different aspects of an event, giving a more rounded understanding. • Sources can sometimes contain errors, biases, or omissions, and crossreferencing multiple sources can help identify and rectify these issues. • Diverse sources can provide a richer context, helping one understand the broader implications and nuances of a historical event. • Relying on a single source may lead to a limited or skewed understanding, while multiple sources can fill gaps and provide a more in-depth exploration of a topic. 25 a Yes b The sample of fish were collected in several different places and one at a time, without large nets. 26 a Samples should focus on ways to reach only athletes without including non-athletes. For example, the sample could be all starting players from both teams or all of the players on the bench for one of the teams.

b Samples that include non-athletes. For example, the ﬁrst 50 spectators would not necessarily be athletes, and a sample that includes all of the people in the stadium, including coaches, oﬃcials, and fans, would have a mixture of athletes and non-athletes. 27 a If 10% of people experience side effects, we might expect about 500 000 residents in Sydney to experience side effects with the medication. b One possible answer: The way we share this prediction is important because causing unnecessary panic could lead to people avoiding the medication, but we also need to be honest about potential risks. c One possible answer: While 1000 people is a decent sample size, it may not be truly representative of Sydney’s diverse population of 5 million, so we should be cautious about making wide-reaching assumptions.

2 a 29

b Ungrouped b 83

3 a

Minimum

c 45

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

b 16

60 65 70 75 80 85 90

c Somewhat symmetrical, with a slight left skew. 8 a The number of books read by students varied by 42 books. b 25%

c 13

d 4th quartile

e 10

9 a The median of the control group is 5 weeks higher than the median of the treatment group. b The treatment group’s range is 3 weeks wider than the control group’s range.

d Yes. The group that received the experimental fertiliser matured faster since it had a lower median, and most plants reached maturity in less than 12 weeks. In the control group, half of the plants took more than 13 weeks. 10 a 290 11 a

c 8

b Based on the ﬁve-number summaries, Set A appears to be symmetric, Set B is skewed right, and Set C is skewed left.

b 60

c 31.0%

Number of attendees

Number of workshops (f )

Cumulative frequency (cf )

d 3

5 a This indicates that the middle 50% of ratings in Set B are spread over a larger range of values compared to the other two data sets.

6 70 seconds

b $610.00 4 a 101

Lower Upper Median Maximum quartile quartile

c The treatment group’s interquartile range is 1 week wider than the control group’s range.

Chapter 12 review 1 a Grouped

7 a

b 8 workshops

c 10 workshops

d 22 attendees 12 a 49

b 45

c 55

13 A 14 a Building X c Building Y

b 2 plants d Yes, Building Y

Answers mathspace.co

779

15 a

Antoine

Benji

Fluency 5 a

Minimum

6.5

6.0

6.9

6.8

Median

7.3

7.4

Teenagers

169

Sprinkles Hot fudge

Total

Running Basketball shoes slides

7.5

7.9

Adults

Maximum

7.8

8.1

Total

Mean (1 d.p.)

7.2

7.3

Range

1.3

2.1

Interquartile range

0.6

1.1

d No, Antoine’s range and interquartile range are still smaller than Benji’s, even after removing the outlier. 16 a All cherries in the farm.

b 58.5%

c Yes

d 30 420

17 a Biased

b Biased

Vanilla

Total

b The sample statistic from the second week is more accurate because sample size is larger, and thus more likely to be representative of the population of beetles.

13.01 Two-way tables Understanding

Fish

Total

Adults

Total

b This value represents the number of cats with feline diabetes who received a special diet at Mohawk Animal Clinic. Vegetarian

Vegan

Neither

Children

Adults

Total

b 8 a 39

Total

b 26

c 47

9 a 11% of junior students and 10% of senior students were very satisﬁed with the canteen food. b The most common rating among junior students is 4, while for senior students it is 3.

b 16

c Outback Animal Care Centre 2 a i Marginal frequency

ii 451

b i Marginal frequency

ii 171

c i Joint frequency

ii 143

d i Joint frequency

ii 388

3 a False

b False

c True

d False

4 a True

b False

c True

d True

e True

c The data suggests that satisfaction with canteen food tends to decrease as students progress from junior to senior years, indicating potential issues with food quality or variety for older students. 10 a 14 trains c 127 trains 11 a 27.63% c

780

Chicken

6 a This value represents the number of cats with feline diabetes treated at Venetian Pet Clinic.

18 a The sample proportions are different because the researchers used samples of very different sizes.

1 a 14

Beef Children

7 a

c Not biased

Total

Chocolate

b Antoine c Given that Benji’s lowest jump is signiﬁcantly shorter than his next lowest jump, it might be considered an off-performance. It would be reasonable to exclude this outlier to get a more representative average of his usual performance in competitions.

Work boots

Mathspace WA Curriculum – Year 10 mathspace.co

b 51 trains d b 44.47% d 34.61%

Reasoning 12 a

Problem-solving Brushtail possum

Honey possum

Fruit

Insects

120

Total

Nuts

110

Total

130

180

310

15 a

No Casual Part-time Full-time job 15

Women 25

Men

b 40 people

c 38.9%

e 21 people

b There were 130 brushtail possums and 180 honey possums, so there were more honey possums observed than brushtail possums.

16 a x = 70

c There were 80 possums observed eating fruit, 120 possums observed eating insects, and 110 possums observed eating nuts. So, there were more possums eating insects than fruit or nuts.

17 Answers will vary. The table provided is one such solution:

d There were 35 brushtail possums observed eating nuts and 70 honey possums observed eating insects, so there were signiﬁcantly more honey possums observed eating insects than brushtail possums observed eating nuts. 13 a True. Both sets of marginal frequencies will add to the total. b False. A single joint frequency can be equal to the marginal frequency, but only if all other joint frequencies in that row are equal to 0. This would only be possible if all values in a row were 0, in which case that row likely wouldn’t be included. 14 a The 60 people include teenagers not just adults so it can’t be concluded that all 42 are adults who said either no or sometimes. b The 24 adults represent the total of adults. Using this information with the 18 adults saying “yes” tells us that 6 adults responded either “no” or “sometimes”. c The statement only indicates the relative distribution among the “yes”, “no”, and “sometimes” responses, not their absolute values. We would need more information to precisely determine the number of each response.

b There are 70 animals in the study that carry the Agouti gene and have dark fur.

Football

Soccer

Total

Period 1

Period 2

Period 3

Total

18 a 650 students b Answers may vary. One possible answer: To investigate the hypothesis, you could design a follow-up survey that looks at two primary variables: 1. Participation in school sports during the hot week (using the same Likert scale: Always, Often, Sometimes, Rarely, Never) 2. Purchase of an ice block from the canteen during the hot week (Yes or No) Your two-way table would look something like:

Bought ice block

Did not buy ice block

Total

Always Often Sometimes Rarely Never Total Students would be surveyed on both their sports participation and their ice block purchases. By comparing the results, you could see if there’s a noticeable difference in sports participation between those who bought ice blocks and those who didn’t. If

Answers mathspace.co

781

the coach’s hypothesis is correct, you might expect higher “Rarely” or “Never” responses in the “Bought ice block” column.

9 a Pregnancy status b

13.02 Associations between categorical variables Understanding 1 a 43%

b No

Weight

Total

30%

70%

100%

Currently postpartum

73%

27%

100%

c Yes, there appears to be an association as the numbers are quite different. Those who are pregnant seem to prefer weights, while people who are postpartum seem to prefer cardio.

c No

2 A column percentage frequency table 3 The horizontal axis

d No, association is not causation. There appears to be an association but we cannot say whether one variable causes the other.

4 No Fluency 5

Adults

Children

Total

Math

42.86%

57.14%

100%

Science

45%

55%

100%

6 a

Red

10 a

Brown Blonde

Black

Total

Straight 5.9%

47%

5.9%

41.2%

100%

Curly

40%

30%

20%

100%

Brown

Blonde

Black

10%

Red Straight

50%

66.7%

25%

77.8%

Curly

50%

33.3%

75%

22.2%

Total

100%

7 a

Comedy Action Drama Horror Total Teens

30%

20%

45%

100%

Children

30%

40%

10%

20%

100%

Lefthanded

Righthanded

Total

Not female

14.5%

85.5%

100%

Female

9.7%

90.3%

100%

Left-handed

Right-handed

Not female

53.3%

42.0%

Female

46.7%

58.0%

Total

100%

11 a Parenting status

b Yes

12 a Fitness level

b Yes

13 a Personality type

b No

14 a Dietary restriction

b B

c No

c No Reasoning

Comedy

Action

Drama

Horror

50%

33.3%

81.8%

20%

Children

50%

66.7%

18.2%

80%

15 It means that there is no clear association between the variables.

Total

100%

16 a Location

Teens

b Column percentage table

8 a Height b

782

Cardio Currently pregnant

c Play tennis

Do not play tennis

Total

Height ≥ 170 cm

31%

69%

100%

Height < 170 cm

30%

70%

100%

Mathspace WA Curriculum – Year 10 mathspace.co

Urban

Regional

High emissions

62%

38%

Low emissions

38%

62%

Total

100%

d Yes, there does appear to be an association as the numbers are signiﬁcantly different.

17 a 40

b 21

Employed

Unemployed

Owns a smartphone

32%

17%

Does not own a smartphone

68%

83%

Total

100%

c Dominant hand d Row frequency table e

Maths Music Science English Total Righthanded children

30%

20%

25%

100%

Lefthanded children

33%

20%

30%

17%

100%

d Yes, association exists because values across the rows are different. e No, association is not causation. There appears to be an association but we cannot say one variable causes another.

f N o, association does not exist because values down the column are the same.

18 a Number of passengers b Column frequency table No. of passengers

Roll over

17.1%

5−9

Total

> 15

10−15

30.0% 28.8% 28.6%

No roll over 82.9% 70.0%

71.2%

71.4%

100.0% 100.0% 100.0% 100.0%

d No, because values across the rows are mostly the same. b Column frequency table Maths

Science

Maths major 39.52% 38.36%

20 0

Art

11.67% 23.33%

Employed Unemployed Employment

b Column percentage frequency table c

Music major 22.75% 22.60% 37.78% 20.56% Art major

7.19%

11.64%

26.67% 42.22%

Total

100%

Problem-solving c Book loans

d No, there does not appear to be an association as the corresponding segments are similar in size. 21 a Employment b Column percentage frequency table

32.14%

22.86%

9.52%

14.71%

4 to 8 hours

7.14%

28.57%

9.52%

26.47%

9 to 15 hours

25.00%

22.86% 38.10%

29.41%

Over 15 hours

35.71%

25.71% 42.86% 29.41%

Total

100%

d 3 or less 100 Percentage

e No, association is not causation. There appears to be an association but we cannot say that one causes the other.

10 years 11 to 20 21 to 30 31 years and under years years or older Less than 3 hours

d Yes, because values across the rows are different.

b June

22 a Age Music

Science major 30.54% 27.40% 23.89% 13.89%

20 a November

80 60

g Yes it does as the segments are different size in each column.

19 a Favourite subject at school c

Doesn’t own smartphone

100 Percentage

Smartphone owners employment Owns smartphone

100%

TV watching per age group 9 to 15 15 or over 4 to 8

80 60 40 20 0 10 & under

11 to 20 21 to 30 31 & older Age group

e Yes it does as the segments in each column are quite different. It suggests that older people tend to watch more hours of TV.

Answers mathspace.co

783

4 a Neither

13.03 Bivariate numerical data

c Decreasing

b Increasing d Decreasing

Understanding Fluency

1 a Univariate data

b Bivariate data

c Univariate data

d Univariate data

5 a i Number of dogs owned

e Bivariate data

f Bivariate data

ii Number of shoes destroyed

g Bivariate data

h Univariate data

2 a

c i Time spent studying

18 16 14 12 10 8 6 4 2

ii Exam mark 6 a Ticket sales affect the revenue made from a show. b Dependent variable - revenue from the show, Independent variable - number of ticket sales 2 4 6 8 10 12 14 16 18

b Dependent variable - height of the kite, Independent variable - speed of the wind

y 18 16 14 12 10 8 6 4 2

8 a Distance from the equator affects temperature. b No

c Both time spent travelling and distance covered affect each other, with speed playing a crucial role in their relationship.

d No

2 4 6 8 10 12 14 16 18

e Time spent working affects wages earned.

f N umber of pages affects time spent to ﬁnish a novel.

18 16 14 12 10 8 6 4 2

g No h Number of pets affects time spent caring for pets (hours).

i T ime spent training (hours) affects athletic performance.

9 a Height of the ball

2 4 6 8 10 12 14 16 18

b Amount of time passed

y 18 16 14 12 10 8 6 4 2 0

3 a Yes

784

7 a The height of the kite and the speed of the wind.

b i Size of a pizza ii Cost of pizza

10 As temperature increases, soups sold decreases. 11 Height has no effect on salary. 12 a The independent variable is the number of immigrants and the dependent variable is the unemployment rate. x 2 4 6 8 10 12 14 16 18

b No

c No

d Yes

Mathspace WA Curriculum – Year 10 mathspace.co

b Unemployment rate in Australia Unemployment rate (%)

15 a Dependent variable - height, Independent variable - time b

5.5

Height 0.8 0.6 0.4 0.2

5 4.5 4

−0.2 −0.4 −0.6 −0.8

3.5 Number of immigrants (thousands) 250

275

300

325

c As the number of immigrants increases, the unemployment rate appears to decrease.

Time 2 4 6 8 10 12 14 16 18 20 22

16 a 180 y 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90

13 a The independent variable is CO2 emissions and the dependent variable is the global temperature anomaly. b Climate change Global temperature anomaly (°C) 0.9

0.8

x 2

0.7

b 105 cm 0.6

Reasoning CO2 emissions (million MT) 35 000 36 000 37 000 38 000 39 000

c It appears that as CO2 emissions increase, global temperature anomalies also increase. 14 a 18 y 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

17 a 10 trees b No. There is no connection between the number of people in a student’s family and the number of trees in their garden because there is no trend in the data in the scatterplot. 18 a Online shopper survey averages Average amount spent (dollars)

x 1 2 3 4 5 6 7 8 9 10 11 12 13

b 11

Average time spent (minutes)

Answers mathspace.co

785

b Based on the scatterplot, the statement does not seem to be accurate. In fact, the data suggests the opposite: as the average time spent on a website increases, the average amount spent also increases. 19 a

100 90 80 70 60 50 40 30 20 10 0

Exam mark

13.04 Associations between numerical variables Understanding 1 a No

b Yes

c Yes

d Yes

2 a True

b False

c True

d False

e True 3 a The strength and direction of a scatterplot describe the correlation. b When two variables are related, we say there is an association. Books read per year 10

b 50 c Yes, the teacher is correct. Notice how the points follow a line from the bottom left corner to the top right corner. This means that if a student reads more books they receive a higher mark in their yearly maths exam.

c To show causation, we must do a carefully designed study. d When correlation is not causation, some possible explanations are coincidence and contributing (third) variables. Fluency 4 a Linear relationship b Non-linear relationship c Non-linear relationship

Problem-solving

d Non-linear relationship

20 Looking at the scatterplot, the number that fulﬁls the required characteristics is 4.

e Linear relationship

21 a Student B

b Student C

c Student A

22 a Answers may vary. One possible answer is the relationship between the amount of study time and test scores. These variables may have a relationship because the more time a student spends studying, the better they might perform on a test. b Answers may vary. One possible answer is that the amount of study time is the independent variable because it can be controlled by the student, while the test scores are the dependent variable because they can change based on the amount of study time. c Answers may vary. One possible answer is the relationship between the colour of a car and its speed. These variables likely do not have a relationship because the colour of a car does not affect how fast it can go.

f Linear relationship 5 a Positive and strong b Positive and weak c Positive and moderate-weak d Negative and strong 6 a As air temperature increases, sea temperature increases. b Strong, positive, linear 7 Negative 8 a

Sport 90 80 70 60 50 40 Maths 40 50 60 70 80 90

b Weak, positive, linear c Student 10’s

786

Mathspace WA Curriculum – Year 10 mathspace.co

9 a

b 10

Accidents

c Weak, negative, linear

d 8 °C, 69 seconds

13 a

t 70

60 50

20 Age 25

40 30

b Strong, negative, linear

c No

10 a

Hours

n 1

b Increases

c Strong, positive, linear

d No

14 Strongly negative

15 Strongly positive

Age 25

b Strong, negative, linear

b Yes

c No

d No

e No

f Yes

g Yes

h Yes

17 This data suggests that as height increases, there is an increase in the ability to play soccer. However, it does not necessarily mean that if you are tall, you are automatically better at soccer. Correlation does not imply causation, and other factors may also inﬂuence soccer skills.

Price 16 15 14 13 Supply (kg) 600 700 800 900 1000

18 a The explanatory variable is the number of questions and the response variable is the test result. b 100 Test result

b Strong, negative, linear c When very few bananas are available to be sold. 12 a

16 a No

Reasoning

c No 11 a

Time 65 60 55 50

95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15

Number of questions 10 15 20 25 30 35 40 45 50 55

45 Temp. 1 2 3 4 5 6 7 8 9 10

c Strong, positive, linear correlation; the larger the number of completed revision questions, the higher the mark on the test.

Answers mathspace.co

787

d The relationship is not perfectly linear because of the different abilities of the students.

24 Answers may vary. One possible answer: Time spent studying and test scores could lead to a weak positive relationship.

19 a It appears that y can change signiﬁcantly without much corresponding change in x. 20 Answers may vary. For example, if there is a relationship between the average temperature and migration, the scatterplot might show a pattern where the points tend to increase or decrease together, indicating a positive or negative relationship between the two variables.

It could be that while additional study hours slightly benefit most students, other factors play a significant role in determining their final score. For instance, some students might have natural aptitude, meaning even with fewer study hours, they can achieve similar or even better scores than their peers who study longer. Others might benefit from private tutoring, have access to better resources, or engage in practical applications of their knowledge that aren’t directly measured by the study hours.

Problem-solving

Here is a table of values and scatterplot that would lead to a weak positive association:

b i Yes

ii Yes

21 a D 22 a i No

iii Yes

iv No

b B ii Yes

iii No

b Region 2 23 a True. For example, if we know that influenza causes patients to develop a fever then there is a correlation between having influenza and developing a fever. However, it is possible for the correlation between influenza and developing a fever to be a strong or weak correlation. b False. For example, if there is a correlation between the number of people drowning in pools and number of people going to the movie theatre, that does not mean that an increase in pool drownings causes more people to go to the movie theatre or vice versa. c False. A positive association means that as the value of the independent variable increases, the value of the dependent variable also increases, but does not mean that changes in one cause changes in the other. For example, suppose data collected from a study found that people who exercised more were also found to be at higher risk for skin cancer. These quantities demonstrate a positive association, but that does not mean exercising more causes a higher risk of skin cancer, there are likely other underlying factors influencing the relationship.

788

Mathspace WA Curriculum – Year 10 mathspace.co

Hours Studied

Test Score

Student A

Student B

Student C

Student D

Student E

Student F

Student G

Student H

Student I

Student J

Test score

55 Hours studied 1

13.05 Line of good fit Understanding 1 Decreasing 2 Line 2 3 a No 4 a Negative

b No

c Yes b C

d Yes

Fluency

d i

5 19 y

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15 10 5 x 12

ii y = −1.3x + 29.5 x

e i

1 2 3 4 5 6 7 8 9 10 11 1213141516171819

6 a i

8 4

x 0

5 x 0

f i

y 10

4 2

x 0

5 x 0

38 36 34 32 30 28 26 24 22 20

20 15 10 5 x 5

ii y = 0.9x + 0.5

7 Example answer:

ii y = 0.56x + 5 c i

ii y = 1.5x − 17.5

ii y = −x + 21 b i

x 10 20 30 40 50 60 70 80 90

ii y = 0.9x + 1.5

Answers mathspace.co

789

190 180 170 160 150 140 130 120 110 100 90

Reasoning

Copies sold

13 a 27.7 kg b The y-intercept indicates that when a child’s height is 0 cm, their average weight is 27.7 kg. c No, a child cannot have a height of 0 cm. 14 a A scatterplot can help visualise the relationship between the quantities, but is not necessary for a complete response.

Price 18 20 22 24 26 28 30 32 34 36

9 a i 4.62

100

ii Positive

iii Increases by 4.62 units

iv 7.58

b i −7.68

ii Negative

iii Decreases by 7.68 units

iv 5.49

c i −5.69

ii Negative

iii Decreases by 5.69 units iv 7.84 ii Positive iii Increases by 3.16 units iv −2.72 10 a The vertical intercept indicates that when a family has no income, $27 is its average spending on dining out. b No

c 0.3

d $60

Price (thousands of dollars) 24 20

12 8 4

Age (years) 2

b Answers will vary if calculated by hand, but should be around eighteen. c A car that is 0 years old costs $18 400. d Answers should be around 11 years.

790

A student who spends 0 minutes on LookAtMe is predicted to have a 99.6% grade in maths based on the line of good fit. b According to the linear model, for every additional minute spent on LookAtMe, a student’s maths grade is expected to drop 0.25% points. Or, for every additional 100 minutes spent on LookAtMe, a student’s grade is predicted to drop 25%. 15 a There are only 2 points above the line and 18 points below it, so this line does not go through the middle of the data. It is too heavily impacted by the outlier, so is not a line of good fit.

12 a −4

Minutes 50 100 150 200 250 300 350 400 450

There is a strong, negative, linear association between minutes spent on LookAtMe and maths grades.

d i 3.16

11 a

Grades

b 3865

Mathspace WA Curriculum – Year 10 mathspace.co

b An outlier should not be completely ignored, but the line should go through the majority of the data. Lorena’s line has 9 points above it and 10 points below it, and goes through the middle of the data, so it is a line of good fit.

Problem-solving

5 y = −52.61, Extrapolation

16 a

6 a 3 seconds

Ice blocks 140

7 a 20 years

120

8 a

100 80 60 40 20 Temperature

225 200 175 150 125 100 75 50 25

ii Approx. 149

c Increases 17 a A

b x = 3682

18 a y = 0.5x + 4

b 7.45

c 11 years

Temperature °C 12 14 16 18 20 22 24 26 28

b 150 people

c 23 °C

d 140 people

e 19 °C

9 a 2448 Pa

b Interpolation

c Yes d i Yes

19 a A line of good fit: y = 3x + 15

b 1300 ﬁsh

People

14 16 18 20 22 24 26 28 30 32 34 36 38 40

b i Approx. 91

b 2.5 hours

ii No

10 a 90 c i No

H% 30

12 a

iii No

iv No

b 6500 m

c 300 seconds

iv No

b 22°C ii Yes

11 a 8500 m

iii No

d 1800 seconds b c = 100

c $400 000

13 a Negative gradient

b −2.5

c y = −2.5x + 105

d y = 95

Y 1

Reasoning

b The new line of good fit would have a smaller slope and a higher y-intercept.

13.06 Interpolation and extrapolation Understanding 1 a False

b False

e False

f True

c False

d True

2 a Interpolation

b Extrapolation

c Extrapolation

d Interpolation

3 a y = 8.45

b y = −17

c y = −7.4

Fluency 4 a Interpolation c Interpolation

b Extrapolation

14 a 52.5 m b Yes, because the prediction is determined using interpolation. 15 a Interpolation. The speed of 70 km/h falls within the range of known data points, which is from 40 km/h to 100 km/h. b Extrapolation. The prediction for the revenue for the next year is beyond the range of known data points (2015 to 2019). c Extrapolation. The number of hours of study at 15 hours does not fall within the range of known data points, which spans from 2 to 14 hours. 16 Yes, if we examine the scatterplot and compare with the other points. It follows the same relationship.

Answers mathspace.co

791

17 Interpolation and extrapolation can be limited by the accuracy and validity of the data, as well as the assumption that the relationship between variables remains constant. In the context of climate change, factors such as human intervention, natural disasters, and technological advancements can inﬂuence the relationship between variables, making predictions less reliable. Problem-solving 18 a y = −0.005x + 49

b y = 29 °C

d $50 260 e Yes, because it is an extrapolation close to the data range where the cost is likely to increase along with distance driven, and the correlation coeﬃcient is high. f Y es, a car that is driven more will have more wear and tear and would be likely to have greater maintenance costs. 22 a 95 m

9 8 7 6 5 4 3 2 1

b A = 100 hectares

c No, because it is using extrapolation. 20 a

Litter collected (kg) 5 4

24 Neither interpolation, not extrapolation would be overly reliable. This data set is more nonlinear, so a non-linear model would be more appropriate for making predictions.

1 Weeks 0

2 3 4 5 6 7 8 9

b Example answer:

2 3 4 5 6 7 8 9

13.07 Misrepresentations of data

Litter collected (kg) 5

Understanding

1 a 100%

2 a Not representative, convenience bias and likely missing certain age demographics or those without reliable internet.

Weeks 0

b 193%

c No

b Not representative

1 1

c Representative

2 3 4 5 6 7 8 9

c Example answer: y = −0.2x + 4 d Approximately 2.4 kg e No, because it is using extrapolation. 21 a For each 1000 km increase in distance driven, the least-square line predicts that the maintenance costs increase by $990. b A car that is not driven at all is predicted to cost $14 620 to maintain. c No, because for a car that hasn’t been driven, it should have little to no cost to maintain.

792

c Option B

23 Any scatterplot with weak correlation or very few points. For example:

c Yes, because it is using interpolation. 19 a A = −25T + 750

b 50 m

Mathspace WA Curriculum – Year 10 mathspace.co

d Representative 3 a False 4

b False

Date

Cases

October 15

October 18

200

October 21

174

October 24

339

October 27

251

October 30

398

c True

d True

Fluency

Reasoning

5 a Causation and correlation

17 Kang has used only the statistics that are favourable to him.

b Over-generalisation c Comparing apples and oranges d Cherry-picking 6 a False

b False

c True

d False

7 a Yes

b No

c No

d Yes

8 a Yes

b No

c Strategic choice of measure of centre (mode) d Yes 9 a Cherry-picking

b C

10 The claim is directed to any person that uses the product. However, the sample size is small and not representative of all people that might use the product. There is no information on whether the product was properly tested. There is no information on the sampling method used or the method for collecting the data. 11 a Biased question b When giving customers the menu, the servers can simply inform the customers that the mushroom risotto is a new menu item. 12 The tick marks on the vertical axis are not evenly spaced. 13 They do not correspond to the tourist numbers they are supposed to represent. In particular, on the corners the sizes are exaggerated. 14 The title is persuasive and the vertical axis scale doesn’t start at 0. It also does not give an idea of the trend overall. 15 By using a logarithmic scale, the graph appears to show a slight increase after taking the drug, and a steady decline of the drug in the blood over time. This may lead people to think that the drug concentrations in the blood are relatively stable over time. When converted to a normal scale, the initial increase would be much more signiﬁcant. 16 a Russia

b U.S.

18 By compressing or expanding the horizontal axis it can appear as if the trend is gradual or quite abrupt. 19 a The axis is broken too close to the top of the Mar 95 column, so it is not clear where the scale break is. b It is not recommended for graphs such as column graphs where the viewer takes in the comparative areas of the different categories, using a truncated axis will exaggerate any differences in the data. c A line graph with clear indication of a broken axis below the lowest point in March. 20 a The graph with the vertical axis that goes from 13 to 15. b The graph with the vertical axis that goes from 0 to 20. c Since 1920, the temperature has increased by around 0.8 °C. The claim states that a further increase by 0.7 °C in the next few decades could lead to an increase in catastrophic events. This shows that increases by less than 1 °C are actually signiﬁcant increases in temperature. 21 a One possible answer: The survey claims to use over 2000 samples, but doesn’t clarify the selection process. This raises doubts about its accurate representation of all frequent smartphone users. b One possible answer: The survey, conducted by an anti-ageing skin company, suggests half of the regular smartphone users are showing early ageing signs. It’s unsurprising that the study recommends reduced screen time and speciﬁc skincare products. c One possible answer: The article’s headline is sensationalised, potentially causing unnecessary worry as it lacks context on the study’s broader use.

c No

Answers mathspace.co

793

Problem-solving

Fluency

22 Graph 1 is misleading because the vertical axis does not start at zero therefore the difference in the number of days is exaggerated.

6 a Strong positive linear relationship

Number of COVID-19 tests per million people

23 a The y-axis scale is not shown and therefore the Argentinian column appears taller than it should be. There is actually a large difference in the rates of the USA and Argentina. COVID-19 Testing by Country

25000

22 300

20000

15 700 14 100

15000

c Weak positive linear relationship d No linear relationship e Weak negative linear relationship f Moderate negative linear relationship g A perfect positive correlation h A perfect negative correlation 7 a i Strong positive linear relationship ii Between 0.8 and 1

10000

7000

5000 0

b Moderate positive linear relationship

258

b i Moderate positive linear relationship

330

Brazil Argentina

USA

Italy

Germany Norway

24 a It has made the increases in the 4-year intervals 1992-1996 and 1996-2000 appear faster than they really are (relative to the rate in the 2-year interval 1990-1992). b Example answer: Although the scale on the y-axis does not begin from 0, it is reasonable because the difference in the data is small. If the scale began from 0, it would be diﬃcult to choose a scale that would allow the reader to identify the exact values for each year.

ii Between 0.4 and 0.6 c i No linear relationship ii Between −0.2 and 0.2 d i Moderate negative linear relationship ii Between −0.6 and −0.4 e i Strong negative linear relationship ii Between −1 and −0.8 f i Weak negative linear relationship ii Between −0.2 and −0.4 8 a i 0.99 ii Strong positive linear relationship b i −0.95

13.08 Correlation coefficient

ii Strong negative linear relationship

Understanding

ii Weak positive linear correlation

c i 0.47 1 a r2 = 0.8836 2

c r =1

b r2 = 0.25 2

d r = 0.7225

d i 0.07 ii No linear relationship e i −0.39

2 Data set B

ii Weak negative linear relationship

3 a False

b False

c True

d False

4 a i Yes

ii x and z

b i Yes

ii x and y

c i No

ii x and y

d i No

ii x and z

5 a There is a strong, negative correlation between the life expectancy of a group of men and cigarettes they smoke a day. b There is a very weak, negative linear correlation between the number of years in education a person completes and the number of pets.

794

Mathspace WA Curriculum – Year 10 mathspace.co

9 a 40

Coffees

15 10

Max temp 22 24 26 28 30 32 34 36 38

b r = 0.84 c As the temperature increases, the sales tend to increase.

10 a r2 = 0.37

b r2 = 0.19

11 52% 12 a r = 0.8124

b Strong

13 a Yes

b No 2

14 a i r = 0.81

ii r = −0.90

b 81% c While drinking and driving is dangerous and there is a correlation between amount of alcohol and driving score, this conclusion is not valid. 15 a i r2 = 1.00

ii r = −1.00

b 100% c While this claim is supported by a strong relationship between the two variables, we cannot state causality as there may be other factors influencing the outcome. 16 a i r2 = 0.47

ii r = 0.69

b 47% of the variation in the number of viewers who pressed “Skip” is explained by the length of the advertisement. c While this claim is supported by a moderate relationship between the two variables, we cannot state causality as there may be other factors, like ad content, influencing the outcome. Reasoning 17 a Positive b No. Although there is a very strong correlation between these two variables (0.99), we cannot establish causality because there may be other factors affecting the outcome. 18 The relationship graphed is not linear and the outlier will cause the result to be biased. 19 a As the heights of women increases, the probability of being turned down for a promotion decreases. b As the population of a city increases the number of speeding fines recorded increases. c There is no evidence of a linear relationship between length of hair and length of fingernails.

d As the number of bylaws a council has about dog breeding increases the number of dogs available for adoption at the local shelter tends to increase. Problem-solving 20 a A linear model is more appropriate for this data set since when we graph it, there is a linear pattern. b n = −4.4t − 58.2 c From part (a), we found that a linear model better fits our available data, so the correlation coefficient can be used to interpret the strength and association of our data. The correlation coefficient, r, is equal to −0.89 This suggests that there is a strong linear negative association, which means that as the temperature increases, the number of hot cocoas sold will decrease. 21 a r = 0.93 b r2 = 0.8649 c 86.49% d No. Although r and r2 are very strong, other factors, like where a teacher works, influence their salary. 22 a r = −0.908 b Strong negative linear relationship c y = −0.06x + 6.37 d 6.07 kg e The prediction is reliable due to interpolation and a strong correlation. 23 A ny data set with a strong negative correlation. For example: x

14 12 10 8 6 4 2

0 1 2 3 4 5 6 7 8 9

Answers mathspace.co

795

Chapter 13 review 1 a

8 a Sedan

SUV

Total

Over 30

Total

4 3 2

b 2 a 354 students c 27.94%

1 5

b d 8941 students

3 a Living area b Yes, there appears to be an association. 4 a Time of day preference

Sick days 25

b Yes, the statement is correct. The scatterplot shows that the data forms a decreasing, linear relationship, meaning that as a student exercises more, they get sick less.

10 a

b Negative and weak

Sales (in thousands) 25

b Column percentage frequency table High School or less

9 a Positive and strong

b Yes

5 a Education level c

Days of exercise

Truck Under 30

20 Some Bachelor’s Graduate university degree degree

Less than 48.08% 2 hours

36.36%

6.52%

3.51%

3 to 38.46% 6 hours

36.36%

21.74%

21.05%

7 to 13.46% 12 hours

21.82%

54.35%

54.39%

Over 12 hours

0.00%

5.45%

17.39%

21.05%

c No

Total

100%

d No, correlation does not imply causation.

6 a January

10 5 0

Month 1 2 3 4 5 6 7 8 9 10 11 12

b Moderate, positive, linear

11 a i

d Yes b February

c Instrument sales d No, there does not appear to be an association as the corresponding segments are similar in size. 7 a Yes, time spent studying a subject is the independent variable and the number of chapters is the dependent variable. b No c Yes, the duration of sunlight exposure is the independent variable, and plant growth is the dependent variable.

796

Mathspace WA Curriculum – Year 10 mathspace.co

y 25 20 15 10 5 x 0

ii y = x + 4

iii Positive

b i

16 a Cherry-picking

b The sector for Prefers AlphaModel is much smaller than the other two sectors, but it is only 3% smaller than the other two percentages.

16 14 12 10 8 6 4 2 0

x 2 4 6 8 10 12 14 16 18

iii Negative

12 a The y-intercept suggests that when the radish is 0 days old, it has no height above the ground. b Yes, it is plausible that a seed will not sprout above the ground in the same day it is planted. c 19 days 13 a y = 33.75

c A column chart would be more appropriate because it would show that all three phones are preferred by approximately the same percentage of people. 17 a i Strong negative linear relationship ii Between −1 and −0.8 b i Weak negative linear relationship ii Between −0.2 and −0.4 18 a r = 0.8832 b Strong 19 a i r2 = 0.74

ii r = 0.86

b Strong positive linear relationship b Interpolation

14 a M = $941.40 b No, because it is using extrapolation. c 25 days d Yes, because there is a strong correlation and interpolation was used for the prediction.

c 74% of the variation in consumer satisfaction is explained by the battery size of the smartphone. d While there is a strong correlation between the two variables, we cannot state causality as there may be other factors influencing the outcome.

15 a Mistaking causation and correlation b Over-generalisation c Strategic choice of measure of centre

Answers mathspace.co

797