Issuu

1 Equations & Inequalities Big ideas • Expressions are the building blocks of algebra. They can be used to represent and interpret realworld situations. • The properties of real numbers can be applied to many types of expressions. • An equals sign indicates an equivalent relationship between two expressions. • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06

Algebraic expressions Properties of real numbers Properties of equality Multistep equations Literal equations Multistep inequalities

4 13 17 30 40 46

Exploration In order to write an expression that can be used to model the total cost of buying new school supplies, Mr. Okware deﬁnes the following variables: Let x represent the cost of a folder, y represent the cost of a calculator, and z represent the cost of a pencil pack. 1.

What could the following expressions represent in this context? • x+y • 2x + 10z • x+y+z

• x + 3y + 4z

• 5y

• 4(4x + y + 2z)

In this context, what do the coeﬃcients describe?

What expressions could we write that wouldn’t make sense in this context?

Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions. We can use algebra tiles to help us visualize algebraic expressions. The tile x represents an unknown number. The tile +1 represents adding one unit and −1 represents subtracting one unit. Positive +x

Variable tiles

Unit tiles

Negative −x

−x

−1

This table demonstrates how expressions can be built using the tiles: Word expression

Algebraic expression

Five more than twice x

2x + 5

The sum of negative x, four, and double x

−x + 4 + 2x

Three times the difference of x and four

3(x − 4)

Representation with algebra tiles +x

−x

−1

1.01 Algebraic expressions mathspace.co

Example 1 Create a model using algebra tiles for the following algebraic expression: −7 + 5(1 − 3x)

Create a strategy Algebra tiles use tiles with +1, −1, +x, and −x to represent the individual terms. The number outside of the parentheses for the distributive property represents how many groups of identical expressions will be made.

Apply the idea We will need the −x, +1, and −1 tiles to make our model. The 5 outside of the parentheses means we will need 5 groups of 1 − 3x. −1 −1 −1

−x

−1

−x

−1

−x

−1

−x

−1

−x

Example 2 Vincenzo runs a removalist company that charges $37.50 per hour plus a one-off truck hire fee of $150.00. Write an expression that models how much he charges for a job that lasts a hours.

Create a strategy We need to look at the two values that affect the price of the job; the cost per hour of $37.50 and the truck hire fee of $150.00. For each hour worked, Vincenzo charges an additional $37.50. Let’s consider a few cases: Cost of working 1 hour: $150 + $37.50 Cost of working 2 hours: $150 + $37.50 + $37.50 Cost of working 3 hours: $150 + $37.50 + $37.50 + $37.50 Notice that for each additional hour worked, we add an additional $37.50. Multiplication is repeated addition, so we can multiply $37.50 by the number of hours instead of adding repeatedly.

Apply the idea

Reflect and check

Since the $150 is a one-time fee, this value will remain constant. Next, we multiply $37.50 by the number of hours which is a.

For this problem, we would replace a with the number of hours Vincenzo works on a particular job, and the result would be the amount of money he makes on that job.

Cost: 37.5a + 150

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 If the area of a square is given by the expression (2x − 1)2, explain what 2x − 1 represents in the context of the problem.

Create a strategy First, we need to remember the formula for the area of a square. We know that the area of a square is given by s2 where s represents the side length.

Apply the idea

Reflect and check Area = s

2 2

= (2x − 1)

From here, we could easily use algebraic expressions to determine the perimeter or other useful measurements of the square.

Therefore, we have: s = 2x − 1 This shows that s which represents the side length is given by 2x − 1. Thus, the side length of the square is 2x − 1.

Idea summary Expressions can be used to represent mathematical relationships. In an expression, sums often represent totals, coefficients, and factors represent multiplication, and exponents represent repeated multiplication. When interpreting an expression in context, we can use the units to help understand the meaning.

Evaluate expressions In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks. The order of operations tells us the steps to evaluate expressions with multiple operations, so that the same numerical result is achieved. The order goes: 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values ∣…∣. If there are grouping symbols within other grouping symbols, do the innermost operation first. 2. Evaluate all exponents, such as squares and cubes. 3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right. We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the expression to yield a numerical result. As an example, a concession stand sells bags of popcorn for $2.50 each and hot dogs for $1.50 each. The total cost of buying p bags of popcorn and h hot dogs can be represented by the expression 2.50p + 1.50h. Riley goes to the concession stand every Saturday and buys 2 bags of popcorn and 4 hot dogs for her friends, and wants to determine the total cost of her order 3 Saturdays in a row.

1.01 Algebraic expressions mathspace.co

Since the same order will be repeated 3 times, the distributive property will be used to rewrite the expression. By substituting the values p = 2 and h = 4 into the new expression, we see that Total Spent = 3(2.50p + 1.50h)

Rewrite expression.

= 3(2.50 ⋅ 2 + 1.50 ⋅ 4)

Substitute values

= 3(5 + 6)

Evaluate the multiplication

= 3 ⋅ 11 Evaluate the addition The total cost of Riley’s purchase is $33.

Example 4 Evaluate: (u + v) (w − y) when u = 5, v = 8, w = 2, and y = 10.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea (u + v) (w − y) = (5 + 8) (2 − 10)

Substitute u = 5, v = 8, w = 2, and y = 10

= (13) (− 8)

Evaluate the operations inside the parentheses

= −104

Evaluate the multiplication

Example 5 Find the value of:

when x = −4 and y = 3.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea Substitute x = −4 and y = 3

Evaluate the exponents

Find a common denominator

Evaluate the multiplication

Evaluate the addition

Mathspace Virginia SOL Algebra 1 mathspace.co

Reflect and check We can apply the order of operations to problems with any type of real number such as integers, fractions, or decimals.

Example 6 For x = 5 and y = 4, evaluate: correct to two decimal places.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea Substitute x = 5 and y = 4

Evaluate the exponent

Evaluate the multiplication

Evaluate the addition

Use a calculator to calculate the square root

Idea summary Substitute the given value for each variable and then apply the order of operations: 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values ∣…∣. If there are grouping symbols within other grouping symbols, do the innermost operation first. 2. Evaluate all exponents such as squares and cubes. 3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right.

Practice What do you remember? 1

Match the following terms with their deﬁnitions: a

Constant

Symbol that represents an unknown number

Variable

A purely numeric term in an algebraic expression

Coeﬃcient

iii

The value that indicates how many of a variable in a term

Algebraic term

Term including a variable

Write the expression 9x in words.

State whether the following expressions are like terms: a e

11p and 3p 5h and 5hk

b f

2p and 15p 4

4h and 3h

9p and 6q

3 4

4 3

8h k and 9k h

12 and 8

6hk and 7kh

1.01 Algebraic expressions mathspace.co

For the following algebraic expressions: i

State the number of terms.

Identify the numerical coeﬃcient of the ﬁrst term.

iii

Identify the constant term.

Determine if the expression contains like terms.

2x + 4

3x + 2y − 8x + 9

7y + 3 + 5x

8p + 5

7x + 8 + x − 11

Consider the expression x + 6. a

Find the value when x = 4.

Find the value when x = 9.

What happens to the value of x + 6 as we substitute in different values of x?

Evaluate the following expressions to one decimal place: a

14.5 − 4x when x = 4.2

18.6 − 3x when x = 4.1

m2 + 9n

If m = −3 and n = 4, evaluate the following expressions: a

mn − (m − n)

Let’s practice 8

Write an algebraic expression for the following diagrams: a

−x −x +1

+x −1 −1 −1 −1

−x

−1

+x +x

Use the following algebra tiles to draw a diagram that represents each expression: Positive +x

Variable tiles

Unit tiles a

x + 12 − 6x

Negative −x

−x

−1

Mathspace Virginia SOL Algebra 1 mathspace.co

−6 + 4(1 + 2x)

2(−x − 2) + 3x

Evaluate the following expressions: a

when m = 5

when x = 3

when a = 5 and b = 2

when a = 3 and

s∣−12 + t∣ when s = −3 and t = 10

when p = −7 and q = −8

g i k m

when a = −4 when r = 6 and s = 7 when x = 2 and y = 5 and b = 2

when

when y = −2 and

s 11

when a = 5 and b = 12

when p = 6 and q = −2

h j

when p = 2 and q = 4 when x = −2 and y = −2

when a = −9 and b = 10

n p

5y − z2 + 9 when y = 7 and z = 4

y3 + 4yz − 2x2 when x = −4, y = 2 and z = 1

(a + b) (c − d) when a = 6, b = 9, c = 4 and d = 14

Dylan purchased 3 pens, 4 pencils and a single note pad. The total cost of all these items was 3x + 4y + 5 dollars. a

Interpret the variable y in context of the problem.

Interpret the variable x represent.

Interpret what the term 5 represents.

Explain how this expression could be revised to represent purchasing an unknown number of notepads.

Deborah earns $25 per hour of work and is paid double for every hour worked on the weekend. At the end of a week, Deborah calculates her pay for that week to be 25x + 50y dollars. a

Identify an expression that represents the number of hours Deborah worked during the weekdays.

Identify an expression that represents the amount of money Deborah earned from working on the weekend.

Mohamad and Valentina are throwing a party. Mohamad wants to reserve all street parking within a certain distance from their house. They are located 150 ft down a street. Write an expression for the endpoints of the parking area based on the distance on the road.

Brad and Patricia are making paper cranes to decorate their room. Brad can make m an hour while Patricia can make n an hour. Brad spends 6 hours making cranes while Patricia spends 5 hours over the weekend. Together they make a total of 6m + 5n paper cranes over the weekend. a

Interpret the meaning of the term 6m.

Interpret the meaning of the term 5n.

The side length of a square box is 3x + 1 yards. Write an expression for its area.

Roxy is 5 inches taller than Jane, while Katrina is 9 inches shorter than Jane. Write expressions that can be used to represent the height of each person.

Valerie is with a mobile phone provider that charges $0.26 per minute plus a connection fee of $0.45. Write an expression that models how much she will be charged for a call that lasts a minutes.

1.01 Algebraic expressions mathspace.co

An expression for the surface area of cube shown in the image is 6s2. a

Interpret the meaning of the coeﬃcient.

Interpret the meaning of the s2.

Let’s extend our thinking 19

For each of the following, determine if the statement is true or false. Explain your reasoning. a

An expression must include a variable.

Any expression with more than one term can be simpliﬁed.

Tobias has two times as many books as Isabelle does, and Isabelle has four less than triple the number of books that William does. Explain a method that can be used to write an expression for the number of books that Tobias has.

The width of a rectangle is 19 cm less than double the length.

Construct an expression that models the width of the rectangle.

Revise the expression in part (a) to create a model for the perimeter of the rectangle and a model for the area of the rectangle.

Determine whether there are any values of the length of the rectangle that are not viable. Justify your reasoning.

Four friends share the cost of a pizza. The toppings for the pizza added $4.75 to its base price for plain cheese. a

Construct an expression that models the price for one friend’s share of the cost.

The following table has the prices of various pizza sizes: Small $13

Medium $15

Large $17

Determine which size pizza the friends could fairly split if friend 1 has $5, friend 2 has $5.50, friend 3 has $5, and friend 4 has $6.50. Justify your reasoning.

Mathspace Virginia SOL Algebra 1 mathspace.co

1.02 Properties of real numbers After this lesson, you will be able to… • use the properties of real numbers to rewrite expressions or solve equations.

Review: Properties of real numbers Recall that the numbers that we use regularly for counting and measuring are called the real numbers. They include zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in between. Natural numbers

Rational numbers

The counting numbers, starting from 1.

The set of numbers that can be expressed in the where a and b are integers and b ≠ 0.

form

Whole numbers The counting numbers, starting from 0.

Irrational numbers

Integers

The set of numbers that cannot be written in the

A set of numbers that include positive whole numbers (natural numbers), their negative counterparts, and zero.

form

where a and b are integers.

Real Number System Rational Integers Whole Natural … −4 −3 −2 −1 0 1 2 3 4 …

−112 17

−3 2

−3.12

1 2

1.3

2.6

13 3

Irrational − 102 5

−3 2

1+ 5 2

The real numbers are so familiar to us that we hardly notice that they have special properties that involve the addition and multiplication operations. For any real numbers a, b, and c, the following properties are always true. Commutative property Associative property

Addition a+b=b+a (a + b) + c = a + (b + c)

Inverse property

a + (−a) = 0

Identity property Distributive property

a+0=a

Multiplication a⋅b=b⋅a a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

a⋅1=a a (b + c) = a ⋅ b + a ⋅ c

Notice the distributive property only applies to distributing the operation of multiplication. If the positions of the addition and multiplication signs are swapped so that we have a + (b ⋅ c), a similar distributive rule is not true. This is one situation in which we must observe the correct order of operations when simplifying expressions. 1.02 Properties of real numbers mathspace.co

Example 1 Verify the distributive property a(b + c) = a ⋅ b + a ⋅ c for the values a = 3, b = 13, and c = −1.

Create a strategy We need to substitute the values for a = 3, b = 13, and c = −1 into the distributive property equation a (b + c) = a ⋅ b + a ⋅ c. If both sides of the equation evaluate to the same number, then the property is true for those values.

Apply the idea Evaluate the expressions. a(b + c) = 3 (13 + (−1))

Substitute a = 3, b = 13, and c = −1

= 3 (12)

Evaluate the subtraction inside the parentheses

= 36

Evaluate the multiplication

a ⋅ b + a ⋅ c = 3 ⋅ 13 + 3 ⋅ (−1)

Substitute a = 3, b = 13, and c = −1

= 39 − 3

Evaluate the multiplication

= 36

Evaluate the subtraction

So, the expressions are the same for these numbers and the distributive property is true in this case.

Reﬂect and check How might you apply the same strategy to verify the other properties of real numbers?

Example 2 Using the properties of real numbers, rewrite 3 (x + 4) = 9 in another way.

Create a strategy First identify the property we will be using. Given the addition and multiplication, the distributive property a (b + c) = a ⋅ b + a ⋅ c seems like a good ﬁt.

Apply the idea The distributive property tells us we can multiply each term in the parentheses individually by the 3 outside the parentheses as shown: 3 (x + 4) = 3 ⋅ x + 3 ⋅ 4 = 3x + 12 Therefore, the expression 3 (x + 4) is equivalent to 3x + 12 by the distributive property. So we may rewrite the equation as 3x + 12 = 9.

Reﬂect and check We will use the distributive property many times to expand and simplify expressions. Can you think of any ways this may be helpful for us?

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 If 5 ⋅

= 1, use the properties of real numbers to solve for x.

Create a strategy First identify the property we will be using. Given the multiplication by an inverse, the inverse property of multiplication seems like a good ﬁt.

Apply the idea The inverse property of multiplication tells us that for every real number a, there exists a number b such that a ⋅ b = 1. The numbers a and b are called multiplicative inverses of one another. We usually write That is, we can rewrite the property as

instead of b.

This looks a lot like our problem. Let a = 5 for this property. Then, we have by the inverse property of multiplication.

. This tells us x = 5 in our example

Reﬂect and check See if you can challenge yourself to ﬁgure out what the value of x must be in this equation:

Idea summary The properties of real numbers can be used to simplify expressions and solve equations. Commutative property Associative property

Addition a+b=b+a (a + b) + c = a + (b + c)

Inverse property

a + (−a) = 0

Identity property Distributive property

a+0=a

Multiplication a⋅b=b⋅a a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

a⋅1=a a (b + c) = a ⋅ b + a ⋅ c

Practice What do you remember? 1

Determine whether each equation is true or false for every value of x. If the equation is true, state the property that proves it to be true. a

x + 1 = 1 + x

x⋅3=3⋅x

1 ⋅ 8x = 18x

(x + y) + z = x + ( y + x)

(x ⋅ 7) ⋅ 3 = x ⋅ (7 ⋅ 3)

What is the inverse of 3x?

Use the distributive property to rewrite 5 (2x − y) in another way.

1.02 Properties of real numbers mathspace.co

Let’s Practice 4

Is this number sentence true or false for every positive value of v? v÷3=3÷v

When asked to simplify the expression 7 + 6 (8x + 3), a student provides this work and gets the correct answer. 1

7 + 6 (8x + 3) = 7 + (48x + 18)

= 7 + (18 + 48x)

= (7 + 18) + 48x

= 25 + 48x

= 48x + 25

Identify the algebraic property used in: a 6

Step 1

Step 2

Step 3

Step 5

For each statement: i

Identify which property needs to be used to complete the statement.

Complete the statement by ﬁnding the missing value.

−7 + 8 = 8 + ⬚

(9 + 7) + 6 = 9 + (⬚ + 6)

b d

−6 ⋅ 5 = ⬚ ⋅ (−6)

6 ⋅ (9 ⋅ 7) = (6 ⋅ ⬚) ⋅ 7

Let’s extend our thinking 7

Is this equation true or false if a is not equal to b? Either explain why it is true using a property from the lesson or ﬁnd numbers for a and b that show it is false. a−b=b−a

Find the missing term that would make the number sentence true. a c e

9 10

0 + ⬚ = 9m

⬚ − 0 = 3d

d(4 + 3) = (⬚ + 5) ⋅ d

If 4( y + z) = 16, then z + y = ⬚.

b d f

4b ⋅ ⬚ = 1

a ⋅ 19 = 19 ⋅ ⬚

(4y2 − 3x) ⋅ ⬚ = −3x + 4y2

For this statement:

a + (−a) = 0 Explain why this is true for all values of a. 11

For this equation:

Use one of the properties in the lesson to ﬁnd the expression that y is equal to. 12

How could the distributive property help us ﬁgure out the solution to 396 ÷ 4 without using long division or a calculator?

Mathspace Virginia SOL Algebra 1 mathspace.co

1.03 Properties of equality After this lesson, you will be able to… • justify the steps for solving an equation using the properties of equality, identity, and inverse operations.

Properties of equality An equation is a mathematical relation statement where two equivalent expressions and values are separated by an equal sign. The solutions to an equation are the values of the variable(s) that make the equation true. Equivalent equations are equations that have the same solutions. Equations, particularly in real-world contexts, are sometimes referred to as constraints as they describe restrictions or limitations of the given situation. One familiar type of equation is a linear equation. Linear equation An equation that contains a variable term with an exponent of 1, and no variable terms with exponents other than 1. Example: 2x + 3 = 5 Equations are often used to solve mathematical and real-world problems. To solve equations, we use a variety of inverse operations to “undo” what was done to a variable. For instance, if a variable was multiplied by a number, we would use division to get the variable by itself. The distributive property is an important property that we use frequently to simplify and solve equations:

a (b + c) = ab + ac a, b, c are real numbers

Exploration Consider the undeniably true statement 5 = 5. Perform each of the following operations: • Add 3 to both sides

• Multiply −2 to both sides

• Subtract 1 from both sides

• Divide both sides by 3

What do you notice about the equation that results from performing each operation?

Suppose we started with the true equation x = 5. Would the equation still remain true after performing each of the operations? Explain.

Properties of equality are facts about equations. They describe different operations that can be performed on an equation that would maintain the truth of the equation statement. The following are the properties of equality and identity: Symmetric property of equality Transitive property of equality Addition property of equality Subtraction property of equality Multiplication property of equality Division property of equality Substitution property of equality

If a = b, then b = a If a = b and b = c, then a = c If a = b, then a + c = b + c If a = b, then a − c = b − c If a = b, then ac = bc If a = b and c ≠ 0, then If a = b, then b may be substituted for a in any expression 1.03 Properties of equality mathspace.co

If a = b, then a + 0 = b and a = b + 0 If a = b, then a · 1 = b and a = b · 1

Additive identity Multiplicative identity

Using algebra tiles, we can represent these properties to show the balance of the two sides. The balance below represents −2 = 3x + 2(−x + 1). +x

−1

−x

Keeping the two sides balanced, we want to add or remove tiles to work towards a single x tile. We see from the balance that 2(−x + 1) represents −2x + 2. −1

−1

By eliminating the zero pairs on the right side, we are left with −2 = x + 2. By using the subtraction property of equality, we will make zero pairs with the +2 on the right side of the equation to isolate the variable.

A zero pair on the right side leaves us with x + 0 and using the additive identity we are left with just x. −1

−1

−2 − 2 = x + 0

−4 = x

Solutions can be verified a variety of ways, including visually using algebra tiles. After solving −2 = 3x + 2(−x + 1), we want to verify that x = −4. Each +x tile will be replaced with four −1 tiles. A −x tile will change the four −1 tiles to +1 tiles. −1 −1

−1

−1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1

−1 −1 +1 +1

Substituting x = −4 into −2 = 3x + 2(−x + 1)

Mathspace Virginia SOL Algebra 1 mathspace.co

Zero pairs keep the equation balanced at −2 = −2

Example 1 Viola and Akim were asked to solve the equation −10(3x + 7) + 4 = 54. Viola solved it like this: −10(3x + 7) + 4 = 54

Given equation

−30x − 70 + 4 = 54 −30x − 66 = 54 −30x − 66 + 66 = 54 + 66 −30x + 0 = 120

Evaluate the addition

−30x = 120 −30x ÷ (−30) = 120 ÷ (−30) 1 × x = −4

Evaluate the division

x = −4 Akim solved the equation like this:

Given equation

Evaluate the addition

Evaluate the division

Evaluate the addition

Evaluate the division

a Use properties of equality and identities to justify each missing step of their work.

Create a strategy From line to line, identify what changed and what operation was used, then ﬁnd the corresponding property or identity.

1.03 Properties of equality mathspace.co

Apply the idea Viola’s work: −10(3x + 7) + 4 = 54

Given equation

−30x − 70 + 4 = 54

Distributive property

−30x − 66 = 54

Evaluate the addition

−30x − 66 + 66 = 54 + 66 −30x + 0 = 120 −30x = 120 −30x ÷ (−30) = 120 ÷ (−30)

Addition property of equality Evaluate the addition Additive identity Division property of equality

1 × x = −4

Evaluate the division

x = −4

Multiplicative identity

Akim’s work: Given equation

Subtraction property of equality

Evaluate the addition

Additive identity

Division property of equality

Evaluate the division

Multiplicative identity

Subtraction property of equality

Evaluate the addition

Additive identity

Division property of equality

Evaluate the division

Multiplicative identity

Reﬂect and check Although Viola and Akim solved the problem in different ways, they both used inverse operations to arrive at the same answer.

b Compare their strategies.

Create a strategy

Apply the idea

We want to consider how their strategies are similar and how they are different. We can look at how many steps were taken and which properties were used.

Viola decided to expand and simplify the left side ﬁrst, then use inverse operations and apply properties of equality. Akim solved the equation by directly using inverse operations from the start. They both arrived at the same answer, but Viola used less steps.

Reﬂect and check We can notice that Viola and Akim used different notation for simple operations. For example, Viola represented division using the ÷ symbol, however, Akim used fractions. Also, Viola used explicit multiplication in their second last step of their working, whereas Akim used implicit multiplication. In both cases, the difference between notation has no impact on the properties or the results of the identities they use.

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Verify that the x = 3 is a solution to the equation 3 − 6x + 2x = −9.

Create a strategy A value is a solution if it can be substituted into the equation and make the equation true.

Apply the idea We will substitute x = 3 into the equation and see if both sides of the equation are the same after evaluating. 3 − 6 ⋅ (3) + 2 ⋅ (3) = −9

Substitute x = 3

3 − 18 + 6 = −9

Evaluate the multiplication

−15 + 6 = −9

Evaluate the subtraction

−9 = −9

Evaluate the addition

Since the two sides of the equation are equal, x = 3 is a solution.

Reﬂect and check Algebra tiles can also be used to verify that a value is a solution to an equation. The original equation 3 − 6x + 2x = −9 is shown. +1

−x

−1

Each +x tile will be replaced by three +1 tiles and −x tiles will change the +1 to −1. +1

−1

A solution will show the same value on both sides of the equals sign after zero pairs cancel out. +1

−1

Since the same number of −1 tiles are shown on each side, x = 3 is a solution to 3 − 6x + 2x = −9.

1.03 Properties of equality mathspace.co

Example 3 Solve the following equations and justify each step. a

Create a strategy Consider how the expression was constructed, starting from the variable. Once this is identiﬁed, we can solve the equation by applying the inverse operations in the reverse order.

Apply the idea Given equation

Subtraction property of equality

Evaluate the subtraction

Additive identity

Division property of equality

Evaluate the division

Multiplicative identity

Multiplication property of equality

Evaluate the multiplication

Multiplicative identity

Reﬂect and check Sometimes, it is easier to simplify the problem before solving for a. In this case, we could have multiplied by 2 first, then solved the equation. Either way, we get the same answer. Given equation

Evaluate the multiplication

Subtraction property of equality

Evaluate the subtraction

Additive identity

Division property of equality

Evaluate the division

Multiplicative identity

Mathspace Virginia SOL Algebra 1 mathspace.co

Create a strategy Since each term on the left hand side of the equation has the variable b in it, we can start by removing the fraction so we can collect like terms.

Apply the idea Given equation

Multiplication property of equality

Evaluate the multiplication

Multiplicative identity

Collect like terms

Multiplication property of equality

Evaluate the multiplication

Reflect and check You may find that the working could have been condensed to something like this: Given equation

Multiplication property of equality

Collect like terms

Multiplication property of equality

Often, steps such as evaluating sums or products, adding 0, or multiplying by 1, are combined into one step.

Create a strategy It is easiest to work with whole numbers, so we can multiply everything by 3 to eliminate the fraction.

Apply the idea

Reflect and check Given equation

Multiplication property of equality

Combine like terms

Subtraction property of equality

Division property of equality

We can check our answer by substituting it back into the original equation. Substitute

Evaluate the multiplication

Evaluate the addition in the numerator

Evaluate the division

Evaluate the addition

Evaluate the division

1.03 Properties of equality mathspace.co

d 0.5x + 2(1.2x + 3) = 11.8

Create a strategy Before solving for x, we need to simplify the left side of the equation.

Apply the idea 0.5x + 2 (1.2x + 3) = 11.8

Given equation

0.5x + 2.4x + 6 = 11.8

Distributive property

2.9x + 6 = 11.8

Combine like terms

2.9x = 5.8 x=2

Subtraction property of equality Division property of equality

Reflect and check If you prefer working with whole numbers instead of decimals, you could have multiplied everything by 10 after the second step. 0.5x + 2(1.2x + 3) = 11.8

Given equation

0.5x + 2.4x + 6 = 11.8

Distributive property

5x + 24x + 60 = 118

Multiplication property of equality

29x + 60 = 118 29x = 58 x=2

Combine like terms Subtraction property of equality Division property of equality

Example 4 Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total tips she received was $32, $27, and $26. She earned twice as much in tips on the fourth night compared to the fifth night. The average amount of tips received per night for the week was $29. If the amount she received on the fifth night was $k, determine how much she received that night.

Create a strategy The average is equal to the sum of values divided by the number of values. We can use this to build an equation to represent the constraint in terms of k about the average tips Yolanda received. Then, we can solve the equation for k.

Apply the idea We can represent her earnings on the fourth night as 2k, as it is twice her earnings on the ﬁfth night, k. Since $29 is equal to the average of her tips over ﬁve nights, we can write an equation representing the average of her tips in terms of k:

where 2k and k represent her earnings on the fourth and ﬁfth night, respectively.

Mathspace Virginia SOL Algebra 1 mathspace.co

Now, we can solve the equation for k. Start with equation in terms of k

Combine like terms in numerator

Commutative property of addition

Multiplication property of equality

Evaluate product on both sides of equation

Multiplicative identity

Subtraction property of equality

Combine like terms on both sides of equation

Additive identity

Division property of equality

Evaluate the division on both sides of the equation

Multiplicative identity

Symmetric property of equality

Yolanda received $20 on the ﬁfth night.

Reflect and check We can check that the solution, k = 20, is correct and makes sense in this context. Firstly, since Yolanda’s earnings from tips on the fifth night is equal to $20, her earnings on the fourth night is equal to $40. If we use these numbers to calculate the average over the five nights, we get: Calculate average over five nights

Collect like terms on numerator

Evaluate the division

We can see that this gives us the same value for the average tips over the five nights as specified in the question. Notice that both of the values for the tips earned on the fourth and fifth night are positive numbers, which makes sense within the context of the question. We may have questioned our solution if we returned a negative value for tips or the average.

Idea summary To identify the properties of equalities needed to solve an expression, consider how an expression was constructed, starting from the variable. Then, solve the equation by applying the inverse operations in the reverse order and matching the operation to the correct property. • Symmetric property of equality: if a = b, then b = a. • Transitive property of equality: if a = b and b = c, then a = c. • Addition property of equality: if a = b, then a + c = b + c. • Subtraction property of equality: if a = b, then a − c = b − c. • Multiplication property of equality: if a = b, then ac = bc. • Division property of equality: if a = b and c ≠ 0, then a ÷ c = b ÷ c. • Substitution property of equality: if a = b, then b may be substituted for a in any expression containing a.

1.03 Properties of equality mathspace.co

Practice What do you remember? 1

Solve the following equations. Justify each step using properties of equality. a

4x = 8

5m = −15

2.5k = −15

Determine the property that justiﬁes why the equations 9x = −27 and x = −3 are equivalent.

Solve the following equations. Justify each step using properties of equality. a

8m + 9 = 65

−10 + 3k = 5

7 − 8t = 15

3x − 4 = 2

8 = 2(1 − x)

iii

−3x + 5x = −8

3(−x + 2) = −3

Match the pictorial model with its equation a

−x +1 +1 +1 +1 +1 +1

−x

−x +1 +1 −x

−x

−1 −1

−1

+x +x

−1

+x +x

−x

−1

−x

−1

Mathspace Virginia SOL Algebra 1 mathspace.co

Determine the property illustrated for each statement. a

If a = d, then a + 3 = d + 3.

Given that b = 16, then b − 5 = 11.

If f = g and g = h, then f = h

Given

it is true that

Write the new equation produced for each of the following scenarios: a

Adding 1 to both sides of x = 9

Multiplying

to both sides of 5m = 3

Subtracting 7 from both sides of x = 14

Dividing by 9 on both sides of 45x = −54

Fill in the blank so that each resulting statement is true. a b

The Addition property of equality states that if a = b, then a + c = ⬚.

The multiplication property of equality states that if a = b and c ≠ 0, then ac = ⬚.

Let’s practice 8

Solve the following equations. Justify each step using properties of equality. a c

6x − 3 = 4x + 7

2.5x + 3x = 13.5 + x

Given the equation: x + 2(x + 3) = 18

Use algebra tiles to draw a model of the equation.

Solve the equation, using the model to justify steps.

A student incorrectly used the distributive property and wrote 7(4x + 3) = 28x + 3. Explain how to correct the error.

Consider the equation a

Dylan started to solve the equation as follows:

Dylan continued to solve the equation as shown: 8x + 48 = 144 8x = 96 Determine the property that justiﬁes this step of work.

Determine the property that justiﬁes Dylan’s ﬁrst step of work. c

Dylan finished solving the equation as follows: 8x = 96 x = 12 Determine the property that justifies Dylan’s final step of work.

For each of the following statements: i

Write the statement as an equation in which x represents the number.

Solve the equation for x, justifying each step using properties of equality.

The sum of a number and 7 is 17.

Seven more than twice a number is 23.

Fifteen minus three-quarters of a number is 9.

he product of 5 and the sum of a number and T 7 equals 50.

The quotient of a number and −3 is −20.

1.03 Properties of equality mathspace.co

Consider the given triangle which has a perimeter of 171 cm. Solve for the value of x.

8x cm

5x + 76 cm

6x + 19 cm

Consider the equation 21 = x + 13. a

Solve for x.

Kathleen started to solve the equation as follows:

Kathleen ﬁnished solving the equation as shown: 21 − 13 = x + 13 − 13

21 = x + 13

8=x+0

21 − 13 = x + 13 − 13

8=x

Determine the property that justiﬁes this step of work.

Determine the property that justiﬁes Kathleen’s ﬁnal step of work.

Consider the equation a b

Solve for x. i Irene started solving the equation as follows:

Determine the property that justiﬁes this step of work. iii

Irene continued solving the equation as follows:

Determine the property that justiﬁes this step of work. iv Irene ﬁnished solving the equation as follows: 1x = 63 x = 63

Determine the property that justiﬁes this step of work. 16

Determine the property that justiﬁes this step of work

Ursula is solving for the missing leg length, x, of an isosceles right triangle with an area of 200 cm2. She solves the problem as follows:

Since 202 = 400, x = 20 cm Determine the property that justiﬁes Ursula’s ﬁnal step.

Mathspace Virginia SOL Algebra 1 mathspace.co

Consider the equation a

Solve for x.

Susana started solving the equation as follows:

Determine the property that justiﬁes Susana’s ﬁrst step of work. c

Susana continued to solve the equation as shown:

Determine the property that justiﬁes this step of work. d

Susana continued to solve the equation as shown: 4(x + 6) = 4 x+6=1 Determine the property that justiﬁes this step of work.

Susana finished solving the equation as follows: x+6=1 x = −5 Determine the property that justifies Susana’s final step of work.

Let’s extend our thinking 18

Athena and Emilio want to ride go-karts. It costs 50 cents per lap of the course. a

Write an equation to solve for the number of laps Athena and Emilio can afford if they have $12.

Solve for the number of laps they can afford. Justify each step using properties of equality.

Explain how that equation changes if they have to put a deposit of $1 on each cart used.

Give examples of the following properties using equations: a

Addition property of equality or Subtraction property of equality

Distributive property of equality

Transitive property of equality

Symmetric property of equality

Multiplication property of equality or Division property of equality

Use the Substitution property of equality to solve for x given y = x: a

14x − 2y = 36

−0.6x + 1.70y = 22

1.03 Properties of equality mathspace.co

If the equation simplifies to variables with the same coefficients and the same constants on each side of the equality, it will have infinitely many solutions. +x

+1 +1

+1 +1 +1

+1 +1

+1 +1 +1

The equation represented is 3(x + 2) = x + 2(x + 3), which has infinite solutions. When each side is simplified, the equation becomes 3x + 6 = 3x + 6, shown on the balance by three +x tiles and six +1 tiles. Regardless of what value of x is chosen for the +x tile, there are equal numbers of +x tiles on each side that will be added to the constants. The number of +1 tiles are also equal on each side. This creates a balanced scale regardless of the value chosen for x. If the equation simplifies to variables with the same coefficients but different constants, it will have no solution. The equation represented is 3(x + 2) = 3x − 2, which has no solution.

+1 +1

−1

+1 +1

−1

Similar to the previous example, regardless of the value of x chosen, the value represented by the +x tiles on each side of the equality will always be the same. However, the constants on either side of the equality are not the same. While the value of the +x tiles will always match, the constants keep the two sides from balancing.

If the equation simplifies to variables with different coefficients, there will be a unique solution regardless of constant values. +x

+1 +1

−1 −1 −1

+1 +1

The equation represented is 3(x + 2) = 4x − 3, which has a unique solution of x = 9.

+1 +1

When simplified, the equation is 3x + 6 = 4x − 3. In our other examples, attempting to add or remove tiles to isolate a variable would eliminate the variable entirely. Each side of this equation has a different number of +x tiles, so we can add and remove tiles equally to work towards isolating the variable to find the solution.

While there is more than one way to algebraically solve an equation, it is important to keep both sides of an equation equivalent by remembering that what you do to one side, you must do to the other.

1.04 Multistep equations mathspace.co

Example 1 Verify that the m = −4 is a solution to the equation 14 − 7m = −3(−2 + 3m).

Create a strategy Both sides of an equation should be equivalent after substituting a solution for the variable and evaluating.

Apply the idea Substitute m = −4 and use the order of operations to determine if both sides are equivalent. 14 − 7 ⋅ (−4) = −3(−2 + 3 ⋅ (−4))

Substitute m = −4

14 + 28 = −3(−2 − 12)

Evaluate the multiplication

14 + 28 = −3 ⋅ (−14)

Evaluate the subtraction

42 = 42

Evaluate the multiplication

Since both sides are equivalent, m = −4 is a solution to 14 − 7m = −3(−2 + 3m).

Example 2 Use a balance scale and algebra tiles to solve 3(x − 4) = 2(−2x + 1).

Create a strategy A balance scale represents the algebraic expressions on each side of an equation using +x, −x, +1, and −1 tiles. We work towards isolating a variable by adding and removing tiles on each side equally.

Apply the idea The original equation 3(x − 4) = 2(−2x + 1) is represented by the balance scale: +x

−1 −1 −1 −1

−x

−1 −1 −1 −1

−x

Adding +4x to each side creates zero pairs for the −4x on the right side. +x +x +x +x

−1 −1 −1 −1

−x

−1 −1 −1 −1

−x

Mathspace Virginia SOL Algebra 1 mathspace.co

Next, we remove the zero pairs on the right side, and the scale stays balanced. +x +x +x +x +x

−1 −1 −1 −1

Adding +12 to each side creates zero pairs for the −12 on the left side. +x +x

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

Remove the zero pairs from the left and the scale stays balanced. Since there are seven +x tiles on the left side, splitting the +1 tiles on the right into seven equal groups will tell us what each +x tile represents. +x +x

+1 +1 +1 +1 +1 +1 +1

Remove +x tiles from the left and two +1 tiles from the right at the same time to keep the scale balanced. Continue doing this until only one +x tile remains. +x

+1 +1

We can see that x = 2.

1.04 Multistep equations mathspace.co

Reflect and check Balance scales can also be used to verify solutions to equations. For this example, to verify that x = 2 is the solution to 3(x − 4) = 2(−2x + 1), each +x will be replaced with two +1 tiles and each −x will be replaced with two −1 tiles. +1 +1

−1 −1

+1 +1

−1 −1

−1 −1 −1

−1 +1

+1 +1

−1 −1

−1 −1 −1

−1 +1

After making zero pairs, a solution of an equation should show the same number of tiles left on each side of the balance. −1 −1 −1

−1 −1 −1

Since there are six −1 tiles on each side of the balance, x = 2 is a solution to 3(x − 4) = 2(−2x + 1).

Example 3 Determine how many solutions the following equations have without solving. a 4(x − 9) = x + 6

Create a strategy Start by comparing both sides of the equation. We can see both sides are different and that there is an x on both sides. The coeﬃcient on the left side is 4, and the coeﬃcient on the right side is 1.

Apply the idea

Reﬂect and check

The equation will have one unique solution.

We can verify our answer by solving the equation. 4(x − 9) = x + 6

Original equation

4x − 36 = x + 6

Distributive property

3x − 36 = 6

Subtraction property of equality

3x = 42

Addition property of equality

x = 14

Division property of equality

There is only one solution to this equation.

Mathspace Virginia SOL Algebra 1 mathspace.co

b 2x − 5 = 0.5(4x + 10)

Create a strategy The first thing we notice is that both sides seem to have different values. But if we distribute the coefficient on the right side, the first term would be 2x, and the second term would be 5. The equation now has the same variables with the same coefficients on both sides, but the constant values are different.

Apply the idea

Reﬂect and check

No solution

We can verify our answer by solving the equation. 2x − 5 = 0.5(4x + 10)

Original equation

2x − 5 = 2x + 5

Distributive property

−5 = 5

Subtraction property of equality

The resulting statement is not true. This is how we know there is no solution to this equation.

Example 4 Solve the following equations: a 4(5x + 1) = −3(5x − 5)

Create a strategy Looking at the equation, we see that the variables have different coefficients on both sides of the equation. The coefficient on the left side is 4 ⋅ 5 = 20, and the coefficient on the right side is −3 ⋅ 5 = −15. This means it will have a unique solution.

Apply the idea 4(5x + 1) = −3(5x − 5) Given equation 20x + 4 = −15x + 15 Distributive property 35x + 4 = 15 Addition property of equality 35x = 11 Subtraction property of equality x=

Division property of equality

Reﬂect and check We can verify the solution by substituting it back into the equation to see if it makes the equation true. Original equation Substitute x =

Multiply 5 ⋅

Create common denominators

Evaluate the addition

Evaluate the multiplication

What results is a true equation, so we know

is the correct solution.

1.04 Multistep equations mathspace.co

Create a strategy This equation has a mix of rational numbers, and the variables are in multiple terms. If we multiply both sides of the equation by the lowest common denominator, that will remove fractions and only leave us with decimals. Then, we can work on collecting all the variables.

Apply the idea The denominators are 2 and 4, so the lowest common denominator is 4. Original equation

Multiplication property of equality

Distributive property

Combine like terms

Subtraction property of equality

Division property of equality

Reﬂect and check There are many other ways we could have started solving this equation, but eliminating the fractions ﬁrst made it much easier to solve.

Example 5 Right now, Bianca’s father is 48 years older than Bianca. Two years ago, her father was 5 times as old as she was. Solve for y, Bianca’s current age.

Create a strategy We want to write expressions representing Bianca’s age and her father’s age. Then we relate them with an equation and solve for y. Bianca’s father is currently y + 48 years old. Two years ago, Bianca was y − 2 years old. Her father was y + 48 − 2 = y + 46 years old. Her father’s age was ﬁve times her age at this time, which produces our equation: y + 46 = 5( y − 2)

Apply the idea

Reﬂect and check

y + 46 = 5( y − 2)

Original equation

y + 46 = 5y – 10

Distributive property

y + 56 = 5y

Addition property of equality

56 = 4y

Subtraction property of equality

14 = y

Division property of equality

Bianca is currently 14 years old.

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s check by referring back to the information given in the problem. Bianca’s father is 48 years older, so he is 14 + 48 = 62 years old. Two years ago, he was 5 times as old. Two years ago, Bianca would have been 12, and her father would have been 60. Is this 5 times Bianca’s age? Yes, this is correct.

Idea summary A solution to an equation is any value that can replace the variable and make a true statement. A fully simplified equation in one variable will take one of the following three forms, corresponding to how many solutions the equation has: • x = a, where a is a number (a unique solution) • a = a, where a is a number (inﬁnitely many solutions) • a = b, where a and b are different numbers (no solutions) We can simplify an equation using the distributive property or the multiplication property of equality to eliminate fractions or decimals. After simplifying, we can continue using properties of equality to solve for the unknown.

Practice What do you remember? 1

Solve the following equations: a

m + m + 3 + 12 = 13 + m + m + m

x+x+x+x+2+5=x+4+x+x

p+p−3+5=p−p−p+1–5

−k − k − k − 6 = −k − k − k − k + 6

Determine the number of solutions of the following equations: a

7(7 + x) = 2x + 28

c 3

10(5 + x) = 10(x + 5)

−31 − 3(x − 9) = 2(x − 2) − 5x

Consider the equation 9(n + 5) = 4n + 50 for n = 1. a

Find the value of the left-hand side of the equation when n = 1.

Find the value of the right-hand side of the equation when n = 1.

Consider your answers from parts (a) and (b). Is n = 1 a solution of 9(n + 5) = 4n + 50?

Consider the equation 0.28n − 0.4 = 2.3n − 3.03 for n = 0.8. a

Find the value of the left-hand side of the equation when n = 0.8.

Find the value of the right-hand side of the equation when n = 0.8.

Consider your answers from parts (a) and (b). Is n = 0.8 a solution of the equation 0.28n − 0.4 = 2.3n − 3.03?

Solve the following equations: a

3f − 8 = f

10r + 4 = 6r

4w + 24 = w + 15

−72 − 9p = −32 − p

e g 6

3.2y + 17 = 52 − 1.8y

f h

Determine the value that makes each equation true. a e

6 − 7p = 41

b f

−11 = 2x − 5

c g

35 = 5(n − 15)

−2(p − 8) = 16

1 = −( y + 10)

1.04 Multistep equations mathspace.co

A number is multiplied by 5 and then 2 is added. Then the result is multiplied by 6. This is equal to 10 times the number minus 8. a

Form an equation for this problem.

Solve the equation to ﬁnd the number.

Consider the given rectangle with a perimeter of 126 + 3y centimeters. 4y + 8

4y + 3

Write an equation and solve for the value of y. 18

A Payroll Oﬃcer has been told to distribute a bonus to the employees of a company worth 15% of the company’s net income. Since the bonus is an expense to the company, it must be subtracted from the income to determine the net income. If the company has an income of $120 000 before the bonus, then the Payroll Oﬃcer must solve the following for B: B = 0.15 (120 000 − B) Find the bonus, B.

A rectangle with a length of 2.5x + 4 cm and a width of x − 1 cm has the same perimeter as a square with side length

Write an equation and solve for the value of x.

Let’s extend our thinking 20

Three consecutive integers are such that the sum of the first and twice the second is 15 more than twice the third. a

Let x be the smallest of the integers. Solve for x.

State the three consecutive integers.

Are there any other sets of three consecutive integers that could fulfill the requirements? Explain how you know.

Determine whether the following statements are true or false. Explain your thinking. a

Any equation with variables on both sides must have multiple solutions.

Two equations will always have the same solution if one is a multiple of the other.

Explain the criteria for identifying the number of solutions that an equation has, and provide an example for each.

1.04 Multistep equations mathspace.co

1.05 Literal equations After this lesson, you will be able to… • solve linear equations in one variable, including equations with coefficients represented by letters. • solve for a specific variable in a formula.

Literal equations Literal equation

Formula

A formula or equation that consists primarily of variables.

A type of literal equation that describes a relationship between real-world quantities. Example: A = l ⋅ w where A is area, l is length, and w is width.

There are many different formulas in science, mathematics, business, and other subjects that allow us to measure quantities such as area, volume, speed, etc. We can use the properties of equality to isolate any variable in a literal equation or formula. The same variable might be used to represent different quantities across different formulas. For example, in the formula for the area of a rectangle, w is used to represent the width of the rectangle. However, in another context, w might be used to represent a weight or other value. To avoid any confusion, formulas will always state what the variables represent. A variable can also act as a placeholder for other expressions when a formula applies in a variety of different situations. For example, the formula for the volume of both a cone and a pyramid is

where B represents the area of the two-dimensional shape at the base and h is the vertical height from the base. To find the volume of each specific figure, we replace the B with the area formulas for the base shapes.

height (h) h r

A = s2 s

Mathspace Virginia SOL Algebra 1 mathspace.co

A = π r2

Exploration Consider the formula for distance:

d = rt d

distance r

rate t

time

Use the distance formula to solve each of the following: • The speed at which a person travels if they drive 100 mi in 75 min • The distance traveled when walking 5 km in 30 min • The distance traveled by a car driving 65 mph for 5 hr • The time it will take to travel 1000 km if you walk 80 m/min • How fast a plane is traveling if it can ﬂy 2789 mi in 6 hr • How long it will take to walk to the store 2 mi away if you walk 176 ft/min 1.

Which ones took the most effort to solve?

How could we reduce the effort when making repeated calculations for the same variable?

Using the division property of equality, we can rearrange the equation relating distance, rate, and time to be or By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, depending on which variable is unknown.

Example 1 Ohm’s law states: V = IR where V is voltage, I is current, and R is resistance. Write the formula for current.

Create a strategy The formula for current is Ohm’s law with I isolated. We use inverse operations and properties of equality to get the solution.

Apply the idea Given equation

Division property of equality

Symmetric property of equality

Example 2 Solve for x in the following equation:

Create a strategy We need to rearrange the equation to isolate x. We can use the properties of equality and inverse operations to solve literal equations for a variable, just as we would for linear equations.

1.05 Literal equations mathspace.co

Apply the idea

Reﬂect and check Given equation

Division property of equality

Subtraction property of equality

Multiplication property of equality

Symmetric property of equality

Remember, when rearranging an equation, we reverse the operations acting on the variable we want to isolate, in the reverse order of operations. Whatever is done to one side of the equation, must be done to the other to keep the equation balanced.

Example 3 To find the sum, S, of the interior angles of any polygon with n sides, the following equation can be used: S = 180(n − 2) a A polygon’s angles sum to 900°. Use the sum of interior angles formula to determine its number of sides.

Create a strategy Since S is the sum of the angles, replace the S in S = 180(n − 2) with 900 and solve.

Apply the idea

Reﬂect and check

Since the angles sum to 900°, the S in the formula will be An alternative but equivalent way to solve for n once S is repleaced with 900 and we will solve for n. replaced with 900 is to distribute first then solve. Replace S with 900

Divide both sides by 180

Add 2 to each side

Simplify

Replace S with 900

Distribute

Add 360 to both side

Divide by 180

A polygon whose angles sum to 900° has 7 sides b Write an equation to solve for n using the properties of equality.

Create a strategy We need to rearrange the equation to isolate n. We can use the properties of equality and inverse operations to get the solution.

Apply the idea

Reﬂect and check Given equation

Equations can be represented multiple ways, and the

Distributive property

simplified equation

Addition property of equality

equation found in the second to last step,

Division property of equality

After applying the reflexive property,

Reflexive property

The equation S = (n − 2) solved for n is

Mathspace Virginia SOL Algebra 1 mathspace.co

is also equivalent to the

represents an equivalent and correct way to solve for n.

c Use the equation from part (b) to find the number of sides of a polygon whose angles sum to 1440°.

Create a strategy Replace S with 1440 and evaluate.

Apply the idea

Reﬂect and check Original equation

Using the original equation equation S = 180(n − 2) can also be used to find the number of sides, and will give the

Substitute S = 1440

same value as

Divide

Original equation

Simplify

Replace S with 1440

A polygon whose angles sum to 1440° has 10 sides.

Distribute

Add 360 to both sides

Divide both sides by 180

Simplify

Using the original equation helps confirm the correct steps were taken to isolate the needed variable.

Idea summary In the same way we solve one-variable equations, we can use inverse operations and the properties of equality to isolate a variable in a literal equation.

Practice What do you remember? 1

What beneﬁts come from rewriting literal equations?

Which of the following are literal equations? A D

3x + 6y = 18

C = 2π r

A = 1 5π

List the variables in each equation: a

12 = 4x − (−6 + 2x)

P = 4s

V = π r2h

The formula for the area of a triangle is

180(n − 2) = s

, where b is the base length of the triangle, and h is the height.

Find the area of the triangle with a base length of 6 cm and a height of 14 cm.

Find the height of a triangle with an area of 45 cm2 and a base length of 15 cm.

Find the base length of a triangle with a height of 12.4 cm and an area of 62 cm2.

Solve for x in each of the following equations: a

x−7=y

−2x = 4m

8y = x + 2z

1.05 Literal equations mathspace.co

Write an equation that could be used to ﬁnd v0.

The displacement of an object is given by

Write an equation that could be used to solve for r2 for each of the following formulas:

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

The volume of a cone is given by the formula

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

When the heater in a house is on a setting of s, the temperature, T, of the house within the ﬁrst 30 minutes can be estimated by using the formula: Where a is the initial temperature and t is the number of minutes since turning the heater on.

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

Find the temperature setting required to reach a room temperature of 70 °F after 25 minutes when the initial temperature is 50 °F.

The surface area of a rectangular prism is given by formula S = 2(lw + wh + lh), where l, w and h are the dimensions of the prism. a

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

Find the length of a rectangular prism with a width of 4 cm, a height of 3.5 cm and a surface area of 73 cm2.

Let’s extend our thinking 21

The resistance of a parallel circuit is given by the formula a

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

Find the value of b, correct to two decimal places, if a = 8 and c = 23.

Solve for k in the following equations: a

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

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

Determine whether Rufino is correct. Explain each correct step of his work, otherwise fix his error.

1.05 Literal equations mathspace.co

Example 1 Consider the inequality

a Solve the inequality.

Create a strategy We want to isolate x on one side of the inequality and a number on the other.

Apply the idea Original inequality

Multiplication property of inequality

Addition property of inequality

Division property of inequality

The solution can be written as x ≥ −6, {x | x ≥ −6}, or [−6, ∞).

Reﬂect and check Solving an inequality is similar to solving an equation. However, we need to reverse the direction of the inequality when multiplying or dividing by a negative number. b Plot the inequality on a number line.

Apply the idea Plot the solution set of the inequality x ≥ −6. Note that since we include −6 the point should be ﬁlled. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Reflect and check What if the solution was x > −6? Endpoints included in the solution are filled points. Endpoints not included in the solution are unfilled points. c Is x = 3 a viable or nonviable solution to the inequality?

Create a strategy We can determine if x = 3 is viable or non-viable by using the number line or algebraically substituting the solution into the inequality.

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea Original inequality

Substitute x = 3

Evaluate the multiplication

Evaluate the subtraction

Evaluate the division

Since x = 3 leads to a true statement, we can conﬁrm that x = 3 is a viable solution to the inequality.

Reﬂect and check By using the number line, we can see that the point x = 3 is in the solution set of the inequality, meaning it is a viable solution to the inequality. Any points that are not in the solution set are considered non-viable and will lead to a false statement when substituted into the inequality algebraically.

Example 2 Calandra charges $37.72 to style hair, as well as an additional $6 per foil. Pauline would like the total cost for her styling to be no more than $95.86. a Write an inequality that represents the number of foils Pauline could get.

Create a strategy Pauline has no more than $95.86 to spend. “No more than” means “less than or equal to.”

Apply the idea We can write an inequality in words that represents the cost to style Pauline’s hair: cost of styling + cost per foil ⋅ number of foils ≤ total Pauline can spend Translating that into an algebraic expression we get: 37.72 + 6N ≤ 95.86 where N represents the number of foils.

b How many foils could Pauline get and still afford the styling?

Create a strategy Solve the inequality and then write the solution set.

Apply the idea 37.72 + 6N ≤ 95.86

Original inequality

6N ≤ 58.14

Subtraction property of inequality

N ≤ 9.69

Division property of inequality

According to the solution, Pauline could get 9.69 foils or fewer. However, since she can’t get a partial foil, a more realistic solution is that she can get 9 foils or fewer.

1.06 Multistep inequalities mathspace.co

c Determine whether N = −2 is a viable solution to the inequality in the context of the question.

Create a strategy Keep in mind it is not realistic to get part of a foil or a negative number of foils.

Apply the idea Pauline can get a maximum of 9 foils and a minimum of 0 foils, so while −2 is mathematically part of the solution set for the inequality N ≤ 9.69 it is not a viable solution in this context.

Reﬂect and check Unlike a value that is not in a solution set of an inequality, this is an example of a solution that was mathematically valid and part of the original solution set but when considering the context we have found that it is non-viable.

Example 3 Solve the inequality 4(x + 5) < 3(2 − x).

Create a strategy To solve this inequality, we need to simplify each side, isolate x on one side, and then use the inequality to find the solution. We can start by distributing the 4 and 3 on each side of the inequality.

Apply the idea 4(x + 5) < 3(2 − x)

Original inequality

4x + 20 < 6 − 3x

Distributive property

4x + 3x < 6 − 20

Add 3x to both sides and subtract 20 from both sides

7x < −14

Simplify both sides

x < −2

Divide both sides by 7

The solution to the inequality can be written as x < −2, {x | x < −2}, or (−∞, −2).

Reflect and check To check our solution, we can substitute a value less than −2 into the original inequality and see if it holds true. Let’s take x = −3 as an example. 4 (− 3 + 5) < 3(2 − (−3)) 4⋅2<3⋅5 8 < 15

Substitute x = −3 into the original inequality Simplify the expressions Continue simplifying the expressions

The inequality 8 < 15 is true, so our solution x < −2 is correct.

Mathspace Virginia SOL Algebra 1 mathspace.co

Idea summary Just like the properties of equality, the properties of inequality can justify how we solve inequalities. The multiplication and division properties of inequality change the meaning of an inequality when multiplying or dividing by a negative number, meaning we have to reverse the inequality symbol when applying the property: • If a > b and c < 0, then a ⋅ c < b ⋅ c • If a > b and c < 0, then The solution set of an inequality is the set of values that makes the inequality true. Because inequalities can have infinitely many solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.

Practice What do you remember? 1

Plot the following inequalities on a number line: a

x ≥ 29 x ≤ −4

x ≤ −1

x < −29

x≤1

x≥4

2 3 4 5

−5 −4 −3 −2 −1 0

Which of the following number lines represent the solution for 4x − 7 < 5? A

x ≤ 15

Which of the following inequalities represents the solution for x in 10 ≤ 6 − 4x? A

Describe the range of values that satisfy each inequality. a

x > −9

−5 −4 −3 −2 −1 0

2 3 4 5

−5 −4 −3 −2 −1 0

2 3 4 5

Write each of the following relations as an inequality using mathematical symbols: a

The sum of 3 groups of p, and 9, is less than 24.

The sum of 5 times x, and 3 is at least 23.

Six more than the value of x is at least seven.

Half of x is no more than ﬁve.

The product of negative four and x is at most three.

For the following inequalities: i

Solve for x.

Plot the solutions on a number line.

3x − 7 < 8

−6x − 7 ≤ 5

1.5x + 8 > 12.5

2 − 3.6x < 20.9

4 < 6x − 2

2(x − 3) < −16

Which of the following inequalities represents the solution for r in “5 more than 2r is less than 39”? A

r > 17

r < 17

r > 22

r < 22

1.06 Multistep inequalities mathspace.co

Let’s practice 8

Consider the inequality: 5(x + 3) ≤ 35. a

Solve for x.

State whether the following solutions are viable or nonviable: i

x = −4

iii

x=4

x=0

x=2

Consider the inequality: 4 − 2x < 3x − 2. a

Solve for x.

State whether the following solutions are viable or nonviable: ii

i 10

x=8

x=1

iii

Which of the following are values in the solution set of the inequality shown? Select all that apply. −3.5(2 − 6x) < −27 + x

−1

5x − 40 ≥ 50

SOL

−2

−8 − m > 3

−8(x + 8) ≥ −40

d h

Find the solution to each inequality, writing each answer in set builder notation. a

Solve the following inequalities and justify each step using properties of inequality: a

−m + 7(−3m + 4) ≥ 4 + 2m

−1 + 4b ≤ 2b − 18 + b

Solve each inequality, writing each answer in interval notation. a

42 − 3y ≥ 74 − y

9x − (12 − x) ≥ − 6x

27 + 3a < −5 + 2(7a − 6)

Consider the situation: “3 less than 3 groups of p is no more than 24”. a

Write the relation as an inequality.

Solve the inequality.

Find the largest value p can take.

Graph the solution to the inequality on the number line provided. 3 − (x + 11) ≥ −5x −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9 10

Skye has a budget for school stationary of $43, but has already spent $21.14 on books and folders. Let p represent the amount that Skye can spend on other stationery. Write the budget constraint as an inequality and solve for p.

James is saving up to buy a laptop that is selling for $550. He has $410 in his bank account and expects a nice sum of money for his birthday next month.

Write the inequality that models the situation in which James can afford the laptop. Let x represents the amount he is to receive for his birthday.

Plot the solution to the inequality on a number line.

Mathspace Virginia SOL Algebra 1 mathspace.co

To get a grade of C, Uther must obtain an average score of at least 75 over his four exams. So far he has taken the ﬁrst three exams and achieved scores of 68, 60, and 86. a

Write the inequality that models the situation for the score, x, Uther must obtain on his last exam to get a C or better.

Solve for x.

Describe the solution regarding Uther’s score.

Let’s extend our thinking 19

Ryan wants to save up enough money so that he can buy a new sports equipment set, which costs $40.00. Ryan has $22.10 that he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows for $2 per window. a

Let w be the number of windows that Ryan washes. Write an inequality to represent the situation.

Solve for w, correct to two decimal places.

State whether each statement is correct. Explain your thinking. i

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.

iii If Ryan washed 8 windows, and 95% of another window, he could afford the equipment. iv The number of windows Ryan must wash to be able to afford the equipment must be greater than or equal to 9. 20

Rochelle tried to solve the following inequality but made a mistake in her work: Step 0: −4 − 2x > 10 Step 1:

−2x > 14

Step 2:

x > −7

Determine which step is incorrect and explain the error. 21

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 errors and explain how to rectify them.

1.06 Multistep inequalities mathspace.co

2 Functions & Relations Big ideas • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. • Functions provide a representation for how related quantities vary. This makes functions a good way to represent many real world situations.

Chapter outline 2.01 2.02 2.03 2.04

Functions and relations Domain and range Evaluating functions Characteristics of functions

56 71 82 93

2.01 Functions and relations After this lesson, you will be able to… • determine whether a relation is a function.

Relations A relation is a set of ordered pairs which represent a relationship. For example, we can think of the names of people in a math class and their ages as ordered pairs, like (Bob, 13). These pairs of information represent a relation. If we chose a speciﬁc age (like 13), we could list all the names of the people who are this age. It could be one person, Bob, or it could be multiple. If a teacher wanted to look for the person who was 13 years old, that description might ﬁt four people which means there’s not one clear answer. We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph. A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below: A mapping of a relation

−1

We can write an input-output table from the mapping, making sure that each pair is represented. Remember that the ﬁrst value of a relation is an input value and the second value is the output value. The input is the value of x that is applied to the relation. The output is the y, or the answer that is received as a result of putting x into the relation. A table can be laid out horizontally (like the one shown below) or vertically. x y

−1 2

0 0

1 2

2 4

This also corresponds to the set of ordered pairs {(−1, 2), (0, 0), (1, 2), (2, 4)}, which can be graphed in the coordinate plane, as shown below. 4

A graph of the relation represents the (x, y) pairs in the coordinate plane.

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 Write the relation {(2, 2), (4, 4), (6, 3), (7, 5)} in the table below. x y

Create a strategy

Apply the idea

Write the second coordinate of each ordered pair in the y row, below the x-value it corresponds to.

x y

2 2

4 4

6 3

7 5

Example 2 Consider the relation: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}. Represent the relation on the coordinate plane.

Create a strategy

Apply the idea

The ﬁrst value of each ordered pair tells us how to move along the x-axis, while the second value tells us how to move along the y-axis.

6 4 2 −8

−6

−4

−2

x 2

−2 −4 −6 −8

−10

Example 3 A relation is deﬁned as follows: y = −4 if x is positive and y = 4 if x is 0 or negative. a Complete the table. x y

−4

−3

−2

−1

Create a strategy

Apply the idea

For each positive x-value: y = −4, otherwise y = 4.

When x = 1, 2, 3, 4: y = −4. When x = 0, −1, −2, −3, −4: y = 4. x y

−4 4

−3 4

−2 4

−1 4

0 4

1 −4

2 −4

3 −4

4 −4

2.01 Functions and relations mathspace.co

Graphs can also show one input (x-value) paired with one output (y-value ). By graphing the previous set of ordered pairs, we see the graph of a function and the graph not representing a function. 4

−4 −3 −2 −1 −1

−2

−2 −3

−3

−4

Function 4

Not a function

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Function

x 1

Not a function

Example 4 x

This mapping shows the relation F.

1 2 3 4

a Find the output when x = 1.

Create a strategy

Apply the idea

We start from the input oval labeled x and follow the line(s) from 1 to the output value(s).

When x = 1, y = −1.

b Determine if F is a function.

Create a strategy For a function, each x-value maps to a unique y-value.

Mathspace Virginia SOL Algebra 1 mathspace.co

y −1 0 1 2

Apply the idea

Reflect and check

If we input x = 2 into the mapping, we get both y = 0 and y = 1. This means that F is not a function.

To be a function, each input value should only map to one output value. For example, G would be a function: x 1 2 3 4

y −1 0 2

Example 5 The pairs of values in the table represent a relation between x and y. x y

−8 8

−7 13

−6 −18

−3 −16

2 −15

7 −2

9 −4

9 11

10 −9

Do they represent a function?

Create a strategy

Apply the idea

The relation is a function if for every x-value, there is exactly one y-value.

The x-value of 9 yields the y-values of −4 and 11. So, the points do not represent a function.

Example 6 Oprah makes scarves to sell at the market. It costs her $2 to produce each one, and she sells them for $5. a Complete the graph of the points representing the relation between the number of scarves she manages to sell and her total proﬁt for when 1, 2, 3, 4 and 5 scarves are sold. The ﬁrst point has been plotted for you.

Profit

16 14 12 10 8 6 4 2 −8 −6 −4 −2

Quantity 2 4 6 8 10

Create a strategy The total proﬁt can be found using the formula: Total profit = Total revenue − Total cost

2.01 Functions and relations mathspace.co

Apply the idea Total revenue = Number of scarves sold ⋅ $5 Total cost = Number of scarves sold ⋅ $2 Profit for 1 scarf = 1 ⋅ 5 − 1 ⋅ 2

Substitute the number of scarves

= $3

Evaluate

Profit for 2 scarves = 2 ⋅ 5 − 2 ⋅ 2

Substitute the number of scarves

= $6

Evaluate

Profit for 3 scarves = 3 ⋅ 5 − 3 ⋅ 2

Substitute the number of scarves

= $9

Evaluate

Profit for 4 scarves = 4 ⋅ 5 − 4 ⋅ 2

Substitute the number of scarves

= $12

Evaluate

Profit for 5 scarves = 5 ⋅ 5 − 5 ⋅ 2

Substitute the number of scarves

= $15

Evaluate

Plot the pairs of values found. 18

Profit

(5, 15)

14 12

(4, 12)

(3, 9)

8 6

(2, 6)

(1, 3)

Quantity 1

2 3 4 5 6

7 8 9

b Is this relation a function?

Create a strategy This relation is a function if for every quantity sold, there is exactly one total profit.

Apply the idea Checking each pair of values in the graph, each quantity of scarves sold is associated with only one total profit, so this relation does represent a function.

Idea summary A relation is a function if and only if each element in the domain is paired with a unique element of the range.

Mathspace Virginia SOL Algebra 1 mathspace.co

The vertical line test Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 2.01 to answer these questions. 1.

What similarities did you notice in the graphs that were labeled as functions?

How did the vertical line help you determine which graphs represented functions?

Would a horizontal line be useful in determining if a relation is a function?

Sometimes it is easier to investigate the graph of a relation to determine whether or not it is a function. When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then it is not a function. Vertical line test The graph of a relation is a function if a vertical line intersects the graph of a relation at exactly one point across the entire graph. Here are two examples of relations being checked with the vertical line test. A function is said to “pass the vertical line test” while a relation that is not a function “fails the vertical line test.” 7 6 5 4 3 2 1 −7−6−5−4−3−2 −1 −1

y 4 3 2 x

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Fails the vertical line test (is not a function)

1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Passes the vertical line test (is a function)

One pair of points is enough to decide that a relation is not a function, but it is not enough to decide that a relation is a function. We must keep checking points on the graph until it either fails the test or we have checked for all x-values. When classifying, remember that every function is a relation, but not every relation is a function.

2.01 Functions and relations mathspace.co

Example 7 Determine whether the following graphs show functions. a

6 5 4 3 2 1

−6 −5 −4 −3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

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

x 1 2 3 4 5 6

Each vertical line passes through the graph only once, so the graph is a function.

y 4 3 2 1 −8 −6 −4 −2

−1

x 2

−2 −3 −4

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea 4

3 2 1 −8 −6 −4 −2

x 2

−1

−2 −3 −4

The vertical lines shown cross the graph at multiple points, so the graph is not a function.

c 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

Apply the idea 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Each vertical line passes through the graph only once, so the graph is a function.

2.01 Functions and relations mathspace.co

Idea summary While all functions are relations, not all relations are functions. The vertical line test for functions: 5 4 3 2 1

When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then it is not a function.

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Practice What do you remember? 1

−3

−2

State the inputs.

−1

−2

Complete the following table: x y

−3

Consider the following mapping:

−1

State the outputs.

If the input is −1, what is the corresponding output?

Represent each relationship of the inputs and outputs as a set of ordered pairs.

Input

Output

Determine whether the following statements are true or false: a

When working with a function, substituting a certain value of x into the formula gives only one value of y for that value of x.

All relations are functions.

Some relations are functions.

All functions are relations.

No functions are relations.

No relations are functions.

A relation always passes the vertical line test.

The graph of every straight line is a function.

The graph of every non-vertical straight line is a function.

There is no straight line graph that is a function.

The graph of every straight line is a relation.

The graph of every non-horizontal straight line is a function.

Mathspace Virginia SOL Algebra 1 mathspace.co

Identify the name used to describe a graph where for some value of x, there exists two or more different values of y.

Express the following relations in a table of values: a

For each of the following relations represented by a table, represent the relation on a coordinate plane: a

{(1, 1), (3, 4), (5, 6), (7, 8)}

x 5 10 15 20 y 15 30 45 60

25 75

−4 −4

x y

−3 3

−2 −2

−1 1

0 0

1 1

2 −2

3 3

4 −4

Express each of the following relations as a series of points on the coordinate plane: a

{(2, 5), (2, 7), (−3, −4), (−9, 13)}

{(1, 5), (7, −2), (−5, −10), (13, −13)}

Let’s practice 8

Determine whether each of the following sets of points represents a function or not.: a

{(2, 5), (7, −3), (5, 2), (−4, −9)}

{(2, 5), (2, 7), (−3, −4), (−9, 13)}

{(1, 5), (1, 1), (7, −2), (−5, −10)}

{(1, 5), (7, −2), (−5, −10), (13, −13)}

{(1, 5), (1, 7), (−2, −5), (−5, −10)}

{(−1, −9), (0, 0), (1, 9), (2, 18)}

{(−2, 4), (−1, 1), (2, 4), (6, 36)}

The pairs of values in the table represent a relation between x and y. Determine whether they represent a function: a

x −4 −3 −2 −1 0 y −4 −3 −2 −1 0

x −9 −7 −6 −5 −3 −2 y 10 10 10 10 10 10

2 3 4 −2 −3 −4

3 10

5 10

10 10

x y

−8 −7 −6 −3 2 7 9 8 13 −18 −16 −15 −2 −4

9 3

9 11

10 −9

Consider the points in the table: −4 4

x y a 11

1 −1

−3 3

−1 1

0 2

0 0

1 1

2 2

3 3

Plot the points on a coordinate plane.

4 4 Do they represent a function?

A relation is deﬁned as follows: y = 1 if x is positive and y = −1 if x is 0 or negative. a

Complete the table for this relation: x y

−4

−3

−2

−1

Plot the points on a coordinate plane.

Do these values represent a function?

2.01 Functions and relations mathspace.co

Determine whether each of the following graphs shows a function: a

4 2

x −6

−4

−2

−6

−4

−2

−4

−6

2 x

−6

−4

−2

−4

−6

−8 −6 −4 −2 −2

−4

−6

−8

h 8

2 x 2 −2 −4 −6

−8 −6 −4 −2 −2

−4

−2

−4

−2 −2

−6

−4

−2

x −6

Mathspace Virginia SOL Algebra 1 mathspace.co

2 −8 −6 −4 −2 −2 −4 −6 −8

1 −4 −3 −2 −1

−1

x 1

x −6

−4

−2

−4

−3

−6

−4

Determine whether each mapping is a function or not. a

5 9

4 5 6 7 8

−5 1

−4

−3 −2 4 5 6

−2

−1 3 8

Let’s extend our thinking 14

A particular store is offering one free t-shirt for every two t-shirts purchased. The relationship between the total cost and the number of t-shirts is shown in the following table. a

Is the relationship between the total cost and the number of t-shirts a function?

Based on the table, how much will you pay for one t-shirt?

Based on the table, how much will you pay for 3 t-shirts?

Using this relation, how much will you pay for 6 t-shirts?

The following shows the relationship of the the total distance traveled (in kilometers), by Valentina in t hours.

Number of t-shirts

Total cost (dollars)

1 2 3 4 5

19 38 38 57 76

Distance (km) 575

Express the relation as a set of ordered pairs.

Is this relation a function?

460

If Valentina’s car is traveling at a constant speed what might be the total distance traveled after 6 minutes?

345 230 115 t 1

2.01 Functions and relations mathspace.co

A domain made up of a single connected interval of values is said to be a continuous domain. The function shown has a continuous domain. It is deﬁned for every x-value in an interval. 4

The domain and range of this function can be written in interval or set-builder notation. Shown below is the set-builder notation:

Domain: {x−∞ < x < ∞}

2 1 −4 −3 −2 −1 −1

x 1

Range: {y−3 ≤ y < ∞} Inequality notation is similar to set builder notation but we only include the inequalities.

−2

Domain: −∞ < x < ∞

−3

Range: −3 ≤ y < ∞

−4

Example 1 Consider the function shown in the graph.

y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

a State whether the function has a discrete or continuous domain.

Apply the idea The function is deﬁned at every value of x across an interval, so it has a continuous domain.

b Determine the domain of the function using set-builder notation.

Apply the idea

Reﬂect and check

We can see that the function is deﬁned for every x-value between −6 and 8, including −6 but not including 8.

The domain of the function written in inequality notation is

So the domain of the function can be written as Domain: {x−6 ≤ x < 8}

Mathspace Virginia SOL Algebra 1 mathspace.co

Domain: −6 ≤ x < 8

c Determine the range of the function using set-builder notation.

Apply the idea

Reﬂect and check

We can see that the function reaches every y-value between −6 and 4, including 4 but not including −6.

The range of the function written in inequality notation is Range: −6 < y ≤ 4

So the range of the function can be written as Range: {y−6 < y ≤ 4}

Example 2 Consider the function shown in the graph.

y 5

a Determine the domain of the function using interval notation.

4 3 2 1 −1

−1

x 1

−2 −3

Apply the idea

Reﬂect and check

We can see that the function is defined for every x-value between negative infinity and positive infinity.

The domain could have been written in inequality notation as:

So the domain of the function can be written as Domain: (−∞, ∞)

Domain: −∞ < x < ∞ This domain is also referred to as the set of all real numbers.

b Determine the range of the function using interval notation.

Apply the idea

Reﬂect and check

We can see that the function is defined for every y-value between, and including, −2, to positive infinity.

The range could have also been written using inequality notation:

Range: [−2, ∞)

Range: y ≥ −2

Idea summary Different notations help us represent discrete and continuous functions: Set notation (discrete): {1, 2, 3, 4, 5} Set notation (continous): {x− 4 ≤ x < 10} Inequality notation: −4 ≤ x < 10 Interval notation: (−5, 6]

2.02 Domain and range mathspace.co

Domain and range in context Understanding the limitations on the domain and range of a function in context are important for interpreting situations. Depending on the context, a discrete function may be appropriate for a situation or a continuous function could be better suited to the scenario. The choice of whether rational numbers or speciﬁcally integers or whole numbers should also be considered when given a real-world situation for interpretation. Domain constraint

Dependent variable

A limitation or restriction of the possible x-values, usually written as an equation, inequality, or in set-builder notation

The output of a function whose value depends on the independent variable

Independent variable The input of a function whose value determines the value of other variables

Example 3 Consider the relationship between the cost of a hotel stay and the length of the stay. Suppose the hotel charges $75 per night and the stay last 7 nights. a State the independent and dependent variables.

Apply the idea Since the total cost of the hotel room depends on the number of nights at the hotel, the number of nights is the independent variable and the cost is the dependent variable.

b Determine an appropriate domain and range and explain your reasoning.

Create a strategy The domain and range may be discrete or continuous, and the types of real numbers in the domain and range also need to be considered.

Apply the idea An appropriate domain for the function would be based on the number of nights a person plans to stay at the hotel, and it makes sense for the domain to be discrete whole numbers in set notation because payment is counted in full days. Domain: {1, 2, 3, 4, 5, 6, 7} Based on the choice for the domain, the range will also have discrete whole number values. Range: {75, 150, 225, 300, 375, 450, 525}

Reﬂect and check It doesn’t make sense to determine the cost of staying at the hotel for 1.5 nights for instance, because the hotel would need to be booked for 2 nights in order to stay longer than a night.

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 4 Dog Weight

Consider the graph of the function of a dog’s weight over time:

Weight (pounds) 80 60 40 20 Time (months) 5

a Determine if this graph represents a function.

Create a strategy The graph of a relation represents a function if it passes the vertical line test.

Apply the idea While the graph appears almost vertical close to the y-axis, it still passes the vertical line test. Dog Weight Weight (pounds) 80 60 40 20 Time (months) 5

Since it passes the vertical line test, the graph is a function.

b Explain why the domain of this function is continuous.

Create a strategy The domain of a function is either continuous or discrete, and the context of the problem gives us information about which type of domain makes the most sense.

Apply the idea Since a dog’s weight can be measured at any time, such as 4.5 months as opposed to only at each month mark based on the graph, the function is continuous.

2.02 Domain and range mathspace.co

c State an appropriate domain and range of the function based on the information in the graph. Justify your solution.

Create a strategy Since the graph is continuous, the domain and range may be given in interval or set-builder notation.

Apply the idea An appropriate domain for the relationship would be from 0 to around the age when the dog stops growing because the graph will eventually ﬂatten. Domain: {x  0 ≤ x ≤ 24} An appropriate range for the function would also begin at 0 up to the maximum weight when it stops growing. It doesn’t make sense for a puppy to be born weighing 0 pounds exactly so we will exclude 0 from our interval, however, a puppy could weigh between 0 and 1 pound (think of puppies whose weight is measured in ounces). Based on the graph, the dog may grow up to 80 pounds. Range: {y  0 < y ≤ 80}

Idea summary Discrete domains apply to problems where the independent variable only includes certain values in an interval, whereas continuous domains apply to problems where the independent variable includes all values in an interval.

Practice What do you remember? 1

State the deﬁnition for the: a

Range

Discrete

State when the domain of a function is: a

Domain

Continuous

Determine whether each relation is a function. If the relation is a function, determine its domain and range in set notation. a

x 2 3 2 3 2 y −1 −2 −3 −4 −5 x 5 7 1 0 y 2 8 2 −4

4 1

{(−2, 3), (4, 3), (6, 0), (0, 6)}

9 7

Mathspace Virginia SOL Algebra 1 mathspace.co

10 −8 2

For each of the following graphs: i

State the domain of the relation using interval notation.

State the range of the relation using interval notation.

2 −8 −6 −4 −2 −2

−4

−6

−8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

For each of the following graphs: i

State the domain of the relation using set notation.

State the range of the relation using set notation.

2 −8 −6 −4 −2 −2

x 2

2 −8 −6 −4 −2 −2

−4

−6

−8

x 2

2.02 Domain and range mathspace.co

2 −8 −6 −4 −2 −2

−8 −6 −4 −2 −2

−4

−6

−8

For each of the following graphs, identify whether the domain of the function is discrete or continuous: a

1 −4 −3 −2 −1

x 2

−1

x 1

−4 −3 −2 −1

−1

−2

−3

−4

The following graph represents the amount of gas in a fuel tank over time during a roadtrip: a

State the domain and range of the relation using set notation.

Interpret the meaning of the domain and range within the context of the problem.

13 12 11 10 9 8 7 6 5 4 3 2 1

x 1

y (gallons)

x (hrs) 6 7

Let’s practice 8

State the domain and range of the collection of points shown on the graph.

6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

Mathspace Virginia SOL Algebra 1 mathspace.co

x 2

Determine whether each relation is a function. If the relation is a function, determine its domain and range in set notation. a

y 4

4 3

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2

−3

−4

−6 −5 −4 −3 −2 −1 −1

−1

−2

x −4 −3 −2 −1

1 2 3 4 5

−1

−2

−3

−4

Determine for each graph whether or not it has a domain consisting of all the real numbers. If it does not, state the domain. a

2 −8 −6 −4 −2 −2 −4 −6 −8

x 2

8 6 4 2 −8 −6 −4 −2 −2

x 2

2.02 Domain and range mathspace.co

Determine for each graph whether or not it has a range consisting of all the real numbers. If it does not, state the range. a

−8 −6 −4 −2 −2

−4

−6

−8

For each of the following: i

Identify an independent variable and a dependent variable.

State the domain and range for the scenario.

Uma earns between $200 and $450 selling between 35 and 50 burritos, inclusive.

Jermaine starts reading his book from page 112. Over the course of 17 days, he reads up to and including page 237.

Brigid Kosgei completes a 26.2 mile marathon in 134 minutes.

Let’s extend our thinking 13

State the domain and range for each of the following graphs: a

−8 −6 −4 −2 −2

8 10

−2 −4

−6 −8

−6

−10

−8

y 8 6 4 2 x −8 −6 −4 −2

−2 −4

2 −4 −2

−4

Mathspace Virginia SOL Algebra 1 mathspace.co

x 2

The graph of a function is shown. a

State the range of the function.

A ball is thrown from an apartment window in a high-rise building. The height of the ball above ground over time can be modelled by the function shown in the graph, where the ball is thrown at x = 0. State the range of the function for this context.

y 125 100 75 50 25 x −6 −4 −2

−25

A comet is travelling through the solar system. It passes by the Earth before curving around the Sun and heading back out into deep space. The distance of the comet from Earth, y, is tracked by satellite telescopes and expressed as a function of time, x, since the comet was ﬁrst spotted. a

State the domain of this function.

The satellites detect that, at its closest point, the comet was a distance of 0.14 AU (astronomical units) from Earth. State the range of the function.

2.02 Domain and range mathspace.co

2.03 Evaluating functions After this lesson, you will be able to… • evaluate functions for valid inputs. • interpret statements that use function notation in a real-world context.

Evaluating functions Recall that a function maps each input of a relation to exactly one output. Functions are typically represented in function notation, so the relationship between inputs and outputs are clear. Input

Function notation

The independent variable of a function; usually the x-value

A notation that describes a function. For a function f when x is the input, the symbol f (x) denotes the corresponding output.

Output The dependent variable of a function; usually the y-value

We have seen the equation for a linear function y = mx + b. By naming a linear function f, you can also write the function using function notation: f (x) = mx + b. This is a way of saying that mx + b is a function of x. This is useful because it quickly tells us that we are working with a function where y can represent a relation that is not a function. The notation f (x) is another name for y. If f is a function, and x is in its domain, then f (x) represents the output of f corresponding to the input x. You can use letters other than f to name a function, such as g or h.

f (x) = y f x y

is the name of function is the input is the output

To evaluate a function at a point is to calculate the output value at a particular input value: If f (x) = −7x + 9, then determine the value of f (1). This is the same as stating to evaluate the function y = −7x + 9 when x = 1. f (1) = −7(1) + 9 f (1) = −7 + 9 = 2 Therefore, f (1) = 2 for the function f (x) = −7x + 9.

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 Consider the function

where x is the independent variable. a Construct a table of values for the function at x = −3, 0, 9, 12, 27.

Create a strategy In order to construct a table of values, we will need to evaluate the function at the given values of x.

Apply the idea Substituting x = −3, we have:

Substituting x = 0, we have:

Substituting x = 9, we have:

Substituting x = 12, we have:

Substituting x = 27, we have:

A completed table for the function at the given values of x is: x f (x)

−3 −6

0 −5

9 −2

12 −1

27 4

b Evaluate the function for f (2).

Apply the idea Substituting x = 2, we have:

2.03 Evaluating functions mathspace.co

Example 2 Consider the graph:

5 4 3 2 1

a Evaluate the function for f (−1).

−3 −2 −1 −1 −2 −3 −4 −5

f (x)

x 1

Create a strategy

Apply the idea

We can find the value of the function on the graph when x = −1 by moving down the y-axis until we intersect the curve.

From x = −1, move down until we intersect the curve. 5

f (x)

4 3 2 1 −3 −2 −1

−1

x 1

−2 −3 −4 −5

When x = −1, we can see on the graph that f (x) = −2. b Determine the value of x when f (x) = 4.

Create a strategy

Apply the idea

To find the value of x on the graph where f (x) = 4, move horizontally to the left from f (x) = 4 until you intersect the curve, then move vertically downward until you reach the x-axis.

f (x)

4 3 2 1 −3 −2 −1

−1

x 1

−2 −3 −4 −5

When f (x) = 4, we can see on the graph that x = −3.

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 Consider the function f (x) = 3x − 5 to answer the following: a Find the range when the domain is {−3, 0, 11}.

Create a strategy We need to find the range (the values of f (x)) when the domain (the x-values) are {−3, 0, 11}. Substitute x-values and solve for f (x).

Apply the idea Substitute x = −3 for x in the function. f (x) = 3x − 5

Original function

f (−3) = 3(−3) − 5

Substitute x = −3

= −9 − 5

Evaluate the multiplication

= −14

Evaluate the subtraction

Substitute x = 0 for x in the function. f (x) = 3x − 5

Original function

f (0) = 3(0) − 5

Substitute x = 0

=0−5

Evaluate the multiplication

= −5

Evaluate the subtraction

Substitute x = 11 for x in the function. f (x) = 3x − 5

Original function

f (11) = 3(11) − 5

Substitute x = 11

= 33 − 5

Evaluate the multiplication

= 28

Evaluate the subtraction

When the domain is {−3, 0, 11}, this function has a range of {−14, −5, 28},

Reflect and check This could be written in function notation as {f (−3), f (0), f (11)} = {−14, −5, 28}. b Find the domain when the range is {−2, 4, 7}.

Create a strategy We need to find the domain (the x-values) when the range (the values of f (x) is {−2, 4, 7}. Substitute the f (x) values and solve for x.

Apply the idea Replace f (x) with −2 in the equation and solve for x. f (x) = 3x − 5

Original function

−2 = 3x − 5

Substitute f (x) = −2

3 = 3x

Add 5 to both sides

1=x

Divide both sides by 3

2.03 Evaluating functions mathspace.co

Replace f (x) with 4 in the equation and solve for x. f (x) = 3x − 5

Original function

4 = 3x − 5

Substitute f (x) = 4

9 = 3x

Add 5 to both sides

3=x

Divide both sides by 3

Replace f (x) with 7 in the equation and solve for x. f (x) = 3x − 5

Original function

7 = 3x − 5

Substitute f (x) = 7

12 = 3x

Add 5 to both sides

4=x

Divide both sides by 3

When this function has a range of {−2, 4, 7}, the domain is {1, 3, 4}.

Reflect and check We can write this using function notation as {f (−2), f (4), f (7)} = {1, 3, 4}. c Evaluate f (7) − f (2)

Create a strategy Find the values of f (7) and f (2), then subtract the value of f (2) from the value of f (7).

Apply the idea First, let’s evaluate f (7). f (x) = 3x − 5

Original function

f (7) = 3(7) − 5

Substitute x = 7

= 21 − 5

Evaluate the multiplication

= 16

Evaluate the subtraction

Now evaluate f (2). f (x) = 3x − 5

Original function

f (2) = 3(2) − 5

Substitute x = 2

=6−5

Evaluate the multiplication

Evaluate the subtraction

Now we can find f (7) − f (2) by substituting their values. f (7) − f (2) = 16 − 1 = 15

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 4 Let f (x) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

Plant Growth 20 18 16 14 12 10 8 6 4 2

a Interpret the meaning of f (10) = 8.

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Create a strategy We can use the units of the given information and the graph to help with the interpretation.

Apply the idea We’re given that f (x) represents the height of a growing plant in inches, so to interpret f (10), we need to determine what an input of x = 10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted. We also know that all of this is equal to 8. This is the output, or what our function f (x) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall. Based on the graph, when x = 10, y = 8 so f (10) = 8 is represented by the ordered pair (10, 8) on the graph. The plant has a height of 8 inches 10 days after being planted. b Interpret the meaning of f (6).

Apply the idea

Reflect and check

We know that x represents the time in days since the plant was planted and x = 6. So this means that 6 days have passed since the plant was planted.

Using the graph, we can find the actual height of the plant after 6 days.

Since f (x) represents the height of a growing plant in inches, f (6) represents the height of the plant 6 days after being planted.

Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

2.03 Evaluating functions mathspace.co

c Interpret the meaning of f (x) = 12.

Apply the idea We know that f (x) represents the height of a growing plant in inches, so if f (x) = 12, then the height of the plant is 12 inches x days after being planted.

Reflect and check By using the graph, we can find the number of days when the height of the plant is 12 inches. Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Idea summary An equation where the output variable is isolated like y = mx + b can be written as a function in the form, f (x) = mx + b. We evaluate a function, written in function notation as f (c), by replacing all values of x with c and evaluating the expression.

f (x) = y f is the name of function x is the input y is the output

Practice What do you remember? 1

Evaluate each expression for the following values of x: i

x=2

x = −3

4x + 1

6x − 2

x2 − 3x − 2

For the function notation y = f (x), identify: a

Independent variable

Dependent variable

Input

Output

Mathspace Virginia SOL Algebra 1 mathspace.co

(x + 1) (x + 2)

SOL

Suppose that (7, −6) is an ordered pair that satisﬁes the function g. Write this situation using function notation.

For the function

Construct a table of values for the function at x = −6, −3, 0, 18, 24, 72.

Evaluate the function for x = −27.

What is the value of this expression when a = 27, b = 25, and c = −7?

A SOL

, where x is the independent variable:

−12

154

−55

−125

f (0)

f (−5)

f (−4)

g(−4)

f (−5)

If f (x) = x2 + 8x, evaluate f (−11).

y − 4x2 = 5 + x

What is the value of this expression when z = −18? −5z + 7 A

125

Let’s practice 7

For each function, find the value of f (x). a

If f (x) = −6x + 4, find: i

If f (x) = 4x + 4, find: i

, find: g(5) , find:

If i

f (3)

If i

f (2)

If f (x) = 3x − 1, find: i

f (4)

Evaluate the functions for each given value. a

If f (x) = 2x2 − 2x + 5, evaluate

If p(x) = x2 + 8, evaluate p(20).

Consider each of the following equations. i

Rewrite the equation using function notation f (x).

Find the value of f (3).

x + 3y = 6

−6x + 5y = 7

y + 6x2 = 3 − x

2.03 Evaluating functions mathspace.co

For each graphed function, find the value of f (x) for each value of x. a

f (1)

f (3)

7 6 5 4 3 2 1

5 4 3 2 1 −2 −1

x 1

−1

f (−1)

5 4 3 2 1

4 3

−3 −2

−1

x 1

−3 −4

x 1 2 3 4 5

For each function, find the value of x for each value of g(x). a

g(x) = 3x − 8, g(x) = 7

For each graphed function, find the value of x for each value of f (x). a

f (x) = −5

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−2

f (0)

−3 −2 −1 −1 −2 −3

−2

x 1 2 3 4 5

Mathspace Virginia SOL Algebra 1 mathspace.co

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

x 1 2 3 4 5

f (x) = 6

f (x) = −4

y 9 8 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−5 −4 −3 −2 −1 −1 −2 −3

5 4 3 2 1

1 2 3 4 5

Given the graph of p(x), find each input of the corresponding outputs. 100

p(x)

80 60 40 20 −20

x 1

p(x) = 110

−40 −60 −80 −100

p(x) = −100

p(x) = −60

p(x) = 50

For the function f (x) = 5x2 − 2x + 6, find the range given the domain is {−5, 0, 4}.

For the function f (x) = −4x − 4, find the domain when the range is {−8, 16, 24}.

Sharon works at a pizza shop and makes $15 per hour. Her wages can be represented by the function W(t) = 15t, where t represents the number of hours that she works. a b

Explain what the expression W(19) represents. Find her wages after working 26 hours.

Mindy can model her tomato plant’s height in inches over time using the function h(t) = 2t, where t is the number of weeks after Mindy planted it. a

Explain what the equation h(4) = 16 represents.

Find the height of Mindy’s tomato plant after 6 weeks.

Let’s extend our thinking 18

You are designing a square garden with a quadratic function representing its area, A(x) = x2, where x is the length of one side. a

What does A(x) = 36 mean?

If the area of the garden is 36 square meters, find the length of each side.

What is the significance of x in A(x) = 36?

2.03 Evaluating functions mathspace.co

Consider the function f (x) = x2 − 17x + 4. a

Form an expression for f (b).

Simplify the expression for f (2).

If f (2) = 15, find the value of k.

Consider the functions f (x) = 7x + 2 and g(x) = 15 − 2x. Find the value of the following: a

Consider the function f (x) = x2 + 2x + k. a

Form an expression for f (a).

f (1) + g(2)

f (−2) − g(7)

Han is painting a mural which requires

f (0) ⋅ g(0)

gallons of paint per square foot. Han also needs 3 additional gallons

of paint to go over the outline of the mural once they are ﬁnished.

Express the relationship of the scenario using function notation, letting x represent the square footage of the mural.

Find the number of gallons of paint that Han will need if the mural has a size of 24 square feet.

A function describing the strictly increasing relation between temperature and a person’s average resting heart rate has a domain of 50 ≤ T ≤ 105 (Fahrenheit) and a range of 40 ≤ f (T ) ≤ 90 (beats per minute), based on experimental results. a

Determine the average resting heart rate at a temperature of 50° F.

State the temperature that is expected if a person has an average resting heart rate of 90 beats per minute.

Explain whether or not the function output f (10) be a reliable estimate for the corresponding real life scenario.

Mathspace Virginia SOL Algebra 1 mathspace.co

9 8 7 6 5 4 3 2 1 −9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7 8 9

• Domain: {x∣ − 3 ≤ x ≤ 4} • Range: {y∣ − 6 ≤ y ≤ 8} • x-intercept: (1, 0) • Zeros: (1, 0) • y-intercept: (0, 2) • Maximum: (−3, 8) • Minimum: (4, −6)

Example 1 Consider the function shown in the graph:

14 12

10 8 6 4 2 −2 −1 −2 −4 −6

a Identify whether the function has a maximum or minimum value and state this value.

Create a strategy We need to find the lowest point on the graph and use the y-value to indicate how low it is.

Apply the idea 14 12

10 8 6 4 2 −2 −1 −2 −4 −6

This function has a minimum value of −4.

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

2 3 4 5 6 7 8

x 1

2 3 4 5 6 7 8

b State the range of the function.

Create a strategy

Apply the idea

In part (a), we identiﬁed that the function has a minimum value of −4, so we know that the function can’t take values smaller than −4.

Looking at the function, we can see that it stretches up towards inﬁnity on both sides of the minimum point, so the function can take any value greater than or equal to −4. That is, the range of the function is: Range: {y  y ≥ −4}

c State the x-intercept(s) of the function.

Create a strategy The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case, by looking at the graph, we can see that there are two x-intercepts.

Apply the idea 14 12

10 8 6 4 2 −2 −1 −2 −4 −6

x 1

2 3 4 5 6 7 8

The x-intercepts of this function are the points (1, 0) and (5, 0).

Example 2 A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is graphed against time. 50 40 30 20 10 −10 −20 −30 −40 −50

h(ft)

t(mins) 5

Use the key features of the graph to describe the penguin’s time spent hunting. Be as detailed as possible. 2.04 Characteristics of functions mathspace.co

Create a strategy We can see that the graph has key features like intercepts, a minimum point, a domain, and a range. To interpret the graph in context, we can use the axes of the graph to match key features to their real-world meaning.

Apply the idea The y-intercept is (0, 30), meaning that the penguin is 30 ft above sea level at 0 minutes into its hunting time. The x-intercepts are (5, 0) and (20, 0), meaning that the penguin is exactly at sea level at 5 and 20 minutes into its hunting time. The minimum of the graph is approximately (12.5, −41), so the penguin’s lowest point is about 41 ft below sea level at about 12.5 minutes into its hunting time. If we combine this information, we can make a description of the penguin’s hunting time. For example: When the penguin needs to hunt, it leaves its nest, which is 30 ft above sea level. The penguin makes its way down to the water and dives into the water 5 minutes after leaving the nest. The penguin swims down to a depth of around 41 ft below sea level, reaching its deepest point around 12.5 minutes into its hunting time before returning to the water’s surface at 20 minutes. The penguin spends 15 minutes underwater in total. The penguin then spends the last 5 minutes of its hunting time climbing back up to its nest, ﬁnishing a bit higher than where it started.

Reﬂect and check We only need to make sure that the description matches the key features of the graph, so there are many possible examples. For example, it is completely valid to say that the penguin dives into the water using a submarine as long as the deepest point is still 41 ft below sea level.

Example 3 A hiker’s elevation over a given period of time is graphed: a What could the zeros of the function represent in this situation?

Elevation

Time

Create a strategy The zeros of a function represent where f (x) = 0.

Apply the idea The y-axis represents elevation and the x-axis represents time. The zeros of this function represent the times where the hiker reached ground level.

Mathspace Virginia SOL Algebra 1 mathspace.co

b Write a description of the meaning of the maximum of the function in relation to the hiker’s elevation over time.

Create a strategy

Apply the idea

The maximum is the highest point of the function.

The function relates the elevation of the hiker over a period of time. The maximum represents the time when the hiker reached their highest elevation on their journey.

c Would the domain or range tell us how long the hiker traveled?

Create a strategy

Apply the idea

The domain represents the inputs, or x-values of a function, while range represents the y-values.

The x-axis is labeled as time. The domain represents how long the hiker traveled.

Idea summary The key features of a function and how to describe them are as follows: • •

We can write intercepts, zeros, minimums, and maximums as values or ordered pairs. Domain and range: write in inequality, set notation, and interval notation.

Practice What do you remember? 1

State the deﬁnition for the following function characteristics: a

Maximum

Minimum

Describe the connection between zeros and x-intercepts.

For each graph of the function y = f (x):

x-intercept

Identify whether the function has a maximum or minimum and state its value.

State the range of the function.

iii

State the domain of the function

−4 −2 −2

4 2 −6 −4 −2 −2

x 2

y-intercept

x 2

10 12

−4 −6 −8

−4

−10

−6

−12

2.04 Characteristics of functions mathspace.co

d 8

4 −10 −8 −6 −4 −2

−8 −6 −4 −2

−4

−2

−8

−4

−12

−6

For each function: i

State the coordinates of the x-intercept.

−5 −4 −3 −2 −1

8 6 4

−1

−5 −4 −3 −2 −1 −2

2 3 4 5

−2

−4

−3 −4

−6 −8

−5

−10

State the coordinates of the x-intercept(s). State the coordinates of the y-intercept(s).

x −2 −1 0 1 y 8 6 4 2

x y

2 0

3 2

0 2

1 5

2 8

x 0 1 2 3 y 5 0 −3 −4

4 −3

5 0

2 10

3 15

x y

−2 10

−2 −1

−1 5

−1 0

0 0

Mathspace Virginia SOL Algebra 1 mathspace.co

1 5

3 2

−3 −4

State the coordinates of the y-intercept.

For each table:

x 1

2 3 4 5

Let’s practice 6

For each graph, identify the number of: x-intercepts

Zeros

b 8

−8 −6 −4 −2

2 3 4 5

−4

−3

−6

−4

−8

d 4

−1

2 3 4 5

−2

x 1

−5 −4 −3 −2 −1 −1

x 2

−2

−5 −4 −3 −2 −1 −1

−4 −3 −2 −1

y-intercepts

iii

−3

−4

For each graph, describe the location of any maximum and minimum points. a

4 −6 −4 −2 −4 −8 −12

x 2

6 4 2 −4 −3 −2 −1 −2

−16

−4

−20

−6

x 1

2.04 Characteristics of functions mathspace.co

For each graph, state the following: i

Domain

iii

x- and y-intercepts

Range

1 −2 −1

x 1

−1

−2

−3

−4

−5

−1

−6

−1

x 1

1 −4 −3 −2 −1

−3 −2 −1

x 1

−1

−1 −2

−2

−3

−4

Grover counts the number of people in his class, x, and the number of empty seats, y. He puts his results into a table: x y

22 0

20 2

17 5

13 9

11 11

8 14

3 19

0 22

Identify and describe the following features of the table and explain what each means in this context: a

x-intercept

y-intercept

Domain

Range

Let’s extend our thinking 10

The graph shows the height (in meters) of a soccer ball against the time (in seconds) that has passed after it has been kicked.

Height 12

Find the coordinates of the y-intercept.

Interpret the y-value of the y-intercept in this context.

Find the coordinates of the x-intercept.

Interpret the x-value of x-intercept in this context.

6 4 2 Time 1

100

Mathspace Virginia SOL Algebra 1 mathspace.co

Two construction workers are competing to see who can lay the most bricks in one hour. The graph shows the number of bricks laid and the time, in minutes.

320

Bricks laid

280

Determine who can lay more bricks in 60 minutes.

240

Explain why the y-intercept of both lines is 0.

200

Charlie

160

Neville

120 80 40

Minutes 10 20 30 40 50 60 70 80

Immanuel records his speed as he runs around the neighborhood and graphs the results as speed, y, against time, x. 9

y (mph)

8 7 6 5 4 3 2 1

x (mins) 2

Identify the maximum point and describe what it means in the context of Immanuel’s run.

What do the domain and range tell us about Immanuel’s run?

2.04 Characteristics of functions mathspace.co

101

3 Linear Functions Big ideas • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made.

Chapter outline 3.01 3.02 3.03 3.04 3.05 3.06

Slope Transformations of linear functions Slope-intercept form Standard form Point-slope form Equations of parallel and perpendicular lines

104 116 128 145 161 173

3.01 Slope After this lesson, you will be able to… • determine the slope of a line from two points or a graph.

Identify slope from a graph Recall the slope of a line is a value that describes the line’s steepness. Slope A rate of change in a proportional relationship between two quantities We can find the slope of a line by identifying the vertical and horizontal change or:

There are four types of slope: 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Zero

104

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1 2 3 4 5

Negative

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Positive 5 4 3 2 1

5 4 3 2 1

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

x 1 2 3 4 5

Undefined

Apply the idea rise = 8 – 0 = 8 units run = 4 – 0 = 4 units

Subtract the y-coordinates of points A and B Evaluate Subtract the x-coordinates of points A and B Evaluate

From A, move 8 units up and 4 units to the right. b Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a : b.

Create a strategy Use the slope formula: m =

, then convert to ratio.

Apply the idea Substitute the value of the rise and run

Evaluate

Express as a ratio

c Complete the directions that explain how to move from point A to point C. From A, move ⬚ units up and ⬚ units to the right.

Create a strategy

We need to ﬁnd the difference in the y-values (i.e. rise) between points A and C, and the difference between their x-values (i.e. run).

Apply the idea rise = 12 – 0 = 12 units run = 6 – 0 = 6 units

Subtract the y-coordinates of points A and C Evaluate Subtract the x-coordinates of points A and C Evaluate

From A, move 12 units up and 6 units to the right. d Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a : b.

Create a strategy Use the slope formula: m =

, then convert to ratio.

Apply the idea Substitute the value of the rise and run

Evaluate

Express as a ratio

106

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Consider the graph shown: 10

8 6 4 2 −8 −6 −4 −2 −2 −4

x 2

−6 −8

Find the slope of the line represented by the graph.

Create a strategy To find the slope of a line represented by using the formula: Slope =

Apply the idea From point A(−2, 10) to point B(2, 2), move 8 units down and 4 units to the right. A

8 6 4 2

−8 −6 −4 −2 −2 −4

B x 2

−6 −8

The ratio of the rise to the run is

simplified into −2, the slope of the line.

The slope of the line is m = −2.

Idea summary The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).

3.01 Slope mathspace.co

107

The slope formula Finding the ratio of the rise and run of the line works when it’s easy to see the graph, with clearly marked points on the line. We can extend this thinking to use the coordinates of two points and construct a general formula. The rise of the line is found with points that lie on a vertical line. These points share the same x-value. The distance between them is the difference in the y-values.

y (x2, y2)

rise = y2 − y1 y2 (x1, y1)

The run of the line is found with two points that lie on a horizontal line. These points share the same y-value. The distance between them is the difference in the x-values.

y (x2, y2)

run = x2 − x1 x1

(x1, y1) x

We ﬁnd the slope by using the formula:

slope

(x1, y1)

a point on the line

(x2, y2)

a second point on the line

Example 3 What is the slope of a line that passes through the points A(3, 5) and B(−2, 10)?

Create a strategy Use the slope formula: m =

. Let point A = (x1, y1) and point B = (x2, y2).

Apply the idea Use the slope formula

Substitute x1 = 3, y1 = 5, x2 = −2, and y2 = 10

Evaluate the subtraction

Evaluate the division

108

Mathspace Virginia SOL Algebra 1 mathspace.co

Reﬂect and check By graphing the two points and sketching the line through them, we can see that it does in fact have a negative slope.

y 10 8 6 4 2 −2 −2

x 2

Example 4 Gasoline costs a certain amount per gallon. The table shows the cost of various amounts of gasoline in dollars: Number of gallons (x) Cost of gasoline ( y)

0 0

10 26.40

20 52.80

30 79.20

40 105.60

How much does gasoline cost per gallon?

Create a strategy The cost per gallon of gasoline is the unit rate. This is the same as the ratio of the change in the cost of gasoline ( y) to the change in the number of gallons (x). We can ﬁnd the unit rate by using the slope formula, m =

, with any two input-output pairs in the table.

Apply the idea

Write the slope formula

Substitue two input-output pairs

Evaluate the subtraction

Evaluate the division

Reﬂect and check The relationship between the cost of gasoline and the number of gallons purchased can be modeled by the equation y = 2.64x. If we use technology to graph this line, we can see that all the points in the table lie on the line.

110 100 90 80 70 60 50 40 30 20 10

y (40, 150.60) (30, 79.20) (20, 52.80) (10, 26.40) x 5 10 15 20 25 30 35 40 45 50

3.01 Slope mathspace.co

109

Example 5 Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body. For the liquid form, the amount of the medication in the bloodstream is presented in the graph shown.

A 35 30 25 20 15 10 5

The results for the capsule form are presented in the table below. Time (mins), t 4 7 10 13

t 1

Amount in blood (mgs), A 24.6 42.3 60 77.7

a At what rate, in milligrams per minute, is the liquid form absorbed?

Create a strategy Choose any two points that lie on the line, and use the formula for slope.

Apply the idea We use the points (0, 0) and (2, 8) that lie on the line.

Write the formula for slope

Substitute the values

Evaluate

b At what rate, in milligrams per minute, is the capsule form absorbed?

Create a strategy Choose any two points from the table and use the slope formula.

Apply the idea We can use the points (4, 24.6) and (7, 42.3) from the table. Write the formula for slope

Substitute the values

Evaluate

110

Mathspace Virginia SOL Algebra 1 mathspace.co

2 3 4 5 6 7 8 9

c In which form is the medication absorbed more rapidly?

Create a strategy Use the fact that the medication that is more quickly absorbed will have a higher rate.

Apply the idea 5.9 mg/min > 4 mg/min Comparing the two rates from part (a) and part (b), the capsule form is more quickly absorbed than the liquid form as it has a higher rate of absorption of 5.9 mg/min.

Idea summary We ﬁnd the slope of a line by using the formula:

(x1, y1)

are the coordinates of the lower points

(x2, y2)

are the coordinates of the upper points

We can ﬁnd the slope in different ways depending on how it is represented: • • •

From a graph: Count the slope as the rise over the run, or ﬁnd two points and use the slope formula. Table: Find the difference in two consecutive outputs and divide by the difference in the corresponding inputs, or ﬁnd two input-output pairs and use the slope formula. Description: The slope is the rate given in the problem.

Practice What do you remember? 1

Determine whether the following is a correct description of slope: a

Classify the slope of each line as positive, negative, zero, or undefined: a

14 12 10 8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

x 2

14 12 10 8 6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

x 2

3.01 Slope mathspace.co

111

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

For each segments containing A and B: i

Find the rise going from A and B.

iii

Find the slope of the line.

A (−4, 0) and B (0, 2) y 5 4 3 2 B 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

112

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

Find the run going from A and B.

A (0, 4) and B (1, 0) y 5 A 4 3 2 1

x 1 2 3 4 5

Mathspace Virginia SOL Algebra 1 mathspace.co

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x B 1 2 3 4 5

If we have two points and the slope formula m = (x2, y2)? Explain.

Consider the points A, B, and C on the coordinate plane: a

Describe how to move from point A to the following points: i

, does it matter which point is (x1, y1) and which point is

Point B

Point C

Express the direction of each movements as a simpliﬁed ratio, comparing vertical movement to horizontal movement. i

From point A to point B

From point A to point C

A 6 4

2 −6 −4 −2 −2

Determine the slope of the line.

Graph the y-intercept, and label it D.

2 4 6 8 10 12 14

−4

Let’s practice 6

Consider the straight line that passes through the points A, B, C, and D: 14 12 10 8 6 4 2

y C B

−8 −6 −4 −2 −2 −4 A −6 −8

Find the slope of the line using the points A and D.

Find the slope of the line using the points B and C.

What do you notice about the slopes? Will this be true for any two points?

Consider the points A and B on the straight line: A

6 4 2 −4 −3 −2 −1 B1 −2

x 2

−4 −6 −8

Describe how to move from point A to point B.

Express the direction of the movement in part (a) as a simpliﬁed ratio, comparing vertical movement to horizontal movement.

3.01 Slope mathspace.co

113

Identify the slope m for each line: a

−8 −6 −4 −2 −2 −4 −6 −8

Find the slope of the line that passes through the given points: a

(2, 0) and (0, 3)

(4, 7) and (1, 10)

(6, 4) and (3, 4)

(10, 5) and (15, −5)

(−8, 20) and (−4, 10)

(11, −17) and (3, −9)

(−15, −35) and (−30, −25)

(−2.25, −8) and (−1.75, 6.5)

(2.3, 4.8) and (5.6, 1.2)

Complete the table.

For each table of values: i

Find the slope.

x 1 2 3 4 5 y −3 −6 −9 −12

x −2 −1 0 1 2 y −10 −5 10

3 15

Two slopes are calculated. • Line 1 slope = 3 • Line 2 slope = 5 Which line is steeper? Explain why.

114

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

Mathspace Virginia SOL Algebra 1 mathspace.co

4 20

x y

−1

1 82

2 74

3 66

4 58

x y

−4 −9

−3

−2 −8

–1

0 −7

2 −6

Let’s extend our thinking 12

Pair up these points to create 3 lines, one with a positive slope, one with a negative slope and one with a slope of 0. A(3, 5), B(4, 6), C(−1, 8), D(7, 5), E(−3, 10), F(4, 12)

A ski resort has two ski runs as shown in the diagram: a

Find the slope of Run A, correct to two decimal places.

Find the slope of Run B, correct to two decimal places.

What is the slope of the steeper run?

Run A 21 m Run B

20 m 29 m

Consider the ramp shown in the diagram: a

Find the slope of this skateboard ramp if it rises 0.9 yd above the ground and runs 1 yd horizontally at the base.

The ramp can only be used as a ‘beginner’s ramp’ if for every 1 yd horizontal run, it has a rise of at most 0.5 yd. Can it be used as a ‘beginner’s ramp’?

Rise Run

Hiro believes that the slope of the line passing through the points (0, 0) and (−2, 5) and the slope of the line passing through the points (0, 0) and (2, −5) are the same. a

Plot the points (0, 0), (−2, 5), and (2, −5) on the coordinate plane.

Is Hiro correct? Explain how you know. y 7 6 5 A 4 3 B 2 1

Consider the points A and B on the straight line: a

Now, find y if the coordinates of C are (1, y) and C also lies on the same line.

Now, find x if the coordinates of C are (x, −19) and C also lies on the same line.

−5 −4 −3 −2 −1 −1 −2 −3

Bonnie recorded her savings (in dollars) over a few months in the table.

Months Savings

1 20

x 1 2 3 4 5

2 40

3 60

4 80

Plot the values in the coordinate plane where x corresponds to the months and y corresponds to the savings in dollars. Draw a line to connect the points from the origin.

Find the slope of the line.

Write an equation in the form y = mx representing the relationship between the number of months, x, and Bonnie’s monthly savings, y.

Using the equation in (c), ﬁnd the amount of Bonnie’s savings in one year.

The points A(26, m − 24), B(−1, m) and C(−10, 9) are plotted on the coordinate plane. Find m, given that A, B and C all lie on the same line.

Complete the coordinates using the points and slope provided. a c

(4, −3) and (1, ⬚), slope = −2

(5, 3) and (⬚, 63), slope = 4

b d

(5, 3) and (2, ⬚), slope = −4

(11, ⬚) and (−20, 16), slope =

3.01 Slope mathspace.co

115

A vertical translation, or shift, to the y-intercept of f (x) represents the transformed function f (x) + k.

In this example, the solid line represents the function f (x) = x.

The dashed line represents a shift 2 units down from f (x).

1 −4 −3 −2 −1

x 1

−1

The transformed function is represented by f (x) = x − 2.

−2 −3 −4

A horizontal translation, or shift, to the x-intercept of f (x) represents the transformed function f (x + k).

3 2 1 −4 −3 −2 −1

x 1

−1

The solid line represents the function f (x) = x, which is the same as above. Notice we can achieve the same transformed function by shifting the parent function 2 units right. The dashed line represents a shift 2 units right from f (x). The transformed function is represented by f (x − 2) = x.

−2 −3 −4

The slope of a line can be transformed by a vertical dilation, represented as kf (x) with k > 0. The line will either compress or stretch depending on the factor used on the slope, m.

3 2 1 −4 −3 −2 −1

x 1

−1

The black dashed line represents a vertical dilation (stretch) by a factor of 2 on f (x). The equation of the new line becomes f (x) = 2x. The blue dashed line represents a vertical dilation (compression) by a factor of . The equation of the new line becomes

−2

−3 −4

The dashed line represents another transformation on the parent function.

A vertical reflection occurs for kf (x) with k < 0.

2 1 −4 −3 −2 −1

−1

x 1

A change in the slope of f (x) = x from 1 to −1 represents a reﬂection of the line, which becomes f (x) = −x. This vertical reﬂection will transform f (x) = x into f (x) = −x.

−2 −3 −4

Transformations on the parent function f (x) = x can be used to graph and write equations.

3.02 Transformations of linear functions mathspace.co

117

Example 1 Determine what transformation has occured from the parent function f (x) = x. 4

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

Create a strategy

Apply the idea

First, determine whether the transformation can be achieved through a translation, dilation, or reflection.

Since the slope is negative, a vertical reflection has occurred, which means it is in the form kf (x) with k < 0. The line has also been stretched by a scale factor of 2, so k = −2. For our equation, f (x) = −2x.

Example 2 Consider the graph of the parent function f (x) = x. Graph the function after a vertical stretch of a factor of 3, and a vertical translation of −2 units. Write the equation of the transformed line. 4

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy A dilation will affect the slope of the line, and a translation will affect the y-intercept.

118

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea A scale factor of 3 for the vertical stretch means the original slope of 1 will be multiplied by 3. This creates the equation y = 3x, which is shown in the graph:

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

The second shift is 2 negative units, which means the y-intercept will change by −2, so the graph becomes:

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

The equation of the transformed line is y = 3x − 2.

Example 3 By ﬁrst identifying the slope and y-intercept, describe the transformations of the following lines from the parent line f (x) = x. a g (x) = 5x

Create a strategy

Apply the idea

The functions f (x) and g (x) have the same y-intercept, but different slopes. This means a vertical dilation has occurred.

The slope is 5.

• When the scale factor is greater than 1, a stretch has occurred. • When the scale factor is between 0 and 1, a compression has occurred.

The y-intercept is 0. f (x) = x has been vertically stretched (made steeper) by a factor of 5.

b g(x) = x − 2

Create a strategy

Apply the idea

The functions f (x) and g (x) have the same slope, but different y-intercepts. This means a vertical translation has occurred.

The slope is 1. The y-intercept is −2. f (x) = x has been vertically translated (shifted) 2 units down.

3.02 Transformations of linear functions mathspace.co

119

Create a strategy

Apply the idea

The functions f (x) and g (x) have different slopes and different y-intercepts. This means a vertical dilation and a translation have occurred. In addition, the new slope is negative, which means a reﬂection has occurred.

The slope is

The y-intercept is 4. f (x) = x has been reﬂected across the x-axis, compressed vertically (made less steep) by a factor of , and vertically translated 4 units up.

Example 4 The drama club is raising money for a ﬁeld trip to see a Broadway musical. To raise the money, they plan to set up a face-painting stand during the high-school football game, and charge $4 per person. The function R (x) = 4x represents their revenue in dollars where x represents the number of faces painted. a The club members spent $45 on face-painting supplies. Write the function P (x) that represents their proﬁt.

Create a strategy

Apply the idea

Proﬁt is calculated by subtracting cost from the revenue.

P (x) = 4x − 45

b Describe the transformation applied to R (x) to get P (x).

Create a strategy We have subtracted 45 from the revenue function which can be represented by P (x) = R (x) − 45. The two functions have the same slope, but their y-intercepts are different, meaning a translation has occurred.

Apply the idea

Reﬂect and check

R (x) has been translated down 45 units to get P (x).

In context, this means that the revenue is decreased by the costs of the supplies, and that is how we get the proﬁt function.

c The drama team realizes that they will need to paint 11 faces to break even at the current rate, so they decide to increase the cost per person to $8. The function N (x) = 8x represents their new revenue function. Describe the transformation from the original revenue function, R (x), to the new revenue function, N (x).

Create a strategy When comparing R (x) = 4x and N (x) = 8x, we can see the only difference is a change in the slope. This means a dilation has occurred.

Apply the idea The slope of N (x) is double the slope of R (x). This means R (x) has been vertically stretched by a factor of 2 to get N (x).

120

Mathspace Virginia SOL Algebra 1 mathspace.co

Reﬂect and check Using technology to compare the graphs of R (x) and N (x), we can see that N (x) does represent a vertical stretch, and its outputs are double the outputs of R (x). 16

12 8 4 −3 −2

−1

−4

x 1

−8 −12 −16

Example 5 Use transformations from the parent function f (x) = x to graph the function f (x) = 3x − 4.

Create a strategy First, identify the transformations from the parent function. Then, use the transformations to graph the function.

Apply the idea 6 5 4 3 2 1

y f (x) = x

The resulting function is f (x) = 3x. x 1 2 3 4 5 6

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6 f (x) = 3x 6 5 4 3 2 1

First, we can perform the vertical dilation by a scale factor of 3. Every y-value will be 3 times the value of the y-value of the parent function.

Next, we can perform the vertical translation down 4 units. Every y-value will be 4 less than the y-value of the function y = 3x. The resulting function is f (x) = 3x − 4.

x −6 −5 −4 −3 −2−1 1 2 3 4 5 6 −1 −2 f (x) = 3x −3 −4 f (x) = 3x − 4 −5 −6

3.02 Transformations of linear functions mathspace.co

121

Idea summary The slope of a line is represented by

m (x1, y1) (x2, y2)

slope a point on the line a second point on the line

The parent function f (x) = x can be transformed to write new functions with changes to the slope and y-intercept. A vertical shift represented by f (x) + k will translate the graph of a total of k units up if k > 0 or k units down if k < 0. A horizontal shift represented by f (x + k) will translate the graph of a total of k units right if k > 0 or k units left if k < 0. A transformation of a vertical dilation kf (x) will stretch a graph’s slope if k > 1 or compress the graph if 0 < k < 1. A vertical reflection of k will change the sign of the slope.

Practice What do you remember? 1

Write the new equation of the function f (x) = x after a transformation of: a

a vertical shift 3 units down.

a vertical dilation by a factor of .

a vertical shift 2 units up.

a vertical dilation by a scale factor of 5.

The graph of y = x is shown with a dashed line. Does the solid line show a vertical stretch or a vertical compression? a

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−2

−3 −4

122

−1

−3

−1

x 1

−2

−4

−4 −3 −2 −1

x 1

−4 −3 −2 −1

−1

−2

−3

−4

Mathspace Virginia SOL Algebra 1 mathspace.co

Match the type of transformation that has been applied to the parent function f (x) to achieve each graph. Answers may be used more than once. i

Translation

Vertical Dilation

Vertical Reflection

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

iii

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Match the type of transformation that has been applied to the parent function f (x) to achieve each function. Answers may be used more than once. i

Translation

Vertical Dilation

iii

Vertical Reflection

f (x) =

f (x) = x + 4

f (x) = 5x

f (x) = −x

For each type of transformation applied to y = x, explain whether the following key features change. If they change, explain how. • Slope • y-intercept • Domain a

Vertical dilation

Reﬂection across the x-axis

Determine which slope represents the steepest line: A

Vertical translation

−7

−1

Let f (x) = 2x and g(x) = 10x − 2. Compare the statements to compare the graph of g(x) to the graph of f (x). a b

The graph of g(x) is shifted ⬚ from the graph of f (x).

The graph of g(x) is ⬚ than the graph of f (x). A up

down

steeper

less steep

3.02 Transformations of linear functions mathspace.co

123

Let’s practice 8

The functions f (x) and g (x) = f (x) + k have been graphed on the coordinate plane. Determine the value of k for each graph. a

y 6

2 −8 −6 −4 −2 −2 f (x)

x 2

f (x)

−4

g(x) −8

−10

f (x)

d 8

x 4

f (x) x

−6

−8

The functions f (x) and g (x) = kf (x) have been graphed on the coordinate plane. Determine the value of k for each graph. a

y 8

b 8

g(x)

−8 −6 −4 −2 −2

4 f (x) 2

x 6

−6

f (x)

−8

y g(x)

−6 −8

Mathspace Virginia SOL Algebra 1 mathspace.co

x 2

−8

−8 −6 −4 −2 −2 f (x) −4

−6

−4 −3 −2 −1 −2 g(x) −4

−4

4 2

124

8 10

−8 −6 −4 −2 −2 g(x) −4

6 g(x)

−6 −8

x 2

−4

−6

−10 −8 −6 −4 −2 −2 g(x) −4

−6 −4 −2 −2

−8 −6 −4 −2 −2 g(x) −4 f (x)

−6 −8

x 2

Describe the transformations of the given line from the parent function f (x) = x. a

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2

−3

−4

−4 −3 −2 −1

−1

x 1

−1

−2

x 1

−4 −3 −2 −1

−1

−2

−3

−4

Match each graph to the equation: i a

−1

iii

b 4

−1

−4 −3 −2 −1

−2

−3 −4

−2

−3

−1

−4

−4 −3 −2 −1

x 1

y = 3x + 1

1 −4 −3 −2 −1

y = −x

x 1

−4 −3 −2 −1

−1

x 1

−2

−3

−4

3.02 Transformations of linear functions mathspace.co

125

For each equation: i

Describe the transformations of the given line from the parent function f (x) = x.

Find the value of the y-intercept.

iii

Find the domain.

Find the range.

g (x) = x − 5

g (x) = x + 3

g(x) = −5 + 3x

g(x) = − x − 4

g (x) = 2x + 1

y = − 2x + 3

f (x) = −5x + 3

Use transformations from the parent function f (x) = x to graph each function. a

g(x) = 3 −

f (x) = 3x

f (x) =

f (x) = −x − 4

Find the resulting equation when y = x undergoes these transformations. a

A dilation by a factor of 10

A dilation by a factor of

A vertical translation of 10 units up

A reﬂection about the x-axis and a vertical translation of 7 units down

A vertical compression by a factor of 0.75 and a shift of 2 units upwards

and a reﬂection across the x-axis

Explain the difference between the graphs of the equations y = 2x and y = −2x.

Let’s extend our thinking 16

Consider the graph of the function f (x) = x. Sketch the graph of the transformed function if: 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 f (x) = x

−3 −4

the slope is tripled.

the slope is multiplied by −1.

the y-intercept is moved up 4 units.

the slope is halved, and the y-intercept is moved down 2 units.

A personal tutor charges a transportation fee of $10 plus $20 per hour. The tutor’s fees can be modeled by f (x) = 20x + 10. For the beginning of the school year, the tutor is offering a reduced rate of $10 per hour, plus the transportation fee. The new rate can be modeled by r (x) = 10x + 10. Describe the transformation from the original fees, f (x), to the new fees, r (x).

A café owner models his daily coffee sales with the function C (x), where x is the number of cups of coffee sold. After a marketing campaign, the sales increase by a factor of 1.5. If he originally sold 100 cups in one day, how many cups will he sell in a day after the campaign?

126

Mathspace Virginia SOL Algebra 1 mathspace.co

Oreste draws the graph of g (x) = −f (x), as shown. Describe the error Oreste made. 4

3 2

g(x)

f (x)

−4 −3 −2 −1

x 1

−1

−2 −3 −4

Stavros has a bank account with $100. Every week, Stavros deposits an additional $50 into the account. A graph of Stavros’ bank balance reveals that the growth of his account is linear. Balance 500 400 300 200 100 Weeks 1

Suppose that Stavros was instead depositing an additional $100 every week. Explain whether or not this could be represented by a vertical stretch of the current graph.

3.02 Transformations of linear functions mathspace.co

127

Given the information about a linear function, we can write the equation in slope-intercept form. 36 33 30 27 24 21 18 15 12 9 6 3

Consider this graph of a leaky faucet that drips at a rate of 4 cups per hour. It has already dripped 15 cups of water.

Let x represent the number of hours and y represent the number of cups of water that have leaked from the faucet. First, identify or solve for the slope, which represents the m in y = mx + b. In the graph, we can use a slope triangle to see the slope is . We can also see the rate of change in the context is 4. x 1

2 3 4 5 6 7 8 9

Next, identify or solve for the y-intercept, which represents the b in y = mx + b. In the graph, the y-intercept is at (0, 15), so b = 15. From the context, we can see the initial value of the leaked water is 15 cups. Lastly, rewrite y = mx + b, substituting the solved values of m and b. Since we have found m = 4 and b = 15, we can write the equation: y = 4x + 15 Slope-intercept form is especially helpful when we want to graph a linear function. The graph of a line represents the set of points that satisﬁes the equation of a line. 4

Consider the equation y = 2x − 1.

First, identify the value of b, and plot this as the y-intercept. Since the y-intercept is −1, plot a point at (0, −1).

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3

Then, identify the value of m, and the value of the rise and the run. From the y-intercept, use the rise and run to create slope triangles to plot additional points. Each point on the graph is a solution of the equation, so every point satisﬁes y = 2x − 1.

−4

Example 1 Consider the following graph of a line:

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

x 1 2 3 4 5

a What is the slope of the line shown in the graph?

Create a strategy The slope of a line is equal to the vertical ‘rise’ divided by horizontal ‘run’. It is the ratio of the vertical change to the horizontal change.

3.03 Slope-intercept form mathspace.co

129

Apply the idea Considering two points from the graph: (0, 1) and (5, 0), from (0, 1) requires moving 1 unit down and 5 units to the right. The slope is

b What is the y-value of the y-intercept of the line shown in the graph?

Create a strategy

Apply the idea

The y-intercept is the point where the line intersects the y-axis.

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

x 1 2 3 4 5

Looking at the graph, the line intersects the y-axis at point (0, 1). Thus, the y-value of the y-intercept is y = 1.

c Write the equation of the line in slope-intercept form.

Create a strategy Substitute the values of the slope and y-intercept in the slope-intercept form of the equation of a line.

Apply the idea Start with the slope-intercept form

Substitute

and b = 1

Reflect and check We can check our equation using points from the graph. Since each point must be a solution to our equation, choose any point on the line. For this example, choose (5, 0). Substitute x = 5 and y = 0

Simplify

Since it is true that 0 = 0, this point satisfies our equation.

130

Mathspace Virginia SOL Algebra 1 mathspace.co

d Find the domain and range of the line.

Create a strategy The domain of a line in the graph represents all possible x-values that the line covers. The range of a line in the graph represents all possible y-values that the line covers.

Apply the idea

Reflect and check

Domain: −∞ < x < ∞

We can see from the graph that the line extends without any breaks or endpoints along the x and y-axis, confirming that the domain includes all real numbers.

Range: −∞ < y < ∞

The domain in interval notation is: (−∞, ∞) The domain in set notation is: {x ∣ x∈} The range in interval notation is: (−∞, ∞) The range in set notation is: {y ∣ y∈}

Example 2 Find the equation in slope-intercept form for: a A line with a slope of

and a y-intercept of −3.

Create a strategy Substitute the values of the slope and y-intercept in the slope-intercept form

Apply the idea Start with the slope-intercept form

Substitute

and b = −3

b A line with a slope of −3 and passes through the point (2, 3).

Create a strategy Substitute the slope and point in the slope-intercept form, and solve for the y-intercept before rewriting.

Apply the idea y = mx + b

Start with the slope-intercept form

3 = −3(2) + b

Substitute y = 3, m = −3, and x = 2

3 = −6 + b

Evaluate the multiplication

9=b

Add 6 to both sides

y = −3x + 9

Rewrite using slope-intercept form

3.03 Slope-intercept form mathspace.co

131

c A line that passes through the points (−4, 5) and (8, 8).

Create a strategy Substitute the given points into the slope formula to ﬁnd slope. Use the slope and one point to solve for the y-intercept before rewriting in slope-intercept form.

Apply the idea Start with the slope formula

Substitute (x1, y1) = (5, −4) and (x2, y2) = (8, 8)

Evaluate

Simplify Choose (x1, y1) or (x2, y2) to substitute with m into y = mx + b

Evaluate the multiplication

Subtract 2 from both sides

Rewrite using slope-intercept form

Example 3 Write each of the following equations in slope-intercept form. a −4y = 12 − 8x

Create a strategy Slope-intercept form is y = mx + b, where m is slope and b is the y-intercept.

Apply the idea

Reflect and check

Each term will be divided by −4 and simplified.

Slope is the simplified ratio of

. By

Original equation

looking at the graph, we could see a change of

Divide each term by −4

counting down and left between points. This rate of

Simplify and reorder terms

change matches the simplified ratio of , or a slope of 2.

The slope-intercept form of −4y = 12 − 8x is y = 2x − 3.

5 4 3 2 1 −2

132

Mathspace Virginia SOL Algebra 1 mathspace.co

−1

−1 −2 −3 −4 −5

x 1

b 7x − 14y = −28

Create a strategy Isolate the y and rearrange terms to write in slope-intercept form, y = mx + b.

Apply the idea The x term will be moved first then divide by the coefficient of y. Original equation

Subtract 7x from each side

Write with x term before constant

Divide each term by −14

Simplify The slope-intercept form of 7x − 14y = −28 is

Reflect and check Other forms of equations will create the same graphs as slope-intercept form but may highlight different information. By creating a table of values and graphing, we see that 7x − 14y = −28 has the same slope and y-intercept as . 4

3 2 y-intercept 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy

Apply the idea

Slope-intercept form is y = mx + b. Distribute any values outside parentheses and rearrange terms to isolate y.

We will first distribute the then isolate the y to write the equation in slope-intercept form. Original equation

Distribute

Subtract 2

Simplify

The slope-intercept form of

and multiply

3.03 Slope-intercept form mathspace.co

133

Example 4 A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water. The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes. Time in minutes (x) Water remaining in gallons ( y)

0 30

1 28

2 26

3 24

a Determine the linear equation in slope-intercept form that represents this situation.

Create a strategy We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the table of values.

Apply the idea Find the slope using the values (0, 30) and (1, 28): Slope formula

Substitute (x1, y1) = (1, 30) and (x2, y2) = (1, 28)

Evaluate

Notice that the initial value, or y-intercept is given in the table as (0, 30). Writing in the form y = mx + b the equation that represents this situation is y = −2x + 30.

Reﬂect and check If we had not noticed that the y-intercept was given, we could have substituted in any pair of values for x and y, and solved for b.

b Draw the graph of this linear relationship with a clearly labeled scale. Only show the viable solutions.

Create a strategy We can’t have a negative time (x ≥ 0) and we should end the graph when the tub is empty ( y = 0). To plan our graph, we need to ﬁnd when the tub is empty.

Apply the idea To ﬁnd when the tub is empty: y = −2x + 30

Original equation

0 = −2x + 30

Substitute y = 0

−30 = −2x 15 = x

134

Subtraction property of equality Division property of equality

Mathspace Virginia SOL Algebra 1 mathspace.co

This tells us that we have the restriction that 0 ≤ x ≤ 15 and 0 ≤ y ≤ 30, which will help us choose the appropriate axes and scale. 30

Water in gallons

25 20 15 10 5 5

Time in minutes 10 15

We know that the bathtub begins with 30 gallons which is represented by the y-intercept at (0, 30) and that the slope of the linear equation is −2 which means that the tub loses 2 gallons of water every minute until there is no water in the tub at 15 minutes. 30

Water in gallons

25 20 15 10 5 5

Time in minutes 10 15

Reﬂect and check We can also graph the slope by using the idea of

. Since the slope is −2, we can write it as a fraction

and

identify that the change in y-values (or rise) is −2 and that the change in x-values (or run) is 1.

c Find the domain and range.

Create a strategy

Apply the idea

The domain of a function represents all possible x-values. Domain: 0 ≤ x ≤ 15 The range of a function represents all possible y-values. Consider values that may fit the pattern but do not fit the context.

Notice that even though negative x-values could fit the pattern, it does not make sense with the context to have a negative amount of time. Range: 0 ≤ y ≤ 30 In the context, water is never added so it can never be above the starting amount of 30 gallons. Continuing the pattern, the y-value will never drop below empty, or 0 gallons.

3.03 Slope-intercept form mathspace.co

135

Idea summary The slope-intercept form of a line is:

y = mx + b m slope b

y-intercept

Slope-intercept form is useful when we know, or want to know the slope of the line and the y-intercept of the line.

Practice What do you remember? 1

What general equation represents slope-intercept form? What do m and b represent?

Consider the graph of each linear function. i

Determine the slope, m.

Determine the y-intercept, b.

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−1

−2

−3

−4

x 1

In each of the equations, identify the value of m and b. a

y = 15x − 10

y = 6 + 8x

y = 7x

y=6−x

Determine whether each equation is in slope-intercept form. a

y = 2x − 1

y = 5x

4x + 3y = 0

y = 2y − 4

x = 2y

Find the equation in slope-intercept form for: a

A line with a slope of 2 and y-intercept of 5.

A line with a slope of

and y-intercept of 3.

A line with a slope of −2 and y-intercept of 0.

A line with a slope of

and y-intercept of −3.

3x − 2y = −4

Rewrite each equation in slope-intercept form: a

y = 5 − 4x

y − 1 = 5 (x − 4)

y + 7 = −3x

3.03 Slope-intercept form mathspace.co

137

Let’s practice 7

For each graph: i

What is the y-intercept?

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2 −3

−4

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

For the line shown: y 5 4 3 2 1 −2

−1

x 1

−2

138

What are the values of the slope, m, and the y-intercept, b?

Write the equation of the line in slope-intercept form.

Find the value of y when x = 27

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

−1

−3

6 5 4 3 2 1

−2

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

What is the slope?

x 1 2 3 4 5 6

This graph shows the relationship between the time (x) in minutes and distance ( y) in kilometers of a highspeed train ride at a constant speed. y (kilometers) 5 4 3 2 1 1

x (minutes) 4 5

Find the equation of the line in slope-intercept form.

Find how far the train traveled, y, after 29 minutes.

Find the equation of the line in slope-intercept form. a

5 4

−2

1 −1

−1

x 1

−1

3 −2

−2

1 −2

−1

x 1

−1

= 3x − 2

−1

−1 −2

−2

−3 −3

−4

By rewriting the equation in the form y = mx + b: i

What is the slope, m?

What is the value of the y-intercept, b?

y−7=

9x − y − 8 = 0

3y = 12x − 15

y = 3 (4x − 3)

Consider the line passing through the point (7, 1) that has a slope of 4. a

Find the value of b, the y-intercept.

Use the slope and y-intercept to write the equation in slope-intercept form. 3.03 Slope-intercept form mathspace.co

139

Find the equation in slope-intercept form for: a

A line with a slope of 8 and passes through the point (0, −4).

A line with a slope of −1 and passes through the point (0, 0).

A line with gradient

A line with gradient 8 and passes through the point (0, −4)

and passes through the point (0, 3)

For each line described: i

Find the slope of the line.

Find the equation of the line in slope-intercept form.

A line passes through the points (−3, 0) and (6, 6).

A line passes through the points (3, −3) and (5, −11).

A line passes through the points (−6, 7) and (−8, −4).

A line passes through the points (−7, 1) and (5, 2).

For each description: i

Find the y-value of the y-intercept.

Write the equation of the line in slope-intercept form.

iii

Find the x-value of the x-intercept of the line.

Sketch the graph of the line.

A line that has slope of

A line that has slope of −2 and passes through the point (3, −8).

and passes through the point (−10, 4).

For each of the following functions: i

Rewrite the equation in slope-intercept form using the function notation f (x).

Find the value of f (0).

iii

Graph the function, labeling each intercept.

y = 12 − 3x

y − (−5) = 5 (x − 2)

2y = −2x − 8

10x − 2y = 20

Sketch the graph of each line described. a

The line with a y-intercept of −2 and slope of −3

The line with a y-intercept of 3 and slope of

The line with equation y = 2x + 5

The line with equation

A cleaner charges an initial fee of $110 plus $20 per hour. a

Identify the slope and y-intercept of the scenario.

Write an equation to represent the total amount charged by the cleaner, y, as a function of the number of hours worked, x.

State the domain of the linear relationship.

A race car starts the race with 60 gallons of fuel. From there, it uses fuel at a rate of 1 gallon per minute. a

Complete the table of values. Number of minutes passed (x) Amount of fuel left in tank f (x)

140

Write the equation relating the number of minutes passed x and the amount of fuel left in the tank f (x).

Graph the function, including clearly labeled axes with an accurate scale.

Explain how the function would change if the car was repaired and now uses 0.9 gallons per minute.

Mathspace Virginia SOL Algebra 1 mathspace.co

For each of the following pairs of linear functions, identify which function has the greater y-intercept: a • Function 1: The line with a slope of 4 that crosses the y-axis at (0, 6). • Function 2: y = x + 4 Function 1: b •

• Function 2:

x 2 4 6 y 2 −2 −6

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

• Function 1: x y

2 19

4 35

2 19

4 31

• Function 2: y = 4x + 6

6 51

Function 1: d • x y

• Function 2: y = 3x + 4

6 43

Function 1: e •

• Function 2:

x 2 4 6 y −5 −13 −21

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

1 2 3 4 5

x 1 2 3 4 5

• Function 1: The line with a slope of 7 that crosses the y-axis at (0, 5). • Function 2: y = (−m) x + 7

For each of the following pairs of linear functions, identify which function has the greater slope. a • b • Function 1: y = −3 + 3x Function 1: y = 1 + 8x • Function 2: • Function 2: x 0 1 2 y 1 6 11

x y

0 −3

1 2

2 7

3.03 Slope-intercept form mathspace.co

141

A carpenter charges an initial fee of $250 plus $35 per hour for labor. a

Write an equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.

Interpret the slope of this linear function.

Interpret the y-intercept of this linear function.

Peta has budgeted $2500 to have a ramp built at the community center to increase accessibilty.

The carpenter works 8 hour days and estimated that the project would take one and a half days. He also priced the materials at $1800. Determine if Peta has budgeted enough for the ramp. Justify your answer.

Let’s extend our thinking 23

Which of the following linear relationships has the smaller corresponding x-value when y = 42? • Function 1: x y

0 2

1 10

2 18

• Function 2: y = 3x + 18 24

Two construction workers are competing to see who can lay the most bricks in one hour. The graph shows the number of bricks laid and the time, in minutes.

280

Use the given graph to compare the construction workers.

240

320

Bricks laid Charlie

200 160 120

Neville

80 40

Minutes 10 20 30 40 50 60 70 80

Edin and Marius are saving money for a trip. Edin saves $100 per month, and Marius saves $80 per month after starting with $60. Both start saving in January. a Write linear equations to represent the total amount ( y) saved by each friend after x months. b Interpret the meaning of the y-intercept of each equation. c Graph both equations on the same coordinate plane. Use an appropriate scale and label the axes. d Determine which friend will have saved more by June.

Some friends decide to go camping for the weekend. They cannot all ﬁt in one car so some of them catch a bus to the campground, which is 450 mi from home. Those in the car started driving at 8:00 am and arrived at the campground at 3:30 pm, driving at a constant speed. The bus also drives at a constant speed and takes the same route as the car. Its distance in miles, y, from home x hours after leaving is given by the equation y = 71x.

142

Determine the speed of the car, in miles per hour.

Determine the speed of the bus, in miles per hour.

Determine which vehicle was traveling faster.

Mathspace Virginia SOL Algebra 1 mathspace.co

For the three linear equations and their corresponding graphs: 10

y = x + 4, y = 2x + 4, y = 4x + 4 a

What do all of the equations have in common?

What do all of the graphs have in common?

What conclusion can be made about all lines that have the form y = mx + 4?

8 6 4 2

−8 −6 −4 −2 −2

−4

Look at the three lines: 8

• Equation 1: y = 2x + 4 • Equation 2: y = 2x + 8 • Equation 3: y = 2x − 4

y = 2x + 8

What do all the equations have in common?

What do all the graphs have in common?

Describe all lines that have the form y = 2x + b.

6 4

y y = 2x + 4

2 −8 −6 −4 −2 −2 −4

x 2

y = 2x − 4

−6 −8

Create a scenario where you would choose to use slope-intercept form, y = mx + b and explain what the slope and y-intercept mean in context.

The table shows the water level of a well that is being emptied at a constant rate with a pump where the time is measured in minutes and the water level in feet. Time (x) Water level ( y) a b c d

3 56.7

7 48.3

10 42

Write an equation to model the scenario. Interpret the key features of this linear function. Find the water level after 18 minutes. State the viable values of x for this scenario. Justify your answer.

This graph shows the relationship between the time (x) in seconds and distance ( y) in meters for a sprint that Jocel ran at a constant speed. a

Find the equation of the line in slope-intercept form.

Find how far Jocel ran, y, after 50 seconds.

10 9 8 7 6 5 4 3 2 1

y (meters)

x (seconds) 1 2 3 4 5 6 7 8 9 10

3.03 Slope-intercept form mathspace.co

143

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth below the surface of the diver over the ﬁrst 5 minutes. Number of minutes passed (x) (min) Depth of diver (y) (ft)

0 0

1 1.63

2 3.26

3 4.89

4 6.52

State the rate of change with units of ft/min.

Write an equation for the relationship between the number of minutes passed (x) and the depth below the surface (y) of the diver.

Determine the depth of the diver after 6 minutes.

Find how long the diver takes to reach 11.41 ft beneath the surface.

The diver has two air tanks, each of which last for 45 minutes. One is used for going down and one is used for returning to the surface. Determine an appropriate domain restriction for the function from part (b).

144

Assuming the diver comes back up at the same rate they go down, state the equation for the linear function which could describe the return to surface including the domain restriction. Sketch the full scenario as a graph.

Mathspace Virginia SOL Algebra 1 mathspace.co

x-intercept: substitute y = 0 and solve:

2x − 3 (0) = 3

2x = 3

2 1 −4 −3 −2 −1

−1

x 1

y-intercept: substitute x = 0 and solve: 2 (0) − 3y = 3

−3y = 3

−2

Graph the intercepts, in this case line through the points.

−3 2x − 3y = 3 −4

and (0, −1), and draw the

We see that the graph still maintains the same slope and y-intercept as the original slope-intercept form, but standard form allows us to focus on other key features of the function.

Example 1 Consider the linear equation:

a Determine whether the equation is written in standard form. If not, rewrite it in standard form.

Create a strategy In order for the equation to be written in standard form, we must conﬁrm that the equation is written in the form Ax + By = C, that A, B and C are integers, and A and B are not both zero.

Apply the idea Since

in the equation, it’s not written in standard form because A must be an integer and integers are the set of

positive and negative whole numbers. To convert the equation to standard form, we can use properties of equality and multiply each term by 2 to change the rational coeﬃcient of x. Original equation

Multiplication property of equality

Evaluate the multiplication

A = 1, B = 8, and C = 32 now meet the requirements for standard form.

b Graph the equation using the standard form from part (a).

Create a strategy To graph the equation, we will ﬁnd the x-intercept and y-intercept and then draw the line passing through these points.

146

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea To ﬁnd the x-intercept, set y = 0 and solve for x: x + 8(0) = 32

Substitute y = 0

x = 32

Simplify

So, the x-intercept is (32, 0). To ﬁnd the y-intercept, set x = 0 and solve for y: 0 + 8y = 32

Substitute x = 0

8y = 32

Simplify

y=4

Divide both sides by 8

So, the y-intercept is (0, 4). y 8 6 4 2 x 2

10 12 14 16 18 20 22 24 26 28 30 32

Reﬂect and check By using the intercepts, we accurately graphed the equation. The line passes through both intercepts (32, 0) and (0, 4).

Example 2 A line with a slope of

passes through the point (−8, 5).

a Write the equation of the line in slope-intercept form.

Create a strategy Use the slope and the point to solve for the y-intercept and write in slope-intercept form.

Apply the idea Slope-intercept form

Substitute y = 5, m = − , and x = −8

Evaluate the multiplication

Simplify

Write in slope-intercept form

3.04 Standard form mathspace.co

147

b Convert this equation to standard form.

Create a strategy Rearrange the terms and multiply coeﬃcients as needed to ﬁt standard form.

Apply the idea Slope-intercept form from part a

Multiply both sides by 4

Evaluate

Move the x term to the other side of the equation.

The standard form of the equation is x + 4y = 12.

c Solve for the x-intercept and y-intercept of the standard form equation.

Create a strategy Substitute 0 and solve to ﬁnd the x-intercept, (x, 0), and the y-intercept, (0, y).

Apply the idea Finding the x-intercept:

Finding the y-intercept

x + 4y = 12

Standard form equation from part b

x + 4y = 12

Standard form equation from part b

x + 4 (0) = 12

Substitute y = 0

(0) + 4y = 12

Substitute x = 0

x + 0 = 12 x = 12

Evaluate the multiplication

4y = 12

Evaluate the addition

Simplify

y=3

Divide both sides by 4

The x-intercept is (12, 0).

The y-intercept is (0, 3).

Example 3 Consider the line shown in the graph shown.

9 8 7 6 5 4 3 2 1 −1

−1

x 1

Write the equation of the line in standard form.

Create a strategy To write the equation of the line in standard form, we will ﬁrst write it in slope-intercept form y = mx + b, then convert it to standard form Ax + By = C where A, B, and C are integers.

148

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea First, we ﬁnd the slope (m) of the line. Write the formula

Substitute y2 = 4, y1 = 1, x2 = 0 and x1 = 2

Evaluate the subtraction

Simplify

The slope of the line is

Now, we use the slope-intercept form with the point (0, 4) to ﬁnd the y-intercept (b): Write the slope-intercept form

Substitute

and b = 4

Now, we convert the slope-intercept form to standard form: Original slope-intercept form

Multiply both sides by 2

Evaluate the multiplication

Add 3x to both sides

Evaluate and rearrange to standard form

The equation of the line in standard form is 3x + 2y = 8.

Reﬂect and check To conﬁrm our equation, we can check the intercepts: For the x-intercept, set y = 0: Substitute y = 0

Evaluate the multiplication

Divide both sides by 3

Evaluate

The x-intercept is

For the y-intercept, set x = 0: Substitute x = 0

Evaluate the multiplication

Divide both sides by 2

Evaluate

The y-intercept is (0, 4). These intercepts match the points given in the original problem, conﬁrming that the equation 3x + 2y = 8 is correct.

3.04 Standard form mathspace.co

149

Example 4 A tour company travels to the Great Smoky Mountains National Park. They use a combination of buses and vans to get tourists to their destination. One bus can take 42 passengers, and one van can take 7 passengers. One day, they have 168 people register for the tour. a Write an equation in standard form that could be used to model the number of buses and vans they could use to transport all the people registered, assuming that each vehicle will be filled.

Create a strategy In words, we can start with the idea that: (Number of people on buses) + (Number of people on vans) = Total number of people and that: Number of people on buses = 42 ⋅ (Number of buses) and: Number of people on vans = 7 ⋅ (Number of vans) We will then need to deﬁne variables and write an equation using them.

Apply the idea Let b represent the number of buses used and let v represent the number of vans used. So the Number of people on buses = 42b and Number of people on vans = 7v which ﬁnally gives us the whole equation: 42b + 7v = 168

Reﬂect and check It is important to declare variables and it can be helpful to use variables that relate to the quantities in the scenario to make sure we don’t mix them up.

b Graph the equation with an appropriate scale and labels.

Create a strategy In this case, there isn’t a clear independent and dependent variable, so we can put b on the horizontal axis and v on the vertical axis. To determine an appropriate scale, we can ﬁrst ﬁnd the values of the intercepts as those can give an idea of the maximum values for each axis.

150

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea Find the b-intercept:

Find the v-intercept:

42b + 7v = 168

Equation from part (a)

42b + 7v = 168

Equation from part (a)

42b + 7 (0) = 168

Substitute v = 0

42 (0) + 7v = 168

Substitute b = 0

42b = 168 b=4

Evaluate the product

7v = 168

Evaluate the product

Division property of equality

v = 24

Division property of equality

26 Number of vans (v) 24 22 20 18 16 14 12 10 8 6 4 2 Number of buses (b) 1

An appropriate scale would be going up by 1 along the b-axis to a maximum of 5 and going up by 2 or 4 along the v-axis to a maximum of 26. This will extend both of our axes just past where we need to plot the intercepts.

Reﬂect and check Notice that it would not make sense to connect these dots with a line. This is because the situation is discrete, not continuous. If we were to include all points that fall on the line, this would mean we could use partial buses or vans to transport people. While the vans or buses could only be partially filled, we would still require a whole number of buses and vans.

c Predict the number of vans that would be required if only 1 bus was available.

Create a strategy We can ﬁnd the point along the b-axis where b = 1 and then up to the line and across to the v-axis to ﬁnd the corresponding value for v, the number of vans.

Apply the idea 26 Number of vans (v) 24 22 20 18 16 14 12 10 8 6 4 2 Number of buses (b) 1

Using the graph, we can see that the point (1, 18) lies on the graph of the equation. If only 1 bus was available, they would need 18 vans.

3.04 Standard form mathspace.co

151

Reﬂect and check We can check using the equation. 42b + 7v = 168

Equation from part (a)

42 (1) + 7v = 168

Substitute b = 1

42 + 7v = 168

Evaluate the product

7v = 126

Subtraction property of equality

v = 18

Division property of equality

Idea summary The standard form of a line is:

Ax + By = C A, B, C

are integers

To write the equation of the line in standard form, we will follow these steps: 1. Use the given information to write the equation in slope-intercept form, y = mx + b. 2. If m is a fraction, multiply each term in the equation by the denominator of m. 3. Move the x term to the y side of the equation. Standard form is useful when we know, or want to know both intercepts of the line.

Horizontal and vertical lines Recall that lines can also be either horizontal or vertical. These types of lines will not look like slope-intercept form. However, they do follow another type of special pattern.

Horizontal lines are the set of all points with a ﬁxed y-value. They are parallel to the x-axis and have equations of the form y = a, where a is a real number. Horizontal lines have a slope of zero.

Shown is the horizontal line y = 2.

1 −4 −3 −2 −1

x 1

−1

Notice this equation is in standard form with A = 0, since the equation is equivalent to 0x + y = 2. What happens to the equation y = mx + b if you substitute 0 for m? We get y = b.

−2 −3 −4

Vertical lines are the set of all points with a ﬁxed x-value. They are parallel to the y-axis and have equations of the form x = a, where a is a real number. Vertical lines have a slope that is undeﬁned.

Shown is the vertical line x = 1.

1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

152

Mathspace Virginia SOL Algebra 1 mathspace.co

Notice this equation is in standard form with B = 0.

Example 5 Consider the line x = −8. a Plot the line on a coordinate plane.

Create a strategy

Apply the idea

We can use the fact that vertical lines have equations of the form x = a.

This will be a vertical line that crosses the x-axis at −8. y 4 2 x −8 −6 −4 −2

−2 −4

b Determine the domain and range of the line.

Create a strategy

Apply the idea

The domain of a vertical line is restricted to the single value of x at which the line is located.

Domain: {−8} Range: (−∞, ∞)

The range of a vertical line is all possible y-values since the line extends inﬁnitely in both the positive and negative y-directions.

Example 6 Write the equation of the line given. a 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Apply the idea Horizontal lines have a ﬁxed y-value and have equations of the form y = a. The ﬁxed y-value for this line is −3, so the equation of the line is: y = −3

3.04 Standard form mathspace.co

153

b An undeﬁned slope through the point (2, −1).

Apply the idea Lines that have an undeﬁned slope are vertical lines which have equations with the form x = a. The x-value of the ordered pair (2, −1) is 2, so the equation can be written as: x=2

Idea summary Horizontal lines are the set of all points with a ﬁxed y-value. They are parallel to the x-axis and have equations of the form y = a, where a is a real number. Vertical lines are the set of all points with a ﬁxed x-value. They are parallel to the y-axis and have equations of the form x = a, where a is a real number.

Practice What do you remember? 1

State whether the each equation are written in standard form: a

y = −x + 1

18x − 6y = 12

3y = 4x

Write each of the following equations in standard form. a

SOL

3x + 2y = 6

−y + 4x = 8

3y = 8 − 4x

Select the equation that is shown on the graph. A

10x + 5y = −1

−10x + 5y = −1

−5x − 10y = −1

−x + 5y = −10

2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

For each equation, identify the slope, the x-intercept, and the y-intercept. a

2x + y = −6

3x − 7y = −21

9x − y = 18

State whether the graphs of the following lines are horizontal or vertical: a

y=2

x = −4

y=3

y=0

x=0

y−

The table shows some points on the line with equation y = 0. Does the line y = 0 represent the y-axis or y-axis?

154

Mathspace Virginia SOL Algebra 1 mathspace.co

x = 17

x y

−6 0

−4 0

1 0

5 0

Consider the sets of points in the following coordinate planes: i

State whether the set of points lies on a vertical or horizontal line.

Find the equation of the line that passes through the set of points.

−8 −6 −4 −2 −1

−8 −6 −4 −2 −2

−2

−4

−3

−6

−4

−8

Let’s practice SOL

Select the graph of the equation 3x + 4y = −2. A

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

For each of the following equations: i

Calculate the x-value of the x-intercept of the line.

iii

Draw the graph of the equation of the line on the coordinate plane.

3x − 5y = −15

5x − 3y = −15

Calculate y-value of the y-intercept of the line.

2x + y = 10 3.04 Standard form mathspace.co

155

Draw the graph of each of the following linear equations: a

2x = 6

6y = −3

3x − 2y = −12

Write the equation of the line.

For each of the following graphs: i

Find the slope.

1 −4 −3 −2 −1

−1

−2

−3

−4

−4 −3 −2 −1

−1

−2

−3

−4

For each of the following tables of values: i

Sketch the graph of the line that passes through the four points.

Find the equation of the line.

x 2 2 2 2 y −2 −1 0 1

x y

y = −2

−4 5

−2 5

0 5

2 5

Plot the following lines on a coordinate plane: a

−1

x 1

−1

−2

−4 −3 −2 −1

5x + 4y = 40

y=8

x=7

y=0

Consider the following line: a

State the y-value of the y-intercept.

Find the slope, m, of the line.

Find the equation of the line in the form y = mx + b.

Rewrite the equation of the line in standard form.

2 1 −4 −3 −2 −1

−1

−2 −3 −4

156

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

For each graph, write the equation of the line in standard form. a

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

x 1

−1

−2

−3

−4

Write the equation of the line in standard form. a

A straight line passes through the point (3, 5) with a slope of

A straight line passes through the point (−2, 0) with a slope of 2

A straight line passes through the point (−3, −1) with a slope of −1

A straight line passes the point (−1, 2) and the point (3, 0)

Marvin went to the bulk food store to buy x lb of granulated sugar for $0.60/lb and y lb of powdered sugar for $0.90/lb. The total cost of all the sugar is $12. a

Select the graph of the linear function that relates the number of pounds of granulated sugar x and the number of pounds of powdered sugar y. A

y 18

6 3

x 3

D 21

6 x 6

y 21

Identify the restrictions on the independent variable.

3.04 Standard form mathspace.co

157

Salvina needs to buy a mix of red and green apples. She is going to spend $15 in total. Green apples cost $1.50 lb and red apples cost $3 lb. a

Write a linear equation in standard form to represent the scenario, deﬁning any variables.

Calculate and interpret the x-intercept of the linear equation from part (a).

Convert the equation from part (a) to slope-intercept form.

Graph the function, including clearly labeled axes with an accurate scale.

Apples

Green $1.50/lb

Red $3/lb

For each of the following pairs of linear functions, identify which function has the greater x-intercept: a • Function 1: The line with a slope of 3 and a y-intercept of −3. • Function 2: x − y = 3 • Function 2: b • Function 1: The line with a slope of −1 and a y-intercept of 4. x 0 1 y −6 −3 c

• Function 2: • Function 1: 5x − y = 15 x y

x y

0 4

1 2

0 3

1 5

• Function 2:

1 2

2 1

5 4 3 2 1 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5 6 7

• Function 1: • Function 2: 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

158

2 4

2 0

e • Function 1: x y

0 6

• Function 2: 4x + y = 16

Function 1: d •

2 0

x y

x 1 2 3 4 5

Mathspace Virginia SOL Algebra 1 mathspace.co

0 −8

1 −4

2 0

Which of the following linear relationships has an x-intercept? • Function 2: • Function 1: y = 3 8

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Let’s extend our thinking 21

The line ax + 4y = 12 passes through the point (8, −3). a

Solve for a.

Write the equation of the line in standard form.

The line kx + 3y = 15 passes through the point (13, −8). a

Find the value of k.

Solve for the x-value of the x-intercept of the line.

Solve for the y-value of the y-intercept of the line.

Given: • Line A: 5x + 3y + 5 = 0 • Line B: 7x + 6y − 3 = 0 Which line is steeper? Justify your answer.

Describe a scenario where using the standard form of a linear equation is more useful than the slope-intercept form.

Create a scenario where you would choose to use standard form, Ax + By = C, and explain what the intercepts mean in context.

Eﬃe is a entomologist and is currently studying mosquitos and spiders. She knows that mosquitos have six legs and spiders have eight legs. In her lab, she has a mix of mosquitos and spiders. Between all the bugs, there is a total of 240 legs. a

Create a model to represent this scenario. Deﬁne any variables and include appropriate labels in your model.

State and describe the possible number of mosquitos.

Explain whether or not every point on the line represents a possible solution.

3.04 Standard form mathspace.co

159

Nakisha is selling the baby outfits her daughter has outgrown and is buying second-hand toddler outfits. She is able to sell each baby outfit for $3 and she buys each toddler outfit for $5. In the end, she wants to make a profit of exactly $30. Profit = revenue − cost Revenue is the money earned. Cost is the money spent.

160

Write an equation to represent the scenario, using x for the number of baby outfits sold, y for the number of toddler outfits bought, and the profit of exactly $30.

Calculate and interpret the x-intercept.

Nakisha has 25 baby outfits available to sell and needs 15 toddler outfits. Determine if she is able to make her desired profit of $30.

Using the additional information from part (c), state and interpret the restrictions on the viable values for x.

Mathspace Virginia SOL Algebra 1 mathspace.co

The same function could also be represented as

3 (−2, 2)

2 1

since the slope remains constant at line (−2, 2).

−4 −3 −2 −1 −1

but uses the point on the

−2 −3 −4

We can also ﬁnd the equation in point-slope form when given the coordinates of two points on the line by following these steps: 1. Choose which ordered pair will represent (x1, y1) and the other (x2, y2).

9 y 8 7 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 −7 −8 −9

x1, y1 x 1

162

x2, y2

9 y 8 7 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 −7 −8 −9

2. Solve for slope using

(3, 5) x 1

Mathspace Virginia SOL Algebra 1 mathspace.co

8 (7, −2)

3. Rewrite y − y1 = m(x − x1), substituting m and (x1, y1).

9 y 8 7 6 5 4 3 2 1

(3, 5)

x 1

−1 −2 −3 −4 −5 −6 −7 −8 −9

8 9 (7, −2)

We are also able to ﬁnd the slope-intercept form of an equation written in point-slope form by solving the equation for y.

Example 1 Given the point (4, −1) and the slope m = 2, write the equation of the line in point-slope form.

Create a strategy To write the equation of the line in point-slope form, we use the formula: y − y1 = m(x − x1) where (x1, y1) is a point on the line and m is the slope.

Apply the idea y − (−1) = 2(x − 4) y + 1 = 2(x − 4)

Substitute x1 = 4, y1 = −1, and m = 2 Simplify

The equation of the line in point-slope form is y + 1 = 2(x − 4).

Example 2 The graph of a linear function is shown. Write the equation of the line in point-slope form for each of the given points on the line.

6 5 4 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

3.05 Point-slope form mathspace.co

163

a (−6, 1)

Apply the idea From the graph, we can choose any two points to ﬁnd the slope. We can either count the . the slope formula

from the graph or use

Start with the slope formula

Substitute (−6, 1) = (x1, y1) and (6, −3) = (x2, y2)

Evaluate the subtraction

Simplify

Find the equation of the line in point-slope form: Start with the point-slope form

Substitute y1 = 1, m =

, and x1 = −6

b (3, −2)

Apply the idea The slope from the previous part was

and will use (3, −2) as (x1, y1).

Given y − y1 = m(x − x1), the resulting equation will be

Example 3 A line passes through the two points (−3, 7) and (2, −3). a Write the equation of the line in point-slope form.

Create a strategy We will ﬁrst ﬁnd the slope of the line using the two points. Then we will pick one of the points to substitute into the point-slope equation.

Apply the idea Find the slope of the line: Start with the slope formula

Substitute (x1, y1) = (−3, 7) and (x2, y2) = (2, −3)

Evaluate the subtraction

Simplify

Find the equation of the line in point-slope form:

164

y − y1 = m (x − x1)

Start with the point-slope formula

y − 7 = −2 (x − (−3))

Substitute y1 = 7, m = −2, and x1 = −3

y − 7 = −2 (x + 3)

Simplify

Mathspace Virginia SOL Algebra 1 mathspace.co

Reﬂect and check We can conﬁrm we have correctly written this in point-slope form: y − y1 = m (x − x1), as we can see the coordinates (−3, 7), and a slope of −2 are represented correctly. This is easier to see in the previous line: y − 7 = −2 (x − (−3))

b Determine whether the ordered pair (−10, 21) lies on the same line as (−3, 7) and (2, −3).

Create a strategy We can substitute the ordered pair into the equation we wrote in part (a) and determine whether the statement is true. If so, we can conﬁrm whether the point lies on the same line as (−3, 7) and (2, −3).

Apply the idea We have y − 7 = −2(x + 3)

Point-slope form from part (a)

(21) − 7 = −2((−10) + 3)

Substitute y = 21 and x = −10

14 = −2(−7)

Evaluate the subtraction and addition

14 = 14

Evaluate the multiplication

Since the resulting equation is true, the ordered pair (−10, 21) satisﬁes the equation and as a result is on the same line as (−3, 7) and (2, −3).

Reﬂect and check Any ordered pair that satisﬁes the equation will be on the same line as (−3, 7) and (2, −3).

Example 4 A carpenter charges for a day’s work using the given equation, where y is the cost and x is the number of hours worked: y − 125 = 50 (x − 2) a Interpret the meaning of each number in the equation.

Create a strategy Recall the point-slope form: y − y1 = m (x − x1)

Apply the idea The slope is represented by the number 50. In this context this is the rate the carpenter charges for one hour of work. So, the carpenter charges $50 for each hour of work. Using the point slope formula, we also know the point (2, 125) ﬁts our context. This means after 2 hours of work, the total cost is $125.

3.05 Point-slope form mathspace.co

165

b Draw the graph of the linear equation from the point-slope form. Clearly label the axes with labels, units, and an appropriate scale.

Create a strategy We need two points to plot a line. We can read from the equation that one point on the line is (2, 125). To get another point, we can use the slope from the given point or substitute in an x-value in the domain and solve for y. Since the x-axis will be the number of hours worked and generally an 8-hour work day is reasonable, showing the graph for 0 ≤ x ≤ 8 would be a good scale.

Apply the idea Before we graph, we label our axes. To do that, we need to know what our maximum and minimum values should be. We can use the slope or the equation to determine the y-value when x = 8. We can ﬁnd that the y-value when x = 8 is y = 425. 450

Charge in $ ( y)

450

Charge in $ ( y)

400

400 350

350

300

250

200

150

100

Hours (x) 1

Hours (x)

A reasonable scale for the x-axis is from 0 to 8, going up by 1. A reasonable scale for the y-axis is from 0 to 450, going up by 50, with one tick between each label at the 25s.

The given point is (2, 125). Since the slope is 50, we can go up 50 units and right 1 unit from our given point to plot another point on the line.

c Predict the charge for 6 hours of work using the graph.

Create a strategy We can go to x = 6 on the x-axis and then go up to the line and across to the y-axis to make our prediction.

Apply the idea 450

Charge in $ ( y)

400 350 300 250 200 150 100 50

Hours (x) 1

The charge for 6 hours of work will be $325.

166

Mathspace Virginia SOL Algebra 1 mathspace.co

d Give an example of a non-viable solution if the carpenter only uses this model for a maximum of 10 hours per day. Explain your answer.

Create a strategy Since the carpenter only uses this model between 0 and 10 hours, any solution outside of this interval will not be viable.

Apply the idea One possible non-viable solution would be (11, 575) which would represent working 11 hours and getting paid $575, but the model only applies to a maximum of 10 hours, so we don’t actually know what would happen for 11 hours.

Reﬂect and check Any solution with x < 0 or x > 10 would be non-viable.

Idea summary The point-slope form of a line is:

y − y1 = m (x − x1) m x1 y1

slope x-coordinate of a point on the line y-coordinate of the same point

Point-slope form is useful when we know or want to know the slope of the line and a point on the line.

Practice What do you remember? 1

State the point-slope form of a linear equation.

Consider a straight line with slope 1 going through point A(2, 1). What is the slope between A and any other point on the line?

For each of the following equations, state the form they are written in:

y = −4x + 5

y − 1 = 5(x − 4)

5x − 3y = 8

y = −2

True or false? The line with the equation y − 5 = 4(x + 6) contains the point (−5, 6).

3.05 Point-slope form mathspace.co

167

Which graph correctly represents the function y + 3 = −2(x − 1)? A

1 −4 −3 −2 −1 −1

x 1

−3

−3 −4

−2

−4

−4 −3 −2 −1 −1

−2

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Select the linear equation that could represent the following graph. A

y + 3 = 2(x + 3)

y − 7 = 2(x − 2)

y − 7 = 2(x − 5)

y − 4 = 2(x − 5)

4 2 −8 −6 −4 −2 −2 −4 −6 −8

Let’s Practice 7

168

Using the point-slope formula, ﬁnd the equation of the following lines: a

A line passes through (−5, 9) and has a slope of 2.

A line passes through (−2, −1) and has a slope of .

A line passes through (8, 2) and has a slope of −3.

A line passing through the point (7, 1) has a slope of

A line passing through the point (4, 0) has a slope of −5.

Mathspace Virginia SOL Algebra 1 mathspace.co

x 2

For each of the following lines: i

Find the slope, m, of the line.

Write the equation of the line in point-slope form.

A line passes through the two points (7, 6) and (9, 12).

A line passes through the two points (4, −8) and (2, 0).

For each of the following table of values: i

Find the slope, m, of the line represented in the table.

Write the equation of the line in point-slope form.

x 1 2 3 4 y −3 2 7 12

x y

1 9

2 7

3 5

4 3

Write the equation of each line in point-slope form, using the point shown on the graph. a

5 4

3 2 1 1

−2 −3

5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1

5 4

1 −3 −2 −1 −1 −2 −3

x 1

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

−5 −4 −3 −2 −1 −1

7 6 5 4 3 2 1

1 2 3 4 5 6 7

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7 −8 −9 −10

3.05 Point-slope form mathspace.co

169

A linear function is defined by the equation a

Select the graph that could represent the equation. A y

5 x

−5

x −5

5 −5

C y

5 x

−5

x −5

5 −5

Identify the domain of the function.

Identify the range of the function.

Sketch the graph of the lines on the coordinate plane given the following equations: d

y + 1 = −2(x + 1)

2x + 3y = 12

y − 2 = 3 (x − 7)

y + 7 = −3(x + 2)

y−7=

(x − 5)

y − y1 = m(x − x1 )

A race car uses fuel at the rate of 0.9 gallons per minute. After running for 12 minutes, the car has 48 gallons of fuel left in the tank. a

170

Rewrite the following linear equations in standard form: a

y + 8 = 3(x + 2)

Rewrite the following linear equations in slope-intercept form: a

−5

a 13

Select the linear equation that could describe the relationship between the number of minutes x the car is running and the number of gallons of fuel y left in the tank. A y + 12 = −0.9 (x + 48)

y + 48 = 0.9 (x + 12)

C y − 12 = 0.9 (x − 48)

y − 48 = −0.9 (x − 12)

Graph the function chosen in part (a). Explain how the y-intercept relates to the context.

Mathspace Virginia SOL Algebra 1 mathspace.co

A plumber charges a fixed amount for a call out fee plus $30 per hour of work. He charged a total of $80 for 2 hours of work. He cannot work for more than 8 hours in any given day. a

Select the graph of the linear function that relates the number of hours of work x and the total charge y for the day. A y

20 x 1

C y

20 x 1

Identify the domain of the function.

Let’s extend our thinking 17

Create a scenario where you would choose to use each of the following forms and explain why you would use that form over the others: a

Point-slope form, (y − y1) = m(x − x1)

Standard form, Ax + By = C

Slope-intercept form, y = mx + b

Sally receives an order to make key chains for an upcoming festival. She is told to make at least 50 and at most 100 key chains, and will be paid $8 for each key chain she makes. Sally makes 80 key chains and is paid $665 altogether. a

Let x represent the number of key chains Sally makes and y represent the amount she is paid. Write an equation in point-slope form relating x and y.

Convert the equation from part (a) to slope-intercept form.

Interpret the y-intercept in the context of the problem.

Explain whether or not we can use the equation in part (a) to predict the amount paid for 10 key chains.

3.05 Point-slope form mathspace.co

171

Paige is planning an event and the brochure for the caterer at the venue shows the following formulas for the meal costs. Point-slope form: y − 500 = 12.5(x − 10) Slope-intercept form: y = 12.5x + 375 Compare and contrast the information presented by each equation.

Create a scenario where you would choose to use point-slope form, y − y1 = m(x − x1), and explain what the point and slope mean in context.

172

Mathspace Virginia SOL Algebra 1 mathspace.co

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1−1

Perpendicular lines have slopes with opposite signs and they are reciprocals of one another.

y y = 3x − 2

These are called negative reciprocals (or opposite reciprocals). x

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

The product of slopes of perpendicular lines is −1 unless one of the lines has an undeﬁned slope. The lines y = 3x − 2 and y =

x + 4 are perpendicular.

Perpendicular lines may have the same y-intercept.

Consider a line perpendicular to y = 3x − 2 with the same y-intercept. We know the slope must be the negative reciprocal which is m =

and the y-intercept must be the same, which is b = −2.

Therefore, the equation of a perpendicular line with the same y-intercept is y =

x − 2.

Example 1 The line AB passes through the points (−2, 9) and (3, −21). a Write the equation of the line.

Create a strategy To find the equation, we need to know the slope and the y-intercept. We will find the slope using m = will find the y-intercept using y = mx + b.

, and we

Apply the idea Slope formula

Substitute (x1, y1) and (x2, y2)

Evaluate the subtraction

Evaluate the division

The slope of the line is m = −6. Now, we will use y = mx + b with the slope we found and one of the points. We can use either point because either will result in the same answer. y = mx + b

Slope-intercept form of a linear equation

9 = −6(−2) + b

Substitute m = −6 and (x1, y1)

9 = 12 + b

Evaluate the multiplication

−3 = b

Subtraction property of equality

b = −3

Reﬂexive property of equality

This means the y-intercept is at (0, −3). Substituting m = −6 and b = −3 into slope-intercept form of a linear equation, we ﬁnd the equation of the line to be y = −6x − 3.

Reﬂect and check The equation of the line in standard form is 6x + y = −3.

174

Mathspace Virginia SOL Algebra 1 mathspace.co

b Find the equation of the line that passes through (1, 5) and is parallel to the line AB.

Create a strategy Since this line is parallel to the line AB, we know that it will have the same slope as to ﬁnd the y-intercept of the parallel line.

which was −6. We only need

Apply the idea Just like we did in the previous part, we will substitute the slope, and the x- and y-values of the point into y = mx + b. y = mx + b

Slope-intercept form of a linear equation

5 = −6(1) + b

Substitute m = −6 and the point (1, 5)

5 = −6 + b

Evaluate the multiplication

11 = b

Addition property of equality

b = 11

Reﬂexive property of equality

The equation of the parallel line is y = −6x + 11.

Reﬂect and check

Using technology to graph the lines, we can see that they are parallel, and they pass through the speciﬁed points from parts (a) and (b).

10 5 x −3 −2 −1 −5

−10 −15 −20

Example 2 Consider the line 4x − 3y = −6. a Find the equation of the line that is perpendicular to the given line and has the same y-intercept.

Create a strategy For the new line to be perpendicular to the given line, its slope must be the opposite reciprocal of the slope of the given line. To ﬁnd the y-intercept, we can either substitute x = 0 or rearrange to slope-intercept form.

Apply the idea Rearranging the equation of the given line to slope-intercept form gives us: Given equation

Subtract 4x from both sides

Divide both sides by −3

The slope of the given line is

3.06 Equations of parallel and perpendicular lines mathspace.co

175

Example 3 A mirror is placed along the x-axis. A laser beam is projected along the line y = −x + 4 which reﬂects off the mirror.

6 5 4 3 2 1 −1

x 1

−1

a A normal is a line which is perpendicular to the surface of the mirror at the point of reﬂection. Find the equation of the normal.

Create a strategy

Apply the idea

We can do a quick sketch of the normal to help:

Since the mirror is a horizontal line, the normal must be a vertical line if it is to be perpendicular. This means it will be of the form x = a.

Since it goes through the point where the laser hits the mirror, (4, 0), the equation of the normal will be x = 4.

6 Normal

5 4

Laser

3 2 1 −1

−1

x 1

2 3 4 5 6 7 Surface of the mirror

b The angles that the laser and its reflection make with the normal will be congruent. If the angle between the laser beam and the normal is 45°, find the equation of the path of the reflection.

Create a strategy Since the angles are congruent, know that the angle formed between the normal and the reﬂection will also be 45°. We can label this on our diagram: 7

6 Normal

5 4 3 2

45° 45°

1 −1

−1

x 1

3.06 Equations of parallel and perpendicular lines mathspace.co

177

Given the graphs of the lines:

• y=x+5 • y=x+2 • y=x−3

Determine whether the following statements are true or false:

They all have the same slope.

They all have positive y-intercepts.

They all have positive x-intercepts.

They are all parallel to the line y = x.

They all have slope equal to 1.

x −5 y=x+5 y=x+2 −5 y=x−3

State whether each of the following lines are parallel, perpendicular, or neither to the line a

3x − 4y = −12

8x + 6y = 6

6x − 2y = 4

Let’s practice 5

Determine which, if any, of the lines are parallel. A

y = 3x − 4

y = −3x + 1

3x + y = 4

2x − 3y = 4

Determine which, if any, of the lines are perpendicular. A

y = 2x − 4

y = −3x + 1

3x + 2y = 0

2x + 4y = 4

Find the equations of the lines that are: a

Parallel to x = −2, and 4 units away from the line x = −2.

Parallel to y = −4, and 2 units away from the line y = −4.

Perpendicular to y = 4, and 3 units away from the line x = 2.

Perpendicular to x = 0, and 1 unit away from the line y = 1.

Find the equation of the line, in slope-intercept form, that is: a

Parallel to the line y = 8x − 3 and cuts the y-axis at 5.

Parallel to the line y = −2x + 9 and passes through the point (−3, 1).

Parallel to the line y =

x − 1 and passes through the point (0, 0).

Parallel to the line y =

and passes through the point (3, 6).

Perpendicular to y =

+ 7, and goes through the point (0, 6).

Perpendicular to the line y = −2x and passes through the point (2, 1).

Perpendicular to the line y =

Perpendicular to the line y = 5x + 1 and intercepts the y-axis at y = 2.

x − 2 and passes through the point (−2, 4).

Find the equation of the line, in standard form, that is: a

Parallel to the line 3x + 6y = −2 and intercepts the y-axis at y = 3.

Parallel to the line 4x − 2y = 0 and intercepts the y-axis at y = −1.

Perpendicular to the line x − 2y = 3 and intercepts the y-axis at y = 1.

Perpendicular to the line 4x + y = −2 and intercepts the y-axis at y = 0. 3.06 Equations of parallel and perpendicular lines mathspace.co

179

For each graph: i

Write an equation for the line parallel to line l that goes through point P.

Write an equation for the line perpendicular to line l that goes through point P.

y 7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 P −5 −6 −7

7 6 5 4 3 2 1

P x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Consider the equation f (x) =

−7 −6 −5 −4 −3 −2 −1−1 P

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

+ 2.

Graph a line parallel to f (x) that passes through the point (4, −3).

Graph a line perpendicular to f (x) that passes through the point (1, −2).

Consider the line x = 0. a

Determine the equation of a line that is perpendicular to x = 0.

Determine if there is more than one possible answer for part (a). Explain.

Describe and correct the error in writing an equation of the line that passes through (4, 2) and is parallel to the line y =

180

−2 −3 −4 −5 −6 −7

−7 −6 −5 −4 −3 −2 −1−1

x + 2.

Mathspace Virginia SOL Algebra 1 mathspace.co

Systems of Equations & 4 Inequalities Big ideas • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 4.01 4.02 4.03 4.04 4.05

Write and graph linear systems Substitution method Elimination method Two variable linear inequalities Systems of linear inequalities

184 198 208 220 236

The number of people signing up to join the soccer team can be represented by the equation p(x) = 3x + 11 where x represents the number of days the signup has been open.

y 25 20

The number of people signing up to join the lacrosse team next season can be represented by the equation r(x) = 7 + 3x.

r(x)

15 p(x) 10 5

x 1

When two lines are parallel and distinct, they have no points of intersection. The corresponding system of equations has no solutions, which means the teams will never have the same number of players.

The amount of money Fernando makes at his construction job is shown by the equation F(x) = 10 + 2x and the amount of money Heather makes at McDonalds is represented by the equation H(x) = 2(x + 5) where x represents the number of hours they work.

y 25 20

When two lines are identical, they intersect at every point. The corresponding system of equations has infinitely many solutions. This means that Fernando and Heather make the same amount of money at any point in time.

H(x)

15 10 F(x) 5

x 1

The number of solutions can be determined by looking at the equations without graphing. • One solution: the lines have different slopes so they intersect at one point • No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect • Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line A solution to a system of equations in a given context is said to be viable if the solution makes sense in the context and non-viable if it does not make sense within the context, even if it would otherwise be algebraically valid.

Example 1 Graph each system of equations and state the solution.

Create a strategy We can convert the second equation in the system to slope-intercept form and graph both equations using the y-intercept and slope.

Apply the idea Second equation

Divide equation by 4

Evaluate the division

4.01 Write and graph linear systems mathspace.co

185

Time (weeks) 8 9 10

Car Mileage (Rodica) 200(8) + 9000 = 10 600 200(9) + 9000 = 10 800 200(10) + 9000 = 11 000

Car Mileage (Yuwei) 500(8) + 6000 = 10 000 500(9) + 6000 = 10 500 500(10) + 6000 = 11 000

When we get to 10 weeks in the table we can see that the mileage for both cars is 11 000. So we know the mileage will be the same after 10 weeks. c Graph the system of equations. Choose an appropriate scale for the axes.

Create a strategy Based on the solution in part (b), we know that the cars will have the same mileage of 11 000 miles after 10 weeks. This will help us decide on an appropriate scale.

Apply the idea Since the number of weeks is the x-value, we can count by 1 along the x-axis to slightly beyond the point of intersection at 10 weeks. Since the mileage is the y-value, we can count by 500 along the y-axis to slightly beyond the point of intersection at 11 000 miles.

Used car mileage 11 000 10 000 9000 8000 7000 6000 5000 4000 3000 2000 1000 −1

Mileage

Time (weeks) 1 2 3 4 5 6 7 8 9 10 11

d Identify the solution for the system of equations and intrepret it in the context of the problem.

Create a strategy We can use the graph from part (c) to find the point of intersection and use the labels on the axes to interpret it.

Apply the idea Used car mileage

The point of intersection from the graph is (10, 11 000): The x-axis measures time so 10 represents 10 weeks. The y-axis measures mileage so 11 000 represents 11 000 miles. The solution (10, 11 000) means that after 10 weeks both cars will have the same mileage of 11 000 miles.

11 000 10 000 9000 8000 7000 6000 5000 4000 3000 2000 1000 −1

188

Mathspace Virginia SOL Algebra 1 mathspace.co

Mileage

Time (weeks) 1 2 3 4 5 6 7 8 9 10 11

Example 3 Bixia is saving up her quarters and dimes in a jar. She has a total of $24.50 in 125 coins. a Write a system of equations that models this situation.

Create a strategy We’ll need to write a system of equations with the information provided in the problem, and we’re only given two numbers. The two totals give us information about each equation: one of the equations utilizes the units of a dollar amount throughout, and the other equation must include the number of coins as its units. We should deﬁne the unknown variables. Since we don’t know how many of each type of coin Bixia has, we can say that x = the number of quarters and y = the number of dimes.

Apply the idea

Reﬂect and check We can conﬁrm that the system of equations makes sense by checking the units of each equation. Since the total of x + y = 125 is the total number of coins, it should make sense that the number of quarters added to the number of dimes is equal to the total number of coins, which is what is being represented in this equation. The total of 0.25x + 0.10y = 24.50 represents the fact that Bixia has $24.50. The expression 0.25x represents the value of 1 quarter multiplied by how many quarters are in the jar. This term is the dollar amount of all the quarters combined. The expression 0.10x represents the value of 1 dime multiplied by how many dimes are in the jar. This term is the dollar amount of all the dimes combined. When these two expressions are added together, the result is the total value of all the coins, or $24.50.

b Graph the system of equations. Use appropriate axes, labels, and scales.

Create a strategy Based on the equation, x represents the number of quarters because we know that a quarter is $0.25, and y represents the number of dimes because we know that a dime is $0.10. We can calculate the intercepts of both equations and use those to determine the maximum values on our x- and y-axes.

Apply the idea First, we’ll calculate the x-intercept of x + y = 125: x + y = 125

First equation

x + (0) = 125

Substitute y = 0

x = 125

Evaluate the addition

Now, we’ll calculate the y-intercept of x + y = 125: x + y = 125

First equation

(0) + y = 125

Substitute x = 0

y = 125

Evaluate the addition

4.01 Write and graph linear systems mathspace.co

189

The x-intercept of x + y = 125 is (125, 0) and the y-intercept is (0, 125). Next, we’ll calculate the x-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

0.25x + 0.10(0) = 24.50

Substitute y = 0

0.25x = 24.50

Evaluate the multiplication

x = 98

Divide both sides by 0.25

Finally, we’ll calculate the y-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

0.25(0) + 0.10y = 24.50

Substitute x = 0

0.10y = 24.50

Evaluate the multiplication

y = 245

Divide both sides by 0.10

The x-intercept of 0.25x + 0.10y = 24.50 is (98, 0) and the y-intercept is (0, 245). Since the x-intercepts have a maximum value of 125, we can draw an x-axis up to 130 and count by 10. Since the y-intercepts have a maximum value of 245, we can draw a y-axis up to 250 and count by 50.

Bixia’s coin jar Number of dimes 200 150 100 50 Number of quarters 10

110

c Interpret the solution to the system of equations.

Create a strategy

Apply the idea

We can use the graph to determine the solution to the system and the axes labels to interpret the meaning of the point of intersection. Since the solution to the system is not clear on the graph, we can use technology to graph the system of equations.

The solution to the system of equations is (80, 45). Since x is the number of quarters, we know that Bixia had 80 quarters, and since y is the number of dimes, we know that Bixia had 45 dimes in the jar.

Example 4 Tyson is saving money in order to purchase a new smartphone for $800 when the latest model is released. He currently has $350 saved up and is able to put away $100 each month. a Write a system of equations to represent the situation.

Create a strategy

Apply the idea

To write a system of equations, we will need to deﬁne some variables. Let’s choose y to represent an amount of money (in dollars), and x to represent the number of months that have passed.

Using these variables, the amount of money Tyson has saved over time can be represented by y = 350 + 100x. The price of the smartphone can be represented by y = 800.

190

Mathspace Virginia SOL Algebra 1 mathspace.co

b Sketch the two lines representing these equations on the coordinate plane.

Create a strategy All of the values involved in the question are multiples of $50, so we can use this for the scale of the y-axis. Also, both x and y only make sense for positive values in this context, so we only need to think about the ﬁrst quadrant.

Apply the idea Tyson’s cell phone savings 900

Amount of money

800 700 600 500 400 300 200 100 −1

Time (months) 1

c If the new phone is to be released in 5 months’ time, determine if Tyson will be able to afford it on release.

Apply the idea

Reﬂect and check

The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months, Tyson will have saved $800. Therefore Tyson will have saved enough money before the phone is released.

While the model equation of Tyson’s savings is linear, in reality, he probably puts money away once per month or once per week, depending on how often he gets paid. So although the point of intersection is at x = 4.5 months, Tyson might not actually reach $800 in savings until the end of the 5th month.

Example 5 Gordiano made two trips to a flower shop to purchase roses and sunflowers. On his first trip, he purchased 4 roses and 4 sunflowers and paid $12. The following day, Gordiano went back to the flower shop and purchased 12 roses and 8 sunflowers for $16. Without graphing, determine the number of solutions.

Create a strategy First, define the variables. Based on the problem, we see that each trip to the flower shop is represented in its own equation. Gordiano buys 4 roses, then 12, so the expressions 4x and 12x indicate the cost for the roses. Gordiano buys 4 sunflowers, then 8 sunflowers, so the terms 4y and 8y indicate the cost of the sunflowers. Let x = the cost per rose and let y = the cost per sunflower. Now that we have defined the variables, we can convert the equations to slope-intercept form and determine the number of solutions. Lines that have the same slope will always have one solution. Lines that have the same slope with different y-intercepts will have zero solutions, and parallel lines with the same y-intercept will have infinite solutions.

4.01 Write and graph linear systems mathspace.co

191

Idea summary The solution to a system of linear equations is the ordered pair of the point of intersection of the lines. Systems of equations may have one solution, no solutions, or infinitely many solutions. Graphing a system can help to determine the number of solutions a system will have: • • •

One solution: the lines have different slopes so they intersect at one point No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line

Practice What do you remember? 1

For each graph, select the solution to the system of equations. a

A (−1, −7)

(−6, −7)

C (−7, −6)

(1, −6)

y 4 2

E No solution

x −8

−6

−4

−2

−2 −4 −6 −8

A (−3, 0)

(−3, 5)

C (3, 5)

Inﬁnitely many solutions

E No solution

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

4.01 Write and graph linear systems mathspace.co

193

State the number of solutions for the systems of equations: a

6 5 4 3 2 1

−6−5−4−3−2 −1 −1

6 5 4 3 2 1

2 1

−4 −3 −2 −1 −1

1 2 3 4 5 6

−2 −3 −4 −5 −6

−2 −3 −4

Solve each system of equations graphed below: a

4 2

−4 −2 −2 −4 −6 −8 −10 −12 −14 −16 −18

x 2

10 12

9 y

−3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

8 y 7 6 5 4 3 2 1 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8

8 7 6 5 4 3 2 1

194

1 2 3 4 5

−6−5−4−3−2 −1 −1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5 6

−2 −3 −4 −5 −6

5 4 3 2 1

x 1 2 3 4 5 6 7 8 9

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

2 3 4 5

9 y 8 7 6 5 4 3 2 1 −9−8 −7−6−5−4−3−2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

x 1 2 3 4

Consider the equations y = 2x and y = 28 − 2x which have the following tables of values: y = 2x x

y = 28 − 2x

x 3 y 22

4 20

5 18

6 16

7 14

State the values for x and y which satisfy both equations. 5

Consider the equations x + y = 3 and y = x − 5 which have the following tables of values: x + y = 3 y = x − 5 x y

4 −1

5 −2

x y

6 −3

4 −1

5 0

6 1

y = 4x − 11 −3 1 5 9 13

State the values for x and y which satisfy both equations. 6

Consider the equations y = 4x − 11 and y = −2x + 13 which have the following table of values:

2 3 4 5 6

State the values for x and y which satisfy both equations.

y = −2x + 13 9 7 5 3 1

Let’s practice 7

Use the graph to solve the systems of equations: 5 4 3 2 1 −2

a 8

−1

y=x−2

−1 −2 −3 −4 −5

x −2y = 4 2

y = −8 + 2x

For each of the following system of equations: i

Sketch the two lines representing the equations on the coordinate plane.

Solve the system of equations using the graph.

Consider the system of equations:

Sketch the two lines representing these equations on the coordinate plane.

State how many solutions this system of equations has. 4.01 Write and graph linear systems mathspace.co

195

Consider the equations y = −2x and y = −12 − 4x. a

Find the solution to the system of equations.

Solve the system of equations.

(−4, 8)

No solution

(1, −5)

(−8, −5)

Consider the equations y = 3x and y = −35 − 4x. a

Represent each function using a table of values.

Consider the system of equations:

Select the solution to this system of equations.

(0, 0)

Inﬁnitely many solutions

(−4, 0)

Consider the system of equations:

Select the solution to this system of equations.

SOL

(0, −5)

No solution

(0, −8)

The system of linear equations is graphed as shown.

9 8 7 6 5 4 3 2 1

What is the solution to this system of equations?

(0, −3)

(1, −4)

(2, −2)

(3, 0)

−9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

For each of the following equations: i

Rewrite the equation as a system of equations.

Create a graph to represent each equation.

iii

Solve the system of equations.

x − 2 = −3x – 6

x 1 2 3 4 5 6 7 8 9

−5x − 22 = 7x + 38

Sarah is determining the dimensions of a rectangular frame. The length, l, of the frame is 5 cm more than the width, w. The perimeter of the frame is equal to 50 cm. Sarah came up with the following equations to solve for the length and width of the frame:

Determine the correct dimensions of the rectangular frame.

Interpret the solution in terms of the 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 + 2

A system of linear equations has infinitely many solutions. One of the equations is y = 4x − 5. Determine if the following can be the other equation: a

196

y = −3x − 3

y = 8x − 10

Mathspace Virginia SOL Algebra 1 mathspace.co

y = 4x + 5

3y = 12x − 15

2y = 8x − 10

The sum of two mystery numbers is 4. The difference of the two numbers is −2. a

Write a system of equations to represent the situation.

Find the solution to the sytem of equations.

Rochelle and Mohamad are sister and brother. Rochelle’s age is 11 more than 4 times the age of Mohamad. The sum of their ages is 21. a

Write a system of equations to represent the situation.

Determine how old Rochelle and Mohamad are.

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 9 pieces of work to do for another subject. a

Write a system of equations to represent the situation.

Solve the system of equations and interpret the result in terms of the context.

Is it possible to have a pair of linear equations with exactly two solutions? Explain why or why not.

Consider the system of equations:

Without graphing, is there a solution? Explain your answer.

Let’s extend our thinking 24

Write a scenario to represent the system of equations and its solution. Explain what the solution to the system means in terms of the scenario.

18 16 14 12 10 8 6 4 2

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.

x 2 4 6 8 10 12 14 16 18 20 22 24

25 20 15 10 5 −40 −30 −20 −10 −5 −10 y1 −15 −20 y2 −25

x 10 20 30

Describe a situation where it would not be reasonable to solve a system of equations by graphing.

Homer plans to start taking an aerobics class. Nonmembers pay $4 per class. Members pay a one-time $8 sign-up fee but only have to pay $2 per class. Should Homer join as a member or take classes as a nonmember? Create and analyze a model, then use it to justify your response. 4.01 Write and graph linear systems mathspace.co

197

The following steps can be used to solve a system of equations using substitution: Steps

Example x − y = 5, 2x + 3y = 6 x−y=5 x=5+y 2(5 + y) + 3y = 6 10 + 2y + 3y = 6

Given system 1. Isolate a variable in one of the equations 2. Substitute the resulting expression into the other equation

10 + 5y = 6 3. Solve the equation for the variable

5y = −4

4. Substitute the value into one of the original equations 5. Solve for the remaining variable 6. Write the solution as an ordered pair

Recall that a solution to a system of equations is the set of ordered pairs that make all equations in the system true and that a system of linear equations can have three types of solutions: y

1 −4 −3 −2 −1

−1

x 1

−2

−4 −3 −2 −1

−1

x 1

−2

−4 −3 −2 −1

−1

−3

−4

A system of linear equations with no solution.

Solving algebraically will result in an Solving algebraically will result in equations of the form x = a and y = b, equation of the form a = b, where a and b are different numbers. where a and b could be the same number.

−2

−4

A system of linear equations with one solution.

x 1

A system of linear equations with inﬁnitely many solutions. Solving algebraically will result in equations of the form a = a, where a is a number.

When the solution to a system of equations does not consist of integer values it is diﬃcult to determine the exact solution by graphing, so solving algebraically using a method like substitution is necessary.

4.02 Substitution method mathspace.co

199

Example 1 Solve the following systems of equations using the substitution method. a

Create a strategy We ﬁrst want to number our equations to make them easier to work with.

y = x + 11

y = 3x + 19

Since both equations already have y isolated, we can start by substituting equation 1 into equation 2 to eliminate y from the equation. We can then solve for x, and substitute this value back into one of the equations to solve for y.

Apply the idea Now we will use x = −4 to solve for y:

First we will solve for x: y = 3x + 19

Equation 2

y = x + 11

Equation 1

Substitute y = x + 11

y = −4 + 11

Substitute x = −4

11 = 2x + 19

Subtract x from both sides

y=7

Evaluate the addition

−8 = 2x

Subtract 19 from both sides

−4 = x

Divide both sides by 2

x + 11 = 3x + 19

So the solution to the system of equations is x = −4, y = 7 and written as (−4, 7).

Reflect and check Substitute the value of x and y back into both of the equations from the original system. Evaluate to check if both equations are true. Equation 1: y = x + 11

Write the first equation

7 = −4 + 11

Substitute x = −4, y = 7

7=7

Evaluate the addition

This is a true statement so the ordered pair is a solution to the first equation. Equation 2: y = 3x + 19

Write the second equation

7 = 3(−4) + 19 Substitute x = −4, y = 7 7 = −12 + 19

Evaluate the multiplication

7=7

Evaluate the addition

This is also a true statement so the ordered pair is a solution to the second equation. Since the ordered pair is a solution to both equations it is a solution to the system of equations.

200

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 The length of a rectangle is 3 inches less than twice its width. If the perimeter of the rectangle is 48 inches, ﬁnd the length.

Create a strategy For this real-world situation, we should start by deﬁning the variables, then build a system of equations to model the situation. The length of the rectangle in inches is unknown, so we can give it the variable l. Since the length and width make up the perimeter of the rectangle and the width in inches is also unknown, we can give it the variable w.

Apply the idea Based on the ﬁrst sentence in the problem, we know that the length is equal to 3 inches less than twice its width. The equation that models this relationship is l = 2w − 3. We know that the perimeter of a rectangle is equal to the sum of twice the length and twice the width, so the equation that models this relationship is 2l + 2w = 48. We can write our system of equations as:

Solve for w:

Solve for l: 2l + 2w = 48

Second equation

l = 2w – 3

2(2w − 3) + 2w = 48

Substitute l = 2w – 3

l = 2(9) – 3 Substitute w = 9

4w − 6 + 2w = 48

Distributive property

l = 15

6w − 6 = 48

Combine like terms

6w = 54

Add 6 to both sides

w=9

Divide both sides by 6

First equation Evaluate the multiplication and subtraction

The length of the rectangle is 15 inches.

Reflect and check To confirm that our solution is correct, we need to check both the criteria provided in the problem. First, we verify that the length is 3 inches less than twice the width. We found the length to be 15 inches and the width to be 9 inches. “Twice the width” is 2 ⋅ 9 = 18 and “3 inches less” than that is 18 − 3 = 15. So this criteria is satisfied. Next, we verify that the perimeter is 48 inches. Remember perimeter is found by adding all of the sides (both lengths and both widths for a rectangle). 9 + 9 + 15 + 15 = 48 so this criteria is also satisfied.

202

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 A theater club at a high school charges a student rate and an adult rate to attend the spring musical. The cost for a student ticket is $3 and the cost for an adult ticket is $7. If 200 people attended the show and the theater club at the school raised $700, determine how many students and how many adults attended.

Create a strategy A system of equations will help us solve for the unknown variables, which are the number of students, x, and the number of adults, y. Based on the total amounts provided in the problem, we will write two equations in standard form to model the situation and solve the system to ﬁnd the solution.

Apply the idea A system of equations that models the situation follows:

To solve the system using the substitution method, at least one equation must have an isolated variable. We can easily isolate either x or y in the ﬁrst equation because they do not have coeﬃcients. x + y = 200

First equation

x = 200 – y

Subtract y from both sides

We will substitute the expression 200 − y for x in the second equation to solve for y. 3x + 7y = 700

Second equation

3(200 − y) + 7y = 700

Substitute x = 200 − y

600 − 3y + 7y = 700

Distributive property

4y = 100

Combine like terms

y = 25

Divide both sides by 4

Now we will solve for x using y = 25. x + y = 200

First equation

x + 25 = 200

Substitute y = 25

x = 175

Subtract 25 from both sides

So the solution to the system of equations is x = 175, y = 25 and written as (175, 25). Based on the context, this means that 175 students attended the spring musical and 25 adults attended.

Idea summary Solving systems of equations using the substitution method leads to one of three solutions: • • •

One solution: When solving algebraically an equation of the form x = a or y = b is reached No solutions: When solving algebraically an equation of the form a = b is reached Inﬁnitely many solutions: When solving algebraically an equation of the form a = a is reached

4.02 Substitution method mathspace.co

203

Practice What do you remember? 1

Select the solution to this system of equations. a A

(−1, −5)

(1, 5)

(−5, −1)

(5, 1)

(−8, −2)

(−2, 2)

(−2, −8)

(2, −2)

b A 2

State whether these statements are true or false. a

The method of substitution involves isolating one variable in one equation and substituting this expression into the other equation.

The substitution method can be used to solve any system of linear equations.

Once you solve one of the equations for one variable, you must substitute it back into the first equation.

Solving a system by substitution will always result in the same solution, regardless of which variable you solve for first.

A system of linear equations can have exactly two solutions, no solutions, or infinitely many solutions.

Solve the following systems of equations by substitution: a

Identify the number of solutions for each of the following systems of equations: a

A group of students were solving the equations:

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 end up with and comment on the efficiency of solving each. 6

If you solve a system of linear 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 a system of equations by substitution and get a true statement, such as 0 = 0 or 5 = 5?

204

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s practice 8

Solve the following systems of equations by substitution: a

Solve the equation 3x − 7y = 4 for x.

Substitute your expression for x from part (a) into −12x + 28y = −16 and solve for y.

Solve the system of equations using the substitution method:

Verify each solution by graphing.

For each system of equations: a

Find the solution to the system.

Check the solution algebraically.

For this system of linear equations:

iii

For the system of equations:

Explain the steps you would take to solve the system of equations and why you would do it in that way. 13

Do the equations y = 3x + 5 and y = 3(x + 5) have infinitely many solutions? Explain your answer.

A car rental agency offers two types of cars for rent: compact cars and SUVs. Today, the agency rented out a total of 50 cars, generating $3000 in revenue. The rental fee for a compact car is $40 per day, and the rental fee for an SUV is $60 per day. a

Write a system of equations to represent the context.

How many compact cars and how many SUVs were rented today?

Judy and Jorge both walk from their houses to the bus stop every morning. Judy walks 1.1 mi further, and together they walk 3.1 mi. Find the distance that Jorge walks.

At a bookstore, Stella purchased a novel and a magazine for a total cost of $59. The price of the novel, n, is $10 less than twice the price of the magazine, m. This situation can be represented by the following system of equations:

Find the solution to the system of equations.

Interpret the solution in terms of the context.

4.02 Substitution method mathspace.co

205

The total cost of 5 rulers and 3 books is $15.30. If the cost of a ruler is a and a book is $2.70 more expensive, find the cost of one ruler.

A rectangular garden bed has a perimeter of 13.8 m. The length of the garden bed is 2.5 m longer than the width. a

Find the width of the garden bed.

Find the length of the garden bed.

Rustam bought some fresh produce from the farmers’ market. He picked up 4 oranges and 5 plums. The total cost of Rustam’s fruit was $13.33. Valentina went to the same shop and bought 1 orange and 1 plum. The total cost of Valentina’s fruit was $2.89. a

Write a system of equations where f represents the price of an orange and g represents the price of a plum.

Solve for the prices of oranges and plums at the farmers’ market.

A man is ﬁve times as old as his son. Four years ago the man was thirteen times as old as his son. a

Write a system of equations where x represents the age of the man and y represents the age of his son.

Solve for x and y.

Determine whether the solution makes sense in terms of the context. Explain your answer.

The number of new jobs created in Miami varies greatly each year. The number of jobs created in 2013 was 440 000 less than double the number of jobs created in 2004. This is equivalent to an increase of 10 000 jobs created from 2004 to 2013. a

Write a system of equations to represent the context.

Find the solution to the system of equations.

Determine whether the solution is viable in terms of the context. Explain your answer.

Valentina has $2000 to invest, and wants to split it up between two accounts: Account A earns 8% annual interest, while Account B earns 9% annual interest. Her target is to earn $177 total interest from the two accounts in one year. a

Write a system of equations to represent the context.

Solve the system of equations.

State if you would make the same investment as Valentina. Explain your answer.

The function f (x) = 0.47x + 8.9 represents the US annual bottled water consumption (in billions of gallons) and the function g(x) = −0.17x + 14.2 represents the US annual soda consumption (in billions of gallons). For both functions, x is the number of years since 2009. a

Determine the year in which the bottled water and soda consumption in the US is the same.

Determine whether the solution is viable in terms of the context. Explain your answer.

Let’s extend our thinking 24

A pizza shop offers two types of pizzas: Margherita and Pepperoni. Today, the shop sold a total of 100 pizzas, generating $2000 in revenue. The price for a Margherita pizza is $20, and the price for a Pepperoni pizza is also $20. How many Margherita pizzas and how many Pepperoni pizzas were sold today?

At Soul Food Express you can buy two orders of oxtail stew and a slice of sweet potato pie for $38.49, or you can get ﬁve orders of oxtail stew and three slices of sweet potato pie for $99.22. Construct a system of equations to model the situation and use it to determine the cost of 4 oxtail stews and 2 slices of sweet potato pie.

206

Mathspace Virginia SOL Algebra 1 mathspace.co

Edilene has to read a book for class and only has 8 days before the test. So far, Edilene has read 90 pages. She is planning on reading 15 pages each day between now and the test. The book is 215 pages long. a

Construct a system of equations to model the situation.

Construct models to represent the system of equations and determine whether she will ﬁnish her book before the test.

Create a scenario for each system of equations where the solution is viable in terms of the scenario. Explain your answer. a

Shufang spent $55.25 to purchase 9 flowers. He bought two different types of flowers. Rhododendron cost $6.45 each and chrysanthemums cost $5.75 each. Construct a system of equations to model the scenario and use the model to determine how many of each type of flower Shufang purchased.

4.02 Substitution method mathspace.co

207

4.03 Elimination method After this lesson, you will be able to… • find solutions to a system of equations algebraically using the elimination method. • interpret solutions to a system of equations in a real-world context. • determine the number of solutions to a system of equations. • create systems of equations to represent constraints from a real-world context.

The elimination method The elimination method is an algebraic method for solving systems of equations where like terms are aligned. In this system, the equations are aligned: 3x + 4y = 12 x + y = 11

and in this system, the equations are not aligned:

3x = 12 − 4y

x + y = 11

Elimination method

A method of solving a system of equations by adding or subtracting the equations until only one variable remains

Exploration Begin by graphing the system of equations and identifying the solution.

3x + 2y = 10

x + 2y = 8 Next, perform each operation using the original equations and consider the result. What do you notice? Record your observations. 1.

Divide the second equation by 2 and regraph the system. What do you notice?

Multiply the ﬁrst equation by 2 and regraph the system. What do you notice?

Add the equations together and regraph the system. What do you notice?

Multiply the ﬁrst equation by −1 and add the equations together. Graph the new equation with the system. What do you notice?

Multiply the second equation by 3 and add the equations together. Graph the new equation with the system. What do you notice?

Multiply the second equation by −3 and add the equations together. Graph the new equation with the system. What do you notice?

What happens to the solution to a system of linear equations if one equation is multiplied by a number and then added to the other equation?

208

Mathspace Virginia SOL Algebra 1 mathspace.co

c Does the solution make sense in terms of the context? Explain your answer.

Apply the idea Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.

Example 4 Determine the number of solutions to each system without solving. a

Create a strategy Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are different representations of the same line. If the coefficients of the x and y terms are multiples but the constant is not a multiple, the system has no solutions because they represent parallel lines and will never cross. Lastly, if the x and y terms are not multiples then the lines intersect and have one solution.

Apply the idea Since neither the entire equation nor the coefficients of the x and y terms are multiples, these lines intersect exactly once. Therefore, this system has one solution.

Create a strategy Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are different representations of the same line.

Apply the idea We can manipulate the second equation by dividing each term by 2: Original equation

Divide each term by 2

This is exaclty the same as the first equation. Therefore, both equations represent the same line, and there are infinite solutions.

Apply the idea In this system, the coefficients of the x and y terms are multiples but the constant is not a multiple. Therefore, the system has no solutions because they represent parallel lines and will never cross. 4.03 Elimination method mathspace.co

215

Let’s practice 5

Consider the following system of equations:

What value can we multiply each equation by in order to eliminate the variable y?

Solve for x and y.

For the system of 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.

Solve each system of equations using the elimination method: a

Consider each system of equations: i

Solve the system of equations.

Verify each solution algebraically.

Consider the following systems of equations: i

Use the elimination method to solve each pair of linear equations:

Verify each solution by graphing.

a 10

Consider the following system of equations:

Explain how to eliminate the variable y.

Explain why eliminating y would help solve the system of equations.

Consider each pair of systems of equations and state whether they are equivalent. a

4.03 Elimination method mathspace.co

217

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 following system of equations:

Select the correct number of four-passenger cabs and six-passenger cabs. A x = −2, y = 17

b 13

x = 12, y = 3

x = 17, y = −2

When comparing her test results, Judy noticed that the sum of her Geography test score and Math test score was 137, and that their difference was 29. Judy scored higher on her Geography test than her math test. a

Write a system of equations where x represents Judy’s Geography score and y represents her Math score.

Solve for Judy’s Geography score using the elimination method.

Now, solve for Judy’s Math score.

12 pens and 5 rulers cost $70 while 3 pens and 25 rulers cost $65. a

Write a system of equations where x represents the price of a pen and y represents the cost of a ruler.

Solve for the price of a pen using the elimination method.

Now, solve for the price of a ruler.

Fred spent $55.25 to purchase 9 ﬂowers. He bought rhododendrons which cost $6.45 each and chrysanthemums which cost $5.75 each. a

If R is the number of rhododendrons and C is the number of chrysanthemums that Fred bought, construct two equations describing the total number of ﬂowers bought and the total amount spent in dollars.

Solve for the number of rhododendrons and chrysanthemums that Fred purchased.

For each system of equations, explain which strategy you would use and why: b

For each system of equations: i

State whether the substitution method or the elimination method would be more efficient for solving.

Explain why you chose the method in part (i).

x = 3, y = 12

Determine whether the solution is viable in terms of the context.

a 17

Identify the number of solutions for each of the following systems of equations: a

Let’s extend our thinking 19

Describe a situation where it would be more efficient to use the elimination method rather than the substitution method to solve a system of equations and explain why.

218

Mathspace Virginia SOL Algebra 1 mathspace.co

Consider the following system of equations:

Explain how to rewrite the system of equations so that it has integer coeﬃcients. 21

Ricardo solved the following system of equations used to model wildlife populations in a wildlife sanctuary, where r represents the number of rhinos and h represents the number of hippopotami.

Explain why the solution to the system of equations is not viable.

Explain how you might interpret the solution to the system of equations to make sense in terms of the context.

Elias solved a nonlinear system of 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.

Original equations

Multiply equation 2 by 3

Add the equations, solve for y

Substitute y, solve for x

The solution is (1, −3). 23

Without solving, how can you determine the number of solutions to the system of equations?

Kwabena bought some fresh produce. He picked up 2 oranges, and 3 bananas. The cost of the Kwabena’s shopping was $18.30. Amra also went to the same shop and bought 5 oranges and 7 bananas. The cost of the Amra’s shopping was $44.03.

Construct a system of equations to model the scenario.

Explain two different ways to approach ﬁnding the cost per orange and the cost per banana.

At a donut shop, two types of donuts are available: classic donuts and premium donuts. The donut shop offers two deals for these donuts: Deal A with 3 classic donuts and 5 premium donuts for $6; Deal B with 6 classic donuts and 10 premium donuts for $12. a

Write the systems of equation where c represent the price of one classic donut and p represent the price of one premium donut.

Solve for the prices of the classic and premium donuts using the elimination method.

Omeida’s piggy bank contains 70 nickels and dimes with a total value of $3.85. a

Construct a system of equations to model the scenario and use it to determine how many of each coin Omeida has.

Suppose the coins in the piggy bank were only dimes and quarters. Revise the model in part (a) for this change. Explain how this changes the solution. 4.03 Elimination method mathspace.co

219

Example 1 Consider the inequality y ≤ −2x + 5 a Graph the boundary line y = −2x + 5.

Create a strategy We can graph the equation of the boundary line using the y-intercept and slope.

Apply the idea 5

y = −2x + 5

3 2 1 −5 −4 −3 −2 −1 −1

x 1

2 3 4 5

−2 −3 −4 −5

b Determine whether the point (4, 2) is a solution to the inequality y ≤ −2x + 5.

Create a strategy We will substitute the ordered pair into the inequality to determine whether the statement is true.

Apply the idea y ≤ −2x + 5

Given inequality

2 ≤ −2(4) + 5

Substitute (4, 2)

2 ≤ −3

Evaluate the multiplication and addition

Since 2 is not less than or equal to −3 the statement is false and the point (4, 2) is not a solution to the inequality y ≤ −2x + 5.

c Determine whether the point (0, 0) is a solution to the inequality y ≤ −2x + 5.

Create a strategy We will substitute the ordered pair into the inequality to determine whether the statement is true.

Apply the idea y ≤ −2x + 5

Given inequality

0 ≤ −2(0) + 5

Substitute (0, 0)

0≤5

Evaluate the multiplication and addition

Since the statement is true, the point (0, 0) is a solution to the inequality y ≤ −2x + 5.

222

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 4 A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs 8 pounds and each box of grapefruits weighs 12 pounds. a Write an inequality to represent the number of boxes of oranges and grapefruit that can be in the truck.

Create a strategy

Apply the idea

Our unknown values are the number of boxes of oranges and the number of boxes of grapefruits. Let x represent the number of boxes of oranges in the truck. Let y represent the number of boxes of grapefruit in the truck.

The weight of all the orange boxes is the product of the weight of one box, 8, and the number of boxes, x.

We should consider the structure of the inequality. The maximum weight capacity is 3000 pounds, so we know that the total weight of the boxes ≤ 3000. From there, we can use the deﬁned variables and numbers from the problem to build the other part of the inequality.

Weight of orange boxes: 8x pounds The weight of all the grapefruit boxes is the product of the weight of one box, 12, and the number of boxes, y. Weight of grapefruit boxes: 12y pounds The total weight is the sum of the weights of orange boxes and grapefruit boxes. Total weight: 8x + 12y pounds The truck can carry at most 3000 pounds, so we get our inequality: 8x + 12y ≤ 3000

b Create a graph of the region containing the points corresponding to all the different numbers of orange and grapefruit boxes that can be loaded into the truck.

Create a strategy We will graph the region representing 8x + 12y ≤ 3000. We need to identify our boundary line, then graph it. Then, we need to decide which side of the boundary line to shade.

Apply the idea The boundary line is 8x + 12y = 3000. This is a line in standard form, so we can graph it by ﬁnding the intercepts. Find the x-intercept by setting y = 0 and solving: 8x + 12y = 3000

Given equation

8x − 12(0) = 3000

Substitute y = 0

8x = 3000

Evaluate the multiplication

x = 375

Divide both sides by 8

4.04 Two variable linear inequalities mathspace.co

227

Let’s practice 10

The line y = x − 5 is shown. 5

4 3 2 1 −5 −4 −3 −2 −1 −1

x 1

2 3 4 5

−2 −3 −4 −5

Graph the solution to y > x − 5.

Graph the solution to y ≤ x − 5.

For each of the following inequalities: i

Determine whether the point (2, 3) satisfies the inequality or not.

Graph the region that satisﬁes the inequality.

y ≤ 3x + 5

y ≥ 2x + 4

y < 3x + 2

Does the point satisfy the inequality? a

x − y > 10 i

232

(15, 20)

iii

(16, 7)

(10, −6)

(12, 20)

(−2, −3)

iii

(0, 0)

(2, 7)

(−2, 4)

iii

(−4, 30)

(2, −13)

y > −8x − 3 i

y > 3x i

(−2, −9)

(6, −50)

For each of the following inequalities: i

State the coordinates of the x- and y-intercepts of the boundary line.

Graph the region that satisﬁes the inequality.

3x + 2y < 12

Mathspace Virginia SOL Algebra 1 mathspace.co

5x + 2y ≥ 10

c Suppose the maximum of the verbal reasoning test was a score of 50. Is the solution (15, 56) a viable solution in the context?

Apply the idea No. This would mean that the score for the quantitative reasoning test was 15 and the verbal reasoning test 56. By viewing the graph, we can see that this point technically satisﬁes both inequalities, but we now know that the maximum possible score on the verbal reasoning test is a 50.

d Update the system of inequalities that models the new information about the tests.

Create a strategy We now know that the maximum of the verbal reasoning test was a score of 50, and that y represents the score on the verbal reasoning test.

Apply the idea Since the score for the verbal reasoning test cannot be above a score of 50, the inequality y ≤ 50 may be included in the system of inequalities modeling the relationship:

Idea summary The solution to a system of inequalities lies in the region where the solutions of more than one linear inequality overlaps. Since solutions to systems of inequalities can have many solutions, we use a graph to show the solution set.

4.05 Systems of linear inequalities mathspace.co

241

For each system of inequalities: i

Find the x-coordinate of the point at which the boundary lines intersect.

Find the y-coordinate of the point of intersection.

iii

Graph the solution set.

x ≤ 5 and y < 3

y ≤ 3x − 4 and y > −4x + 1

3x + y > 5 and 3x + y < 7

y < 4x − 2 and y < 3x − 3

Sheila sells popcorn to raise money during summer break. The popcorn comes in two ﬂavors, classic butter which costs $2 each and cheese which costs $3 each. Sheila needs to sell at least $50 worth of popcorn with at least 8 of the cheese popcorn. Let b be the number of butter popcorn and c be the number of cheddar popcorn. This situation can be represented by the following system of equations:

Select all the viable solutions.

b = 6, c = 13

b = 18, c = 6

b = 10, c = 10

b = 11, c = 9

b = 13, c = 8

Applicants for a particular university are asked to sit a numeracy test and verbal reasoning test. Successful applicants must obtain a minimum score of 17 on the numeracy test and a minimum combined score of 37 for both tests. Let x and y represent an applicant’s score on the numeracy and verbal reasoning test respectively. a

It’s not possible to get a negative score therefore y ≥ 0 is one inequality. Write two more inequalities from the information in terms of x and y.

Graph the solution to the system of inequalities.

Which points represent scores that would make the applicant successful? i

(14, 16)

(33, 18)

iii

(20, 13)

(17, 22)

If an applicant obtains a score of 24 in the numeracy test, find the minimum integer score they need to obtain in the verbal reasoning test to be successful.

Yreka Bakery makes two types of cookies: plain and iced. They have enough oven space to bake 15 dozen cookies each day. Each dozen iced cookies requires 0.8 pounds of icing. Yreka Bakery can make no more than 32 pounds of icing per day. If x is the number of dozens of iced cookies and y is the number of dozens of plain cookies, write a system of inequalities to represent the scenario.

Throughout university, Jimmy works as a barista, getting paid $10 per hour, and as a mentor getting paid $13 per hour. The number of hours he works in each job can vary from week to week, but he never works more than 27 hours in total each week, and he needs to be able to at least cover his weekly expenses of $260. If b represents the number of hours worked as a barista and m represents the number of hours worked as a mentor, write a system of inequalities to represent the scenario.

4.05 Systems of linear inequalities mathspace.co

247

Let’s extend our thinking 18

Without graphing, determine whether each system of inequalities has no solutions or infinitely many solutions. Explain your answer. a

y > 4 and y < 4

x ≤ −2 and x ≥ −2

9x − y < 7 and 9x − y > 7

2x − y ≤ 7 and 2x − y ≥ 7

Is it possible for an ordered pair to satisfy one part of the linear inequality pair but not the other? Explain with a diagram what this would look like.

David’s Pizza makes two types of pizzas: Vegan and Meat Lover’s. Based on recent sales they know they will sell at least twice as many Meat Lover’s Pizzas as they do Vegan Pizzas. David’s Pizza can use 42 pounds of vegan cheese each day, and each Vegan Pizza requires 0.6 pounds of vegan cheese.

248

If v is the number of Vegan Pizzas and m is the number of Meat Lover’s Pizzas, write a system of inequalities to represent the scenario.

Is (28, 70) a viable solution in terms of the scenario? Explain.

Is (25.25, 71) a viable solution in terms of the scenario? Explain.

In a team crossﬁt competition, each team is to be made up of no more than 10 people. For a particular challenge, each team member must complete the course and the team’s total time in completing the course must be under 20 minutes. Women take on average 2.4 minutes and men take on average 1.5 minutes to complete the course. a

If w is the number of women and m be the number of men on the team, write a system of inequalities to represent the scenario.

Graph the system of inequalities.

Give an example of one viable and one non-viable solution to the number of men and women on the team. Explain your answer for both.

Throughout university, Tom works as a mentor, getting paid $15 per hour, and as a ﬁtness instructor getting paid $18 per hour. The number of hours he works in each job can vary from week to week, but he never works more than 29 hours in total each week, and he needs to be able to at least cover his weekly expenses of $270. a

If x represents the number of hours Tom works as a mentor and y represents the number of hours he works as a ﬁtness instructor, construct a system of inequalities to represent the scenario.

Graph the system of inequalities.

If Tom works 6 hours as a mentor in one week, what is the minimum number of hours he can work as a ﬁtness instructor so that he can cover his expenses?

Is it possible for him to work the same number of hours in both jobs and still be able to cover his expenses?

Mathspace Virginia SOL Algebra 1 mathspace.co

A bicycle manufacturer employs a mechanic and a painter to construct two types of bikes: racing bikes and mountain bikes. The time each worker spends on each bike is shown (assume both workers can work on the same bike at the same time). Let x represent the number of racing bikes built and y represent the number of mountain bikes built. Each worker can work at most 40 hours in a week. Mechanic

Painter

Racing Bike

5 hours

10 hours

Mountain Bike

8 hours

4 hours

Write an inequality relating x and y to the total time spent working by the mechanic.

Write an inequality relating x and y to the total time spent working by the painter.

Exponents, Radicals, & 5 Exponential Functions Big ideas • Changing the form of an expression or equation can reveal information that was previously unknown. • Expressions are the building blocks of algebra. They can be used to represent and interpret realworld situations. • The properties of real numbers can be applied to many types of expressions. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models.

Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09

Product rule Power rule Quotient rule Zero and negative exponents Rational exponents Simplify radicals Operations with numerical radicals Characteristics of exponential functions Graphs of exponential functions

252 257 262 269 277 284 289 296 305

5.01 Product rule After this lesson, you will be able to… • derive the product of powers from patterns. • use the product of powers law to simplify algebraic expressions.

Product rule Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.01 to answer these questions. 1.

When a3 is written in expanded form, how many a’s are being multiplied?

When a3 ⋅ a2 is written in expanded form, how many a’s are being multiplied?

By counting the number of a’s being multiplied in expanded form, what is a3 ⋅ a2 in exponential form?

By counting the number of a’s being multiplied in expanded form, what is a4 ⋅ a5 in exponential form?

Is there a faster way to multiply terms with the same base?

When multiplying a number by itself repeatedly, we are able to use exponent notation to write the expression more simply. Here, we are going to look at a rule that allows us to simplify products of expressions with exponents. Consider the expression a5 ⋅ a3. Notice that the terms share like bases. Let’s think about what this would look like if we distributed the expression: a5 ⋅ a3 = (a ⋅ a ⋅ a ⋅ a ⋅ a)(a ⋅ a ⋅ a) a 5 ⋅ a3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a a 5 ⋅ a 3 = a8 We can see that there are eight a’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. We can write this in exponential form as a8, where a is the base and 8 is the power. So, in our example above, a5 ⋅ a3 = a5 + 3 = a8 We can avoid having to write each expression in expanded form by using the product rule. It states that when multiplying two powers with the same base, we add the powers. For any base number a, and any numbers m and n as powers, am ⋅ an = am + n That is, when multiplying terms with a common base: • Keep the same base • Find the sum of the exponents

252

Mathspace Virginia SOL Algebra 1 mathspace.co

When using the product rule for exponents, the coefficients are handled separately from the exponents. Let’s take a look at an example. 3a2 ⋅ 4a5 • Multiply the numerical coefficients: 3 ⋅ 4 = 12 • Apply the product rule to the exponents: a2 ⋅ a5 = a2 + 5 = a7 The final result is: 3a2 ⋅ 4a5 = 12a7

Example 1 Fill in the blank to make the equation true: b2 ⋅ b⬚ = b2 + 3

Create a strategy We can use the exponent law: am ⋅ an = am + n

Apply the idea Since we know that we can add powers when two common bases are being multiplied together, we know that the blank box must be the power that is being added to 2 on the right hand side of the equation. Therefore, we know that 3 must go into the blank box to make the equation true. b2 ⋅ b3 = b2 + 3

Example 2 Simplify m2 ⋅ m7 + r3 ⋅ r2, giving your answer in exponential form.

Create a strategy We can use the exponent law: am ⋅ an = am + n

Apply the idea m2 ⋅ m7 + r3 ⋅ r2 = m2 + 7 + r3 + 2 9

=m +r

Add the powers of the bases m and r Simplify the powers

Example 3 Simplify: a 35 ⋅ 39

Create a strategy Use the product rule to simplify the expression.

Apply the idea 35 ⋅ 39 = 35+9 14

Apply the product rule Simplify

5.01 Product rule mathspace.co

253

b c7 ⋅ c6

Create a strategy Use the product rule to simplify the expression.

Apply the idea c7 ⋅ c6 = c7+6

Apply the product rule

Simplify

c 5d5 ⋅ 3d3

Create a strategy Multiply the coeﬃcients, then use the product rule with the exponents.

Apply the idea 5d5 ⋅ 3d3 = 5 ⋅ 3 ⋅ d5 ⋅ d3

Group the coeﬃcients and variables

5+3

= 15 ⋅ d

Multiply the coeﬃcients and apply the product rule

= 15d

Simplify

d −3m2n5 ⋅ 7m3n

Create a strategy Multiply the coeﬃcients, then use the product rule to simplify the exponents. Work with one base at a time.

Apply the idea −3m2n5 ⋅ 7m3n = −3 ⋅ 7 ⋅ m2 ⋅ m3 ⋅ n5 ⋅ n 2+3

= −21 ⋅ m

5+1

⋅n

Group the coeﬃcients and variables Multiply the coeﬃcients and apply the product rule

5 6

= −21m n

Simplify

Example 4 Multiply and write the answer in scientiﬁc notation. (2.7 × 105) (6.04 × 1013)

Create a strategy Multiply coeﬃcients and use product rule for exponents.

Apply the idea (2.7 × 105) (6.04 × 1013) = 2.7 × 105 × 6.04 × 1013 5

= 2.7 × 6.04 × 10 × 10 5+13

254

Rewrite parentheses as multiplication Commutative property of multiplication

= 16.308 × 10

Product rule of exponents

= 16.308 × 1018

Simplify

= 1.6308 × 1019

Rewrite in scientific notation

Mathspace Virginia SOL Algebra 1 mathspace.co

Idea summary For any base number a, and any numbers m and n as powers, am ⋅ an = am + n When multiplying terms with like bases, we add the powers.

Practice What do you remember? 1

When multiplying two powers with the same base, do we add, subtract, multiply, or divide the powers?

Is each expression equivalent to 6p8? c

p2 ⋅ p2 ⋅ p2 ⋅ 6p2

a ⋅ a ⋅ b ⋅ b

3⋅3⋅3⋅3⋅y⋅y⋅y

3 ⋅ u ⋅ u ⋅ u ⋅ 5 ⋅ v ⋅ v ⋅ v

−3 ⋅ c ⋅ c ⋅ b ⋅ 5⋅ c ⋅ c ⋅ a ⋅ b

(3p) ⋅ (3p) ⋅ (3p)

g ⋅ x13 ⋅ ⬚ = g2 ⋅ x21

x9y8 ⋅ ⬚ = x11y15

y2 ⋅ y6

x4 ⋅ 10x3

8y9 ⋅ 5y7

8y4 ⋅ 8y3

a 3

6p2 ⋅ 6p4

6p4 ⋅ p2

6p4 ⋅ p4

Write each expression in exponential form: a

Let’s practice 4

Write the term that would make each statement true. a e

a3 ⋅ ⬚ = a6 16 6

b 22 13

x y ⋅⬚=x y

Simplify each expression in exponential form. a

22 ⋅ 23

4y5 ⋅ 3y2

9m ⋅ 6m

m 4y8 ⋅ 6y6 ⋅ 3y3 6

q 2 ⋅ q3 5

39 ⋅ 310

4y3 ⋅ 6y 3

7y ⋅ 5y

u ⋅u ⋅u

6y7 ⋅ 6y5 ⋅ 6y3

8y3 ⋅ 9y4

4y2 ⋅ 5y4 ⋅ 6y8

3y7 ⋅ 5y8 ⋅ 2y

d2 ⋅ d6

x4 ⋅ x3

s9 ⋅ s10

y ⋅y

p ⋅p

w ⋅w

a4 ⋅ a3

b 2 ⋅ b2

m2 ⋅ m6 ⋅ m3

f7 ⋅ f5 ⋅ f3

n8 ⋅ n6 ⋅ n3

am ⋅ a n

a3 ⋅ −2a8

Simplify each expression in exponential form. 6y2 ⋅ y5 2

3y6 ⋅ 4y 4

u ⋅u ⋅u

2w ⋅ w ⋅ w

5z ⋅ 3z ⋅ z

6y7 ⋅ 6y5 ⋅ 6y3

4y8 ⋅ 6y6 ⋅ 3y3

5g4h2 ⋅ 8g3h4

6bc4 ⋅ 8b6a ⋅ 0

8m2n4 ⋅ 7m6n

t5 ⋅ t12 + y10 ⋅ y4

k7 ⋅ k8 + w13 ⋅ w3

43y2 ⋅ ⬚ = 415y15

Simplify each expression in exponential form. a

Simplify each expression in exponential form.

a3 ⋅ ⬚ = a4

p2 ⋅ p3 + w9 ⋅ w3

a ⋅ a2 + c

Write the term that would make each statement true. a e

9h5 ⋅ ⬚ = 27h7

-5b1 ⋅ 1 ⋅ ⬚ = 35b1 ⋅ 2

5.01 Product rule mathspace.co

255

Clifton and Duncan are discussing how to simplify the expression x7 ⋅ x4. Clifton thinks that x7 ⋅ x4 = x28, because they think that when multiplying expressions, you also multiply the exponents. Duncan argues that x7 ⋅ x4 = x11, as they believe the exponents should be added when multiplying expressions with the same base. Who is correct, Clifton or Duncan? Explain your reasoning.

Explain why the expression a4 ⋅ b4 cannot be simplified using the product rule.

Let’s extend our thinking 12

Multiply the expressions. Write your answer using scientific notation. a

(2 × 103) (4 × 104)

(2 × 104) (6 × 107)

(4.6 × 106) (6.37 × 103)

What quantity multiplied by itself results in x100?

The length of the base of a triangle is 3c4 and the length of its perpendicular height is 6c8. What is the expression for the area of the triangle?

Fill in the missing number using positive integers: a

256

3mc⬚ ⋅ ⬚bm⬚ ⋅ 2b⬚c3 = −6b4c5m7

If 2x ⋅ 2 y = 210, x ≥ y. a

How many positive integer values of x and y satisfy the inequality?

Why can we not list the possible solutions if we allow negative integers?

Mathspace Virginia SOL Algebra 1 mathspace.co

5.02 Power rule After this lesson, you will be able to… • derive the power of power law from patterns. • use the power of a power law to simplify algebraic expressions.

Power rule Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.02 to answer these questions. 1.

After clicking “Show expanded form”, how many groups of a2’s are there?

Click “Show answer”. What do you notice about the exponent in the simplified form?

Adjust the sliders to change the base’s exponent and the exponent outside the parentheses. What happens to the simplified form (am)n?

The power rule states that for any base number, a, and any numbers m and n as power, (am)n = am ⋅ n That is, when simplifying a term with a power that itself has a power: • Keep the same base • Multiply the exponents (a2)3 = (a2) (a2) (a2) (a2)3 = (a ⋅ a) (a ⋅ a) (a ⋅ a) (a2)3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a (a2)3 = a6 When using the power rule for exponents, the coefficient is handled separately from the exponents. Let’s take a look at an example. (3x2)4

• Raise the coefficient to the power: 34 = 81 • Multiply the exponents, applying the power rule: (x2)4 = x2 ⋅ 4 = x8 Therefore, (3x2)4 simplifies to: 81x8

5.02 Power rule mathspace.co

257

Example 1 Simplify (a5)3.

Create a strategy Use the power rule.

Apply the idea (a5)3 = a5 ⋅ 3 =a

Multiply the powers of the variable Evaluate the multiplication

Idea summary For any base number a, and any numbers m and n as power, (am )n = am ⋅ n That is, when simplifying a term with a power that itself has a power: • •

Keep the same base Multiply the exponents

Power of a product rule Exploration Consider the following expressions: (2 ⋅ 3)2 (4 ⋅ 5)3 (7 ⋅ 2)4 (3 ⋅ 6)5

and and and and

22 ⋅ 32 43 ⋅ 53 74 ⋅ 24 35 ⋅ 65

Evaluate each expression and record your results in the table: Expression (2 ⋅ 3)2 22 ⋅ 32 (4 ⋅ 5)3 43 ⋅ 53 (7 ⋅ 2)4 74 ⋅ 24 (3 ⋅ 6)5 35 ⋅ 65

258

Result

What patterns do you notice in the results of the expressions?

Can you form a rule or law based on your observations?

Mathspace Virginia SOL Algebra 1 mathspace.co

For the product of any numbers a and b in the base, and for any number n in the power, (ab)n = anbn The power of a product rule states that a product raised to a power is equivalent to the product of the two factors each raised to the same power. (ab)4 = (a4) (b4) (ab)4 = (a ⋅ a ⋅ a ⋅ a) (b ⋅ b ⋅ b ⋅ b) (ab)4 = a4b4

Example 2 Simplify (a9 ⋅ b3)4

Create a strategy We can use the rule: (ab)n = anbn

Apply the idea (a9 ⋅ b3)4 = (a9)4 (b3)4 Start with the power of a product rule = a9 ⋅ 4 ⋅ b3 ⋅ 4 Multiply the powers Evaluate the powers = a36b12

Example 3 Simplify (−2x2)2.

Create a strategy Use the power rule.

Apply the idea (−2x2)2 = (−2)2 x2 ⋅ 2 Multiply the powers of the variable Evaluate the multiplication and coefficient = 4x4

Idea summary The power of a product rule states that for the product of any numbers a and b in the base, and for any number n in the power, (ab)n = anbn In other words, a product raised to a power is equivalent to the product of the two factors each raised to the same power.

5.02 Power rule mathspace.co

259

Practice What do you remember? 1

When simplifying a term with a power that itself has a power, do we add, subtract, multiply, or divide the exponents?

Consider (r2)4. a

Is each expression equivalent to (r2)4? i

r2 ⋅ r4 4

iii (r ⋅ r) v b

(r ⋅ r) ⋅ (r ⋅ r ⋅ r ⋅ r)

(r ⋅ r) ⋅ (r ⋅ r) ⋅ (r ⋅ r) ⋅ (r ⋅ r)

(r2)4 = r2 ⋅ 4

r2 ⋅ r2 ⋅ r2 ⋅ r2

Is each statement correct? i

(r2)4 = r2 + 4

Fill in the box to complete the rule: (r2)4 = r⬚

Let’s practice 3

Consider the following terms which form a pattern: (a2)1, (a2)2, (a2)3, (a2)4, … What would the ﬁfth term be? Simplify your answer in exponential form.

Consider the following terms which form a pattern: (x2)3, (x2)6, (x2)9, (x2)12, … What would the eighth term be? Simplify your answer in exponential form.

Simplify each expression: a

(−x3)4

(4y4)3

(−2x2)2

(−3p2)5

260

d h

(2r2s)4

Simplify each expression in exponential form: a

(8y6)2

( j 2)5

(c9)2

y8 ⋅ (3y)5

(u6)3 ⋅

6(r6)5 ⋅ 4(r4)7

(x9y)7 ⋅ (xy2)4

m ( p10)6 ⋅ ( p4)3

((x3)4)5

( f 8)6

(w3)4

(w9v6)4

(k5)3 ⋅ (k2)8

10(r7)5 ⋅ (2rs)4

Simplify each expression in exponential form: a

9y9 ⋅ 8(−y)8 ⋅ 7y7

(−5y6) ⋅ (−3y7) ⋅ (−4y7)

(−9b4) ⋅ 4b4

2v2w ⋅ (−5u2v3) ⋅ 3u3w4

(−8q4) ⋅ p2 ⋅ (−7q4) ⋅ p3

Mathspace Virginia SOL Algebra 1 mathspace.co

(−7y5z) ⋅ (−3yx3) ⋅ (−6y2) ⋅ (−4y2)

( p5)9 ⋅ (q6)10

Complete each statement: a

(abc)⬚ = a3b3c3

(0.1a⬚)2 ⋅ (10b4)⬚ = 0.1a6b4

(⬚m⬚n⬚)3 = 8m6n9

(4m2)⬚ ⋅ (⬚b⬚)2 = 256m6b4

[(2a⬚)3]2 = 64a12

In a laboratory experiment, a type of algae quadruples in size every day. A sample starts with a size of 10 units and can be modeled by the expression 10 ⋅ (22)5, where 22 represents the daily quadrupling growth factor and 5 represents the number of days. Simplify the expression to calculate the final size of the algae after 5 days. State the operations or rules applied in each step of your work.

Consider the expression shown: (x6 ⋅ x2)3 ⋅ (x2 ⋅ x5)4 Simplify the expression two different ways: using either the product of powers law or the power to a power law first. Show your work and justify each step.

Consider (33)2. a

Expand and simplify the expression (33)2 in exponential form.

Explain how the simplified expression demonstrates the power rule.

Consider the expressions simplified by two students: Student 1:

Student 2:

(3 + 4)2 = 32 + 42

(3 ⋅ 4)2 = 32 ⋅ 42

= 9 + 16

= 9 ⋅ 16

= 25

= 144

Describe the difference between the two workings and determine if the students’ solutions are correct. Explain your reasoning.

Let’s extend our thinking 13

Dante and Daniella are trying to simplify the expression n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6. • Dante wrote 1 2

n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6 = n6 ⋅ r21 ⋅ r18 = n6 ⋅ r39

• Daniella wrote: 1 2

n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6 = n6 ⋅ r10 ⋅ r9 = n6 ⋅ r19

Whose work is correct? Explain. 14

Write (16 p)4 in the form ab, where a is a prime number.

5.02 Power rule mathspace.co

261

5.03 Quotient rule After this lesson, you will be able to… • derive the quotient of powers law from patterns. • use the quotient of powers law to simplify algebraic expressions.

Quotient rule Exploration Expand each of the expressions for the following values: Expression

Values

Substitute values

Expanded form

Simplified exponential form

m = 5, n = 3 s = 4, t = 2 x = 6, y = 1 1.

What do you notice about the relationship between the exponents in the simplified form and the original exponents?

What happens to the base when you divide two expressions with the same base?

If we wanted to simplify the expression a6 ÷ a2, we could write it as:

We can see that common factors are divided out of the expanded expression, leaving a4. Consider the expression a6 ÷ a2, which can be written as

Let’s think about what this would look like if we expanded the expression:

We can avoid having to write each expression in expanded form by using the quotient rule:

where a is any base number, and m and n are powers. That is, when dividing terms with a common base: • Keep the same base • Find the difference in the power. 262

Mathspace Virginia SOL Algebra 1 mathspace.co

Of course, we can also write this law in the form am ÷ an = am - n When using the quotient rule for exponents, the coefficients are handled separately from the exponents. Let’s take a look at an example.

• Divide the numerical coefficients: 12 ÷ 2 = 6 • Apply the quotient rule to the exponents: The final result is:

Example 1 Simplify the following expressions: a

Create a strategy Expand and simplify the expressions in the quotient.

Apply the idea Expand the expressions

Divide the like terms

Simplify the expression

So,

simplifies to z11.

Reflect and check We can instead use the rule of exponents which states that when we divide terms with the same base, we subtract the exponents.

Subtract the exponents

Simplify the exponent

Create a strategy We can simplify the fractions and use the rule of exponents which states that when we divide terms with the same base, we subtract the exponents.

5.03 Quotient rule mathspace.co

263

Apply the idea Rewrite the quotient as a product of quotients

Divide the coefficients and apply the quotient of powers law

Simplify the exponents

simplifies to 3m5n2.

So,

Idea summary The quotient of powers law for exponents:

a is any base number m is a power n is a power That is, when dividing terms with a common base: • •

Keep the same base Find the difference in the power.

Power of a quotient rule Exploration Consider the mathematical expression

Complete the table below by writing the expanded forms of the expressions and then simplifying your expressions: Expression

264

Expanded form

Simplify as the quotient of two powers

Can you identify any patterns or relationships between the original expression and the expanded form?

Can you write a rule based on your observations?

Mathspace Virginia SOL Algebra 1 mathspace.co

When the power is applied to the entire quotient, we use the power of a power law to write an expression for the power of a quotient rule as:

Consider the expression

This can be expanded as

. Keep in mind we cannot simplify this any further because

a and b are different bases.

Example 2 Simplify the following expressions: a

Create a strategy The power of a quotient rule of exponents states that

. Apply this rule to the given expression.

Apply the idea Apply the power of a quotient rule

Apply the power rule

Simplify the exponents

So, the simplified form of

Reflect and check Notice that the expression cannot be simplified further, as the bases in the numerator and denominator are not the same and the numerical coefficients are fully simplified.

5.03 Quotient rule mathspace.co

265

Create a strategy We can apply the power of a quotient rule and the quotient rule to simplify the expression.

Apply the idea Apply the power of a quotient rule

Apply the quotient rule

Evaluate the subtraction

Therefore,

simplifies to a12b9.

Idea summary When we raise a fraction to a power, we can use the power of a quotient rule:

Practice What do you remember? 1

When dividing two exponential terms with the same base, which operation should be used to simplify the exponents? A

Addition

Multiplication

f8÷f6

y8 ÷ y4

Complete each statement: a

266

Division

Simplify each expression in exponential form. a

Subtraction

Rewrite each expression using a single exponent. a

x⬚ ÷ x3 = x7

Mathspace Virginia SOL Algebra 1 mathspace.co

n5 ÷ n3

w6 ÷ w2

Let’s practice 5

Rewrite each expression using a single exponent. a

a10 ÷ a5

m5 ÷ m2

j 11 ÷ j 4

(240u32) ÷ (8u9) ÷ (5u12) f

15x18 ÷ 15x8 ÷ 15x5

Simplify: a

9r8 ÷ (4r2)

Simplify:

12r30 ÷ 12r8 ÷ 12r7

(m12)9 ÷ (m4)2

A student simplifies the expression 115 ÷ 1112 and gets 117 as the result. Explain the student’s mistake.

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

When simplifying the expression

, Stein’s first line of work was a3 ⋅ a2

Describe what Stein may have done to get this line of work and determine its correctness. 12

Explain why m5 ÷ z3 is not equal to

Let’s extend our thinking 13

Two students were asked to simplify the expression

James simplified the expression as follows:

Neil simplified the expression as follows:

Which student is correct? Explain.

5.03 Quotient rule mathspace.co

267

Find the value of a and b in the equation:

Simplify: a

Complete the work by stating the exponent rule(s) used at each step to simplify the expression:

⬚ and ⬚

Simplify the exponents inside each group

⬚

⬚ and ⬚

268

Mathspace Virginia SOL Algebra 1 mathspace.co

Simplify the exponents

5.04 Zero and negative exponents After this lesson, you will be able to… • simplify algebraic expressions using the zero rule for exponents. • simplify algebraic expressions with negative exponents.

Zero and negative exponents Now let’s look at the quotient of powers rule when m and n are equal.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.04 to answer these questions. 1.

Use the applet to complete the table of values for the number of sections in the paper created based on the number of folds we make: Number of folds (exponent) Number of sections created

As the number of folds decreases, what is the pattern in the number of sections created?

Moving from right to left, complete the table of values using the pattern you found above: 4-2

4-1

42 16

43 64

The zero rule states that a0 = 1 The zero rule tells us any non-zero base raised to the power of zero is equal to 1. For example: Substitute 0 = 3 − 3

Quotient rule

Write in expanded form

Divide out common factors

5.04 Zero and negative exponents mathspace.co

269

The negative exponent rule states:

The negative exponent rule tells us any non-zero base raised to a negative exponent is equal to 1 divided by the same base raised to the opposite positive exponent. Consider the following: Write in expanded form

Group common factors

Divide common factors

Multiplicative identity

Write in exponential form

And also: Quotient rule Therefore,

Evaluate the subtraction .

We can use this rule to write negative exponents as positive exponents or positive exponents as negative exponents.

Example 1 Simplify x5 ÷ x5 by first writing the expression in expanded form.

Create a strategy Write the expression as fraction and write it in expanded form to cancel out the common factors.

Apply the idea

Write the expression as fraction

Write in expanded form

Divide out common factors

Simplify

Reflect and check Recall that am ÷ an = am - n, we could have written x5 ÷ x5 as = x5 - 5 = x0. Now we can see that x0 = 1. This should be no surprise; the initial expression asks us “What do we get when we divide x5 by itself?”, to which the answer is simply 1, since anything divided by itself is equivalent to 1.

270

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Simplify 9p0.

Create a strategy

Apply the idea 0

We can use the zero rule: a = 1

Since the base of p has a power of 0, the whole expression is equal to 9 ⋅ 1. So by simplifying this, we have: 9p0 = 9 ⋅ 1 = 9

Example 3 Write the following with a negative exponent:

Create a strategy We can use the product rule, am ⋅ an = am + n, and the negative exponent rule,

Apply the idea Each fraction is to the power of 1 so we can start by applying the product rule. Apply the product rule

Evaluate the addition

Negative exponent rule

Power rule

Therefore,

expressed with a negative exponent is g-4.

Example 4 Express the following with positive exponents. 3a2 ⋅ x-4 ⋅ 5 ⋅ x-2 ⋅ a6

Create a strategy We can use product of powers rule: am ⋅ an = am + n

5.04 Zero and negative exponents mathspace.co

271

Apply the idea Multiply the coefficients

Commutative property

Product rule

Evaluate the addition

Negative exponent rule

Multiply

Idea summary For any numeric or algebraic expression a, the zero rule tells us that a0 = 1 The negative exponent rule states:

Raise a fraction to a negative exponent We can use this same understanding to raise a fraction to a negative exponent. Let’s first review what a reciprocal is. The reciprocal of

. The reciprocal of

is . The reciprocal of 91 is

So to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator.

Exploration Consider the expression

. We want to write this without negative exponents. Apply the exponent to the terms inside the parentheses

Rewrite as a division

Apply the negative exponent rule

Multiply by the reciprocal

Simplify

Complete the working above by filling in the blanks.

Complete the statement: In general,

When raising a fraction to any negative exponent, we find the reciprocal of the fraction in the parentheses, then apply the power of a quotient law:

272

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 5 Express

using positive exponents.

Create a strategy

Apply the idea

We can use the exponential rule:

Rewrite using the reciprocal

Example 6 Write

with positive exponents.

Create a strategy First rewrite with a positive exponent and then apply the quotient of powers rule.

Apply the idea Apply the negative exponent rule

Apply the quotient of powers rule

Apply the power rule

Reflect and check An alternative method for solving this problem is applying the quotient of powers rule first, then the power rule, finally the negative exponent rule: Apply the quotient of powers rule Apply the power rule Write as a division Negative exponent rule Multiply by the reciprocal Multiply

5.04 Zero and negative exponents mathspace.co

273

Let’s practice 5

Express the following using only positive exponents: a

t-2

e 6

s-6

v-12

y-2x3

a-5b-4c4

Express the following using only negative exponents: a

r-3

Determine the missing exponent so that following equation is always true. (a3b-5)⬚ =a-15b25

Fill in the missing exponent to make the following equations true. a

j16 ÷ j16 = j ⬚

Rewrite with positive exponents: a

a-9

p-2

3x-4

8p-3

2x-8y3

m 3a4b-5c-7

7x-9

p-2q3

m-5n-4p4

4-2k-3l7

Simplify: a

(w0 + s)2

(x7)0 + x - x0

18a0

11a0

(a0)79

9 ⋅ (15x6)0

(3m4)3 ⋅ mq0

(6a)0

Simplify using the appropriate exponent rule(s). Final answer should contain only positive exponents. a

g12 ÷ g12 = g⬚

(4x4y3)3(2x− 4y5)−2

(3x10y7)4(5x4y−5)−3

What is the value of x0? Explain why this remains the same for any value of x. What is 04 equal to?

Explain the difference between 2x0 and (2x)0.

Explain why 0−4 is undefined.

5.04 Zero and negative exponents mathspace.co

275

Let’s extend our thinking 15

Write

Celeste and Donovan are writing an equivalent expression for

at least three different ways.

Celeste writes down

as a single term.

-10

and Donovan writes down x

Who is correct and why? 17

Nigel simplified the expression as follows: 1 2 3 4 His teacher noticed that there was an error in his solution. Identify the error and explain how to correct it.

A student was writing 5a-1 without negative indices and wrote and write the correct answer.

When asked to find an expression that is equivalent to -20, a student responded 1. Is this answer correct? Explain why or why not.

276

Mathspace Virginia SOL Algebra 1 mathspace.co

. Explain why their working is incorrect,

In a similar way, we can look at the value

. Using the product law:

From the definition of a cube root:

Once again, by comparison we see that

These results can be generalized to:

and

Notice that the index of the radical becomes the denominator of the rational exponent. When there is no index shown, it is a square root. Rational exponent

1 3

a =

Index

Radical

Base

Radicand

Exponential form

Radical form

We can use these rules for rewriting radicals along with the laws of exponents to simplify expressions involving radicals and rational exponents. Product rule Quotient rule Power rule Power of a product Power of a quotient Identity exponent Zero rule Negative exponent rule Rational exponent

278

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 For a Rewrite in radical form.

Create a strategy Use the rule

Apply the idea

Reflect and check We can also write this as

, but writing the root without an index is usual for a square root.

b Evaluate

Create a strategy We now know that

is the square root of 36.

Apply the idea From part (a)

Using 36 = 62

Simplify

Reflect and check We could also evaluate this using exponents. Using 36 = 62

Power rule

Simplify

Example 2 Write

in exponential form.

Create a strategy

Apply the idea

We can write this expression in exponential form using the fact:

5.05 Rational exponents mathspace.co

279

Example 3 Evaluate without a calculator. a

Create a strategy We can rewrite −216 with an exponent of 3 since it is a perfect cube, and then use the power rule to simplify.

Apply the idea Rewrite −216 using its factors

Rewrite in exponential form

Use the power rule

Simplify

Reflect and check We may also notice that

Create a strategy We can use the power of a power law, then convert to radical form to evaluate.

Apply the idea Power rule

Rational exponent law

Rewrite 8 using its factors

Rewrite in exponential form

Evaluate the cube root

Evaluate the power

Reflect and check If we had used the power of a power law the other way and got , we would not have been able to evaluate as easily. This is because 85 = 32 768 and it is not easy to identify the cube root of 32 768 without a calculator.

280

Mathspace Virginia SOL Algebra 1 mathspace.co

Create a strategy Use the laws of exponents to simplify, then use rational exponents to simplify.

Apply the idea Use the power rule and quotient laws

Simplify exponents using operations on fractions

Product of a power law

Simplify the exponent

Using 33 = 27

Apply the power rule

Reflect and check All of the exponent laws can be used for rational exponents, not just integer exponents.

Idea summary In addition to the laws of exponents we have seen, we can simplify rational exponents by converting to and from radical form.

and

We can simplify expressions involving rational exponents using the laws of exponents.

Practice What do you remember? 1

Use the properties of exponents to define a rational exponent that would make the statement true:

Explain the similarities between a rational exponent and a radical expression.

Simplify each expression: a

d 5.05 Rational exponents mathspace.co

281

Use the laws of exponents to fully simplify each expression. If possible, evaluate the expression. Otherwise, write in exponential form. a

Select all of the expressions that are equivalent to A

: C

492

Let’s practice 6

Write each expression in an equivalent radical form. a

Write each expression in an equivalent exponential form. a

Use rational exponents to justify that each of the following equations is true: a

Write in radical form.

For each expression: Evaluate.

For each expression: i

282

Write in the form

Mathspace Virginia SOL Algebra 1 mathspace.co

Evaluate.

For each expression: i

Write in the form

a 12

Blair has attempted to fully simplify an expression with positive variables in reduced radical form and showed his work: Rewrite the rational exponent as a radical

Evaluate.

Take the square root of 64

Evaluate the cube

Identify where Blair has made errors and explain the errors.

Fully simplify the expression, showing the correct steps of work.

Identify and correct the error in the work:

Let’s extend our thinking 14

Simplify: a

Aliyah is trying to simplify the expression using the laws of exponents.

Write the expression 52 as a radical in two different ways, one using a square root and one using a cube root. Explain your steps.

Without calculating, which is greater:

A cube has sides of length

. She claims that

. Is Aliyah correct? Justify your answer

? Explain your reasoning.

cm. Find the volume of the cube.

5.05 Rational exponents mathspace.co

283

5.06 Simplify radicals After this lesson, you will be able to… • simplify square roots of whole numbers. • simplify cube roots of integers.

Simplify square and cube roots Radical expressions have many parts as shown in the following diagram: Parts of a Radical Radical symbol

Index

Radicand

Radical

Index

Radicand

The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. The index on a square root is usually not written, but would be 2

The value or expression inside the radical symbol Perfect square A number that is the result of multiplying two of the same integer

Radical A mathematical expression that uses a root, such as a square root , or nth root

Perfect cube A number that is the result of multiplying three of the same integer together

Radical expressions are written in simpliﬁed radical form if the radicand cannot be factored any further. For square roots, this means there are no remaining factors of the radicand that are perfect squares and for cube roots this means there are no remaining factors of the radicand that are perfect cubes. If an expression is in simplified radical form, and there is still a number left in the radicand, the result will be irrational. We can use the following facts to simplify radical expressions, for a, b ≥ 0 and m, n positive integers,

284

Mathspace Virginia SOL Algebra 1 mathspace.co

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

, we want to see if 60 has any factors which are perfect squares. Using the factor tree, we can see that 60 = 22 ⋅ 3 ⋅ 5, so the perfect square factor is 22 = 4, and the remainder is 15.

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

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

Is it possible to get two different prime factorizations?

Can every radical be simplified?

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

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

Prime factorization method We can use a few steps to help us simplify any radical: 1. Find prime factorization of radicand 2.Group factors in groups equal to index of radical expression 3. Use multiplication property of radicals 4. Simplify any rational factors Perfect square method This is the quickest method for simplifying a radical. 1. Find largest perfect square (or cube) factor of radicand 2. Use multiplication property of radicals 3. Use the properties of radicals to factor 4. Simplify any perfect square (or cube) factors

Example 1 Simplify

Create a strategy Split 12 into its prime factors and use

5.06 Simplify radicals mathspace.co

285

Apply the idea Find prime factorization of 12

Rewrite using perfect square

Product of radicals property

Square root of a perfect square

Example 2 Simplify

Create a strategy Split −64 into factors. Notice the negative sign.

Apply the idea

Reflect and check Factor radicand

Rewrite using perfect cube

Cube root of perfect cube

Negative radicands can have rational cube roots. Can the same be said for square roots?

Example 3 Simplify the expression

Create a strategy To simplify the expression, we can first simplify the square root and then multiply the result by the coefficient outside the root.

Apply the idea Factor 18 into 9 ⋅ 2

Use the property of square roots that

Find the square root of 9, which is 3

Multiply the coefficients together

Therefore, the simplified form of

Reflect and check Let’s check our solution by substituting back into the original expression and evaluating both sides. Substitute

Evaluate Substitute

with its decimal approximation 4.24

with its decimal approximation 1.41

Evaluate

The values on both sides are approximately equal, confirming that our solution is correct. The slight difference is due to the rounding off of the square roots to two decimal places.

286

Mathspace Virginia SOL Algebra 1 mathspace.co

SOL

Express each expression in simplest form: a

What is

in simplest form? B

A SOL

−

What is the value of

in simplest radical form?

Let’s extend our thinking 12

Is each expression rational or irrational? a

Find the value of x in the equation

The volume of a cube is 3456 cubic inches. What is the length of each of its sides?

In general, if we take the cube root of x, where x is some negative integer with a rational cube root, what can we say about the resulting expression?

Use your knowledge of simplifying radicals to simplify these variable expressions. a

288

Mathspace Virginia SOL Algebra 1 mathspace.co

5.07 Operations with numerical radicals After this lesson, you will be able to… • add and subtract radicals. • multiply and divide radicals.

Multiplication of radicals We have seen that we can simplify radicals using the property

If we wanted to multiply two radicals together, we could combine them back in the same way. That is

Unlike adding and subtracting radicals, these do not have to be like terms before multiplying. If we want to find the product of radicals that have a coefficient, we can use the properties of multiplication to rearrange the expression. Say we have , which we can rewrite as . Using the commutative property, we can also write . We can simplify this by multiplying the numbers together in one group and the this expression as radicals together in another. This becomes

Although simplifying radicals before starting the question is not necessary with multiplication and division like it is with addition and subtraction, it can still help us get the job done quicker sometimes.

Example 1 Simplify the expression Give your answer in the simplest radical form.

Create a strategy Multiply the radicals together.

Apply the idea Rearrange the terms

Perform the multiplication

We can simplify our answer since 175 = 25 ⋅ 7, where 25 is a perfect square. Write the radical as a product of its factors

Evaluate

Simplify

5.07 Operations with numerical radicals mathspace.co

289

Example 2 Simplify the expression Give your answer in the simplest radical form.

Create a strategy Use the distributive property to multiply and then simplify the result.

Apply the idea

Distributive property

Evaluate the multiplication

Evaluate the perfect square

Idea summary In general, we found that:

Addition and subtraction of radicals We looked at simplifying radicals which related to multiplication and division, but what about addition and subtraction? Let’s explore addition and subtraction of radicals. Let’s look at

16 and 9 are perfect squares, so we can simplify Let’s look at whether

to 4 + 3 = 7.

is the same as

So we can see that, in general,

, and similarly

So how can we add and subtract radicals? Well unfortunately, if the radicands, a and b, are different values, we . But if they were the same number, then just like can not simplify the expression collecting like terms in algebra. We can summarize this as:

Sometimes we are asked to add and subtract radicals that have different radicands (arguments). In this case, we can try to simplify one of the radicals so that we have the same radicands. When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to simplify all radicals ﬁrst.

290

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 Simplify:

Create a strategy Simplify each radical before subtracting.

Apply the idea We can find that 24 = 8 ⋅ 3 where 8 is a perfect cube. Write the radicals as a products of their factors Evaluate Evaluate

Example 4 Fully simplify the expression

Create a strategy Add like radicals.

Apply the idea Group like radicals

Evaluate

Example 5 Consider the rectangle shown.

8 cm

5 50 cm

a Find the exact perimeter of the rectangle. Give your answer in the form

, where a and b are integers.

Create a strategy Add the four side lengths of the rectangle.

Apply the idea

Add the lengths and widths Perform multiplication

5.07 Operations with numerical radicals mathspace.co

291

We can split up both

and

into two factors, with one being a perfect square. Write the radical as a product of its factors and

Evaluate

Evalute the product of the coeﬃcients

Evaluate

b Find the exact area of the rectangle.

Create a strategy Use the formula: Area of rectangle = Length ⋅ Width

Apply the idea Substitute Length

and Width

Write the radical as a product of its factors

Evaluate

Evaluate the product of the coeﬃcients

Evaluate the product of the radicals

Evaluate

and

Idea summary When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to simplify all radicals first.

Practice What do you remember? 1

Express the following as a single radical expression:

a 2

Determine if the two radicals could be written in terms of the same radical: a

292

and

Multiply and simplify: a

Mathspace Virginia SOL Algebra 1 mathspace.co

and

Consider the equation a

b 5

and answer the following questions.

Evaluate each of the following to one decimal place: i

iii

Does

Is the equation

In general does

true or false?

Let’s practice 6

Simplify: a

Simplify completely: a

Complete the equation to create a true statement. Each option may be used more than once. 2

Simplify completely: a

Find the exact area of the rectangle.

7 ft

35 ft

5.07 Operations with numerical radicals mathspace.co

293

Find the area of the trapezoid in simpliﬁed radical form.

Find the perimeter of the triangle in simpliﬁed radical form.

Consider the rectangle: a

Find the exact perimeter of the rectangle. Give your answer in the form where a and b are integers.

Find the exact area of the rectangle.

Iain is participating in a race around the track shown in the diagram, which has dimensions in miles. Find the length of one lap around this race track.

Bob is about to paint the perimeter of a rectangular handball court, which has the dimensions shown in the diagram.

Find the total perimeter he has to paint.

The body surface area of a person in square meters can be modeled by

, where A is the surface

area, h is the height of the person in inches, and w is the weight of the person in pounds. Use the model to ﬁnd the surface area of a person who is 75 inches tall and weighs 172 pounds.

Let’s extend our thinking 17

The surface area of a human body can be approximated by the formula

where A is the surface area of the body, h is the height of the person in cm, and w is the weight of the person in kg. Find the surface area of a person who is 164 cm tall and weighs 63 kg.

294

Mathspace Virginia SOL Algebra 1 mathspace.co

Uma solved for the height of a triangle whose area is

and whose base measures

1 2 3

4 5 Did she make an error? 19

Aviva and Jillian are traveling from the shore at Point A to their home at Point C. Jillian wants to swim home and travels directly from A to C. Aviva wants to avoid the water and decides to run home via point B. a

How far does Jillian have to swim to get home?

How far does Aviva have to run to get home?

How much further did Aviva travel than Jillian?

Give two different pairs of values for k and m which make the following equation true:

Identify and correct the error in the following work:

Sasha is simplifying the expression Explain how you know.

and says that the answer is

. Is she correct?

5.07 Operations with numerical radicals mathspace.co

295

5.08 Characteristics of exponential functions After this lesson, you will be able to… • identify the key features of an exponential function. • interpret key features of an exponential function in context.

Characteristics of exponential functions Exponential relationships include any relations where the outputs increase by a constant factor or decrease by a constant factor for consistent changes in x. An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an exponential function:

f (x) = abx a b x f (x)

Leading coefficient Base where b > 0, b ≠ 1 Independent variable Dependent variable

Exploration Consider the following equations with a = 1: y = 5x x

−2

−1

y = 2x x y

−2

−1

For each of the functions, think about the following questions: 1.

What happens to y as x increases?

Compare y = 5x and y = 2x. How are they similar? How are they different?

What is the y-intercept for each of the functions? How does this relate to the value of a?

Does either function have an x-intercept?

Create a table of values for y = 1x. Does it have the same properties as y = 5x and y = 2x?

We can determine whether a function is exponential by dividing consecutive function values to see if they have a constant factor.

296

Mathspace Virginia SOL Algebra 1 mathspace.co

The base, or constant factor, is the number being multiplied repeatedly. It tells us how quickly the output values are growing or shrinking. We can find the base by dividing a term by the previous term, as shown below: x

f (x)

×3 Constant factor:

×3

9 3 =3 =3 1 3

27 =3 9

In this example, we see that the function is growing exponentially. A function grows exponentially when it increases by a constant factor. Exponential functions change at a faster rate than linear functions. In the table below, we are adding 3 to each term, but the terms do not grow as quickly. x

f (x)

9 8 7 6 5

y = 3x

3 2 1 −4 −3 −2 −1

All exponential functions of the form y = abx have these features in common: • The domain is all real values of x. • The range is y > 0. • For a context, the domain and range may be further restricted. • The y-intercept is at (0, a). • The common factor is b. • There is a horizontal asymptote at y = 0.

x 1

Asymptote

A line that a curve or graph approaches as it heads toward positive or negative infinity.

An exponential function can get infinitely close to an asymptote, but it can never cross it. This means that an exponential function of this form will not have an x-intercept.

5.08 Characteristics of exponential functions mathspace.co

297

Example 1 Consider the table of values for the function y = 3(2)x. x

−3

−2

−1

3072

a Describe the behavior of the function as x increases.

Create a strategy We want to identify if the values of y are increasing or decreasing as x increases.

Apply the idea

Reflect and check

As x increases, the function increases at a faster and faster rate.

We can see that the equation has a constant factor that is greater than 1. This is why the function is increasing.

b Determine the y-intercept of the function.

Create a strategy The y-intercept occurs when x = 0. We can read these coordinates from the table.

Apply the idea

Reflect and check

(0, 3)

We can see that the equation has a leading coefficient of 3. This is the value of the y-intercept, and the result of substituting x = 0 into the equation.

c State the domain of the function.

Create a strategy The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in both horizontal directions.

Apply the idea

Reflect and check

All real values of x.

All exponential equations of the form y = abx have a domain of all real x.

d State the range of the function.

Create a strategy The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, but it approaches an asymptote at y = 0 towards the right.

Apply the idea

Reflect and check

y>0

All exponential equations of the form y = abx have a range of y > 0 for positive values of a.

298

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 A population of bacteria can be modeled with the equation p(t) = 100(2)t, where p(t) is the population after t days. This graph shows the population over time.

2000 1800 1600 1400 1200 1000 800 600 400 200

t 1 2 3 4 5 6 7 8 9

a Identify and interpret the p-intercept.

Create a strategy The p-intercept is the vertical intercept and will occur when t = 0. We can read this off the graph or use the equation.

Apply the idea

Reflect and check

Using the equation: t

Start with the equation

Let t = 0

p(t) = 100(2) p(0) = 100(2) = 100(1)

Use the zero power law

= 100

Evaluate the product

Reading from the graph is not as precise as using the equation, but is still an important approach.

This means that initially there were 100 bacteria.

b Estimate and interpret when p(t) = 1600.

Create a strategy We can create a table of values using the equation or read it off the graph.

Apply the idea 2000 1800 1600 1400 1200 1000 800 600 400 200

Starting from p(t) = 1600, we can go across to the curve and then down to find the t-value for when p(t) = 1600.

t 1 2 3 4 5 6 7 8 9

This means that after 4 days, the population reaches 1600.

5.08 Characteristics of exponential functions mathspace.co

299

Reflect and check We can confirm this by creating a table of values: t

0 100

p(t)

1 200

2 400

3 800

4 1600

c Identify and interpret the domain and range in this context.

Create a strategy We can read the domain and range from the graph, or think about what is possible given the context.

Apply the idea Since t represent time, it cannot be negative. This means the domain is x ≥ 0. The initial population is 100, and it is growing from there, so the range is y ≥ 100.

Reflect and check Without the context, the domain and range of y = 100(2)x would be different.

Idea summary The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by dividing one output by the previous output. All exponential functions of the form y = abx have the following features in common: The domain is −∞ < x < ∞. The range is y > 0. If there is a context, the domain and range may be different based on the realistic constraints. The y-intercept is at (0, a). There is a horizontal asymptote at y = 0.

• • • • •

Practice What do you remember? 1

Would an exponential function generate the values shown in each table? x 1 2 3 4 f (x) 5 25 125 625

6 15 625

x f (x)

1 4

2 9

3 11.5

What type of function would generate values as shown in the table? x f (x)

300

5 3125

1 2

2 3

3 4.8

4 7.6

Approximately exponential

Not exponential

Mathspace Virginia SOL Algebra 1 mathspace.co

5 11.8

6 18 B

Exactly exponential

4 15

5 13.5

6 11

The function n(x) = 40(3)x represents the number of cells after x hours. a

State which characteristic of the graph is represented by 40.

State what 40 represents in this context.

State how the number of cells changes from one hour to the next.

Let’s practice 4

Which one of the following exponential functions rises most steeply? A

y = 2(2)x

For f (x) = 5x, find f (3).

Consider the function y = −(2x). a

y = 2(3)x

y = 2(7)x

Complete the table of values: x y

y = 2(5)x

−5

−4

−3

−2

−1

Can the value of y ever be zero or positive? Explain your answer.

Describe the behavior of the function as x increases.

State the domain of the function.

State the range of the function.

For each function: i

Identify the equation of the horizontal asymptote.

Find the y-intercept.

iii

Find one other point on the curve.

1 −4 −3 −2 −1

8 7

−2

−3

−4

−5

−6 x 1

y −4 −3 −2 −1 −1

−4 −3 −2 −1

x 1

−7 −8

5.08 Characteristics of exponential functions mathspace.co

301

The graph of g(x) = 2x is shown: a

Find the value of g(3).

Find the value of x when g(x) = 32.

36 32 28 24 20 16 12 8 4

−3 −2 −1

Consider the graph of the functions y = 3x and y = 10x:

State the coordinates of the point of intersection of the two curves.

Describe what happens to the values of y for each function as x gets increasingly larger.

Describe what other features these functions have in common.

9 y = 10x

y = 3x

−5 −4 −3 −2 −1 −3

x 1 2 3 4 5

−6

The local seagull population is changing according to the exponential function f (t) = 1000(2)t where t is the number of years that have passed. a

Describe what is happening to the population each year.

Describe what 1000 represents in context.

Find f (3) and describe what it represents in context.

Consider the graphs of the two exponential functions R and S: a

y 18 16 14 12 10 8 6 R 4 2

One of the graphs is of y = 4x and the other graph is of y = 6x. Identify which is the graph of y = 6x. Explain your answer.

For x < 0, determine the relationship between y = 6x and y = 4x. Explain your reasoning.

−3 −2

For each pair of functions, select the function that increases more rapidly for x > 0: a

y = 4x and y = 5x

y = 2x and y = 3(2)x

Do either of the functions y = 9x or y = −(9x) have x-intercepts? Explain your answer.

302

Mathspace Virginia SOL Algebra 1 mathspace.co

−1

x 1

Let’s extend our thinking 14

The number of layers, y, resulting from a rectangular piece of paper being folded in half x times, is shown in the graph. a

Interpret the meaning of the y-intercept in this context.

Describe what happens to the thickness with each fold.

If a rectangular piece of paper is folded 10 times, find the resulting number of layers.

If a rectangular piece of paper of thickness 0.02 mm is folded 11 times, find the total resulting thickness.

10 9 8 7 6 5 4 3 2 1

x 1

Find the missing coordinate in each ordered pair so that the pair is a solution of y = −3x: a

(5, ⬚)

(−1, ⬚)

(⬚, −81)

The graph of f (x) = 3x is shown. Find the length of the line segment PQ.

y P

x −5 −4 −3 −2 −1

Q1 2 3 4 5

A lottery winner has two options: 1. Getting $1 000 000 today 2. Getting $1 today, $2 tomorrow, $4 the next day, and so on with the amount of money each day doubling for a month. The second option can be modeled by A(d) = 2d, where d is the number of days after today (today is d = 0). a

Calculate A(0) and interpret its meaning in the context.

Calculate A(18) and interpret its meaning in the context.

1000000

Using the graph, determine on which day the lottery winner would get more than $1 000 000 using the second option.

900000

Determine which option would give the larger prize. Explain.

600000

A ($)

800000 700000

A (d) = 2d

500000 400000 300000 200000 100000

d 2 4 6 8 10 12 14 16 18 2022242628

5.08 Characteristics of exponential functions mathspace.co

303

Winston and Rasiah want to organize a lunch time dominos 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. Semi-Final

Quarter-Final Final

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)

304

Winner

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.

Over two weeks, they can fit in 8 rounds of play. Determine how many players can they accept into the tournament.

Mathspace Virginia SOL Algebra 1 mathspace.co

For k > 0, the graph will shift up k units. For k < 0, the graph will shift down k units. y

6 f (x) = 2x

4 2 −4 −3 −2 −1

(0, a) 1 2

4 (0, a + 4) 2

x 3

f (x) = 2x + 4

−4 −3 −2 −1

Parent Function f (x) = 2x

Vertical shift 4 units up f (x) = 2x + 4

The key features of an exponential function can be found in both the equation and the graph. Given f (x) = abx, the leading coefficient, a, represents the y-intercept and is plotted as the point (0, a). It also determines the range of the function. • When a > 0, the range is y > 0 • When a < 0, the range is y < 0. The value of a also affects the rate of change of the function. If two functions have the same value for b, a larger value for a will have a greater rate of change. 8

y −2

a > 0, b > 1

−4

−5

−6

1 −1

−3 a < 0, b > 1

−2

−1

x 1

−2

6 4

−1

x 1

y > 0, Increases at an increasing rate

−7 −8

y < 0, Decreases at an increasing rate

The absolute value of a, tells us whether the graph’s height will be made taller or shorter. If ∣a∣ > 1, then every y-coordinate of the function is multiplied by a factor a that is greater than 1. The points on the graph move further away from the x-axis, increasing the steepness of the graph. This is called a vertical stretch. If 0 < ∣a∣ < 1, then every y-coordinate of the function is multiplied by a factor a that is between 0 and 1. The points on the graph move closer to the x-axis, decreasing the steepness of the graph. This is called a vertical compression.

306

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 Consider the exponential function y = 2.5(4)x. a Draw the graph of the function.

Create a strategy We can identify both the y-intercept and constant factor from the equation since it is of the form y = abx. Using these two key features, we can plot other points to the left and right of the y-intercept and connect them with a smooth curve.

Apply the idea The function has a constant factor of 4 since that is the base of the exponent, and a y-intercept at (0, 2.5) as that is the coefficient. We can start by plotting the y-intercept and choosing an appropriate scale. We can see that the function will always be positive, so we don’t need any negative y-values.

y 40 35

We know that the constant factor is 4. If we want to go from −2 to 2 on the x-axis, we need to go up to at least 2.5(4)2 = 40 on the y-axis.

30 25 20

This would be an appropriate scale as it doesn’t have too many labels or tick marks and is easy to read.

15 10 5 −2

−1

Either using a table of values or using the common ratio, we can plot another three points to get a good shape for the graph. Then we can connect the points with a smooth curve.

y 40 35 30 25 20 15 10 5 −2

−1

x 1

5.09 Graphs of exponential functions mathspace.co

307

b Check the graph from part (a) using technology.

Create a strategy When using a graphing calculator, there will typically be an input bar for functions. We just need to type out the function in the input bar. We need to be careful that we format our input correctly by using the correct buttons. The GeoGebra calculator has a math input keyboard, but we can also use a computer keyboard. • Math input keyboard y = 2.5 (4)x

• Computer keyboard y = 2.5 (4)x

(

)

y = 2.5 (4)x

Apply the idea The graph should look the same as in part (a). We may need to change the axes or zoom settings to check.

c Without graphing, how would you expect this graph to look different than the graph of y = (4)x

Create a strategy

Apply the idea

Remember how the leading coefficient and vertical shift can impact the graph.

Since the leading coefficient, a, is greater than 1, we can expect the graph to have a greater rate of change and increase more quickly.

308

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 In 2010, Bob counted 3 rabbits in his backyard. He noticed they double every week. Let f (x) describe this scenario, and use it to answer the questions below. a Find the y-intercept.

Create a strategy Imagine time in weeks as x, the independent variable. The y-intercept occurs when x = 0 at week 0 of counting.

Apply the idea y=3 The y-value of the y-intercept is the starting number of rabbits which is 3. The y-intercept is (0, 3).

b Write an equation describing this scenario.

Create a strategy

Apply the idea

Since the rabbits double every months, we can start by identifying the constant factor.

We know our y-intercept, or a = 3. Our constant factor b = 2. We can write our equation as f (x) = 3(2)x.

c Graph the function.

Create a strategy

Apply the idea

Draw the curve using the equation from part (b).

y 24 21 18 15

y = 3(2)x

12 9 6 3 −1

x 1

Example 3 Consider the function y = −4x − 1: a Find the y-intercept.

Create a strategy Substitute x = 0 into the function.

Apply the idea y = −4x − 1

Write the function

= −40 − 1

Substitute x = 0

= −2

Evaluate

5.09 Graphs of exponential functions mathspace.co

309

b Graph the function as a transformation of the function f (x) = −4x.

Create a strategy

Apply the idea x

Start by graphing the function f (x) = −4 .

Draw the curve passing through each plotted point.

−5 −4 −3 −2 −1 −2

−5 −4 −3 −2 −1 y = −(4x) −1 −2

x 1

−4

y = −(4x)

−6

−6 −8

−8

−10

−12

−14

−16

This curve of the function y = −(4x) − 1 must be shifted down 1 unit from the function f (x) = −(4x).

Idea summary We can use the y-intercept, the constant factor, b, and the vertical shift, k. to graph an exponential function in the form y = abx + k and identify key features: • • • •

When a > 0 and b > 1, the function is increasing at an increasing rate. When a < 0 and b > 1, the function is decreasing at an increasing rate. k > 0, will shift the graph k units up. k < 0, will shift the graph k units down.

Practice What do you remember? 1

Match each graph to one of the equations: i

y = 2x

y = −2x

b 8

−4 −3 −2 −1 −2

x 1

2 −4 −3 −2 −1 −2

−4

−6

−8

Mathspace Virginia SOL Algebra 1 mathspace.co

y = −2(2)x

310

iii

x 1

2 −4 −3 −2 −1 −2

x 1

−4 −3 −2 −1 −2

−4

−6

−8

For f (x) = 3x, will the transformation change the equation of the horizontal asymptote? a

A vertical dilation by a factor of 5

A translation down 2 units

A refection across the x-axis

A vertical dilation by a factor of

Describe how the graph of y = 2x is transformed to get each graph: a

y = 2x + 4

y = 2x − 6

y = 5(2)x

Let’s practice 4

Draw the graphs of the functions y = 2x, y = 3x and y = 5x by hand or using technology, then answer the following questions: a

State whether the following statements are true for all of the functions: 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.

State the y-intercept of each curve.

Describe what happens to the values of y as x gets increasingly larger.

Consider the function y = 4(2x). a

Find the y-value of the y-intercept of the curve.

Can the function values ever be negative?

State an appropriate scale for the axes to graph y = 4(2x). Justify your choice.

Graph y = 4(2x) by hand or using technology.

List the domain and range for the function.

Consider the functions f (x) = 4(2)x and g(x) = 2(4)x. a

Graph the two functions using technology.

Compare the domain and range of f (x) and g(x).

Consider the given graph of y = 5x. a

10 8 6 4 2

Complete the table of values for y = −5x. x y

−2

−1

Graph y = 5x and y = −5x on the same coordinate plane.

Compare the domain and range of y = 5x and y = −5x.

Describe a transformation of the graph of f (x) = 5x that would obtain g(x) = −5x.

−3 −2

−1

−2 −4 −6 −8 −10

y = 5x x 1

5.09 Graphs of exponential functions mathspace.co

311

Gabby starts by saving 6 pennies in her piggy bank. She decides to triple the amount she saves every week. a

Write an equation to match the scenario.

Graph the function.

Use your function to determine how much money Gaby will have after 10 weeks.

The graph of f (x) = 3x is shown.

5 4 3 2 1

Describe a transformation of the graph of f (x) = 3 that would obtain g(x) = 3x − 5.

Sketch the graph of g(x) = 3x − 5 on the same set of axes as f (x) = 3x.

f (x) = 3x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

The graph of f (x) = 8x is shown. a b

Describe transformation of the graph of f (x) = 8 that would obtain g(x) = 0.5(8)x.

1 2 3 4 5

Sketch the graph of g(x) = 0.5(8)x on the same set of axes as f (x) = 8x.

f (x) = 8x

−4 −3 −2 −1 −2

−4 −6 −8

Consider a graph of y = 5x:

y x

Describe a transformation of the graph of y = 5 that would obtain the graph of y = 5x + 1.

Sketch the graph of y = 5x + 1.

25 20 15 10 5 x −3 −2

Of the two functions y = 4x and

312

Mathspace Virginia SOL Algebra 1 mathspace.co

, which is increasing more rapidly for x > 0?

−1

The graphs of A, B, and y = 6x are shown. a b

What transformation has been applied to y = 6 to get the graph of A? What transformation has been applied to y = 6 to get the graph of B?

10 9 8 7

y = 6x

6 5 4 3

2 1

x 1

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found to double each hour. a

Complete the table of values. Number of hours passed (x) Number of bacteria ( y)

0 1

At this rate, how many bacteria will be present in the petri dish after 15 hours?

Using the table, graph the number of bacteria over time.

Interpret the meaning of the y-intercept in this context.

Let’s extend our thinking 15

Consider the original graph y = 3x. The function values of the graph are multiplied by 2 to form a new graph. a

For each point on the original graph, find the point on the new graph. Point on original graph Point on new graph

(−1, ⬚)

(1, 3)

(2, 9)

(0, ⬚)

(1, ⬚)

(2, ⬚)

State the equation of the new graph.

Graph both functions on the same coordinate plane and compare them.

The function y = 3 (2x) is shown. a

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 modeled by an exponential function, with x representing time and y representing the number of bacteria. Which function should be used? Explain.

(0, 1)

10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10

y y = 3 (2x)

x 2 4 6 8 10

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. 5.09 Graphs of exponential functions mathspace.co

313

6 Polynomials & Factoring Big ideas • The properties of real numbers can be applied to many types of expressions. • A standard algorithm can be applied to rewrite many different kinds of expressions.

Chapter outline 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08

Add and subtract polynomials Multiply polynomials Divide polynomials by a monomial Factor GCF Factor by grouping Factor trinomials Factor using appropriate methods Divide polynomials

316 326 340 345 351 356 363 371

6.01 Add and subtract polynomials After this lesson, you will be able to… • add and subtract polynomials. • represent sums and differences of polynomials using pictorial models, including algebra tiles. • explain why addition and subtraction of polynomials produce another polynomial.

Add and subtract polynomials Polynomial expressions can be added and subtracted much like real numbers. Polynomial The sum or difference of terms which have variables raised to non-negative integer powers and which have coeﬃcients that are constant

Exploration Compare

Create an addition problem like the example provided where the sum of the coeﬃcients is greater than 9. What happens?

Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as numbers when subtracting?

We can use algebra tiles to model sums and differences of polynomials. The difference (6x2 + 4x − 5) − (4x2 − 2x + 3) can be modeled with algebra tiles. Lining up like terms, vertically, we can write:

316

Mathspace Virginia SOL Algebra 1 mathspace.co

The subtraction can be viewed as the expression: (6x2 + 4x − 5) + (−1) (4x2 − 2x + 3) 6x2 + 4x − 5 x2

−

−1

4x2 − 2x + 3 x2

−x

Using the opposites of the expression 4x2 − 2x + 3 with the algebra tiles, we get the expression: (6x2 + 4x − 5) + (−4x2 + 2x − 3) Equivalently, distributing the −1:

Creating zero pairs and combining like terms with the algebra tiles, we are left with the expression: 2x2 + 6x − 8 Therefore, the difference between the revenue from the gaming computers can be modeled by 2x2 + 6x − 8. 6x2 + 4x − 5 x2

−1 −1 −1 −1 −1

−4x2 − 2x + 3 −x2

−x2

−1 −1 −1

2x + 6x − 8 x2 x2

−1 −1 −1 −1 −1 −1 −1 −1

6.01 Add and subtract polynomials mathspace.co

317

Adding and subtracting polynomials creates more polynomials. The following vocabulary is helpful to know when working with polynomials: Standard form (of a polynomial)

Leading coeﬃcient

A way of writing a polynomial expression; anxn + an−1xn−1 + … + a1x + a0, where n is a non-negative integer and each ai is a coeﬃcient.

The coeﬃcient of the leading term Monomial A polynomial with one term

Degree of a polynomial The value of the highest exponent on a variable in the polynomial

Binomial

Leading term

Trinomial

The term in a polynomial with the highest exponent of the variable

A polynomial with three terms

A polynomial with two terms

Example 1 Consider the polynomial 3x − 6 + x2 a Rewrite the expression in standard form.

Create a strategy Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent to the lowest. We can use the commutative property to change the order of the polynomials.

Apply the idea x2 + 3x − 6 b State the degree of the polynomial.

Apply the idea x2 + 3x − 6 is a polynomial of degree 2.

Reﬂect and check Since the polynomial has 3 terms, it may be called a trinomial.

c Identify the quadratic term, the linear term, and the constant term of the polynomial.

Apply the idea

Reﬂect and check 2

• Quadratic term: x • Linear term: 3x • Constant term: −6

318

Mathspace Virginia SOL Algebra 1 mathspace.co

Polynomials of degree 2 are called quadratic polynomials.

Example 2 Consider the polynomials x3 − 6x + 2 and x2 + 9x + 7. a Find the sum of the two polynomials.

Create a strategy Since we want to ﬁnd the sum of the two polynomials, combine the like terms.

Apply the idea Sum = (x3 − 6x + 2) + (x2 + 9x + 7) 3

Add the polynomials together

= x − 6x + 2 + x + 9x + 7 3

Remove the parentheses (Associative Property)

Combine the like terms = x + x + 3x + 9

Reﬂect and check If we want to use the vertical algorithm method, we need to make sure we correctly align the like terms.

b Explain why the sum of two polynomials is also a polynomial.

Apply the idea By deﬁnition, a polynomial is the sum or difference of terms which have variables raised to non-negative integer powers and which have coeﬃcients that may be real or complex. Adding one polynomial to the other is the same as adding more terms to one polynomial. This doesn’t change the fact that it is a polynomial, so the sum of two polynomials will always be a polynomial.

Reﬂect and check We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Example 3 Simplify the expression: (3x2 − 5x + 1) − (x2 + 7x − 10)

Apply the idea (3x2 − 5x + 1) − (x2 + 7x − 10) = 3x2 − 5x + 1 − x2 − 7x + 10 2

= (3x − x ) + (−5x − 7x) + (1 + 10) 2

= 2x − 12x + 11

Distribute the subtraction Group the like terms together Simplify

6.01 Add and subtract polynomials mathspace.co

319

The expression 12x2 + 6x + 6 is the same as having twelve x2 tiles, six x tiles, and 6 unit (or 1) tiles. 12x2 + 6x + 6 x2

x x x x x x

1 1 1

The expression (8x2 + 4) + (12x2 + 6x + 6) can be represented as: 8x2 + 4

12x2 + 6x + 6

x2 1 1 1

1 x x x x x x

1 1 1

Combine the like terms and count each type of tile. x2

x x x x x x

1 1 1

So, there are twenty x2, six x, and ten unit tiles. The result is 20x2 + 6x + 10.

Idea summary We add polynomials by combining like terms. We subtract polynomials by adding the negative terms.

6.01 Add and subtract polynomials mathspace.co

321

Practice What do you remember? 1

Choose the best word from this set: term, coefficient, variable, constant to describe:

The number 3 in the term 3x

The letter x in the term 6x

4x in the expression 4x + 7

The number 7 in the expression 5x + 7

The number −5 in 4 − 5x

The letter u in the term −12u

The number 8 in 6z + 8

−3y in 2 − 3y + 4z

Describe each pair of terms as like or unlike. a

10p and 5p

3 and y

5n2 and 8n

5p and 5

10a and −9a

2ab and 6ba

15p and 15q

8 and 8z

7m2 and 7m

8z and −8z

12b and −21b

13xy and 14yx

Donna earns $15 per hour of work and is paid double for every hour worked on the weekend. At the end of a week, Donna calculates her pay for that week to be 15x + 30y dollars. a

What variable represents the number of hours Donna worked during the weekdays?

What variable represents the number of hours Donna worked during the weekends?

In the expression 3x2 + 5x + 7x2 − 2x, which terms are like terms? Explain your answer.

Write a simplified expression for each set of algebra tiles. a

b −x

−x

−x2

1 2

−1

1 x

−x

−1

Compare and contrast adding two-digit integers and adding binomials.

Let’s practice 7

Simplify: a

2a + 5a

10x + 6x

e i

3c + 4c + 7c

8x − 3x + 7x

j n

4m − 4m

m 2u + 9u 8

322

4b + 3b

12y − 3y

15x − 6x − 2x

19b − 12b − 6b

−3n + 6n + 3n

x+9+7

9x + 4x

12p − 9p

5y + 6y

11 + y

State whether each expression is equal to 11y. a

9 + y + 10y − 9

6y − 5y

9y + 3y − 1

10y + 1

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 26

Simplify: a

8x + 6y − 2y − 4x

11m + 8n + 14m

9xy + 12yx

6p + 8q − 6p

2.5x + 9y − 5x + 10y

7a + 11a − 9b + b

8x − 7y − 6z + 10z

13m − 2.2n − 8m + 1.5n

−3s + 4t − 6t + 9s

Simplify: a

(8a2b2 − 7a2b + 8ab − 11) + (−3a2b2 + 6a2b + 5ab)

(4x2y − 9xy2 + 4) + (−7x2y + 4xy2 + 8)

(3x2y2 − 2xy2 + 4y2) + (−9x2y2 + 3xy2 − 8y2)

Tom simplified the expression 7x + 6p − 4x + 2 and found it to be 3x + 6p + 2. Xanthe simplified the same expression and found it to be 11x + 6p + 2. Explain who is correct and why.

Show that the expression 12ab + 7c − 8ab − 9c is equivalent to the expression −6c + 2ab + 4c + 6ab − 4ba.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning with examples. a

Two polynomials added together will result in a polynomial.

A linear function is a polynomial.

A polynomial added to a non-polynomial will result in a polynomial.

A non-polynomial added to a non-polynomial will result in a polynomial.

Is it ever possible that 8m + 5n = 13mn?

Explain how two trinomials can be added together to produce a binomial.

Given that (ax + 5) + (4x2 − 4x + 4) + (3x + 2) = 4x2 + 4x + 11 for all values of x, solve for a.

Consider [ax2 + (b − 5) x − 1] + [x2 − 5x + 2] = 2x2 + 2x + 1. a

Find the value of a.

Find the value of b.

Consider the following work: (−3x3 + 7x2 − x − 4) − (−2x3 + ax2 + bx + 6) = cx3 + 6x2 + 4x − 10 (−3 − 2) x3 + (7 − a) x2 + − (1 + b) x + (−4 − 6) = cx3 + 6x2 + 4x − 10 −5x3 + (7 − a) x2 + − (1 + b) x + (−10) = cx3 + 6x2 + 4x − 10 7−a=6

−(1 + b) = 4

a=1

−1 − b = 4 b = −3

Therefore, a = 1, b = −3, and c = −5. a

Identify and explain any mistakes.

Give the correct values for a, b, and c.

6.01 Add and subtract polynomials mathspace.co

325

6.02 Multiply polynomials After this lesson, you will be able to… • multiply a monomial and a polynomial. • multiply polynomials. • represent polynomial multiplication using an area model. • explain why multiplication of polynomials produces another polynomial.

Multiply polynomials Exploration Complete the area models for multiplication shown: 7

7x3

3x2

3x2(7x3 + 9)

3(7 + 9) Complete the new area models for multiplication:

2x 20

10 −3

−3

(10 − 3) (20 + 1)

(x − 3) (2x + 1)

What do the area models have in common?

What’s different about the area models?

Make a conjecture about how multiplying polynomials relates to multiplying integers.

Consider the garden plot: 4x + 1

3x − 2

326

Mathspace Virginia SOL Algebra 1 mathspace.co

The length of a rectangular garden plot is 4x + 1 feet. The width of the plot is 3x − 2 feet. The area can be represented as the product (4x + 1) (3x − 2). We can use models to simplify the product as a polynomial. 4x

12x2

−2

−8x

−2

4x + 1 x2

−x −x

3x − 2

An algebra tiles model

A box/area model

Notice the algebra tiles model shows each individual tile, but the box model combines some like terms together. Both models show the product of (4x + 1) (3x − 2) = 12x2 + 3x − 8x − 2. Which we can simplify by combining like terms to 12x2 − 5x − 2. a

The distributive property can be used to multiply two polynomials: (a + b) (c + d) = ac + ad + bc + bd

Area models help us visualize the different terms from the distributive property. They can help us organize the multiplication of polynomials, so we don’t miss any terms. Then, we can combine like terms to get the simplest polynomial.

The product of two polynomials will always result in a new polynomial where • The degree of the new polynomial will be the sum of the degrees of the multiplied polynomials. • The number of terms may vary from the original polynomials depending on how like terms are combined.

Example 1 Multiply 3x (2x2 − 5x + 4).

Apply the idea We can use the distributive property to get the product of the monomial 3x and the trinomial 2x2 − 5x + 4. 3x (2x2 − 5x + 4) = 3x(2x2) + 3x(−5x) + 3x(4) = 6x3 − 15x2 + 12x Since there are no more like terms and the expression is already in standard form, the ﬁnal answer is 6x3 − 15x2 + 12x.

Reﬂect and check 6x3 − 15x2 + 12x is considered a polynomial of degree 3, since 3 is the value of the highest exponent on a variable in the polynomial.

6.02 Multiply polynomials mathspace.co

327

Example 2 Consider the polynomials 7y + 2 and 4y − 5. a Find the product of the two polynomials.

Create a strategy Multiply the polynomials using distribution.

Apply the idea We can create an area model to multiply the two polynomials: 7y

28y2

−5

−35y

−10

Combining each of the terms, we get: 28y2 − 35y + 8y − 10 = 28y2 − 27y − 10

Reﬂect and check We can also use the distributive property to get the product of 7y + 2 and 4y − 5. (7y + 2) (4y − 5) = 4y(7y + 2) − 5(7y + 2) = 4y(7y) + 4y(2) − 5(7y) − 5(2)

Distributive property Distributive property

Distributive property

Combine like terms

= 28y + 8y − 35y – 10 = 28y − 27y – 10

Since the expression is already in standard form, the final answer is 28y2 − 27y – 10.

b Explain why the product of two polynomials is also a polynomial.

Apply the idea A polynomial is a collection of terms in the form mxn where m is a real number and n is a non-negative integer. We know that the product of two algebraic terms with non-negative integer exponents is an algebraic term with non-negative integer exponents. Since multiplying polynomials together results in a sum of such products, by deﬁnition the result is a polynomial expression.

Reﬂect and check We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens if we add two polynomials and multiply this result by another polynomial? What if we multiply three polynomials? Is our result still a polynomial?

328

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x + 1 yards represent the length of each side. a Conﬁrm that the side length will always be odd.

Create a strategy Analyze the given side length.

Apply the idea Since 2x + 1 represents the side lengths of the container, we can note that twice any number is always an even product. If we add an odd number like 1 to that, the side length 2x + 1 will always be odd.

b Write an expression for the surface area of the storage container.

Create a strategy Draw and label a diagram of the storage container ﬁrst, then use it to calculate the surface area.

Apply the idea

4x2

We can calculate the area of one face on the storage container, then multiply the polynomial expression by 6 faces on the cube-shaped container.

Area of one face: 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1 square yards Surface area of container: 6(4x2 + 4x + 1) = 24x2 + 24x + 6 square yards

c Write an expression for the volume of the storage container.

Create a strategy Use the formula for the volume of a cube to calculate the volume of the storage container.

Apply the idea Since the formula for the volume of a cube is V = l ⋅ w ⋅ h, we can calculate the volume of the storage container as shown: V = (2x + 1) (2x + 1) (2x + 1) Substitute expressions for l, w, and h = (2x + 1) [(4x2 + 2x + 2x + 1)] Distributive property Combine like terms = (2x + 1) (4x2 + 4x + 1) Distributive property = (8x3 + 8x2 + 2x) + (4x2 + 4x + 1) = 8x3 + 12x2 + 6x + 1 cubic yards

Combine like terms

6.02 Multiply polynomials mathspace.co

329

Reﬂect and check A labeled diagram of the storage container can help us conceptualize the problem.

Height: (2x + 1) yards

Width: (2x + 1) yards Length: (2x + 1) yards

Example 4 Consider the diagram of the product of the expression (x − 1) (x − 4). (x − 4) x

(x − 1)

−1

a Find the missing values on the diagram to complete the visual representation of multiplying (x − 1) (x − 4).

Create a strategy Use the algebra tiles on the left side and the algebra tiles on the top row to find the area of the tiles with missing values.

Apply the idea The diagram that shows the visual representation of multiplying (x − 1) (x − 4) is given by: (x − 4)

(x − 1)

330

Mathspace Virginia SOL Algebra 1 mathspace.co

−1

−x

−1

−x

b Write the product of (x − 1) (x − 4).

Create a strategy Add all like terms of the algebra tiles from part (a).

Apply the idea (x − 1) (x − 4) = x2 − x − x − x − x − x + 1 + 1 + 1 + 1 2

= x − 5x + 4

Add all the algebra tiles Evaluate

Reﬂect and check You may encounter some products of polynomials that require combining exponents with a degree greater than 1. Recall the expanded form of the product property for exponents that says axm ⋅ bxn = abxm + n. So, we have:

5x3 ⋅ 4x2 = 5 ⋅ x ⋅ x ⋅ x ⋅ 4 ⋅ x ⋅ x =5⋅4⋅x⋅x⋅x⋅x⋅x = 20x5

Idea summary Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials helps keep track of terms.

Special products of binomials For some products of binomials, we can look for patterns to help us simplify more eﬃciently.

Exploration Consider the expansions of the following binomials of the form (a + b) (a + b) = (a + b)2: • (x + 3) (x + 3) = x2 + 6x + 9 2

• (x + 5) (x + 5) = x + 10x + 25

• (x + 6)2 = x2 + 12x + 36 • (2r + 3s)2 = 4r2 + 12rs + 9s2

• (5s − 3)2 = 25s2 − 30s + 9 1.

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

What do you notice about the constant of the product?

Is there a general rule for this type of product?

Consider the expansions of the following binomials of the form (a + b) (a − b): • (x + 3) (x − 3) = x2 – 9

• (2r + 3s) (2r − 3s) = 4r2 − 9s2

• (x − 5) (x + 5) = x2 – 25

• (5s − 3) (5s + 3) = 25s2 − 9

• (x + 6) (x − 6) = x − 36 1.

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

What do you notice about the constant of the product?

Is there a general rule for this type of product?

6.02 Multiply polynomials mathspace.co

331

Some products of binomials follow special patterns. For example, consider the product of a binomial squared, (a + b)2: a

We can expand (a + b)2 to (a + b) (a + b) and represent them with an area model. Evaluating with this model and combining like terms, we get the product a2 + 2ab + b2. So, we have:

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

Now consider the product of a sum and a difference, (a + b) (a − b):

−b2

−b

−b2

Notice that the term ab and −ab are opposites and combine to make zero. We call this a zero pair. So, (a + b) (a − b) = a2 − b2. A binomial of the form a2 − b2 is called a difference of two squares. If we remember the patterns for these special products, we can multiply two polynomials without using the distributive property. For binomials, we have the following special binomial products, which are called identities: Square of a Sum 2

(a + b) = a + 2ab + b

Product of a sum and difference 2

(a + b) (a − b) = a2 − b2

Square of a Difference (a − b)2 = a2 − 2ab + b2

Note: (a + b)2 ≠ a2 + b2 and (a − b)2 ≠ a2 − b2.

Example 5 Multiply and simplify the following binomials. a (x − 4)2

Create a strategy We check ﬁrst whether (x − 4)2 is a special binomial product and identify its form.

332

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea Since (x − 4)2 is a square of a binomial in the form (a − b)2, we use the formula and simplify the expression: (a − b)2 = a2 − 2ab + b2 2

Identity for the square of a binomial 2

(x − 4) = x − 2(x) (4) + 4

Substitute a = x and b = 4

= x − 8x + 16

Evaluate the multiplication and exponent

Reﬂect and check x

−4

We can also use an area model to find the product. Combining like terms, we get: (x − 4)2 = x2 + 16

−4x

−4

−4x

b (x + 4) (x − 4)

Create a strategy We check ﬁrst whether (x + 4) (x − 4) is a special binomial product and identify its form.

Apply the idea Since (x + 4) (x − 4) is a product of a sum and difference, we use the formula and simplify the expression: (a + b) (a − b) = a2 − b2 Identity for the product of a sum and difference (x + 4) (x − 4) = x2 − 42 Substitute a = x and b = 4 = x2 – 16

Evaluate the exponent

Reﬂect and check Using an area model, we see (x + 4) (x − 4) = x2 – 16. 4

−4

−4x

−16

6.02 Multiply polynomials mathspace.co

333

c (2x + 5) (2x − 5)

Create a strategy We check ﬁrst whether (2x + 5) (2x − 5) is a special binomial product and identify its form.

Apply the idea Since (2x + 5) (2x − 5) is a product of a sum and difference, we use the formula and simplify the expression: (a + b) (a − b) = a2 − b2 2

(2x + 5) (2x − 5) = (2x) − 5 2

= 4x – 25

Identity for the product of a sum and difference Substitute a = 2x and b = 5 Evaluate the exponents

d Multiply and simplify: 3(2x + 5y)2

Create a strategy We check ﬁrst whether 3(2x + 5y)2 involves a special binomial product and identify its form.

Apply the idea Since (2x + 5y)2 is a square of a binomial in the form (a + b)2, we use the formula and simplify the expression: Identity for the square of a binomial (a + b)2 = a2 + 2ab + b2 3(2x + 5y)2 = 3 [(2x)2 + 2(2x) (5y) + (5y)2] 2

= 3 [4x + 20xy + 25y ] 2

= 12x + 60xy + 75y

Substitute a = 2x and b = 5y and multiply by 3 Evaluate the exponents and multiplication Distributive property

Idea summary Recognizing the patterns in special binomial factors may be helpful in multiplication problems and upcoming lessons. Remember the patterns: • • •

Square of a Sum: (a + b)2 = a2 + 2ab + b2 Square of a Difference: (a − b)2 = a2 − 2ab + b2 Product of a sum and difference: (a + b) (a − b) = a2 − b2

Practice What do you remember? 1

Complete:

The distributive property states a(b + c) = a ⋅ ⬚ + a ⋅ ⬚

334

Evaluate 5(12 − 6)

Complete 5(12 − 6) = 5 ⋅ ⬚ − 5 ⋅ ⬚

Mathspace Virginia SOL Algebra 1 mathspace.co

Evaluate 5 ⋅ 12 − 5 ⋅ 6

Distribute: a

−m (m + 1)

e 4

y ( y − 9)

7wy ( y + w)

y ( y − 4) + 10

The area of the whole rectangle is 16 ⋅ 20 = 320 units2. a

Find the area of the shaded rectangle.

Find the area of the unshaded rectangle.

Find the sum of the two areas.

Does 16(18 + 2) = 16 ⋅ 18 + 16 ⋅ 2?

The area of this figure is represented by the expression ( y + 3) ( y + 6). a

Find the area of: Rectangle A

Rectangle B

iii Rectangle C

Rectangle D

b 6

Write an equivalent, simplified expression for the area of the figure.

Match each product with an area model. A

9x2

−1

−3x

−1

x 3

9x2

−3

−3x

−9

(3x + 1) (3x − 1)

(−3x + 1) (3x + 1)

iii

−3x

−9x2

−3x

(x + 3) (x − 3)

(x + 3) (x + 3)

6.02 Multiply polynomials mathspace.co

335

Let’s practice 7

Simplify each product: a c

(7y − 6) (6y + 6) 2

(v + 5) (5v − 3v − 5)

e 8

4x(5x(x − 3) + 2x)

(x2 + 3)2

3x(7x − 5y)2

6(8x − 9y) (8x + 9y)

(11 − a) (11 + a) − 10

(3x − 8) (3x + 8)

d h

Complete the expansion of the following perfect squares: a

9( y + 8) ( y + 2)

Multiply and simplify:

i 9

(x − 3)2 = x2 − ⬚ x + ⬚

Simplify:

(x + ⬚)2 = x2 + ⬚ x + 36

4(x + 8) − 2

7 + 5(3x + 4)

5 + 3(x + 4)

8(3x + 4) − 6x

34 + 9(4x − 5)

5x − 8(2x − 3)

9x − 5(2x + 3)

−8(5x − 7) − 9

4y + 5 + 6( y − 9)

4(5(x − 3) + 2)

Use the diagram and areas of rectangles to demonstrate that 4(x + 6) = 4x + 24

This rectangle has width x + 2 and length x + 6. Use the diagram and areas of rectangles to find a simplified expression for (x + 2) (x + 6).

336

Mathspace Virginia SOL Algebra 1 mathspace.co

Area I

Area II

Area III

Area IV

(−7y − 8) (−7y + 8)

Multiply the following binomials using the algebra tiles. i

Find the missing values to complete the visual representation of multiplying the binomials.

Find the simplified expression for the binomials.

The diagram shows the visual representation to simplify (x + 2) (x − 3). (x − 3) x

−1

x (x + 2) 1

ii b

(x + 2) (x − 3) = ⬚

he diagram shows the visual representation to simplify T (x − 4) (x − 3).

(x − 3) −1

−1

(x − 4)

−1

−1 −1

ii c

(x − 4) (x − 3) = ⬚

The diagram shows the visual representation to simplify (x + 2) (x + 5).

(x + 5) x

(x + 2) 1

ii d

(x + 2) (x + 5) = ⬚

he diagram shows the visual representation to T simplify (x + 2) (2x − 1).

(2x − 1) x

−1

x (x + 2)

(x + 2) (2x − 1) = ⬚

−1

6.02 Multiply polynomials mathspace.co

337

Anette decides to use the area model to complete a polynomial multiplication question, (10x + 2) (10x + 3). To warm-up, she decides to try a simple integer example: 12(13) =

(10x + 2) (10x + 3) =

100

100 + 20 + 30 + 6 = 156

10x

100x

20x

30x

100x + 20x + 30x + 6 = 150x + 6

State the error in Anette’s polynomial product. Correct her work and solution.

Show how you would set up the area model to ﬁnd the product (10x + 3) (x2 + 10x + 2). Find the product using any method.

If Mackenzie creates a polynomial with degree 4 and Edgardo creates a polynomial with degree 3, state what we know about the product of their polynomials.

Sonya wants to extend the width and length of their vegetable patch by x ft. Their vegetable patch currently has a width of 5 ft and a length of 7 ft. For each of the following:

Write a simpliﬁed expression in terms of x.

State the unit of the answer to the expression from (i). Explain why you chose this unit.

The new length of the vegetable patch.

The area of the new vegetable patch.

The perimeter of the new vegetable patch.

7 ft

The diagram shows a square with side length x.

What is the area of a square with side length x?

Write an expression for Area I.

Write an expression for Area II.

Show that x (x − 4) + 4x = x2

5 ft

Find the missing values that make each equation true. a b c

338

Area I

A square with side lengths measuring x − 1 centimeters has each side enlarged by a factor of 2. Write a simplified expression for the area of the new square.

Let’s extend our thinking 19

(x + ⬚) (x − 7) = x2 − 3x + ⬚

(2x − 1) (⬚ x − 5) = 2x2 + ⬚ x + 5

(⬚ x + 3) (⬚ x + 6) = 12x2 + 30x + ⬚

Mathspace Virginia SOL Algebra 1 mathspace.co

Area II

Show the following results about special products are valid: a

(a + b)2 = a2 + 2ab + b2

(a + b) (a − b) = a2 − b2

(a − b)2 = a2 − 2ab + b2

Show how you can use one of the special products (a ± b)2 to ﬁnd each perfect square without a calculator. a

212

192

332

472

Write a simplified expression for the product of three consecutive integers, where the middle integer is m.

Consider the rectangle shown:

8st

Write a simplified expression for the area of the rectangle.

2(s + t)

A rectangular garden has a length that is one foot less than twice the width. a

Write and simplify an expression for the area of the rectangle.

A 2 foot border is to be placed all around the garden. Write and simplify an expression for the area of the border.

The landscaper designing the garden has 100 square feet of pavers to use for the border. Determine the largest garden the landscaper can border before running out of pavers. Assume the landscaper will only use positive-integer dimensions.

A flat rate large box from USPS has dimensions of 1 ft × 1 ft × 5.5 in. If Jerome decides to put a layer of insulation in his box x inches thick, write a simplified polynomial expression that models a

The dimensions of the open volume left in the box in inches.

The volume he has left in his box to fill.

Consider the following problem: “Find two binomials whose product results in a polynomial with 5 terms”. a

Yao was given the problem, but they think it’s impossible. State whether you agree or disagree. Explain.

Create two different products of two polynomials that results in an answer with exactly 5 terms.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning with examples. a

Two polynomials multiplied together will result in a polynomial

A term with a negative exponent is a polynomial

A polynomial multiplied by a non-polynomial will result in a polynomial

Explain how the area of a square given a binomial side length will always be a special product.

Polynomial multiplication can be used to ﬁnd the total area of a rectangle that has a border, where the length and width each represent a factor in the product.

Write an algebraic model that can be used to ﬁnd the total area of a square with an unknown side length and a border with an unknown border width.

Use your model to ﬁnd the total area of a square with a side length of 2m + 1 and a border width of 3.

Determine whether the conjectures are true or false. Create algebraic expressions to justify your response. a

The product of any two consecutive even numbers is one more than a perfect square.

The product of any two consecutive odd numbers is one less than a perfect square.

6.02 Multiply polynomials mathspace.co

339

6.03 Divide polynomials by a monomial After this lesson, you will be able to… • divide a polynomial by a monomial.

Divide by a monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. To do this, use the quotient rule and divide coefficients and subtract exponents with the same base. Dividing a polynomial by a monomial

Polynomial division can be modeled with algebra tiles. −2x + 3

−x2

−x

−x2

2x x

−x2

−4x2 + 6x

Create an area model where: 1. The tiles on the inside add up to the dividend (numerator). 2. The tiles on one side add up to the divisor (denominator). 3. The sum of the tiles along the other side must be the quotient (the result of the division). Note: Final answers are usually written without any negative exponents.

Example 1 Simplify the following:

Create a strategy Apply the rule

Apply the idea .

Divide each term by x Simplify Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 3x4 + 4x.

340

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Simplify the following:

Create a strategy Apply the rule

Apply the idea Divide each term by 3y

Simplify

Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 2y2 − 5y + 8.

Reﬂect and check We can check the answer by multiplying it with the monomial in the denominator. The product should be the numerator in the original expression. 3y (2y2 − 5y + 8) = 6y3 − 15y2 + 24y

Check

Example 3 The triangle shown has an area of 13n3 + 11n2 + 29n. Find a simpliﬁed polynomial expression for its height.

Create a strategy Substitute the expressions into the area of triangle formula

Apply the idea Write the area of triangle formula

Substitute A = 13n3 + 11n2 + 29n and b = n

Multiply both sides by 2

Evaluate the multiplication

Divide both sides by n

Evaluate the division

Symmetric property of equality

6.03 Divide polynomials by a monomial mathspace.co

341

Idea summary When dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial then simplify each individual fraction using the rules of exponents.

Practice What do you remember? 1

Determine whether each statement regarding the division of polynomials by monomials is true or false. Justify your conclusion. a

When dividing a polynomial by a monomial, it is possible to reduce the number of terms.

The degree of the polynomial will always decrease after dividing by a monomial

The result of dividing a polynomial by a monomial will have a constant term if the degree of the monomial matches the degree of any term in the polynomial.

The result of dividing a polynomial by a monomial will be another polynomial.

Explain the difference between the two expressions: and

342

Simplify the following: a

Use algebra tiles to model the equation

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s Practice 6

Find the missing length which represents the quotient in each area model. a

−x

−x2

2x −x

−x

−4x −6x2 + 4x

4x2 − 8x

Simplify the following: a

(30j2) ÷ (35j )

(9w3v2 + 45w2v2 + 18wv) ÷ (9wv)

(b3 − 4b2 + 2b) ÷ (4b)

Fill in the blanks: a

Consider the following statement y5 ÷ y = y4 for all nonzero real numbers y. a

Determine whether the statement is true or false.

Is the statement still true when y is zero?

Consider the area of the following rectangles: Find a polynomial expression for its length. a

Area = (4x4 − 8x) square units

Area = (6x3 + 4x2 + 10x + 14) square units

? 2 4x

6.03 Divide polynomials by a monomial mathspace.co

343

The triangle shown has an area of 12n3 + 18n2 + 5n. Find a polynomial expression for its height.

Let’s extend our thinking 12

Fill in the blanks to make a true algebraic statement. a b c

Consider the following statement: x9 ÷ x3 = x3 for all nonzero real numbers x a

What is x9 ÷ x3 actually equal to?

Identify a nonzero real number for which the original statement is true.

Consider the problem

Identify and correct the error in each of the following student’s work. Lawrence:

Marika:

344

Create an example and describe the result when a polynomial is divided by a monomial that: a

Has a coeﬃcient larger than the coeﬃcients of the terms in the polynomial.

Has a variable with a larger degree than the terms in the polynomial.

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 Find the greatest common factor of the given terms. a 60 and 24.

Create a strategy List the prime factorization of 60 and 24, then determine the common factors that comprise the GCF.

Apply the idea The prime factorization of 60 is 2 ⋅ 2 ⋅ 3 ⋅ 5 The prime factorization of 24 is 2 ⋅ 2 ⋅ 2 ⋅ 3. The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12. Therefore, the GCF of 60 and 24 is 12.

Reﬂect and check We can also create factor trees for 60 and 24, then identify the common factors. Factor tree of 60

Factor tree of 24

24 30

2 15

2 3

12 6

2 5

60 = 2 ⋅ 2 ⋅ 3 ⋅ 5

24 = 2 ⋅ 2 ⋅ 2 ⋅ 3

The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12. Therefore, the GCF of 60 and 24 is 12.

b 60x3y2 and 24xy4.

Create a strategy List the whole number factors of the coeﬃcients of 60x3y2 and 24xy4 and ﬁnd the expression with the lowest power of each of the variables.

Apply the idea

Reﬂect and check

We know that the largest whole number that 60 and 24 are divisible by is 12. The expression with the lowest power of each of the variables is xy2.

We can also expand both expressions to find the common factors:

Putting this together, the greatest common factor is 12xy2.

60x3y2 =

2 ⋅ 2 ⋅ 3 ⋅5⋅ x ⋅ x⋅ x⋅ y ⋅ y

24xy4 =

2 ⋅ 2 ⋅2⋅ 3 ⋅ x ⋅ y ⋅ y ⋅y⋅ y

So, the GCF is 2 ⋅ 2 ⋅ 3 ⋅ x ⋅ y ⋅ y = 12xy2.

346

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Factor the expression 8x2 + 4x.

Create a strategy Find the GCF and divide it out of each term.

Apply the idea

Reﬂect and check

The GCF of 8x and 4x is 4x. Dividing out the GCF, we get: 8x2 ÷ 4x = 2x 4x ÷ 4x = 1 So we have: 8x2 + 4x = 4x(2x) + 4x(1)

Although the term 4x is in the original expression when it is factored out the second term does not become zero. Otherwise, when we check the answer by distributing the multiplication, 4x will be lost altogether. We can check our factorization using the distributive property: 4x(2x + 1) = 4x(2x) + 4x(1) = 8x2 + 4x

= 4x(2x + 1)

Example 3 Factor the expression 3x(x − 4) + 7(x − 4).

Create a strategy This time, the expressions are already factored. We can use this to help identify the GCF.

Apply the idea In particular, notice that both terms 3x(x − 4) and 7(x − 4) have a factor of (x − 4). The remaining parts of each expression, 3x and 7, have no factors in common. So the GCF is (x − 4), which we can use to factor the expression:

Idea summary Follow these steps for factoring out a GCF: 1. Identify the GCF 2. Rewrite each term as a product of the GCF and the remaining factors 3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses

6.04 Factor GCF mathspace.co

347

Practice What do you remember? 1

For each of these numbers: i

List the factors.

6 and 12

9 and 24

State the greatest common factor.

14 and 32

28 and 42

For these prime factorizations: 180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 600 = 2⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ Find the greatest common factor of 180 and 600.

For each of these algebraic expressions: i

List the factors.

3x2 and 2x

14 and 7abc

x2y3 and x2y2

8a and 9a 2

3x and 6x2

4b and b

4y2 and 6y2

5p2, 3p2 and p

45n, 55n2 and 20n2

4m2, −7m, 8m and −14m2

−42k, −21k2, −7k2 and −28k3

Simplify the following expressions: a

i 6

12xy4 and 24xy

State the greatest common factor.

Identify the greatest common factor between the following sets of terms: a

(b3 − 4b2 + 2b) ÷ (4b)

d (30j2) ÷ (35j )

Fill in the blanks to make a true algebraic statement.

c 7

Determine which of the following represent a factored form of the expression −12x + 20x2: a

−4 ⋅ 3x + 5

−2(6x − 10)

4x(5 − 3x)

x (−12 + 20x)

Let’s practice 8

Complete each factorization: a d

348

y2 + 5y = y(⬚ + ⬚)

−m + 19m = ⬚ (m − 19)

Factor the following expressions:

2t2 + 2t = 2t(⬚ + ⬚)

−y − 2y = ⬚ ( y + 2)

3y2 + 6y = ⬚ ( y + 2) 8v − v2 = v(⬚ − ⬚)

6v + 30

−2s − 10

−12s + 10

y2 + 4y

2u2 − 8u

4t + 2t2

42x − x2

9z2 – 18z

w(w − 2) − (w − 2)

9(3x + 4) + 4(3x + 4)

8t(t − 3) + 9(t − 3)

r + r + 6r

Mathspace Virginia SOL Algebra 1 mathspace.co

Completely factor each of the following polynomials. a

−6y4 + 14y3 − 10y2

14h4 + 28h6 + 56h3

18x2 − 24x + 36 − 72x4

−3a3 − 9a2 − 15a4

18m7 − 15m3 + 14m + 35m6

−d12 + d8 − d15

0.1h + 0.2h3 − 0.4h5

g 11

Xander was asked to factor the expression 35x2y + 10xy2 − 5xy. Identify his error. Xander: The greatest common factor is 5xy so the factored form of the expression is 5xy (7x + 2y).

Alex and Beth are both asked to factor −5x + 10y. Alex wrote down −5(x − 2y) and Beth wrote down 5(2y − x). Who is correct? Explain.

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 factoring to determine the dimensions of the entire ﬁeld.

Let’s extend our thinking 14

Identify the common factor of any two terms.

Identify the greatest common factor between the following sets of terms: a

45n, 55n2 and 20n2

4m2, −7m, 8m and −14m2

−42k, −21k2, −7k2 and −28k3

45n, 55n2y and 40mn3

3a2bx, 4na4m and 12mba3

e 16

3a mn, 2ya x and 5xma

Fully factor each expression: a

44uv − 8u2v

−8w2 + 3w2y

5k2t + 40k2t2

49p2q − 28pq2

−30w2 − 25w2y

−16a − 18a b

−10u2v + 9uv2

4x + 12 + 16yx

3x + 9 + 12yx

2x + 10x2y + 8yx

4x2y + 8xy2 + 12xz2

5ab2c + 25bc3 + 100abc

5a2b2 + 2ab − 3a2b2 − 4ab

m 30b2c + 10abc + 20c2 o

2 2

5x y + 15yz + 25y + 60y

Create an expression with at least three factorizations. Then, write out each factorization including the fully factored form.

Explain how dividing monomials relates to factoring the greatest common factor.

Explain why the greatest common factor of the variables in any expression has the least possible exponent of any of the terms.

The rectangle shown has an area of 4x2 − 16x square units.

What expression describes the length of the rectangle? 4x

6.04 Factor GCF mathspace.co

349

350

A property management company has a rectangular plot of land available for parking. The area of this plot is represented by the expression 20n + 5n2. a

Determine the dimensions of the block of land.

Given the the length is longer than the width for all n > 1, which expression is the length and which is the width?

The length is further divided into 5 equal sections, and one of these sections is fenced off for storage. What is the length of fencing needed?

You are to design a photo collage made of two large square photos with a side length of x and four smaller rectangular photos that have a height of x and a width of 4 inches. a

Find an algebraic expression for the area of the rectangle formed if the photos are all placed in a single row. Draw an example of what this arrangement would look like.

Fully factor your answer from part (a) and then draw a photo arrangement that would match these dimensions.

Mathspace Virginia SOL Algebra 1 mathspace.co

Follow the steps shown with the example to factor by grouping: 1. Factor out a GCF, if possible

2. Rearrange the terms so that the first pair and the second pair each have a common factor, if possible 3. Factor out a GCF from the binomial expressions 4. Factor out the common binomial expression 5. Verify that the final expression cannot be factored further, otherwise continue factoring

Example 1 Factor the expression 10x2 + 4x + 15x + 6.

Create a strategy We arrange the terms ﬁrst, grouping those with common factors. We factor out the GCF on each pair and the common binomial factor afterward.

Apply the idea 10x2 + 4x + 15y + 6 = (10x2 + 4x) + (15x + 6)

Group based on common factors

= 2x(5x + 2) + 3(5x + 2)

Factor out each GCF (2x and 3)

= (5x + 2) (2x + 3)

Factor out the common binomial factor

Since (5x + 2) (2x + 3) cannot be factored further, it is the ﬁnal answer.

Reﬂect and check We can perform a midway check that we are factoring by grouping appropriately when we factor out the GCF from each set of binomials in the step 2x(5x + 2) + 3(5x + 2). If we factor out the GCF at this step and the binomial factors are not equivalent, then we will want to check that we factored out the GCF correctly. If the factoring is correct, we may need to try a different approach. There may be a better way to arrange the terms from the polynomial. Not every polynomial expression will be factorable, but we can try a few different approaches, checking our work along the way.

352

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 2 Show at least two different ways we can arrange and group the polynomial 4a2 − 10b + 5ab − 8a and factor it.

Create a strategy Determine if the polynomial has common factors between the ﬁrst and second pair of terms, then factor it and rearrange the polynomial so that the ﬁrst or second set of terms has a common factor, then factor it again.

Apply the idea Write the expression as 4a2 + 5ab − 8a − 10b and factor it. 4a2 − 10b + 5ab − 8a = 4a2 + 5ab − 8a − 10b 2

4a + 5ab − 8a − 10b = (4a + 5ab) + (−8a − 10b)

Rearrange the terms Group based on common factors

= a (4a + 5b) − 2(4a + 5b)

Factor out the GCF (a and − 2)

= (4a + 5b) (a − 2)

Factor out the common binomial factor

Since (4a + 5b) (a − 2) cannot be factored further, it is the ﬁnal answer. Write the expression as 4a2 − 8a − 10b + 5ab and factor it. 4a2 − 10b + 5ab − 8a = 4a2 − 8a − 10b + 5ab 2

4a − 8a − 10b + 5ab = (4a − 8a) + (−10b + 5ab)

Rearrange the terms Group based on common factors

= 4a(a − 2) + 5b(−2 + a)

Factor out the GCF (4a and 5b)

= (a − 2) (4a + 5b)

Factor out the common binomial factor

Since (a − 2) (4a + 5b) cannot be factored further, it is the ﬁnal answer.

Reﬂect and check Alternatively, we can group 4a2 and − 8a and 5ab and − 10b together and get the same answer. 4a2 + 5ab − 8a − 10b = (4a2 − 8a) + (5ab − 10b)

Group based on common factors

= 4a(a − 2) + 5b(a − 2)

Factor out the GCF (4a and 5b)

= (a − 2) (4a + 5b)

Factor out the common binomial factor

We can check the answer by multiplying the factored form (4a + 5b) (a − 2). (4a + 5b) (a − 2) = a (4a + 5b) − 2(4a + 5b)

Distributive property

= 4a + 5ab − 8a − 10b

Idea summary Follow these steps when factoring by grouping: 1. Factor out the GCF from the expression, if possible 2. Arrange the terms so that the ﬁrst two have a common factor and the last two have a common factor, if possible 3. Factor out the GCF for each pair of terms 4. Factor out the common binomial expression 5. Conﬁrm that the binomial factors cannot be factored further, otherwise continue factoring

6.05 Factor by grouping mathspace.co

353

Practice What do you remember? 1

For the expression 3(x + 7) + x(x + 7), what is the common factor?

Determine which expression need to be rearranged in order to factor by grouping. Do not factor. a

a2 − 4a − 3a + 12

6x2 − 20 + 8x − 15x

1 + 100x2 − 10x − 10x

Write an algebraic expression that can be factored to (x + 3) (2x − 1).

Factor the following expressions: a

a(a + 6) + 2 (a + 6)

2b(b − 3) − 5 (b − 3)

5x(x + 2) − (x + 2)

2x(x − 7) + 3 (7 − x)

5d(d + 3) + 20 (d + 3)

8y( y − 4) + 10(4 − y)

y( y + 5) + 7( y + 5)

a(a − 4) − 3(a − 4)

p( p − 3) + 6( p − 3)

5(q + 4) − q(q + 4)

5(r − 3) − r(r − 3)

6r(2r − s) − rs(2r − s)

m 7t(t + u) + 2u(t + u)

x( y − z) − w( y − z)

5y(4w + 3x) − z(4w + 3x)

8y( y − 4) + 3(4 − y)

z2 − 7z + 2z − 14

7 + x − 28x − 4x2

2k2 + 12k + k + 6

Let’s practice 5

Factor the following expressions: a

x2 − 5x + 10x − 50

2b + 6b + b + 3

3x − 10x + 3x − 20

−4y + 30y − 5y − 36

x2 − 3x + 8x − 24

x2 + 2x + 5x + 10

20a2 − 12a + 5a − 3

3y2 + 6y + 4y + 8

9t2 + 6t + 12t + 8

m 8x + xz − 16y − 2yz

24 + 3y + 8x + xy

7xy + wx + 7yz + wz

2mp + 6 + 3p + 4m

5mp + 6 + 2p + 15m

2x + 18yz + 12xy + 3z

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

Factor the following expressions: a

q 6

x2 + 5x + 8x + 40

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

Identify and explain the error: 6x − 21x3 + 14x − 4 = 3x(2 − 7x) + 2(7z − 2) = (3x + 2) (2 − 7x)

Complete the factoring process below and explain each step:: 7x + 7 + x + x2 = ⬚ (x + 1) + x (1 + ⬚) = ⬚ (x + 1) + x (⬚ + 1) = (x + 1) (⬚ + x)

The expression for the area of the rectangle shown is 3x2 + 18x + x + 6. Write the expression of the area in factored form.

354

Mathspace Virginia SOL Algebra 1 mathspace.co

3x2

18x

One expression for the area of the rectangle shown 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 factored form.

Find an expression for the total area of the following rectangles in factored form: a

10m

Let’s extend our thinking 12

Factor the following expressions: a

3y3 + 6y2 − 15y − 30

3x3 − x2 + 27x − 9

17x3 + 5x2 + 17x + 5

a3 + 5a2 + a + 5

8z(5x2 + 4y) − (5x2 + 4y)

Factor the following expressions: a

8x(2y + 3w) − z(2y + 3w)

2x + xz − 40y − 20yz

50 + 5y + 10x + xy

12xy + wx + 12yz + wz

6y − yw + w2 − 6w

8xy + 4x2 − 6xy2 − 3x2y

16ab + 6b2 − 32ac − 12bc

The polynomial expression x2 + 9x + 18 is factored by grouping and one of its factors is (x + 6). Rewrite the polynomial in the form x2 + ⬚ x + ⬚ x + 18 and factor the expression. For each polynomial: i

Find three pairs of values that make the polynomial factorable.

Determine what the pairs from part (i) have in common.

x3 − 3x2 + ⬚ x + ⬚

x3 + ⬚ x2 + x + ⬚

By rewriting 4x2 + 17x + 4 as an expression having four terms and factoring in pairs, factor the expression completely.

6.05 Factor by grouping mathspace.co

355

6.06 Factor trinomials After this lesson, you will be able to… • factor trinomials completely.

Factor trinomials Trinomials can be rewritten as polynomials with four terms and factored by grouping.

Exploration Consider the polynomial expressions factored by grouping below:

What patterns do you notice between the original expression and the terms used to rewrite the linear term?

Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the polynomial can still be factored by grouping.

When using the grouping method to factor a trinomial, the coefficients of the terms used to rewrite the linear term have a sum equivalent to the linear coefficient from the original polynomial and a product equivalent to the product of the trinomial’s leading coefficient and constant. Steps in factoring a quadratic trinomial of the form ax2 + bx + c: 1. Factor out any GCF. (If a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b. 3. Rewrite the trinomial with four terms in the form ax2 + rx + sx + c. 4. Factor by grouping. 5. Check whether the answer will not factor further and verify the factored form by multiplication. Remember to include any common factors divided out at the start, so each step results in an equivalent expression. Algebra tiles can also be useful in factoring. Consider the expression 3x2 + 7x − 6 as the area of a rectangle. If we can find the lengths of this rectangle, then we will have two expressions that multiply to 3x2 + 7x − 6 because the area of a rectangle is A = l ⋅ w. We don’t yet know the side lengths of the rectangle, but we will take 3 of the x2 tiles, 7 of the +x tiles, and 6 of the −1 tiles and arrange them as closely into a rectangle as we can.

−1 −1

We will start by lining up all of the x2 tiles, then put the x tiles underneath to match the equal lengths. Finally, put the −1 tiles next to the x tiles to match the equal lengths. Notice we have some empty spaces that need to be filled in. 356

Mathspace Virginia SOL Algebra 1 mathspace.co

−1 −1

−x −x

−1 −1

Notice that x tiles will fit perfectly into the empty spaces. However, we don’t want to change the value of the expression so we need to make sure to add zero pairs. A zero pair is two values that add to 0. x and −x is a zero pair. Since there are 4 empty spaces for x tiles, we can fill 2 spaces with (positive) x tiles and 2 spaces with −x tiles. Technically this represents the expression 3x2 + 9x − 2x − 6 which is equivalent to 3x2 + 7x − 6 by combining like terms. x

−1 −1

−x −x

−1 −1

Now we can use the lengths of the sides of the rectangle to determine the expressions that can be multiplied together to create the original expression 3x2 + 7x − 6. The x2 tile has side lengths of x and x. The x tiles have a shorter side length of 1 and a longer side length of x. The −x tiles have a shorter side length of −1 and a longer side length of x. The shorter side length of the rectangle is x + 3 units and the longer side length is 3x − 2 units. This shows us: 3x2 + 7x − 6 = 3x2 + 9x − 2x + 6 = (x + 3) (3x − 2).

Example 1 Factor x2 + 10x − 24.

Create a strategy Since there are no common factors for all three terms, we proceed with ﬁnding the value of two integers that multiply to ac = (1) (−24) = −24 and add up to b = 10. After ﬁnding these integers, we use them to rewrite the middle term 10x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea The factors of −24 are 1 and −24, −1 and 24, 2 and −12, −2 and 12, 3 and −8, −3 and 8, 4 and −6, −4 and 6. Among these factors, −2 and 12 are the pair that add up to 10.

6.06 Factor trinomials mathspace.co

357

We can use this to rewrite the trinomial and factor by grouping as follows: x2 + 10x − 24 = x2 + 12x − 2x – 24

Rewrite polynomial with four terms

= x(x + 12) − 2(x + 12)

Factor each pair

= (x + 12) (x − 2)

Divide out common factor of (x + 12)

There are no more common factors to be divided out, so the fully factored form of the polynomial is (x + 12) (x − 2).

Reﬂect and check We can perform a midway check that we are factoring by grouping appropriately when we factor out a GCF from each set of binomials in the step x(x + 12) − 2(x + 12). If we factor out a GCF at this step and the binomial factors are not equivalent, we may have split the linear term from x2 + 10x − 24 incorrectly or factored out a GCF incorrectly. This is an important place to stop and check that we are factoring appropriately. Also note that we could have also rewritten the polynomial as x2 − 2x + 12x − 24. This would have resulted in a different middle step in factoring by grouping but the same end result.

Example 2 Factor 3x2 − 27.

Create a strategy We can factor a GCF of 3 out of the polynomial and write the polynomial as 3(x2 − 9). Since the linear term is missing from the polynomial, we can write the polynomial as 3(x2 + 0x − 9). There are no common factors. We will ﬁnd the value of two integers that multiply to ac = (1) (−9) = −9 and add up to b = 0. After ﬁnding these integers, we use them to rewrite the middle term 0x as a sum of two terms. Then factor the trinomial by grouping.

Apply the idea The factors of −9 are 1 and −9, −1 and 9, 3 and −3. Among these factors, 3 and −3 are the pair that add up to 0. We can use this to rewrite the trinomial and factor by grouping as follows: 3(x2 + 0x − 9) = 3(x2 + 3x − 3x − 9) = 3[x(x + 3) − 3(x + 3)]

Rewrite polynomial with four terms Factor each pair

= 3(x + 3) (x − 3) Divide out common factor of (x + 3) There are no more common factors to be divided out, so the fully factored form of the polynomial is 3(x + 3) (x − 3).

Reﬂect and check Recall that the special product of a difference of squares (a + b) (a − b) = a2 − b2. Notice that the factored form of the binomial x2 − 9 = (x + 3) (x − 3).

358

Mathspace Virginia SOL Algebra 1 mathspace.co

This can also be verified using algebra tiles: Algebra tiles key

−x

−1

+1 +1 +1 +1 +1 +1 +1 +1 +1

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

Notice the x terms form a total of 0. So, we know: 3x2 − 27 = (3x + 9) (x − 3) = 3(x + 3) (x − 3)

Example 3 Factor 5x2 − 18x + 9.

Create a strategy Since there are no common factors for all three terms, we proceed with ﬁnding the value of two integers that multiply to ac = 5 ⋅ 9 = 45 and add up to b = −18. After ﬁnding these integers, we use them to rewrite the middle term −18x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea The factor pairs of 45 are 1 and 45, −1 and −45, 3 and 15, −3 and −15, 5 and 9, −5 and −9. Note that since the middle term of the trinomial is negative, we need to consider negative and positive factors. Of these factors, −15 and −3 are the pair that adds up to −18. We can use this to rewrite the trinomial and factor by grouping as follows: 5x2 − 18x + 9 = 5x2 − 15x − 3x + 9

Rewrite polynomial with four terms

= 5x(x − 3) − 3(x − 3)

Factor each pair to leave behind a common binomial

= (x − 3) (5x − 3)

Divide out the common factor of (x − 3)

There are no more factors to be taken out, so the fully factored form of the polynomial is (x − 3) (5x − 3).

6.06 Factor trinomials mathspace.co

359

Reﬂect and check We can check the answer by multiplying the factored form (5x − 3) (x − 3). −

5x2

−3x

−3

−15x

The polynomial 5x2 − 3x − 15x + 9 simpliﬁes to 5x2 − 18x + 9.

Idea summary Steps in factoring a quadratic trinomial: 1. Factor out any GCF. (If a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b. 3. Rewrite the trinomial with four terms, in the form ax2 + rx + sx + c. 4. Factor by grouping. 5. Check whether the answer will not factor further and verify the factored form by multiplication. Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

Practice What do you remember? 1

Complete this statement: To factor x2 + 9x + 18, we need to find two numbers whose product is ⬚ and whose sum is ⬚. Given that a < b, find the values of a and b in each pair of equations: a

Use the diagram and your knowledge of areas of rectangles to write the factored form of x2 + 6x + 4x + 24 x

+ 4

360

Mathspace Virginia SOL Algebra 1 mathspace.co

Factor each quadratic expression completely: a

SOL

x2 + 2x + 3x + 6

3x2 − x + 12x − 4

For each of the following quadratic equations: i

List all factors of the constant term.

Factor the expression completely.

x2 + 11x + 18

x2 + 17x + 72

m−5

x2 + 16x + 64

Which is a factor of 3m2 − 4m − 15? A

3m − 5

3m + 5

m+5

Let’s practice 7

Use the algebra tile model to factor the expression x2 − 2x − 8.

x x

−x −x −x −x

−1 −1 −1 −1 −1 −1 −1 −1

Factor the following quadratic expressions completely: a

x2 − 8x + 15

x2 + 11x + 24

x + x − 90

x2 + 22x + 120

x2 − 34x − 72

x2 + x − 56

3x2 − 27x − 30

35m2 + 140m + 105

2x + 28x + 96

x2 + 10x + 25

x2 + 14x + 49

x2 − 16x + 64

m 81 + 18x + x2

36 − 12x + x2

−3x2 + 12x − 12

o 9

4x + 40x + 100

Brandon claims that the polynomial 3x2 + 15x − 42 will follow the factoring form of trinomials with a leading coeﬃcient a ≠ 1. Explain Brandon’s error, then fully factor the expression.

Draw a set of algebra tiles that shows that (2x + 1) (2x − 1) = 4x2 − 1.

Rewrite the following quadratic expression in factored form (as a product of two linear factors):

4x2 − 32x + 15

5x2 − 47x + 18

2x2 − 19x + 45

10x2 − 77x − 24

24x2 + 22x − 35

6x2 − 19x + 15

−6x2 + 13x − 5

−35x2 + 97x − 66

81x2 + 36x + 4

49x2 − 28x + 4

9x2 + 12x + 3

27 − 123x − 60x2

Fully factor each expression: a

60x2 − 70x − 100

2x4 − 25x3 + 42x2

A square has an area of x2 + 12x + 36. Determine the length of the sides of this square.

A cube has a surface area of 6x2 + 36x + 54. Find an expression for the length of one side of the cube.

6.06 Factor trinomials mathspace.co

361

Fully factor each expression: a

27x2 − 48

x3 + 10x2 + 25x

x4 − 16

2x4 − 1250

Let’s extend our thinking 16

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

19 20

What do the polynomials have in common?

How do you determine if a polynomial where a ≠ 1 is not factorable?

Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a factorable trinomial and provide the factored form: 4x2 + ⬚⬚ x + ⬚ Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a perfect square trinomial: ⬚ x2 + ⬚⬚ x + ⬚⬚

The expression 16x2 − 24x + ⬚ is a perfect square trinomial. Determine the missing value.

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

Why might the photographer use x units to represent the width of the border?

Use factoring to ﬁnd expressions for the dimensions of the photograph with its border if the total area is 4x2 + 24x + 35 square units.

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

Factor the quadratic expression: 6ab2 − 36ab − 162a

Rewrite the quadratic expression in factored form: 3x2 − 24xy + 48y2

Consider the factorable polynomial 12x2 + bx − 6, where b is a positive integer. a

Find the largest possible value for b and list the factors.

Find the smallest possible value for b and list the factors.

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

Gabriella:

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

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

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

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

= (8x + 12) (2x + 3) = 4(2x + 3) (2x + 3) 25

A shape is formed from a square with side lengths of 3x that has a smaller square, with side lengths of 4, cut out from the center. a

Write a simplified expression for the area of this shape.

A rectangle is created to have the same area found in part (a). Determine the dimensions for the rectangle.

4 3x

362

Mathspace Virginia SOL Algebra 1 mathspace.co

For example, consider the product of a binomial squared, (a + b)2: a

We can expand (a + b)2 to (a + b) (a + b) and represent them with an area model. Evaluating with this model and combining like terms, we get the product a2 + 2ab + b2.

So, we have: a

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

Follow these steps for determining if a trinomial is a perfect square trinomial and factoring: 1. Factor out the GCF 2. Determine a from the leading term and b from the constant term 3. Verify whether the linear term is equal to 2ab 4. If yes, use the structure of perfect square trinomials to write the factors Now consider the product of a sum and a difference, (a + b) (a − b): a

−b

−ab

−b2

Notice that the term ab and −ab are opposites and combine to make zero. We call this a zero pair. So, (a + b) (a − b) = a2 − b2. Follow these steps for determining if a binomial is a difference of two squares and factoring: 1. Factor out the GCF 2. Determine a from the leading term and b from the constant term 3. Verify whether a and b are perfect squares by identifying their square roots 4. If yes, use the structure of a difference of squares to write the factors

364

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 1 Factor 3p2 + 12p + 12

Create a strategy We can factor a GCF of 3 out of the polynomial and write the polynomial as 3( p2 + 4p + 4). Determine if the polynomial expression is a perfect square trinomial. Since a = p and b = 2 and the linear term is 2ab = 2( p) (2) = 4p, we can verify that this is a perfect square trinomial and we can use special products to factor.

Apply the idea Since p2 + 4p + 4 is a perfect square trinomial, we use the identity in factoring: a2 + 2ab + b2 = (a + b)2 Identity for a perfect square trinomial = ( p + 2)2 Substitute a = p and b = 2 There are no more common factors to be divided out, so the fully factored form of the polynomial is 3( p + 2)2.

Reflect and check If we instead were to factor by grouping, we will find the value of two integers that multiply to ac = (1) (4) = 4 and add up to b = 4. After finding these integers, we use them to rewrite the middle term 4p as a sum of two terms and then factor the trinomial by grouping. The factors of 4 are 1 and 4, 2 and 2. Among these factors, 2 and 2 are the pair that adds up to 4. We can use this to rewrite the trinomial and factor by grouping as follows: 3( p2 + 4p + 4) = 3( p2 + 2p + 2p + 4)

Rewrite polynomial with four terms

= 3[p( p + 2) + 2( p + 2)]

Factor each pair

= 3( p + 2) ( p + 2)

Divide out the common factor of ( p + 2)

There are no more common factors to be divided out, so the fully factored form of the polynomial is 3( p + 2)2.

Example 2 Fully factor 9x2 − 24x + 16.

Create a strategy We check ﬁrst whether 9x2 − 24x + 16 is a special product and identify its type. Since a = 3x and b = 4 and the linear coeﬃcient is −2ab = 2(3x) (4) = −24x, we can verify that this is a perfect square trinomial and we can use special products to factor.

Apply the idea Since 9x2 − 24x + 16 is a perfect square trinomial, we can use the identity in factoring: Identity for a perfect square trinomial a2 − 2ab + b2 = (a − b)2 Substitute a = 3x and b = 4 = (3x − 4)2

6.07 Factor using appropriate methods mathspace.co

365

Reﬂect and check We can check the answer by multiplying the factored form (3x − 4)2. (3x − 4)2 = 9x2 − 24x + 16

Identity for square of a binomial

Example 3 Factor 1 − x2.

Create a strategy We check ﬁrst whether 1 − x2 is a special product and identify its type. Since it can be rewritten as (1)2 − (x)2 the polynomial is a difference of squares and we can use special products to factor.

Apply the idea Since 1 − x2 is a difference of two squares, we use the identity in factoring: a2 − b2 = (a + b) (a − b) Identity for a difference of two squares = (1 + x) (1 − x) Substitute a = 1 and b = x

Reﬂect and check We can check the answer by multiplying the factored form (1 + x) (1 − x). Identity for product of a sum and difference (1 + x) (1 − x) = 1 − x2 We can also check our answer using algebra tiles: 1

−x

So, (1 − x) (1 + x) is equivalent to 1 − x + x − x2 = 1 − x2.

366

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 4 Factor 4m2 + 40m + 36.

Create a strategy We can factor a GCF of 4 out of the polynomial and write the polynomial as 4(m2 + 10m + 9). Determine if the polynomial expression is a perfect square trinomial. Since a = m and b = 3 and the linear term should be 2ab = 2(m) (3) = 6m and 6m ≠ 10m, we can verify that this trinomial is not a perfect square trinomial and instead factor by grouping.

Apply the idea The factors of 9 are 1 and 9, 3 and 3. Among these factors, 1 and 9 are the factor pair that adds up to 10. We can use this to rewrite the trinomial and factor by grouping as follows: 4(m2 + 10m + 9) = 4(m2 + m + 9m + 9) Rewrite polynomial with four terms = 4[m(m + 1) + 9(m + 1)] Factor each pair = 4(m + 1) (m + 9) Divide out the common factor of (m + 1) There are no more common factors to be divided out, so the fully factored form of the polynomial is 4(m + 1) (m + 9).

Reﬂect and check If the polynomial could not be rewritten as four terms and factored by grouping, we would determine that the polynomial is not factorable.

Idea summary By recognizing the patterns of factoring using special products, we can factor more eﬃciently: • •

Perfect square trinomials: a2 + 2ab + b2 = (a + b)2 or a2 − 2ab + b2 = (a − b)2 Difference of two squares: a2 − b2 = (a + b) (a − b)

Practice What do you remember? 1

For each trinomial in the form ax2 + bx + c: i

Find two integer values that have a sum of b and a product of c.

Write the quadratic in factored form.

x2 + 2x − 24

x2 − 4x − 77

x2 − 17x + 66

x2 + 10x + 24

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

the product of A and B

the signs for A and B: opposite or the same

iii

the product of C and D

the signs for C and D: opposite or the same

15x2 + 14x – 16

−45x2 + 29x − 4

6.07 Factor using appropriate methods mathspace.co

367

A square rug originally had a side length of 2x inches. One of its dimensions is extended by 3 inches. We can model the area of the rug as a collection of rectangles, as shown in the diagram. The square on the left of the diagram has a side length of 2x inches and the short side of each rectangle is 1 inch.

Write a factored expression that represents the area of the rug.

Find the areas of each section from the rug.

Find the total area of the rug in terms of x. Give your answer in the standard form ax2 + bx + c.

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

b 11

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

Area II

Area III

Area IV

Express the area of the quilt in factored form.

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

c is negative.

c is positive and b is negative.

c and b are both positive.

Let’s extend our thinking 12

Determine whether each of the following polynomials can be factored: a

x2 + 12x + 15

−x2 + 17x − 70

x2 + x − 42

x2 + 19x − 90

x2 + 25

x2 + 9x + 5

6x2 − 13x − 5

4x2 − 20x − 25

Find two digits (1 to 9) to ﬁll in the blanks and create a trinomial in the form ax2 + bx + c that a c

Maximizes the value of b. b Minimizes the value of c. (x + ⬚) (x + ⬚) (x + ⬚) (x + ⬚) Minimizes the value of b. (x + ⬚) (x − ⬚)

A quadratic polynomial is of the form ax2 + bx + c, where a is non-zero, and has a special factored form. a

If b = 0, identify the form that the quadratic could be. State what must be true about the sign of c.

If b < 0, identify the form that the quadratic could be. State what must be true about the sign of c.

Factor the following expressions fully using appropriate techniques: a

4b2 − 81c2

x2y2 − 36x2

9a2 + 24ab + 16b2

−2x4 + 2x3 + 24x2

x2 − x2y + xy − x

80x4 + 92x3 + 24x2

15x2y + 50xy − 40y

xy2 + 4x

81 − n

4x + 16x − x − 4

6.07 Factor using appropriate methods mathspace.co

369

6.08 Divide polynomials After this lesson, you will be able to… • divide a polynomial by a binomial or factored divisor.

Divide polynomials Dividing polynomials involves a process known as algebraic manipulation. We can view our dividend as the numerator of a fraction and our divisor as the denominator. . The dividend is 3x2 + 7x − 6 and the divisor is 3x − 2. We can use algebra tiles

Consider the expression:

to model the division. We will create a rectangle to represent the dividend, and the side lengths of the rectangle will represent its factors. x

−1 −1

We already know one of the factors is the divisor so we can make one of the side lengths 3x − 2.

Next we fill in the rectangle with 3 of the x2 tiles, 7 of the positive x tiles, and 6 of the −1 tiles to represent the dividend 3x2 + 7x − 6, making sure to line up tiles with equal lengths. Notice that there are some empty spaces that we need to fill in with zero pairs.

−1 −1

−x −x

The empty spaces are the right size for x tiles. Since we need to add zero pairs so that we don’t change the value of the expression we will fill 2 of the spaces with positive x tiles and the other 2 spaces with −x tiles.

−1 −1

Now we can see that the length of the left side of the rectangle is

−1 −1

x + 3. This means that

−1 −1

6.08 Divide polynomials mathspace.co

371

We can approach this algebraically by following these steps: 1. Completely factor both the numerator and denominator. 2. Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors. 3. Divide out all common factors from the numerator and denominator. 4. Simplify the resulting expression.

Example 1 Factor and simplify:

Create a strategy We need to factor both the numerator and denominator by ﬁrst factoring the greatest common factor of the terms in each expression.

Apply the idea Factor the numerator and denominator

Divide out the common factors

Simplify common factors to 1

Reﬂect and check When we have multiple common factors, it is as simple as dividing out each factor separately.

Example 2 Factor and simplify:

Create a strategy We can use the formula for factoring a difference of two squares: A2 − B2 = ( A + B) ( A − B)

Apply the idea Factor the numerator and denominator

Divide out the common factors and simplify to 1

Simplify common factors to 1

372

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 Factor and simplify:

Create a strategy Start by factoring the numerator. Then, we can divide out common factors.

Apply the idea Factor numerator

Divide out common factors

Simplify common factors to 1

Idea summary To divide polynomials: 1. Completely factor the numerator and denominator 2. Divide out all common factors between the numerator and denominator 3. Simplify the resulting expression (if necessary)

Practice What do you remember? 1

Choose all of the following that are correct steps in dividing a polynomials. a

Divide out all common factors from the numerator and denominator.

Identify all common factors that are present in both the numerator and the denominator.

Completely factor both the numerator and denominator.

Subtract the common factors from the numerator and denominator.

Simplify the resulting expression.

State whether each of the following factors could be divided out of:

a 3

3x + 1

x−1

x2 − 2

n2 − 1

State whether each of the following factors could be divided out of:

n+1

n+2

n−1

6.08 Divide polynomials mathspace.co

373

Assume all variables are non-zero: i

What is the greatest common factor of the numerator and denominator?

Write the numerator and denominator as a product of the greatest common factor and the remainder.

iii

Simplify fully.

a 5

Fill in the blanks: a

Let’s Practice 6

Simplify: a

Which polynomial expression is equivalent to this expression if s ≠ −1?

A 8

374

3s − 5

5 −4s

5 − 4s2

Simplify: a

Factor and simplify:

Simplify:

Mathspace Virginia SOL Algebra 1 mathspace.co

6 − 4s2

Consider the problem

Identify and correct the error in each of the following student’s work.

Andre: 12

Liz:

Find the quotient of the following expressions: a

Ralph rode his bike for a distance of (4d2 − 6d) km over a time of (3d2 + 2d) hours. What is the simpliﬁed expression for the speed he traveled?

Let’s extend our thinking 14

Explain why

Fill in the blanks to make a true algebraic statement. a

does not equal . Illustrate your answer with an example.

What would be the result of dividing 2g2 − 10g + 12 by g2 − 5g + 6?

Mark divided a polynomial using the following steps. Do you agree with his solution? Why or why not? Factor and Simplify: Completely factor the numerator and denominator

Divide out the common factors

Rewrite in simplest form

Find the quotient of the following expressions: a

At a landfill site, a hole in the shape of a rectangular prism is to be dug out. The length of the rectangular cross-section measures 2x meters and the width is (x + 1) meters. If they need the volume of the landfill to be 2x(x2 − 4x − 5) cubic meters, find the expression for the depth of the hole.

Create and solve an example of polynomial division where the degree of the denominator is greater than that of the numerator.

6.08 Divide polynomials mathspace.co

375

7 Quadratic Functions Big ideas • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. • Functions provide a representation for how related quantities vary. This makes functions a good way to represent many real-world situations.

Chapter outline 7.01 7.02 7.03 7.04 7.05

Characteristics of quadratic functions Quadratic functions in factored form Quadratic functions in vertex form Quadratic functions in standard form Compare linear, quadratic, and exponential functions

378 393 409 425 439

Apply the idea

Reﬂect and check 9

Having the vertex in your table is useful, since it tells you where the parabola has a minimum or maximum. Sometimes the table values you select will not include the vertex of the function, depending on the quadratic function being graphed. If you plot your initial table values and find you are unsure where the parabola changes direction, you can add additional values to your table until you can identify the vertex.

8 7 6 f (x)

5 4 3 2 1 −2

−1

x 1

Note that the quadratic function has one x-intercept, at x = 1.

b State the axis of symmetry.

Create a strategy

Apply the idea

The axis of symmetry is a vertical line that passes through the vertex.

8 7 6 5

f (x)

4 3 2 1 −2

−1

x 1

The axis of symmetry is x = 1.

Example 2 Consider the graph of the quadratic function g(x):

9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

a Find the x-intercepts and y-intercept.

Create a strategy To ﬁnd the x-intercepts, locate the places where the parabola crosses the x-axis. To ﬁnd the y-intercept, locate the place where the parabola crosses the y-axis.

380

Mathspace Virginia SOL Algebra 1 mathspace.co

−1 −2 −3

g(x)

x 1

Apply the idea We can identify the intercepts on the graph: 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

−1 −2 −3

g(x)

x 1

From the graph we can see the there are two x-intercepts at (−4, 0) and (2, 0), and there is one y-intercept at (0, 8).

b Determine the domain and range.

Create a strategy To find the domain of g(x), we want to find all possible x-values for which g(x) could be graphed. To find the range, we want to find all possible values of g(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of g(x).

Apply the idea We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues inﬁnitely in either x direction. Domain: {x − ∞ < x < ∞} This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues inﬁnitely in the negative y direction. Range: { y y ≤ 9}

Reflect and check For the domain, we may also see it written as “all real values of x” or in interval notation as “(−∞, ∞)” instead of using inequality or set notation.

7.01 Characteristics of quadratic functions mathspace.co

381

c Describe what happens to the graph as x gets very large and positive.

Create a strategy We can look at the graph and see what is happening for larger and larger values of x. We may need imagine the graph extending beyond what is shown.

Apply the idea As x gets very large, the graph continues down and the function values are negative with a very large size. It is decreasing faster and faster. y 5 −4

−2

−5

g(x) 2

x 4

−10 −15 −20 −25 −30 −35

Reflect and check The function values for g(x) also become large and negative as x becomes large and negative.

Example 3 The graph shows the height, y (in feet), of a softball above ground x seconds after it was thrown in the air.

Softball throw Height in feet, y 14 12 10 8 6 4 2 Time in seconds, x 0.5

1.5

2.5

3.5

a Find the y-intercept and describe what it means in context.

Create a strategy We want to ﬁnd the place where the parabola crosses the y-axis. Once we ﬁnd the y-intercept, we want to connect this to the context of the softball. Since the y-axis represents the height of the softball in feet above the ground, we can use it to identify the height of the softball at 0 seconds.

382

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea We can identify the y-intercept on the graph: Softball throw Height in feet, y 14 12 10 8 6 4 2 0.5

1.5

Time in seconds, x 2.5 3 3.5

The y-intercept is (0, 6). The y-intercept tells us that the softball was thrown from a height of 6 feet above the ground.

b Find the value of the x-intercept and describe what it means in context.

Create a strategy We want to ﬁnd the place where the parabola crosses the x-axis. Once we ﬁnd the x-intercept, we want to connect this to the context of the softball. Since the x-axis represents the time in seconds after being thrown, we can use it to identify how many seconds the softball hits the ground.

Apply the idea We can identify the x-intercept on the graph: Softball throw Height in feet, y 14 12 10 8 6 4 2 Time in seconds, x 0.5

1.5

2.5

3.5

The x-intercept is (3, 0). The x-intercept tells us that the softball hits the ground 3 seconds after it was thrown in the air.

7.01 Characteristics of quadratic functions mathspace.co

383

c Find the value of the vertex and describe what it means in context.

Create a strategy

Apply the idea

In order to ﬁnd the vertex, we want to ﬁnd the maximum point of the parabola.

We can identify the vertex on the graph: Softball throw

Once we ﬁnd the vertex, we want to connect this to the context of the softball. We know that the x-value of the vertex represents time in seconds after the softball is thrown and the y-value of the vertex represents the height of the softball above ground in feet.

Height in feet, y 14 12 10 8 6 4 2 0.5

1.5

Time in seconds, x 2.5 3 3.5

The vertex is (1.25, 12). After 1.25 seconds, the softball reaches a maximum height of 12 feet above the ground.

d State the domain and describe what it means in context.

Create a strategy

Apply the idea

The domain of the context should be reasonable. We can Domain: 0 ≤ x ≤ 3 use the graph of the function to determine the domain The domain of the function starts at x = 0 seconds when and explain its meaning in context. the softball was recorded from where it was initially thrown. The domain of the function ends at x = 3 seconds when the softball lands on the ground.

Idea summary From the graph of a quadratic function, we can identify key features including: • • • • •

384

Domain and range x- and y-intercepts Maximum or minimum function value Vertex Axis of symmetry

Mathspace Virginia SOL Algebra 1 mathspace.co

Practice What do you remember? 1

Which table of values best represents the rule shown? The square of the sum of x and 3 is equal to y. A

x 4 5

y 13 14

x 4 5

y 25 34

x 4 5

y 19 28

x 4 5

y 49 64

Choose the graph that has each set of characteristics: a • Axis of symmetry at x = −1 • x-intercepts: (−7, 0), (5, 0) Select the graph that represents the function. A y 25 20 15 10 5

−6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

x 1 2 3 4 5 6 7 8

C y 25 20 15 10 5

−6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

x 1 2 3 4 5 6 7 8

25 20 15 10 5

−7 −6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

25 20 15 10 5 −7 −6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

1 2 3 4 5 6 7

x 1 2 3 4 5 6 7

7.01 Characteristics of quadratic functions mathspace.co

385

• Vertex is a maximum • x-intercept: (6, 0) • y-intercept: (0, −36) Select the graph that represents the function. x 1 2 3 4 5 6 7 8 9 10 11 12 13 14

y A

y x −12−11−10−9−8−7 −6−5−4 −3−2−1 −5

−5 −10

−10

−15

−20

−25

−30

−35

y x

1 2 3 4 5 6 7 8 9 10 11 12 −5

x −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 y −5

−10

−15

−20

−25

−30

−35

• No x-intercept • Axis of symmetry: x = 10 • Range: y ≤ −6 Select the graph that represents the function. y A

x 1 2 3 4 5 6 7 8 9 1011 121314151617

386

−5

−10

−15

−20

−25

−30

−35

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1 2 3 4 5 6 7 8 9 1011 121314151617

C y

x 1 2 3 4 5 6 7 8 9 1011 121314151617

1 2 3 4 5 6 7 8 9 10 11 121314151617

−5

−10

−15

−20

−25

−30

−35

Consider the graph of the quadratic function. 5 y 4 3 2 1 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

x 1

2 3 4 5 6 7

State the number of x-intercept(s) the quadratic function has based on the graph. 4

For each table, complete the following: i

Graph the quadratic function shown in the following tables.

Find the coordinates of the vertex.

iii

Determine whether the vertex is a maximum or minimum point.

Determine the axis of symmetry.

State the number of x-intercept(s).

x y

0 −7

1 −2

2 1

3 2

4 1

5 −2

6 −7

x y

−7 11

−6 6

−5 3

−4 2

−3 3

−2 6

−1 11

Determine how many x-intercept(s) the equation x2 + 64 = 0 has.

7.01 Characteristics of quadratic functions mathspace.co

387

Let’s practice 6

Consider the graph of the function y = f (x).

State the number of x-intercept(s) of the function.

Determine the domain of the function.

Determine the range of the function.

Describe the behavior of the function for large values of x.

8 6 4 2

−8 −6 −4 −2 −2

−4 −6 −8 −10

Consider the function g(x) = −(x + 5) (x + 1). a

Copy and complete the table. x g(x)

−5

−4

−3

−2

−1

Determine the equation of the axis of symmetry.

Determine if the graph will have a maximum or a minimum.

Graph the function.

Determine the number of x-intercept(s) based on the graph.

Consider the function h(x) = x2 − 4x + 4. a

Copy and complete the table: x h(x)

−6

−1

Find the coordinates of the x- and y-intercepts.

Find the coordinates of the vertex.

Determine the domain and range.

Determine the equation of the axis of symmetry.

Determine if the graph will have a maximum or minimum.

Graph the function.

Zahra jumps off a diving platform and the path of her dive is modeled by the function f (x) = −x2 + 2x + 8, where f (x) is her height in meters above the pool, and x is the horizontal distance in meters from the edge of the diving platform. a

Select the graph of the function: A y

7 6

2 −6

−4

−2

x 2

5 4

−4

−6

−8

x 1

388

Mathspace Virginia SOL Algebra 1 mathspace.co

C y

−6

−4 −6 3

−8

x 2

−2

Find the height of the diving platform.

Find the maximum height of Zahra’s dive.

Determine and interpret the domain and range of the function.

A frisbee is thrown upward and away from the top of a cliff. The height, y meters, of the frisbee at time, x seconds, is given by the equation y = −20(x − 6) (x + 2). a

Select the graph of the function: A y

360

300

240

180

120

x 1

−4

D 320

240

160

−6

−4

−2 −80

−160

−240

−320

Determine the height at which the frisbee is thrown.

Find the maximum height the frisbee reached.

Determine the domain and range of the function.

−2 −80

320

80 −6

C y

−2

2 1

−4

x 2

For each of the following quadratic functions, find the: i

x-intercept(s)

y-intercept

iii

vertex

y = x2 − 4x + 4

y = (x + 4) (x − 2)

y = x2 − 2x − 3

y = −x2 − 2

7.01 Characteristics of quadratic functions mathspace.co

389

For each of the following quadratic functions, find the: i a

y = x + 2x + 1

y = −(x − 5) (x + 1)

Range

y = (x − 3)2

y = −x2 − 2x − 1

Find the range when the domain is {−6, −1, 0, 5, 7} for each of the following quadratic functions: a

Domain

y = x2 − 2x + 3

y = −x2 + 3x − 1

y = −2(x + 1) (x − 4)

y = 3(x − 2)2 + 1

Use the graph of f (x) to evaluate for the following values. a

x = −5

x = −2

f (x) = −9

f (x) = 0

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

The graph shows the height of a soccer ball above ground, in feet, after it is kicked in terms of x seconds. a

Find the y-intercept.

Describe what the y-intercept means in context.

Find the x-intercept.

Describe what the x-intercept means in context.

Find the coordinates of the vertex.

Describe what the vertex means in context.

f (x) x 1

y 10 8 6 4 2 x 1

A clothing company is designing a new jacket and wants to determine how to maximize their proﬁt once the jacket is ready to be sold. The graph represents the total proﬁt, P, the shop will make at each price point, x, the jacket could sell for. y 700 600 500 400 y = P(x)

300 200 100

390

Find the value of the vertex and describe what the vertex means in context.

Determine the domain that results in a proﬁt for the clothing company.

Determine the corresponding range of proﬁt.

The manager believes that selling a jacket at a higher price will always result in a larger proﬁt. Explain how increasing the price of the jacket affects the proﬁt.

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 17

Sketch the graph of quadratic equations with the following key features: a b • Has a maximum function value • Axis of symmetry at x = 1 • x-intercept: (5, 0) • Vertex is a minimum • y-intercept: (0, −25) • x-intercepts: (−2, 0), (4, 0) c

• No x-intercept • Axis of symmetry: x = 8 • Range: y ≤ −9

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 foot. The cost of the ﬂooring in terms of its side length is shown by the quadratic function C(x) = 6.25x2. a

Determine how much Elise should expect to spend on ﬂooring if the room has side lengths of 12 ft.

Determine how much the price would change if the side lengths decreased by 3 ft.

Graph the given quadratic model, C(x) = 6.25x2. Make sure to choose appropriate labels and scale.

Describe what changes and what stays the same about the graph of the quadratic model if the cost per square foot decreases.

Graham, Habib, and Joel throw or kick footballs around the same time. The vertical height of Graham’s football is shown in the graph. The function G(t) represents the vertical distance of the football above the ground, in feet, and t represents time, in seconds. 16 14 12 10 8 6 4 2

y = G(t)

t 1

The vertical height of Habib’s football is shown in the table. H(t) represents the vertical distance of the football above the ground, in feet, and t represents time, in seconds. t H(t)

0.172 0

2 14

3 16

4 14

5 8

5.828 0

The height of Joel’s football can also be modeled with a quadratic function that has the following key features. Let J(t) represent the vertical distance of the football above the ground, in feet, and t represent time, in seconds. • y-intercept: (0, 5) • Vertex at (1, 5.5) • t-intercept: (4.317, 0) Use the above information to complete the following: a

Graph the three quadratic functions on the same coordinate plane.

Determine whose football reached the ground the quickest after being kicked or thrown. Explain your answer.

Determine whose football reached the greatest height. Explain your answer.

Describe what G(0) and H(0) mean in context.

Habib claims that his football reaches the maximum height the quickest. Determine whether or not Habib is correct. Explain your answer.

7.01 Characteristics of quadratic functions mathspace.co

391

Rafael is trying to explain the design for a painting to his friend, over the phone. Describe how Rafael could use key features of quadratic functions to share his idea.

Write the equation of a quadratic function that has the following key features and sketch a graph of the function:

Two x-intercepts that have the same absolute value but opposite signs

One x-intercept that is a fraction and the other is a prime number

One unique x-intercept that is less than 1 but greater than 0

x-intercepts at (7, 0) and (−2, 0), and has a y-intercept at (0, 14)

x-intercepts at (−1, 0) and (−10, 0), and has a y-intercept at (0, 30)

Determine how many unique quadratic equations exist for each of the following key features: a

x-intercepts at (−7, 0) and (10, 0)

One unique x-intercept at (−4, 0) and y-intercept at (0, −20)

Vertex at the origin and passes through (2, 14)

x-intercepts at (15, 0) and (−7, 0), and is symmetric about the y-axis

Ori models his golf shot using the quadratic equation: y = −x2 + 10x − 16 where y is the height of the ball (in yards) and x is the time after placing the ball on the ground (in seconds). Use graphing technology to explore this function. Describe and justify a real-world problem involving this equation which has:

392

One viable x-intercept

No viable x-intercept

Mathspace Virginia SOL Algebra 1 mathspace.co

Two viable x-intercepts

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

−1

−2

−3

−4

If a > 0, then the quadratic function opens upwards and has a minimum value.

x 1

If a < 0 then the quadratic function opens downwards and has a maximum value.

The x-intercepts are the points where f (x) = 0, so we refer to x1 and x2 as the zeros of the function. • (x1, 0) and (x2, 0) are the x-intercepts of the function y = f (x) • x1 and x2 are zeros of the function • (x − x1) and (x − x2) are factors of the function y = f (x) • x1 and x2 are solutions or roots of the equation f (x) = 0

x x1

To draw the graph of a quadratic function, we generally want to ﬁnd three different points on the graph, such as the x- and y-intercepts. 8

Since the graph of a quadratic function has a line of symmetry passing through the vertex, we know the vertex lies halfway between the two x-intercepts.

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

394

Mathspace Virginia SOL Algebra 1 mathspace.co

We can also determine the direction in which the graph opens by identifying if the scale factor, a, is positive or negative.

Example 1 Consider the graph of a quadratic function: 4

3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

a Identify the coordinates of the x- and y-intercepts of the function.

Create a strategy The x-intercepts occur when y = 0 and the y-intercept occurs when x = 0.

Apply the idea 4

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3

The x-intercepts are (−2, 0) and (3, 0).

−4

The y-intercept is (0, 2).

b Find the equation of the quadratic function in factored form.

Create a strategy Substitute the x-intercepts for x1 and x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and solve for a.

Apply the idea Since the x-values of the x-intercepts are −2 and 3, we know that the factored form will be: y = a(x + 2) (x − 3) for some value of a. We can ﬁnd a by substituting in the coordinates of the y-intercept into the function.

7.02 Quadratic functions in factored form mathspace.co

395

To ﬁnd a: Factored form

Substitute (0, 2)

Evaluate the addition and subtraction

Evaluate the multiplication

Divide both sides by −6

The equation of the quadratic function in factored form:

Example 2 Consider the quadratic function: y = 2x2 + 4x − 48 a State the coordinates of the x-intercepts.

Create a strategy In the factored form y = a(x − x1) (x − x2), the values of x1 and x2 are the x-values of the x-intercepts. The y-value of the x-intercepts is y = 0. Factor the quadratic, then determine its x-intercepts. We can factor out a GCF of 2, so that the equation becomes y = 2(x2 + 2x − 24). Since there are no common factors for the remaining three terms and the trinomial is not a perfect square trinomial, we proceed to factor by grouping by ﬁnding the value of two integers that multiply to ac = (1) (−24) = −24 and add up to b = 2. After ﬁnding these integers, we use them to rewrite the middle term 2x as a sum of two terms.

Apply the idea The factor pair whose sum is 2 is −4 and 6. We can use this to rewrite the trinomial and factor by grouping as follows: 2(x2 + 2x − 24) = 2(x2 − 4x + 6x − 24) = 2[x(x − 4) + 6(x − 4)]

Rewrite polynomial with four terms Factor each pair

= 2(x − 4) (x + 6) Divide out common factor of (x − 4) There are no more common factors to be divided out, so the fully factored form of the quadratic function is y = 2(x − 4) (x + 6). The x-intercepts are (4, 0) and (−6, 0).

Reﬂect and check Notice that x + 6 is the same as x − (−6).

396

Mathspace Virginia SOL Algebra 1 mathspace.co

b Determine the coordinates of the y-intercept.

Create a strategy The y-value of the y-intercept is the result when x = 0. We can substitute x = 0 into the factored form to ﬁnd this value.

Apply the idea To ﬁnd the y-value of the y-intercept: y = 2(x − 4) (x + 6)

Given quadratic function

y = 2(0 − 4) (0 + 6)

Substitute x = 0

y = 2(−4) (6)

Evaluate the subtraction and addition

y = −48

Evaluate the multiplication

The y-intercept is (0, −48).

Reﬂect and check The original function shows the y-intercept, which we can identify without making any calculations.

c Determine the coordinates of the vertex.

Create a strategy The vertex lies on the axis of symmetry, so the x-coordinate of the vertex will be exactly in the middle between the two x-intercepts. We can ﬁnd the middle value by taking the average of 4 and −6. We can then substitute this x-coordinate value into the function to ﬁnd the y-coordinate.

Apply the idea To ﬁnd the x-coordinate: The average of 4 and −6 is half way between them. We can calculate that vertex and the axis of symmetry is x = −1. To ﬁnd the y-coordinate: y = 2(x − 4) (x + 6)

Given quadratic function

y = 2(−1 − 4) (−1 + 6)

Substitute x = −1

y = 2(−5) (5)

Evaluate the subtraction and addition

y = −50

Evaluate the multiplication

so the x-coordinate of the

The vertex is (−1, −50). d Draw the graph of the function.

Create a strategy The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through any three points that we know are on it.

7.02 Quadratic functions in factored form mathspace.co

397

Apply the idea

Reflect and check 20 10

−8 −6 −4 −2 −10

Any three points is enough to draw the graph, but knowing where the vertex is can make it easier since the vertex is on the axis of symmetry.

y x 2

−20 −30 −40 −50 −60

Example 3 Identify the characteristics of h(x) =

(3x + 2) (x − 7).

a Identify the factors of the function.

Create a strategy The function is given in factored form y = a(x − x1) (x − x2) where a, (x − x1) and (x − x2) are the factors.

Apply the idea

Reﬂect and check

The factors of h (x) are , (3x + 2) and (x − 7).

is the greatest common factor (GCF) of a (x − x1) (x − x2) but is still a factor of the function.

b Identify the roots of the function.

Create a strategy The roots of the function are the x values, x1 and x2, where h(x) = 0.

Apply the idea To solve for the roots algebraically, set each of the variable factors equal to zero. 3x + 2 = 0 and x − 7 = 0 Rearrange each equation to isolate the term with the variable. 3x = −2 and x = 7 Isolate the variable by dividing by the coeﬃcient of x. x= The roots of h(x) are x =

and x = 7

Reﬂect and check Notice in part (a) we also identiﬁed the factor of a variable will not result in a root because

398

Mathspace Virginia SOL Algebra 1 mathspace.co

but we did not use it to ﬁnd roots. That is because a factor without .

and x = 7 and

We can substitute these values back into the equation to check our answers. If we substitute get h (x) = 0, then we know our roots are correct. Let’s start with the root

. Substitute

Evaluate the multiplication

Evaluate inside the parentheses

Zero product property

Next, let’s try the root x = 7. Substitute x = 7

Evaluate the multiplication

Evaluate inside the parentheses

Zero product property

Evaluating for each root gave an output of 0 conﬁrming that both are in fact roots of the function. c Identify the zeros of the function.

Create a strategy

Apply the idea

The zeros of a function are the same as its roots.

In part (b) we solved for the roots, x1 and x2, and got x = , 7. These are also the zeros of the function.

d State the x-intercepts of the function.

Create a strategy The points (x1, 0) and (x2, 0) are the x-intercepts for h(x).

Apply the idea The zeros or roots of h(x) are

Reﬂect and check We can check our x-intercepts by graphing h(x).

and 7.

h(x)

These are the x-values of the x-intercepts. The y-value of any y-intercept is 0 because the x-axis is at x = 0. The x-intercepts of the function are at

x −2

and (7, 0).

−2 −4 −6 −8

There are two points where the parabola crosses the x-axis, at x =

and x = 7.

7.02 Quadratic functions in factored form mathspace.co

399

Example 4 The graph of a quadratic function has x-intercepts at (−2, 0) and (1, 0) and passes through the point (−3, −2). Write an equation in factored form that models this quadratic.

Create a strategy To write the equation for this quadratic in factored form we need to ﬁrst identify the roots or zeros of the equation. We can then substitute these values for x1 and x2. The x-intercepts of the function are at (−2, 0) and (1, 0), so we know the equation has roots/zeros of x = −2 and x = 1.

Apply the idea Since the zeros are x = −2 and x = 1, we can identify the factors by rearranging those equations so they are equal to 0: By adding 2 to both sides of x = −2 and subtracting 1 from both sides of x = 1 we get: x + 2 = 0 and x − 1 = 0 We can put these in the factored form as the factors: y = a(x + 2) (x − 1) We can ﬁnd a by substituting the coordinates of the additional point, (−3, −2), into the function. To ﬁnd a: y = a(x + 2)(x − 1)

Factored form

−2 = a(−3 + 2)(−3 − 1)

Substitute x = −3 and y = −2

−2 = a(−1)(−4) Evaluate the addition −2 = 4a Evaluate the multiplication = a Divide both sides by 4 Substituting the value we found for a, the equation of the quadratic function in factored form is:

Reﬂect and check Checking the graph of the equation, we can see that it satisﬁes the given information. 10

8 6 4 2 −4 −3 −2 −1 −2 −4 −6 −8

400

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

Example 5 Find the equation that models the graph shown.

y −6

−4

−2

x 2

−2 −4 −6

Create a strategy This quadratic function only has 1 x-intercept, which is also the vertex. When this happens, the function is in the form f (x) = a(x − x1)2. Remember, we need an additional point, like the y-intercept, to ﬁnd the exact equation to this function.

Apply the idea Since the x-intercept is at (−2, 0), the function takes the form f (x) = a(x + 2)2. Next, we can use the y-intercept of (0, −1) to ﬁnd the value of the leading coeﬃcient. Given equation

Substitute x = 0 and y = −1

Evaluate the addition

Evaluate the exponent

Divide both sides by 4

The equation of the graph is

Reﬂect and check We could have used any point on the parabola to solve for the scale factor, a. There is another point at (−6, −4). We would substitute x = −6 and y = −4, then the equation would take the form −4 = a(−6 + 2)2.

Notice that this is the same thing we got earlier because no matter which points we substitute in we will get the same function because they are all points on the same parabola.

7.02 Quadratic functions in factored form mathspace.co

401

Example 6 A cannonball is ﬁred from the edge of a cliff which is 15 meters above sea level. The peak of the cannonball’s arc is 20 meters above sea level and 10 meters horizontally from the cliff edge. The cannonball lands in the sea 30 meters away from the base of the cliff. The path of the cannonball is shown on the following graph, but the axes have not been labeled. a Label the axes of the graph to match the information provided.

Create a strategy To make the graph match the information, we want to make sure that the axes labels and scales accurately represent the path of the cannonball and make sense for the context. Both axes will have meters as their units.

Apply the idea

Reﬂect and check

We can see that the path on the graph starts at a point on the vertical axis and ends at a point on the horizontal axis. So, we can make the vertical axis represent the height, with y = 0 being sea level, and the horizontal axis represent distance, with x = 0 being the edge of the cliff.

Another way to show the scale of the axes is to label some key points. For example:

We can then add values onto the axes to show that the cannonball starts at the edge of the cliff at (0, 15), reaches its peak at (10, 20), and then falls into the sea at (30, 0). 20

y (m)

(10, 20)

(0, 15)

x (m) (30, 0)

y (m)

15 10 5 5

x (m) 30

b Determine the factored equation which models the path of the cannonball.

Create a strategy To match the graph in part (a), we can let x represent the horizontal distance from the cliff, and let y represent the height above sea level. To find the factored equation that models the cannonball, we need to know both x-intercepts and the scale factor. We know that one of the x-intercepts is at x = 30, and that the vertex is at x = 10. Remember that the vertex lies on the axis of symmetry of a quadratic function, so we can use this to find the other x-intercept. We can find the scale factor by substituting any point into the equation (that isn’t an x-intercept) and solving for the scale factor that makes the equation true.

402

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea Since both x-intercepts have the same y-value, they will be mirrored across the axis of symmetry. Since x = 30 is 20 more units than x = 10, the other x-intercept will be at 20 less units than x = 10. So, the other x-intercept is at x = −10. If we let the scale factor of the equation be a, then our equation will be: y = a(x + 10) (x − 30) We can ﬁnd the scale factor by substituting in a point on the graph (let’s use the y-intercept) and solving for a.

Model equation

Substitute (0, 15)

Evaluate the addition and subtraction

Evaluate the multiplication

Divide both sides by −300

So, the equation which models the path of the cannonball is:

c A second cannonball is ﬁred, and this one can be modeled by the equation:

Use this model to predict where the cannonball landed.

Create a strategy We can use what we know about the general factored form of the equation to provide information about the cannonball’s path.

Apply the idea Since the a value is negative, we know this function will open downward. This makes sense for a cannonball, as it will arc upwards to a maximum vertical height before falling back down, due to gravity. The x-intercepts will be at −12 and 27. Since the cannonball is being ﬁred away from the cliff in the positive x-direction, we know that x = −12 is a nonviable solution. So, the second cannonball lands in the sea 27 meters from the base of the cliff.

Idea summary To write the equation of the graph of a quadratic function in factored form, substitute the x-intercepts for x1 and x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and solve for a, the scale factor.

7.02 Quadratic functions in factored form mathspace.co

403

Practice What do you remember? 1

State the factored form for a quadratic equation and describe its graphical features.

Describe the axis of symmetry of a quadratic function.

For each of the following quadratic equations, state the roots: y = (x − 1) (x − 4)

a 4

y = x(x − 4)

y = 2(x − 1) (x + 5)

For each of the following quadratic functions, determine the y-value of the y-intercept: y = 3(x − 1) (x − 8)

a 5

y = (x − 3) (x + 7)

y = x(x + 9)

Consider the following graph of a function.

Select the equation that represents the function: A

y = x(x + 5)

y = x(5 − x)

y = −x(x − 5)

y = x(x − 5)

−2 −1

−1

x 1

−2 −3 −4 −5 −6 −7

Consider the following graph of a function. Select the equation that represents the function: A

h = (2 − t) (t − 4)

h = −(2 − t) (t − 4)

h = (t + 2) (t − 4)

h = −(t − 2) (t + 4)

9 8 7 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2

Use the table of values to graph the function. x y

404

0 12

2 0

4 −4

Mathspace Virginia SOL Algebra 1 mathspace.co

6 0

8 12

t 1

2 3 4

Consider the function Select the graph that represents the function: A

6 5

4 3 2

1 −3 −2 −1 −1

−7 −6 −5 −4 −3 −2 −1 −1

x 1

2 3

−3 −4

−2 −3

−5

−4

−6

3 2 1

4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2

x 1

−2

2 3 4 5 6 7

6 5

−3 −2 −1 x 1

−1

x 1

2 3 4 5 6 7

−2 −3

2 3

−4 −5

−3 −4

−6

Graph a quadratic function with x-intercepts of (5, 0) and (−3, 0), a y-intercept of (0, 60), and its vertex is a maximum.

Let’s practice 10

For each quadratic function: i

State the coordinates of the x-intercepts.

Determine the axis of symmetry.

iii

Determine the coordinates of the vertex.

Graph the function.

y = (x − 2) (x + 4)

y = −2(x − 1) (x − 5)

7.02 Quadratic functions in factored form mathspace.co

405

For each graph: i

State the zeros.

State the y-value of the y-intercept.

iii

Write the equation of the quadratic function in factored form.

2 −2 −1

x 2

−4

−6

−8

For each of the following quadratic functions, find the: i

Solutions

a c

y-intercept

iii

vertex

y = (x − 3) (x + 1)

y = (x + 2) (x − 5)

y = (x − 4) (x + 6)

y = −(x − 1) (x + 2)

Range

For each of the following quadratic functions, find the: i

−2

−8 −6 −4 −2 −2

x 1

Domain

y = (x − 1) (x − 2)

y = −(x + 3) (x − 4)

y = (x − 2) (x − 2)

y = −(x + 1) (x + 1)

Find the range when the domain is {−6, −1, 0, 5, 7} for each of the following quadratic functions: a

y = (x + 4) (x − 1)

y = −(x − 3) (x + 2)

y = −2(x − 2) (x + 3)

y = 3(x + 1) (x − 2)

Use the function f (x) = −2(x + 4) (x − 1) to evaluate for the following values: a

x = −5

x = −2

x =

f (x) = 0

A quadratic function has x-intercepts of (5, 0) and (−3, 0), and a y-intercept of (0, 60). Write the equation of the function in factored form.

A quadratic function passes through the points (−2, 0), (9, 0) and (0, −6). Write the equation of the function in factored form.

The zeros of a quadratic function are x = −4 and x = −7. The graph of the function passes through the point (−3, 12). Write the equation of the function in factored form.

406

Mathspace Virginia SOL Algebra 1 mathspace.co

Satellite dishes follow the shape of a parabola to optimally receive signals. Winston models the cross section of a satellite dish with the points (−2, 0) and (2, 0) being the edges of the satellite and the x-axis representing the top opening of the satellite dish. The satellite dish is a

- foot deep.

Select the graph that represents the problem: A y

1 x

x −2

−1

−2

−1

−2

C y

1 x

−2

b 20

−1

x −2

−1

−2

Find the vertex of the satellite dish.

Xia notices that the Sunshine State Arch is in the shape of a quadratic function. They know that the arch has a height of 110 ft and the feet of the arch are 100 ft apart. Xia chooses to let the x-intercepts of the arch be the origin and (100, 0). a

Determine the coordinates of the vertex of the arch.

Write the equation of the quadratic function for Xia’s representation of the Sunshine State Arch in factored form.

Find the domain and range of the Sunshine State Arch.

Let’s extend our thinking 21

Create a quadratic function in the form f (x) = (x − ⬚) (x − ⬚) that meets the given condition. a

Two positive x-intercepts.

Two negative x-intercepts.

One positive and one negative x-intercept.

One positive x-intercept.

One x-intercept at (0, 0).

7.02 Quadratic functions in factored form mathspace.co

407

Rewrite the quadratic functions from the previous question in the form f (x) = ax2 + bx + c and determine the similarities and differences between the values of a, b, and c depending on the type of solution the quadratic has.

Ami throws a javelin forward in a parabolic arc from the ground. Using photos that her friend Maryellen is taking from the stands, she determines that 10 yards horizontally from where she threw the javelin, it reaches a maximum height 5 yards above the ground. Ami models her throw with the origin of a coordinate plane being the point 20 yards behind her.

For Ami’s model, state the roots.

Write the equation for the quadratic function modeling the path of Ami’s throw in factored form.

Burnell jumps up and off a 4 meter high springboard into the diving pool below. Burnell’s jump can be represented by the equation y = −2(x − 2) (x + 1) where y is Burnell’s height above the water and x is the horizontal distance from the springboard towards the pool (both in meters).

Graph the equation modeling Burnell’s jump. Label any x- and y-intercepts.

Use the model to predict where Burnell will enter the water. Explain your answer.

Wilson tosses an eraser into the air and counts how long it takes for the eraser to return to his hand. He estimates that it takes 2 seconds and that he is tossing the eraser 6 ft into the air. Wilson models the height of the eraser above his hand in feet as a function h of time t in seconds, letting t = 0 be when he tosses the eraser and h = 0 be the height of his hand.

Explain how to ﬁnd the intercepts of the function and state them.

Assuming that the path of the eraser is symmetric going up and coming down, state the coordinates of the vertex of the function.

Graph the function. Choose appropriate labels and scales.

Write the equation of the function in factored form.

A quadratic function has the factored form equation y = k(x − x0) (x − 12). If the function has a vertex at the point (8, 15), determine the values of x0 and k. Justify your answers.

408

Mathspace Virginia SOL Algebra 1 mathspace.co

y 4

y = x2 + 2

3 2

y = x2

−4 −3 −2 −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 together with the translation must be squared together. That is, to translate y = x2 to the left by h units we get y = (x + h)2. 6 5

y y = x2

y= (x + 2)2 4 3 2

This graph shows y = x2 translated horizontally left by 2 to get y = (x + 2)2 and right by 2 to get y = (x − 2)2.

y= (x − 2)2

−4 −3 −2 −1 −1

−2

A parabola can be dilated by multiplying every y-value by a constant number greater than 1. So to expand the parabola y = x2 by a scale factor of a we get y = ax2. We can compress a parabola by using a scale factor between 0 and 1. 6

y = x2

4 y = 2x2 3

This graph shows y = x2 vertically expanded by a scale factor of 2

2 1 −4 −3 −2 −1

x 1

−1

to get y = 2x2 and compressed by a scale factor of 2 to get y =

−2

3 2

y = x2

1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

410

y = −x2

Mathspace Virginia SOL Algebra 1 mathspace.co

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 parabola from opening up to opening down.

The x-value of the vertex, h, represents the horizontal translation; the y-value, k, represents the vertical translation; and the leading coeﬃcient, a, represents the shape of the parabola and the direction it opens. The parent function of a quadratic is f (x) = x2, so writing these translations in function notation becomes af (x − h) + k = a(x − h)2 + k. 4

As an example, consider the graph of y = (x − 2)2 − 3 • Translation of the parent function 3 units down and 2 units right with a vertex at (2, −3) • Axis of symmetry x = 2 • y-intercept at (0, 1) can be calculated by substituting x = 0 into the function

3 2 1 −4 −3 −2 −1

−1

(0, 1) x 1

−2 −3 −4

When writing a quadratic function from its graph, we can begin by identifying the vertex and substituting these values into the function for h and k. Then, we can use another point on the parabola, like the y-intercept, to help us find the value of a.

Example 1 Consider the following function: m(x) =

(x − 2)2 + 8

a Find the vertex.

Create a strategy

Apply the idea 2

The function is given in vertex form y = a (x − h) + k where the vertex is the point (h, k).

In the equation m(x) = (x − 2)2 + 8 the x-coordinate of vertex is h = 2 and the y-coordinate of vertex is k = 8. With h = 2 and k = 8, the ordered pair of the vertex is at (2, 8).

b State the domain.

Create a strategy

Apply the idea

To ﬁnd the domain of m(x), we want to ﬁnd all possible x-values for which m(x) could be graphed.

We know that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues inﬁnitely in either x direction. Domain: {x∣ −∞ < x < ∞}

c State the range.

Create a strategy

Apply the idea

To ﬁnd the range, we want to ﬁnd all possible values of m(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of m(x).

This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues inﬁnitely in the negative y direction. Range: {y∣ y ≤ 8}

7.03 Quadratic functions in vertex form mathspace.co

411

d Draw the graph of the function.

Create a strategy The scale factor is

which is negative, so the graph will open down. We can draw the graph through any three

points that we know are on it.

Apply the idea 8

Start by plotting the vertex, found in part (a), on the graph.

vertex

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

We can solve for the x-intercepts by substituting m(x) = 0 and solving for x.

vertex

6 4 2

−8 −6 −4 −2 −2

−4 −6

x = 2 + 4 and x = 2 − 4

−8

x = 6 and x = −2 We can solve for the y-intercept by substituting x = 0 and solving for m(x).

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

412

Mathspace Virginia SOL Algebra 1 mathspace.co

Then, we can get an additional point by reﬂecting the y-intercept across the axis of symmetry.

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

To ﬁnish the drawing of the graph of the function, connect the points plotted to draw the parabola.

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Reﬂect and check Technically we only need 3 points to graph a parabola, but ﬁnding more points can make our graph more precise.

Example 2 The table of values represents a quadratic function. a Write the function p(x) in vertex form.

x p(x)

−4 −5

−3 0

−2 3

−1 4

0 3

1 0

2 −5

Create a strategy We can use the fact that a quadratic function has symmetry about its vertex to identify the location of the vertex from the table.

Apply the idea Looking at the values of p(x), we can see that it has a maximum value of 4 and falls off symmetrically on either side. So, we know that the vertex is the point (−1, 4), so we can use that to set up the equation p(x) = a(x + 1)2 + 4 We can now ﬁnd the value of a by substituting any other pair of values from the table, such as (0, 3). Doing so, we get 3 = a(0 + 1)2 + 4 which we can solve to get a = −1. So the quadratic function shown in the table of values is p(x) = −(x + 1)2 + 4

7.03 Quadratic functions in vertex form mathspace.co

413

b Determine the value of p(x) when x = 6.

Create a strategy We can substitute x = 6 into the equation that was created in part (a) in order to predict what p(x) will be equal to.

Apply the idea The equation found in part (a) is p(x) = −(x + 1)2 + 4 2

p(6) = −(6 + 1) + 4 2

Original equation Substitute x = 6

= −(7) + 4

Evaluate the addition

= −49 + 4

Evaluate the exponent

= −45

Evaluate the addition

This means that when x = 6, p(x) is equal to −45.

c Determine the x- and y-intercepts of the function.

Create a strategy

Apply the idea

Use the table of values to identify the x-intercepts when y = 0 and the y-intercept when x = 0.

The x-intercepts occur at (−3, 0) and (1, 0). The y-intercept occurs at (0, 3).

Example 3 y

The quadratic function f (x) = 2x2 has been transformed to produce a new quadratic function g(x), as shown in the graph:

f (x)

6 4 g(x)

2 x

−8

−6

−4

−2

2 −2

a Describe the transformation from f (x) to g(x).

Apply the idea

Reﬂect and check

The function g(x) has the same shape and size as f (x), but We could conﬁrm that g(x) is a horizontal shift left by has been shifted to the left. Comparing the vertices of evaluating f (x + 6) = 2(x + 6)2: the two parabolas, we can see that this is a translation of g(−8) = 8 and f (−8 + 6) = 2(−8 + 6)2 = 2(−2)2 = 2(4) = 8 6 units to the left. g(−6) = 0 and f (−6 + 6) = 2(−6 + 6)2 = 2(0)2 = 2(0) = 0 g(−4) = 8 and f (−4 + 6) = 2(−4 + 6)2 = 2(2)2 = 2(4) = 8

414

Mathspace Virginia SOL Algebra 1 mathspace.co

b Write the equation of the function g(x) in vertex form.

Create a strategy

Apply the idea

Remember that vertex form for a quadratic is g(x) = a(x − h)2 + k, where the vertex is at the point (h, k).

f (x) = 2x2 has been translated 6 units to the left to produce g(x), and we can see that its vertex is at (−6, 0). So g(x) has can be written as g(x) = 2(x + 6)2.

c Describe the transformations of the graph of f (x) resulting in the function h(x) = −2(x + 6)2 + 3.

Create a strategy

Apply the idea

Since f (x) opens upward and has a vertex at (0, 0), the vertex form of h(x) gives information about its vertex and direction of its opening.

The function h(x) will open downward and be a reﬂection of f (x) across the x-axis, since the value of a became negative. The vertex of h(x) is (−6, 3). The vertex became a maximum value and shifted the graph up 3 units and to the left 6 units.

d Sketch the graph of h(x)

Create a strategy Use the description of the transformations of g(x) from part (c) and graph of g(x) to sketch h(x).

Apply the idea Start by reﬂecting g(x) across the x-axis using points from its graph. g(x)

Perform a vertical shift 3 units up.

g(x)

8 6

2 −8

−6

−4

−2

y 8

x −8

−2 −4

−6

−4 h(x)

−2

x 2

−4

−6

−8

The graph of h(x) follows: y 8 6 4 2 −8

−6

−4

h(x)

−2

x 2

−4 −6 −8

7.03 Quadratic functions in vertex form mathspace.co

415

Example 4 Meri throws a rock into Crescent Lake. The height of the rock above ground is a quadratic function of time. The rock is thrown from 4.4 ft above ground. After 1.5 seconds, the rock reaches its maximum height of 24 ft. Write the quadratic equation in vertex form.

Create a strategy The maximum height of the rock at 24 feet indicates that this is where the vertex of the function is located. This occurs at 1.5 seconds. Substitute the vertex of the graph and the y-intercept into the vertex form of a quadratic function to determine the equation.

Apply the idea The vertex is located at (1.5, 24), so we can substitute h = 1.5 and k = 24. The rock is thrown from 4.4 feet, so the y-intercept is located at (0, 4.4) and we can substitute x = 0 and y = 4.4. y = a(x − h)2 + k

Vertex form of a quadratic function

4.4 = a(0 − 1.5) + 24 −19.6 = a(−1.5)

Substitute x = 0, y = 4.4, h = 1.5, and k = 24

Subtract 24 from both sides

−19.6 = a(2.25)

Evaluate the exponent

−8.7 = a

Divide by 2.25 on both sides

The quadratic equation in vertex form that models the rock’s height above the ground as a function of time is y = −8.7(x − 1.5)2 + 24.

Idea summary The vertex form of a quadratic function is:

f (x) = a(x − h)2 + k

(h, k) a

coordinates of the vertex scale factor

Changing the values of a, h, and k will transform the graph in different ways: • • •

a: vertical stretch or compression h: horizontal translation k: vertical translation

Completing the square Interactive exploration Explore online to answer the question

mathspace.co Use the interactive exploration in 7.03 to answer the question. 1.

416

What patterns do you notice when working through the process of completing the square?

Mathspace Virginia SOL Algebra 1 mathspace.co

Completing the square is a method we use to rewrite a standard quadratic expression in vertex form. Completing the square allows us to rewrite our equation so that it contains a perfect square trinomial. A perfect square trinomial takes on the form A2 + 2AB + B2 = ( A + B)2, which is the same format we need to have an equation in vertex form. For quadratic equations where a = 1, we can write them in perfect square form by following these steps: 1 2

Rewrite the x term

Add and subtract

Factor the perfect square trinomial

Match the completed square to vertex form

to keep the equation balanced

If a ≠ 1, we can ﬁrst divide through by a to factor it out. The quadratic equation, when rewritten by completing the square, becomes the vertex form of a quadratic equation.

Example 5 Consider the following equation:

y = x2 − 4x + 6

a Rewrite the equation in vertex form by completing the square.

Create a strategy We’ll follow the standard complete the square method and stop working once our equation is in vertex form, y = a(x − h)2 + k.

Apply the idea Original equation

Add and subtract

Evaluate the exponents

Factor x2 − 4x + 4

Evaluate the subtraction

We’ve completed the square and the equation is now in vertex form.

Reﬂect and check We must add and subtract

to the right side of the equation so that the value of the equation does not change.

7.03 Quadratic functions in vertex form mathspace.co

417

b Determine the vertex of the quadratic function and if it is a minimum or maximum.

Create a strategy

Apply the idea

Use the vertex form of the quadratic function from part (a) to determine its vertex and whether it is a minimum or maximum value.

The quadratic function in vertex form is y = (x − 2)2 + 2, meaning the vertex is located at the point (2, 2). Since the value of a is 1, we know that the graph opens upward and the vertex is a minimum value.

c Sketch the graph of the parabola.

Create a strategy We can ﬁnd key points of our parabola to sketch it. In part (a) we found the x-value of the vertex, and in part (b) we found the vertex form of our equation which shows us the y-value of our vertex. We can also use the vertex form to consider the x- and y-intercepts.

Apply the idea The y-value of the vertex is 2 since the vertex form of the equation is y = (x − 2)2 + 2. This means that the vertex is located at (2, 2). We know the scale factor a is 1, so the parabola opens upward. Since the vertex is above the x-axis and opens up, the graph will not cross the x-axis and there are no x-intercepts. Find the y-intercept: y = (x − 2)2 + 2 2

Vertex form of the equation

y = (0 − 2) + 2

Substitute x = 0

y = 6

Evaluate

Therefore, the y-intercept is (0, 6).

We can use the vertex and y-intercept to sketch the parabola, remembering there is an axis of symmetry at the vertex.

7 5 4 3 2 1

x 1

Idea summary A quadratic function in standard form can be converted to vertex form by completing the square: 1 2

Rewrite the x term

Add and subtract

Factor the perfect square trinomial

Match the completed square to vertex form

If a ≠ 1, we can ﬁrst divide through by a to factor it out.

418

Mathspace Virginia SOL Algebra 1 mathspace.co

to keep the equation balanced

Practice What do you remember? 1

Determine the axis of symmetry of this quadratic function.

8 6 4 2

−6 −4 −2 −2

−4 −6 −8

Write an equation for each quadratic function in vertex form. a

−2 −1 −2

12 10

−8

4 2

2 3 4 5 6 7 8

−6

x 1

−4

−8 −7 −6 −5 −4 −3 −2 −1 −2

−10

x 1

−12

x h(x) 0 −6 1 −1 2 2 3 3 4 2 5 −1 6 −6

−14

x −6 −5 −4 −3 −2 −1 0

j(x) 11 6 3 2 3 6 11

For the following quadratic equations: i

Rewrite the equation in vertex form by completing the square.

Identify the coordinates of the vertex.

y = x2 − 4x + 3

y = −x2 + 6x − 2

y = −x2 − 5x – 4

y = −2x2 + 8x − 7

Write the equation of the transformed graph in vertex form. a

y = x2 is horizontally translated 10 units to the right and vertically translated 2 units up

y = x2 is horizontally translated 9 units to the left and is vertically stretched by a factor of 9 units

y = x2 is reﬂected across the x-axis and vertically translated 8 units down

7.03 Quadratic functions in vertex form mathspace.co

419

For each of the following, state whether the transformation from the parent function f (x) = x2 to the function g(x) is a translation up, down, left, or right: g(x) = (x − 8)2

a 6

g(x) = x2 − 5

g(x) = (x + 9)2

g(x) = x2 + 0.75

Consider the following graph of a function.

Select the equation that represents the function: y = −(x + 5)2 + 25

y = (x + 5) − 25

5 x

y = (x − 5)2 + 25

−12 −10 −8 −6 −4 −2

y = −(x + 5) − 25

2 −5

−10 −15 −20 −25

Consider the table of values of a function. x y

1 −6

2 0

3 2

4 0

5 −6

Sketch a graph of the function labeling any points of interest. 8

Consider the function Sketch a graph of the function labeling any points of interest.

Let’s practice 9

For each of the graphs: i

Describe the transformation from y = x2.

Write the equation. y 8

6 6

2 −2

x 2

−2 −4

420

−4

−6 −4 −2

−2

x −2

Mathspace Virginia SOL Algebra 1 mathspace.co

x −6

−4

−2

−6 −4 −2

−2 −4

y 6 4 2 x −2

x 2

−2

−4

For each equation: i

Describe the transformations from y = x2.

Sketch the graph of the parabola.

y = (x − 3)2 − 4

y = 3(x + 1)2 + 4

y = −(x + 3)2 − 6

y = −0.5(x − 6)2 + 1

Each of the following describes the transformations of a function from the parent function f (x) = x2. Write the equation of each transformed function in vertex form. a

Vertical stretch by a scale factor 2 and translated right 1 unit and down 3 units.

Reflection over the x-axis, vertically compressed by a scale factor of 3, and translated right 2 units and up 2 units.

Reflection over the x-axis, vertically stretched by a scale factor of 3, and translated left 5 units and down 4 units.

Vertically stretched by a scale factor of 3, translated left 1 unit, and translated up 2 units.

Sketch the graph of each parabola. a

14 12 10 8 6 4 2

y = (x + 3)2 − 9

y = −(x − 2)2 + 1

y = −(x + 4)2 − 2

Write the vertex form equation that represents each of the following graphs: a

4 2 −2

−2

x 2

−2

−4

−6

−8

2 −2

y 2

x −6 −4 −2

−2

−4

−6

−8

7.03 Quadratic functions in vertex form mathspace.co

421

For each of the following: i a

Rewrite the equation in vertex form. 2

y = x − 4x + 2

y = x + 10x

Sketch the graph of the parabola.

y = x2 + 4x – 4

y = x2 + 5x + 3

Explain how the maximum or minimum value of a quadratic function can be found by completing the square.

For each of the following quadratic functions: i

State whether the transformation from f (x) to g(x) is a horizontal translation, vertical translation, vertical stretch, or vertical compression by k units.

State the value of k.

10 g ( x ) 8

10 8

4 2

g(x) −10 −8 −6 −4 −2 −2 −4 −6

x f ( x) 2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8

10 8 6 4 2 −10 −8 −6 −4 −2 −2 −4 −6

f ( x) x 2 4 6 8 10

−8 −10

−10

f ( x) g( x) x 2 4 6 8 10

−8 −10

For each quadratic function: i

Determine the coordinates of the vertex.

Determine the x-intercept.

iii

Determine the y-intercept.

Draw a graph of the quadratic function.

m(x) = (x − 3)2

p(x) = −(x − 1)2 – 7

x-intercept(s)

y-intercept

iii

vertex

y = (x − 1)2 − 4

y = (x + 2)2 − 1

y = (x − 3)2 + 5

r(x) = 3(x + 5)2

y = −2(x − 2)2 + 3

Range

y = (x − 4)2 − 2

y = −0.5(x + 2)2 + 1

For each of the following quadratic functions, find the: a

Domain 2

y = 2(x − 3) + 4

y = −3(x + 1) + 5

Find the range when the domain is {−6, −1, 0, 5, 7} for each of the following quadratic functions: a

422

n(x) = (x + 4)2 − 1

For each of the following quadratic functions, find the:

y = 3(x + 1)2 − 2

Mathspace Virginia SOL Algebra 1 mathspace.co

y = −2(x − 3)2 + 4

y = 4(x + 2)2 + 1

y = −3(x − 4)2 − 5

For example, if we have the function g(x) = 3x2 + 12x − 15 where a = 3, b = 12, and c = −15 we can start by ﬁnding the x-coordinate: Equation for the x-coordinate of the vertex

Substitute a = 3 and b = 12

Evaluate the multiplication

Evaluate the division

We can substitute the x-coordinate of the vertex into the original equation in order to ﬁnd the y-coordinate of the vertex. g(x) = 3x2 + 12x − 15

Original function

= 3 (−2)2 + 12 (−2) − 15

Substitute x = −2

= 3 (4) + 12 (−2) − 15

Evaluate the exponent

= 12 − 24 − 15

Evaluate the multiplication

= −27 Evaluate the subtraction The coordinates of the vertex of g(x) are (−2, −27). We can conﬁrm this by looking at the graph: −6 −5 −4 −3 −2 −1

y 1

−5 −10 −15 −20 − 25 (−2, −27)

We can also see here that the axis of symmetry is the line:

The axis of symmetry always passes through the vertex.

Example 1 For the quadratic function y = 3x2 − 6x + 8: a Identify the axis of symmetry.

Create a strategy

Apply the idea

We will use the formula

so we need to identify

Equation for axis of symmetry

the values of a and b from the equation. a = 3, b = −6

Substitute b = −6 and a = 3

Evaluate the multiplication

Evaluate the division

Evaluate the multiplication

The axis of symmetry is x = 1.

426

Mathspace Virginia SOL Algebra 1 mathspace.co

b State the coordinates of the vertex.

Create a strategy Once we have the x-coordinate of the vertex from the axis of symmetry, we can substitute it into y = 3x2 − 6x + 8 and evaluate to get y. From part (a), we know that the axis of symmetry is x = 1 so the x-coordinate of the vertex is x = 1.

Apply the idea y = 3x2 − 6x + 8 Given equation Substitute x = 1 y = 3(1)2 − 6(1) + 8 y = 3(1) − 6(1) + 8 Evaluate the exponent y = 3 − 6 + 8 Evaluate the multiplication y = 5

Evaluate the subtraction and addition

The vertex is (1, 5). c State the coordinates of the y-intercept.

Create a strategy

Apply the idea

Since the y-intercept occurs when x = 0, substitute x = 0 into the equation and evaluate y.

In this case, the value of c in the equation is 8.

When we are given an equation in standard form, the y-value of the y-intercept will be y = c.

So we have that the coordinates of the y-intercept are (0, 8).

d Draw a graph of the corresponding parabola.

Create a strategy We have all the key features we need to create a graph. For more accuracy, we can use the axis of symmetry and y-intercept to ﬁnd another point. This point will be a reﬂection of the y-intercept across the axis of symmetry. We know that the parabola will open upwards because a > 0.

Apply the idea Axis of symmetry: x = 1

Vertex: (1, 5) y-intercept: (0, 8)

Another point: (2, 8) 6 4 2 x −1

Reﬂect and check From the graph we can identify that the vertex form of the equation would be: y = 3(x − 1)2 + 5

7.04 Quadratic functions in standard form mathspace.co

427

Example 2 Naomi is playing a game of Kapucha Toli, where to start a play, a ball is thrown into the air. Naomi throws a ball into the air from a height of 6 feet, and the maximum height the ball reaches is 12.25 feet after 1.25 seconds. a Sketch a graph to model the height of the ball over time.

Create a strategy To sketch a graph, we’ll use key points found by using the given information. We’ll also use the units which are given, being feet and seconds. We will let x represent the time since the ball was tossed in seconds. We will let y represent the height of the ball in feet.

Apply the idea It’s given that the ball is thrown from a height of 6 feet. This means that at 0 seconds, the height of the ball is 6 feet. So our y-intercept is (0, 6). We’re told the maximum height of the ball is at 12.25 feet after 1.25 seconds. The maximum height will occur at the vertex of the graph, so the vertex is (1.25, 12.25). This also means that our axis of symmetry is x = 1.25. We can use the axis of symmetry to determine a second point on the graph, the point across the axis of symmetry from the y-intercept. The point is (2.5, 6). We can sketch our graph by plotting the y-intercept, the vertex, and the point found with our axis of symmetry. Now we need to identify an appropriate scale. Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 1 Time in seconds (x) −1

We know that our graph will not go above y = 12.25 feet and that any part of the graph that goes below the x-axis will not be viable, so graphing −1 ≤ y ≤ 13 going up by 1 will show the full picture. We know that time starts at x = 0 and the ball is on the way back down at x = 2.5, so graphing 0 ≤ x ≤ 4 going up by 1 or 0.5 should be suﬃcient. This is an appropriate way to label the axes.

Now we can graph the height of the ball over time. Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 1 Time in seconds (x) −1

428

Mathspace Virginia SOL Algebra 1 mathspace.co

b Predict when the ball will be 3 feet above the ground.

Create a strategy We can use the sketch of our graph to predict when the ball will be at 3 feet.

Apply the idea We can draw a horizontal line from y = 3 across until we reach the graph. After that, we can draw vertical line until we reach the x-axis to determine after how many seconds the ball is at 3 feet.

We hit the x-axis around x = 2.7. Therefore, the ball is 3 ft above the ground after about 2.7 seconds.

Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 Time in seconds (x) 1 −1

Reﬂect and check When reading from a graph, we often have to estimate. Any prediction between 2.6 and 2.9 would be reasonable in this case. c Write a quadratic equation in standard form to model the situation.

Create a strategy To write the equation, we can use the key points and the graph we’ve sketched in previous parts.

Apply the idea Since we know 3 points, we can use the standard form and substitution in order to solve for a, b, and c for our standard form quadratic equation which is of the form y = ax2 + bx + c. We know that c represents the y-value of the y-intercept which is (0, 6): y = ax2 + bx + 6 Now we can substitute our other two points to solve for a and b. Next, we can substitute in (1.25, 12.25). Standard form of a quadratic with c = 6 y = ax2 + bx + 6 12.25 = a(1.25)2 + b(1.25) + 6

Substitute (1.25, 12.25)

12.25 = 1.5625a + 1.25b + 6

Evaluate the exponent

6.25 = 1.5625a + 1.25b Subtract 6 from both sides Since we have two unknowns, we’ll have to use our ﬁnal point to create a second equation. We’ll now substitute in (2.5, 6). Standard form of a quadratic with c = 6 y = ax2 + bx + 6 6 = a(2.5)2 + b(2.5) + 6

Substitute (2.5, 6)

6 = 6.25a + 2.5b + 6 Evaluate the exponent 0 = 6.25a + 2.5b Subtract 6 from both sides

7.04 Quadratic functions in standard form mathspace.co

429

Now we have two equations with two unknowns. We can solve this system using the substitution method. Let’s ﬁrst isolate b in our second equation.

Second equation

Subtract 6.25a from both sides

Divide by 2.5 on both sides

Evaluate the division Now we can use the this in our ﬁrst equation, letting b = −2.5a. 6.25 = 1.5625a + 1.25b First equation 6.25 = 1.5625a + 1.25(−2.5a)

Substitute b = −2.5a

6.25 = 1.5625a − 3.125a Evaluate the multiplication 6.25 = −1.5625a Combine like terms −4 = a Divide both sides by −1.5625 Therefore a = −4. Finally, we can use b = −2.5a to solve for b. b = −2.5a b = −2.5(−4)

Substitute a = −4

b = 10

Evaluate the multiplication

Therefore b = 10. Now it’s time to piece it all together. Since a = −4, b = 10, and c = 6, we know that our equation in standard form is: y = −4x2 + 10x + 6

Reﬂect and check An alternative and simpler solution is to use the vertex to write it in vertex form and then using the intercept to solve for a. Vertex form is y = a(x − h)2 + k. The vertex is (1.25, 12.25), this gives us: y = a(x − 1.25)2 + 12.25 We can then substitute in the point (0, 6) and solve for a.

Vertex form of a quadratic with vertex (1.25, 12.25)

Substitute in (0, 6)

Evaluate the parentheses

Subtract 12.25 from both sides

Divide by 1.5625 on both sides

Evaluate the division

So now we have the equation: y = −4(x − 1.25)2 + 12.25 Now we need to convert to standard form: Equation in vertex form y = −4(x − 1.25)2 + 12.25 y = −4(x2 − 2.5x + 1.5625) + 12.25

Expand the binomial

y = −4x + 10x − 6.25 + 12.25 Distributive property Combine like terms y = −4x2 + 10x + 6 We get the same answer of: y = −4x2 + 10x + 6.

430

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 y

Write the standard form equation of the function shown on the graph.

8 6 4 2 −2 −2

x 2

10 12

−4

Create a strategy Use the vertex to ﬁrst write the equation in vertex form, then use another point on the parabola to solve for a, and ﬁnally convert to standard form.

Apply the idea Vertex form is y = a (x − h)2 + k. The vertex is (8, −2), this gives us: y = a(x − 8)2 + (−2) or y = a(x − 8)2 − 2 We can then substitute in the point (4, 6) and solve for a. Vertex form equation

Substitute in x = 4 and y = 6

Evaluate the subtraction

Evaluate the exponent

Add 2 to both sides

Divide by 16 on both sides

Simplify the fraction

Substituting

we get the equation:

Now we need to convert to standard form:

Equation in vertex form

Expand the binomial Distributive property

Combine like terms

is the standard form equation of the function on the graph.

7.04 Quadratic functions in standard form mathspace.co

431

Reﬂect and check Alternatively, we could have used the zeros and factored form. Using the zeros of 6 and 10, we can write the equation in factored form y = (x − x1) (x − x2).This gives us: y = a(x − 6)(x − 10) We can then substitute in the point (4, 6) and solve for a. Factored form equation

Substitute x = 4 and y = 6

Evaluate the subtraction

Evaluate the multiplication

Divide by 12 on both sides

Simplify the fraction

So now we have the equation:

Now we need to convert to standard form:

Equation in factored form

Multiply the binomials Combine like terms

Distributive property

Example 4 The whale jumps out the water at 3 seconds and reenters the water after 6.5 seconds. The whale reaches a maximum height of 49 feet after 4.75 seconds. Determine the equation in standard form that models the whale’s jump.

Create a strategy If we think of the water level as the x-axis, then the moments where the whale exits and reenters the water would represent the x-intercepts. The maximum height is the vertex of the parabola formed by the whale’s jump path.

Apply the idea Using the x-intercepts of 3 and 6.5, we can create the following equation, where x represents the time in seconds and y represents the height of the jump in feet: y = a(x − 3)(x − 6.5)

432

Mathspace Virginia SOL Algebra 1 mathspace.co

Next, we can substitute the values of the maximum point, which occurs at (4.75, 49), to ﬁnd the value of a. y = a (x − 3) (x − 6.5)

Equation for whale’s path

49 = a(4.75 − 3)(4.75 − 6.5)

Substitute y = 49 and x = 4.75

49 = a(1.75)(−1.75)

Evaluate the subtraction

49 = −3.0625a

Evaluate the multiplication

−16 = a

Division property of equality

Now, we know the factored form of the equation that models the whale’s jump: y = −16(x − 3)(x − 6.5) The last step is to get it into standard form. We can do this by multiplying all the factors together. y = −16 (x − 3) (x − 6.5)

Equation for whale’s path

Distributive property

Combine like terms

y = −16(x − 6.5x − 3x + 19.5) = −16x + 104x + 48x − 312 = −16x + 152x − 312

The equation in standard form that models the whale’s jump is y = −16x2 + 152x − 312.

Reflect and check Notice that the parabola formed by the whale opens downward. If we did not have another point to help us find the value of a, our parabola would have been facing upward. Using the vertex helped us find a negative value for a which is what made the parabola face downward.

Idea summary The standard form of a quadratic equation highlights the y-intercept of a quadratic function.

y = ax2 + bx + c a scale factor b linear coeﬃcient c y-value of the y-intercept The axis of symmetry is the line:

The axis of symmetry is also the x-coordinate of the vertex. To ﬁnd the y-coordinate, you substitute the x-coordinate back into the original function. Therefore, the coordinates of the vertex are:

7.04 Quadratic functions in standard form mathspace.co

433

Practice What do you remember? 1

Describe the basic shape of a parabola.

Given that the standard form of a quadratic is y = ax2 + bx + c: i

What does changing the value of c do?

State the coordinates of the vertex.

y = −x2 − 4x − 9

For each quadratic function: i

Determine the axis of symmetry.

iii

State the coordinates of the y-intercept.

y = x2 − 4x + 8

Rewrite each equation in standard form. y = (3x − 1) (2x + 1)

a 4

What does the sign of a tell us?

y = 2(x − 4)2 + 1

y = (x − 6) (x + 6)

y = 5(x − 1)2 − 5

f (x) = 4x2 − x + 1

For each equation: i

Find f (3).

Find f (−5).

iii

Find

f (x) = 2x2 + 3x − 4

f (x) = −x2 + 2x + 10

f (x) = −3x2 − 5

Consider the table of values of a function. x y

−6 −8

−5 −9

−4 −8

−3 −5

−2 0

−1 7

Select the graph that could represent the function: A

3 −12 −10 −8 −6 −4 −2 −3

−9

−12

−15

−18

434

Mathspace Virginia SOL Algebra 1 mathspace.co

10 12

−6

−9

−12 −10 −8 −6 −4 −2 −3 −6 −9

−2 −3

−6

x 2

3 −2 −3 −6 −9

x 2

10 12

Let’s practice 6

Consider the following graph of a function. 2

Select the equation that represents the function: A

y = x2 − 6x + 4

y = x + 6x − 4

y = x2 − 6x − 4

y = x2 + 6x + 4

−7 −6 −5 −4 −3 −2 −1 −1

x 1

−2 −3 −4 −5 −6

For each quadratic function: i

Determine the axis of symmetry.

State the coordinates of the vertex.

iii

State the coordinates of the y-intercept.

Draw a graph of the corresponding parabola.

y = x2 − 2x + 5

y = 2x2 + 24x + 75 y = 4x2 − 64

y = −x + 6x − 8

y = 2x2 + 2x + 9

For each quadratic function: i

State the coordinates of the y-intercept.

iii

Draw a graph of the corresponding parabola.

a c e 9

Determine the coordinates of the x-intercept(s).

y = x2 − 3x − 10

y = x2 − 9

y = 4 − 3x − x2

y = 2x2 + 12x + 18

y = −6x2 + 10x + 4

y = 4x + 8x − 5

Consider the function y = −2x2 + 10x − 3 where x represents time in minutes and y represents the height of a ball from the ground in feet. a

Draw a graph of the corresponding parabola. Make sure to label the axes.

Using your graph, predict when the ball will be 3 ft above the ground.

7.04 Quadratic functions in standard form mathspace.co

435

Write an equation in standard form to represent the following parabola. a

−1−2 −4 −6 −8 −10

14 12 10 8 6 4 2

x 1

−4 −6 −8 −10

14 12 10 8 6 4 2

4 2 −4 −3 −2 −1 −2

x 1

−4 −3 −2 −1 −2 −4 −6 −8 −10

−4 −6 −8 −10

−4 −3 −2 −1−2

2 3 4 5 6 7 8

Darnell throws a bag of cookies to his friend, Ike, from a height of 3.25 ft. The cookies reach a maximum height of 19.25 ft, 1 second after being thrown. a

Determine the equation in standard form that represents the situation.

Graph the corresponding parabola. Make sure to label the axes.

If Ike caught the cookies after 2 seconds, determine the height of the catch.

Carliss is starting a summer car wash business. Her profit function relates the total profit to the rate she charges for each car wash. The rate and profit are in dollars. P(x) = −x2 + 50x − 95

Carliss wants to make at least $525 this summer. Determine if she could make enough money based on her quadratic proﬁt model. Explain your reasoning.

Draw a graph of P(x).

The table shows points on a quadratic function. 2

Another quadratic function has the equation B(x) = x − 2x − 8.

x A(x)

Determine what the quadratic functions have in common and what is different.

436

Mathspace Virginia SOL Algebra 1 mathspace.co

−3 −5

−2 −8

−1 −9

0 −8

2 0

Beau is competing at his high school swim meet and dives off a springboard that is 3 ft above the pool surface. He reaches a maximum height of 8 feet after 1 second. a

Determine the equation in standard form that represents Beau’s diving path.

Determine after how many seconds Beau will be at the same height as the springboard again.

If the pool is 12 feet deep, determine how many seconds after jumping it will take Beau to reach the bottom of the pool. Explain your reasoning.

Fill in the boxes to create a quadratic equation with the lowest possible minimum value. Use the digits 1 to 5 at most once. y = ⬚ x2 + ⬚ x + ⬚

A Happy Birthday banner is modeled by the quadratic function h(x) = 0.2x2 − x + 1.25 where x is the distance from the left side of the banner and h, is the height above the ground. Both x and h are in feet. Currently, the lowest part of the banner touches the ground. Immanuel is trying to create a virtual card and needs to know how to affect the look of the banner by changing parts of the quadratic function.

438

Determine a domain and range for the quadratic function that models the initial Birthday banner. Explain your reasoning.

Describe to Immanuel how changes to the quadratic function affect the shape of the Birthday banner.

Mathspace Virginia SOL Algebra 1 mathspace.co

7.05 Compare linear, quadratic, and exponential functions After this lesson, you will be able to… compare key characteristics of linear, quadratic, and exponential functions using graphs and tables.

Comparing functions In this lesson, we will use our prior knowledge of linear, quadratic, and exponential functions to identify key features and compare various functions represented in different ways.

Exploration Consider the table: 1 2 3 5 1.

y = 3x 3 9 27 243

y = 3x2 3 12 27 125

y = 3x 3 6 9 15

Compare the three functions and how they change as x increases.

The way a function is represented can affect the characteristics we are able to identify for the function. Different representations can highlight or hide certain characteristics. Remember that key features of functions include: • how the function increases or decreases • domain and range • vertex • x- and y-intercepts • maximum or minimum value(s) One way to compare functions is to look at growth rates as the x-values increase over regular intervals. In order to compare the growth rates of quadratics with those of exponential or linear functions, we will look only at the half of the quadratic that is increasing. y

30 g (x) f (x)

h (x) 10 x −2

−1

Notice starting at x = 0, g(x) is greater than h(x) and is increasing at a greater rate. But, as x continues to increase, the quadratic function g(x) is increasing at a slower rate than the exponential function, and eventually the exponential function will overtake the quadratic function. Notice that no matter what the intercepts are, an exponential growth function will always exceed a linear or quadratic growth function as values of x become larger. 7.05 Compare linear, quadratic, and exponential functions mathspace.co

439

Example 1 Which of the following functions increases the fastest for very large values of x? • y=9⋅x • y = 3x • y = 2x2 • y = 4x

Create a strategy Remember that exponential functions (b > 1) will always have a greater rate of change when compared to linear and quadratic functions as x increases toward infinity. Then, we must compare the constant factor b in the equation y = a ⋅ bx.

Apply the idea

Reﬂect and check

As x increases toward infinity, we know that our greatest rate of change is from one of the exponential functions, y = 3x or y = 4x. Remembering our lesson on characteristics of exponential functions, a greater constant factor, b, will result in a greater rate of change as x continues to increase. Therefore, y = 4x increases the fastest for very large values of x.

How would you approach this problem if exponential equations had 2 different leading coefficients? For example, would y = 2 ⋅ 4x or y = 4 ⋅ 2x have a greater rate of change as x increases toward infinity?

Example 2 Consider the functions shown. Assume that the domain of f is all real numbers. • Function 1:

• Function 2:

−1

f (x)

−3.75

−2

−0.25

1.5

3.25

6.75

10 8 6

g (x)

4 2 −8 −6 −4 −2 −2

x 2

−4

a Determine which function has a higher y-intercept.

Create a strategy Remember that the y-intercept of a function occurs when x = 0. We can use this to evaluate the y-intercept of f and identify the y-intercept of g.

Apply the idea For f, we can see from the table that f (0) = −2. For g, we can see from the graph that g(0) = −3. So the y-intercept of f is the point (0, −2) and the y-intercept of g is the point (0, −3), and therefore f has a higher y-intercept.

440

Mathspace Virginia SOL Algebra 1 mathspace.co

b Determine which function will be greater as x gets very large.

Create a strategy

Apply the idea

We can consider how quickly each function changes to get an idea of how it will behave for very large values of x.

We can see that f (x) has a constant rate of change (slope) of 1.75. This is a linear function. From the graph, we can see that g(x) increases at an increasing rate as x increases. So, we can see that Function 2 will eventually surpass Function 1 as x gets very large.

Example 3 Consider functions representing three options to earn money one of the following ways: You are given $2 each day

Option 2 Days

Total Amount

$16

$25

$36

Option 3 Total Amount (in dollars)

Option 1

60 50 40 30 20 10 1

2 3 4 5 6 7 8 9 10 Days

Note: Option 3 starts with $2 on day one and doubles each day after this. a Find the equation that represents each option, where x is the number of days that have passed.

Create a strategy For each option, we can consider how the total amount of money changes as the days progress and derive an equation to represent the relationship.

Apply the idea We can see that Option 1 has a constant rate of change regardless of the interval we considered. So, Option 1 can be represented by the linear function, f (x) = 2x. Now, observing the table of values for Option 2, we can see that the total amount is just the square of the number of days passed. So, Option 2 can be represented by the function f (x) = x2. Finally, the relationship for Option 3 is represented in the graph, but also described to us. Since we are told that the function starts at $2 and is doubled each day, we can see that Option 3 is just represented by the function f (x) = 2x.

Reﬂect and check If the relationship between the days passed and the total amount weren’t directly obvious in Option 2, we could have tested the data provided in the table to rule out a linear or exponential relationship. For a linear relationship, the rate of change between any two points must be equal. We can check that this wasn’t true for Option 2. So, we could have then tested if it represented an exponential relationship. For an exponential relationship, the ratio of between two points, a unit apart, must be equal. We can see that for Option Therefore, we could see that Option 2 represented neither a linear or exponential relationship.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

441

b Find the value of each option at 8 days, 12 days, and 14 days.

Create a strategy Construct a table of values with the amounts of money gained with each option.

Apply the idea Days

Option 1 Total

Option 2 Total

Option 3 Total

$16

$10

$25

$32

$12

$36

$64

$14

$49

$128

$16

$64

$256

$18

$81

$512

$20

$100

$1024

$22

$121

$2048

$24

$144

$4096

$26

$169

$8192

$28

$196

$16 384

At 8 days, Option 1 will make $16, Option 2 will make $64, and Option 3 will make $256. At 12 days, Option 1 will make $24, Option 2 will make $144, and Option 3 will make $4096. At 14 days, Option 1 will make $28, Option 2 will make $196, and Option 3 will make $16 384.

Reﬂect and check We could calculate the total amount of money on days 8, 12 and 14 using the functions found in part (a), instead of constructing a table.

c Determine which option will be greater for larger and larger values of x.

Create a strategy Use the table comparison from part (b) to determine which option will be greater for larger and larger values of x.

Apply the idea

Reﬂect and check

As x gets larger and larger, we can see that Option 3, the exponential option, will be far greater than Options 1 or 2.

An exponential function will always exceed a linear or quadratic function as values of x become larger.

442

Mathspace Virginia SOL Algebra 1 mathspace.co

Idea summary It is important to be able to compare the key features of functions whether they are represented in similar or different ways: • • •

• •

domain and range x- and y-intercepts maximum or minimum value(s)

how the function increases or decreases vertex

Practice What do you remember? 1

For each pair of functions, determine which function y is changing more rapidly: • Function 1: • Function 2: a x 0 1 2 3 y 3 10 17 24 b

−1 −1

x y

0 3

1 7

• Function 3: • Function 4: y 5

1 1

−5 −4 −3 −2 −1 −1

2 11

−5 −4 −3 −2 −1 −1

2 3 4 5

−2

−3 −4

−5

2 3 4 5

For each pair of table and its graph, determine whether a linear or quadratic function could represent it: a

x −3 −2 −1 0 1 y 3 6 7 6 3

−2

−1

2 1

−2

−5 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

x 1

−2 −3 −4

7.05 Compare linear, quadratic, and exponential functions mathspace.co

443

For large values of x, does the function f (x) = 3x − 1 or g(x) = 3x2 − 1 increase at a faster rate?

The graphs of f (x), g(x) and h(x) are shown. Use the graphs to complete the following:

Identify the graphs as exponential, linear, or quadratic.

Evaluate each function for x = 2.

Which function do you think will have the largest value at x = 100? Explain

Approximate the value of x for each function when the function value is 12.

h (x) f (x)

14 12 g (x)

8 4 2

x 1

The population of two different bacteria, labeled J and K, are given by the shown table of values: Bacteria J: t (Time in days) P (Population)

0 1

1 120

2 480

3 1080

0 1

1 50

2 2500

3 1.25 × 105

4 1920

Bacteria K: t (Time in days) Q (Population)

4 6.25 × 106

The population P of bacteria J at time t can be modeled using the general equation P(t) = at2, where a ≥ 0. By using the table of values, graph P(t) for t > 0.

The population Q of bacteria K at time t can be modeled using the general equation Q(t) = bt, where b > 1. By using the table of values, graph Q(t) for t > 0.

Determine which population of bacteria is growing faster.

Let’s practice 6

For each pair of functions, determine which has the greater y-intercept: • Function 1: • Function 2: a y = 4x2 + 6x + 3 y = 4x + 6 b

• Function 3: • Function 4: y 5

1 −5 −4 −3 −2 −1 −1

444

x 1

2 3 4 5

−5 −4 −3 −2 −1 −1

−2

−3 −4

−5

• Function 1: The line with a slope of 4 that crosses the y-axis at (0, 6). • Function 2: The parabola given by the equation y = x2 + 4.

Mathspace Virginia SOL Algebra 1 mathspace.co

x 1

2 3 4 5

• Function 3:

• Function 4:

x 2 4 6 y 2 −2 −6

y 5 4 3 2 1 x −2

• Function 5: x y

2 19

4 35

−1

• Function 6: y = 4x + 6

6 51

Consider each pair of functions: • Function A: y = −4x + 3

• Function B: y 12 10 8 6 4 2 −4 −2 −2

x 2

10 12

−4

Determine how many x-intercept(s) function A has.

Determine how many x-intercept(s) function B has.

Which function has a smaller value for f (4)?

The parabola C is given by

and the exponential function D is given by y = −8x + 1.

Graph the functions on the same coordinate plane.

Determine which function has the lower y-intercept.

The parabola E is given by y = 12(x − 1)2 − 4 and the exponential function F is given by y = 3x + 4. a

Graph the functions on the same coordinate plane.

State which function increases at a faster rate for very large values of x.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

445

The parabola J is given by y = −9x2 + 20. The table shows the function values for exponential function K: x y

−2

−1

135

Determine if the exponential function K is increasing or decreasing.

Graph the functions J and K on the same coordinate plane.

Determine what transformation we can apply to the function y = 3x to produce function K.

Which function has a larger value of y when x = 6? What is the that y-value?

The line P is given by

and the parabola Q is given by y = −(x − 1) (x − 4).

Graph the line P and the parabola Q on the same coordinate plane.

Identify how many times P and Q intersect.

Identify which function has the higher function value at x = 0.

The table of values for the function P and for the function Q are provided. Function P:

Function Q:

x x −2 −1 0 1 2 y y 9 6 3 0 −3

0 6

1 3

2 2

Determine what type of functions Function P and Function Q are.

Graph the functions on the same coordinate plane.

As x gets very large, determine which function will have the greater value.

3 3

4 6

Let’s extend our thinking 13

446

Some friends decide to go camping for the weekend. They cannot all ﬁt in one car so some of them catch a bus to the campground, which is 450 km from home. Those in the car started driving at 8:00 AM and arrived at the campground at 3:30 PM, driving at a constant speed. The bus also drives at a constant speed and takes the same route as the car. Its distance in kilometers ( y) from home x hours after leaving is given by the equation y = 71x. a

Determine the speed of the car, in kilometers per hour.

Determine the speed of the bus, in kilometers per hour.

Determine which vehicle was traveling faster.

Consider the functions f (x) = 2x, g(x) = 2x2 and h(x) = 2x for x ≥ 0. a

Describe the pattern of how each of the functions increases. Explain how you identiﬁed the patterns.

Compare how the three functions increase as x gets very large.

Mathspace Virginia SOL Algebra 1 mathspace.co

Two companies Crest Corporation and Mint Corporation are operating mines. Crest Corporation’s operations are such that the total amount mined by the nth week is given by the equation C = 10n2. The total amount mined by Mint Corporation over time is shown in the graph. Amount mined (thousands of metric tons) 10 8 6 4 2 5

Week 60

If mining operations for both companies were to only last at most a year, determine which company will have mined the most minerals in that time.

Determine if the two corporations will ever mine the same amount at the same time after the ﬁrst week.

At the point of intersection, the total quantity of minerals remaining in both mines is equal. If both mining companies continue to operate in the same way indeﬁnitely, determine which company will exhaust their mine ﬁrst. Explain your reasoning.

During a sudden viral outbreak, scientists must decide between two antivirals to try and control the situation. In a laboratory, they apply Adravil and Felicium to two samples of the virus, each containing 200 microbes. They keep track of the number of microbes in each sample, and notice that the number of microbes using Adravil is increasing by a constant amount of 12 each hour. The table shows the results for Felicium. Number of hours (t) Number of microbes using Felicium

0 200

3 600

6 1800

9 5400

Determine which antiviral will better control the number of microbes. Explain your choice.

The new antiviral, Tretonin, shows the preliminary results: Number of hours (t) Number of microbes using Tretonin

0 200

3 202

6 208

9 218

If the trends from the ﬁrst 9 hours continue in the future, determine which treatment will be better of the short-term, and which will be better over the long-term. Justify your answer.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

447

8 Quadratic Equations Big ideas • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation.

Chapter outline 8.01 8.02 8.03 8.04 8.05

Solve quadratics using graphs and tables Solve quadratics by factoring Solve quadratics using square roots Solve quadratics using the quadratic formula Solve quadratics using appropriate methods

450 461 467 478 491

In the graph and table shown, we see x = 0 is the only solution. This is because x = 0 is the only value that makes x2 = 0 true. If we tried to ﬁnd the solution to x2 = −2 there would be no real solutions, because squaring any non-zero real number will give a positive result. The roots, or zeros, in a quadratic function occur when f (x) = 0. The method we will use to solve a problem such as x2 = 4 is by creating an equivalent equation by rearranging it so it is equal to 0 and then identifying the x-intercepts. x2 = 4

Given equation

x −4=0

Subtract 4 from both sides

Next, we can replace the 0 in the equation with y to get y = x2 − 4. The graph of this equation is the graph of y = x2 shifted down 4 units so the graph of y = x2 − 4 is: 4

f (x)

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

We can see that the graph crosses the x-axis at −2 and 2, so the solutions to x2 = 4 are −2 and 2. We can check this using substitution: (−2)2 = 4 and (2)2 = 4 We can follow this process to solve any quadratic equation graphically. In other words, for any function f (x) = c, for some real number constant, c, we can write the equivalent equation f (x) − c = 0, and find the x-intercepts of g(x) = f (x) − c to solve for x. A quadratic equation can have one, two or no real solutions.

One real solution

Two real solutions

No real solutions

8.01 Solve quadratics using graphs and tables mathspace.co

451

Example 1 Solve the equation 2x2 = 18.

Create a strategy We can write an equivalent equation set equal to zero, and then use a table to ﬁnd the zeros of the new function.

Apply the idea If we set this equation equal to zero, we would get 2x2 = 18

Given equation

2x − 18 = 0

Subtraction property of equality

When building a table, we want to choose values within a suitable range so we don’t have to do too many calculations. Start by ﬁnding the values in the domain −4 ≤ x ≤ 4. If the y-value (also called the function value) is zero for any of these x-values, then we have found a solution to the corresponding equation. In the table, we are looking for the entries where y = 0. x y

−4 14

−3 0

−2 −10

−1 −16

0 −18

1 −16

2 −10

3 0

4 14

We can see the equation has solutions of x = −3, x = 3, which we can also write as x = ±3.

Reﬂect and check We can see that the table of y-values has both positive and negative values. Whenever this is the case for a function of the form f (x) = ax2 + bx + c, we know that the equation 0 = ax2 + bx + c must have two real solutions, and the corresponding parabola will have two x-intercepts.

Example 2 Consider the function y = (x − 2)2 − 9. a Draw a graph of the function.

Create a strategy The function is given in vertex form so we know the vertex is at (2, −9). We can substitute x = 0 to find the y-intercept at (0, −5). We can find other points on the curve by substituting in other values, and by filling a table of values.

Apply the idea

452

−5

−8

−9

−8

−5

Mathspace Virginia SOL Algebra 1 mathspace.co

4 2 −4−3−2 −1 −2 −4 −6 −8 −10

x 1 2 3 4 5 6 7 8 9

Reﬂect and check It is important when drawing graphs to clearly show the key features such as the vertex and the intercepts by choosing appropriate scales for the axes.

b Determine the solution(s) to the equation 0 = (x − 2)2 − 9.

Create a strategy We can find the solution(s) by looking at the graph from part (a) and identifying where it crosses the x-axis.

Apply the idea The solutions to the equation can be found at the x-intercepts of the graph we drew in part (a).

4 2 −4−3−2 −1 −2

x 1 2 3 4 5 6 7 8 9

−4 −6 −8 −10

The solutions are x = −1 and x = 5.

Reﬂect and check We can verify the solutions by substituting each one into the equation for x and substituting 0 for y. If the right side of the equation evaluates to 0 then it is a solution. First, let’s check the solution x = −1 0 = (x − 2)2 − 9 2

0 = (−1 − 2) − 9 2

Original equation Substitute in x = −1

0 = (−3) − 9

Simplify inside parenthesis

0=9−9

Evaluate the square

0=0

Subtract

0 = 0 is a true statement, so x = −1 is a solution to 0 = (x − 2)2 − 9. Now, let’s check our other solution, x = 5 0 = (x − 2)2 − 9 2

0 = (5 − 2) − 9 2

Original equation Substitute in x = 5

0 = (3) − 9

Simplify inside parenthesis

0=9−9

Evaluate the square

0=0

Subtract

0 = 0 is a true statement, so x = 5 is also a solution to y = (x − 2)2 − 9.

8.01 Solve quadratics using graphs and tables mathspace.co

453

Example 4 Identify the number of real solutions each quadratic function has. a

y 6 5 4 3 2 1 −4 −3 −2 −1

x 1

−1

Create a strategy

Apply the idea

Real solutions correspond with x-intercepts. How many x-intercepts does this function have?

This quadratic never crosses the x-axis so there are no x-intercepts. The function has 0 real solutions.

y 4 3 2 1 −1 −1

x 1

−2 −3

Create a strategy Real solutions correspond with x-intercepts. How many x-intercepts does this function have?

Apply the idea

Reﬂect and check

The quadratic crosses the x-axis in two different spots. y 4 3 2 1 −1 −1

x 1

−2 −3

This quadratic has two real solutions.

456

Mathspace Virginia SOL Algebra 1 mathspace.co

We can determine from the graph the exact value of the two real solutions. The solutions are x = 2 and x = 5.

Idea summary We solve a quadratic equation by creating an equivalent equation by making your equation set equal to 0. The solutions to a quadratic equation are any values that make the equation true. If the equation is equal to 0, the solutions are called roots of the equation or zeros of the function. These correspond to the x-intercepts of the graph. For any function f (x) = c, for some real number constant, c, we can write the equivalent equation f (x) − c = 0, and ﬁnd the x-intercepts of g(x) = f (x) − c to solve for x.

Practice What do you remember? 1

Using the given tables, ﬁnd the solutions to the following equations: a

x2 + 7x + 12 = 0 x y

−4 0

−3 0

−2 2

−1 6

−4 21

−3 0

−2 −15

−1 −24

0 −27

1 −24

2 −15

4 21

x2 + 8x + 12 = 5

4x2 = 4

(x − 3)2 = 7

6x2 = 2x − 9

Using the given graphs, ﬁnd the solutions to the following equations: a

x2 + 2x − 8 = 0

15 10 5 x −7 −6 −5 −4 −3 −2 −1 −5

2x2 − 12x + 16 = 0 18 16 14 12 10 8 6 4 2

1 2 3 4 5

−2 −1 −2 −4

−10

3 0

Rewrite each equation so that it is equal to zero: a

−5 2

3x2 − 27 = 0 x y

−6 6

x 1

2 3 4 5 6 7

For the function y = x2 + 10x + 21: a

Copy and complete the table of values. x y

−8

−7

−6

−5

−4

−3

−2

Draw the graph of the equation.

Use the models from parts (a) and (b) to ﬁnd: i

The value of x when y = −4.

The value of x when y = 5.

8.01 Solve quadratics using graphs and tables mathspace.co

457

Determine the number of real solutions each quadratic function has: a

−2 −1

−1

−2

−1

−3

−1

−4

−2

−4 −3 −2 −1

−2

3 2 1

−1

3 2 1

−4 −3 −2 −1

x 1

−1

−2

−3

−4

−5

−6

Let’s practice 6

For each of the following quadratic functions: i

Complete the table of values.

Set each function equal to zero and use the table to determine the solution(s) to the equation.

a x

−2

−1

c x y e

458

−1

f 4

−1

x y

−1

(x − 5)2 = 4

y = (x − 5) (x − 1) x y

−2

x y

y = x2 − x − 2 x y

−2

Solve the following equations by drawing a graph of the corresponding function: a

x2 − 15x + 54 = 0

−(x + 5)2 + 9 = 0

(x − 3) (x + 2) = 0

(x − 1)2 = 0

Mathspace Virginia SOL Algebra 1 mathspace.co

x2 − 15x + 50 = 0

For each of the following quadratic equations: i

Draw the graph of the corresponding function.

Use the graph to determine the solution(s) to the equation.

0 = x2

0 = −x2 + 9

3x2 = 3

0 = − (x − 3)2

x2 + 12x = −32

(x − 3) (x − 2) = 0

2x2 − 2x = 12

(x − 5)2 = 0

A compass is accidentally thrown upward and out of an air balloon at a height of 100 feet. The height, y, of the compass at time x, in seconds, is given by the equation: y = −20x2 + 80x + 100

Graph the relationship y = −20x2 + 80x + 100.

Find the time it takes for the compass to hit the ground.

When will the compass be 160 feet high?

A frisbee is thrown upward and away from the top of a hill that is 120 yards tall. The height, y, of the frisbee at time x in seconds is given by the equation y = −10x2 + 40x + 120. This equation is graphed.

y 160 140

Determine when the frisbee reaches a height of 120 yards.

120

Determine how many seconds it takes for the frisbee to hit the ground.

100

Determine the height reached by the frisbee after 2 seconds.

Verify your answer to part (b) by substituting values into the equation.

80 60 40 20

x 1

Beth throws a pebble vertically upwards. After t seconds, its height h feet above the ground is given by the formula h = 18t − 2t2. This function has been graphed as shown.

h 40 35

Explain what the point at (4.5, 40.5) represents in context of the problem.

Find the values of t where the graph of h intercepts the horizontal axis.

Explain what the intercepts you found in part (b) represent in context of the problem.

Identify when the pebble is 36 feet above the ground.

30 25 20 15 10 5 t 1

2 3 4 5 6 7 8 9

Use the given table to answer the following: x y

−3 9

−2 4

−1 1

0 0

1 1

2 4

3 9

Find f (−2)

Find x when f (x) = 1

Find the domain when the range is {9, 0}.

Find the range when the domain is {−2, 2, 3}.

8.01 Solve quadratics using graphs and tables mathspace.co

459

Use the graph below to answer the following. a

Find f (−1)

Find x when f (x) = 0

Find the domain when the range is {3, 4}.

Find the range when the domain is {1, −1, −2}.

2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Let’s extend our thinking 14

An object is released 900 meters above ground and falls freely. The distance the object is from the ground is modeled by the formula d = 900 − 4.9t2, where d is the distance in meters that the object falls and t is the time elapsed in seconds. Javier graphed the given equation, the line d = 450, and the point of intersection as shown below.

d 900 800 700 600

Explain the point (9.583, 450) in context of the problem.

500

Use the graph to estimate how many seconds it takes for the object to reach a height of 0.

400

Explain how you could find a more precise answer to part (b). Find a more precise answer.

(9.583, 450)

300 200 100 t 2

The kinetic energy E of a moving object is given by speed in meters per second.

where m is its mass in kilograms and v is its

Graph this equation for a vehicle with a mass of 1400 kg using a calculator or other technology.

Use the graph to estimate the velocity when the kinetic energy is 137 200 J.

The formula for the surface area of a sphere is S = 4π r2, where r is the radius in centimeters. a

Graph the relationship S = 4π r2.

Use your graph to estimate the surface area of a sphere with radius 5.5 cm.

Use your graph to estimate the radius of a sphere with a surface area of 150 cm2.

A rectangle has width a, height b, and area A. The graph shows the A 100 possible values of the area A (vertical axis) plotted against the width a 90 (horizontal axis). The equation corresponding to the graph is A = a (20 − a). a

List the values of a where the graph intercepts the horizontal axis.

Explain why it is not possible to have a rectangle with a width of a = 20.

Find the width and height of the rectangle corresponding with the largest possible value of A.

80 70 60 50 40 30 20 10

a 2 4 6 8 10 12 14 16 18 20

460

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea 3x2 + 3x − 10 = 8 2

3x + 3x − 18 = 0

Given equation Subtraction property of equality

Now, we can solve by factoring. 3(x2 + x − 6) = 0 2

Factor out the GCF

3(x + 3x − 2x − 6) = 0

Rewrite the trinomial as a polynomial with four terms

3[x(x + 3) − 2(x + 3)] = 0

Factor each pair of terms

3(x + 3) (x − 2) = 0 x + 3 = 0 and x − 2 = 0 x = −3 and x = 2

Divide out common factor of (x + 3) Zero product property Addition property of equality

Reﬂect and check We can check our answers by substituting them back into the original equation to see if they make the equation true. We will check x = −3 ﬁrst. 3x2 + 3x − 10 = 8

Original equation

Substitute x = −3

3(−3) + 3(−3) − 10 = 8 27 − 9 − 10 = 8 8=8

Evaluate the multiplication Evaluate the subtraction

This is a solution to the equation. Now, we will check x = 2. 3x2 + 3x − 10 = 8

Original equation

Substitute x = 2

3(2) + 3(2) − 10 = 8 12 + 6 − 10 = 8 8=8

Evaluate the multiplication Evaluate the subtraction

This also satisﬁes the equation, so it is a solution.

Example 2 Luis throws a ball straight into the air. The path of the ball can be modeled by the equation y = −5x2 + 14x + 3 where x represents the time the ball is in the air in seconds and y represents the height of the ball in meters. How long will it take the ball to hit the ground?

Create a strategy The question is asking us to find the time (the x-value) it takes for the ball to hit the ground (the y-value). The ground represents a height of 0. In other words, the question is asking us to solve the equation −5x2 + 14x + 3 = 0. When we factor, we usually have a positive leading coefficient. To begin, we can factor out −1 which will give us a positive leading coefficient.

8.02 Solve quadratics by factoring mathspace.co

463

Apply the idea −(5x2 − 14x − 3) = 0

Factor out −1

5x − 14x − 3 = 0

Divide both sides by −1

Next, we need to ﬁnd two numbers that multiply to ac = 5 ⋅ −3 = −15 and add to b = −14. The factor pair that satisﬁes these conditions is −15 and 1. Rewrite the polynomial with 4 terms

Factor by grouping

Factor out the GCF of (x − 3)

Zero product property

Addition property of equality

Division property of equality

The x-values represent time, so a negative value does not make sense since we cannot go backward in time. This means

is a nonviable solution, and x = 3 is the only viable solution.

The ball hit the ground after 3 seconds.

Reﬂect and check As we can see from the graph,

is an x-intercept.

But because it does not make sense in context, it is not a solution to the problem. We can picture Luis standing at the y-axis when he throws the ball since x = 0 would represent the present moment.

12 10 8 6 4 2 x 1

Idea summary We can use the zero product property to solve quadratic equations by ﬁrst writing the equation in factored form: a (x − x1) (x − x2) = 0 then setting each factor equal to zero and solving for x.

464

Mathspace Virginia SOL Algebra 1 mathspace.co

Practice What do you remember? 1

Solve the following equations by using the zero product property: a

x (x − 9) = 0

2m(m − 8) = 0

c (5c − 12) = 0

(k − 3) (k − 5) = 0

( y − 6) ( y + 11) = 0

(3x − 9) (2x − 5) = 0

5(x + 5) (x − 5) = 0

(7a − 2)2 = 0

x = −5, x = −7

x = 6, x = 2

y = (1 − x) (x + 5)

Consider the quadratic equation:

x2 + 2x − 35 = 0

Select the solution of the quadratic equation. A 3

x = −5, x = 7

x = 5, x = −7

Consider the quadratic equation:

x = 5, x = 7

x2 − 4x − 12 = 0

Select the solution of the quadratic equation. A 4

x = −6, x = −2

x = −6, x = 2

x = 6, x = −2

Determine the x-intercept(s) for each of the following quadratic function: a e

y = (x + 8) (x + 4) y = (x − 6) (x − 24)

b f

y = − (x − 10)2

y = − (x − 8) (x + 2)

y = (x − 3) (x + 2) 2

y = − (x + 13)

y = x (x + 3)

Let’s practice 5

Solve the following equations by factoring: a

6x2 + 54x = 0

4y − 8y2 = 0

x2 − 5x − 14 = 0

f 2 + 6f − 55 = 0

h2 + 19h + 88 = 0

x2 − 20x + 100 = 0

x2 + 8x − 20 = 0

x2 − 13x − 114 = 0

Solve the following equations by ﬁrst rearranging, and then factoring: a

3x2 − 14x = −8

x2 − 3x − 10 = 0

x2 + 7x + 12 = 0

x2 + 3x = 28

x2 − 11x + 19 = −5

x(x + 18) + 80 = 0

x(x + 2) − 48 = 0

m2 = 3m + 10

x2 = 4 − 3x

x2 − 12x = −20

m2 + 5m = 14

−6y = y2 + 8

10y = y2 + 24

x2 + 6x + 8 = 0

−m − 7m = −18

−n − 5n = −84

Solve the quadratic equations and verify your solution(s) by graphing. a

2y2 = 9y + 5

Solve:

i 9

Solve the quadratic equations by factoring. Justify your work. a

x2 − 14 = 5x

x2 + x − 42 = 0

x2 − 9x = −20

x2 − 35 = 2x

Software engineers are designing a self-serve checkout system for a supermarket. They notice that the traﬃc through the store during the day is described by the function C = −t (t − 12) where C is the number of customers and t is the number of hours after the store opens. To meet the peak demand, the engineers allow for an extra checkout machine to automatically turn on when the number of customers ﬁrst reaches 32 people, and to automatically turn off when it next falls below 32 people. a

Find the times t when the number of customers is equal to 32 people.

Determine how many hours it takes the extra checkout machine to turn on after opening.

Determine how many hours the extra machine will be on for. 8.02 Solve quadratics by factoring mathspace.co

465

We can simplify using properties of exponents, properties of radicals, or a perfect square factor: Properties of exponents When we simplified radicals using rational exponents we did the following. First, we converted the radical to a rational exponent. Then, we found the prime factors of 24 and applied properties of exponents to simplify.

Properties of radicals We can follow a similar process for this method, except we can leave the expression in radical form. First, we ﬁnd the prime factors of 24, then we can use properties of radicals to simplify.

Perfect square method This is the quickest method for simplifying a radical. Instead of ﬁnding all the prime factors of 24, we want to ﬁnd the largest perfect square factor of 24. Then, we can use the multiplication property of radicals to simplify.

Example 1 Solve the following equations by using square roots: a x2 = 9

Create a strategy In this equation we have 9 being equal to the square of x. This is equivalent to x being equal to the square root of 9.

Apply the idea x = ±3

Reﬂect and check Checking our answers:

(−3)2 = 9 (3)2 = 9

Both answers satisfy the equation.

468

Mathspace Virginia SOL Algebra 1 mathspace.co

b 4x2 − 27 = 0

Create a strategy We can begin by isolating the variable, but we will need to simplify the radical since 27 is not a perfect square.

Apply the idea Given equation

Addition property of equality

Division property of equality

Square root property

Division property of radicals

From here, we can factor 27. Our goal is to separate it into factors where one is a perfect square. 27 = 9 ⋅ 3 where 9 is a perfect square. Multiplication property of radicals

Evaluate the radicals

Reflect and check For nearly all of our work with solutions to functions and equations, it is standard practice to leave our final expression in exact form. In questions involving applications of quadratics, we may be asked to evaluate the square root at the very end using a calculator, then approximate to a specific number of decimal places.

c (x − 2)2 − 100 = 0

Create a strategy In order to use square roots to solve, the squared expression must be isolated. In this example we want to isolate the term (x − 2)2.

Apply the idea (x − 2)2 − 100 = 0 2

(x − 2) = 100 x − 2 = ±10

Given equation Add 100 to both sides Take the square root of both sides

This leaves us with two equations x − 2 = 10 and x − 2 = −10. Add 2 to solve both equations and we ﬁnd that the solutions are x = −8, x = 12.

8.03 Solve quadratics using square roots mathspace.co

469

Now checking the next solution: Original equation

Substitute

Evaluate the multiplication

Evaluate the subtraction

Evaluate the exponent

Both answers satisfy the original equation.

Example 2 State a quadratic equation that has the given solutions. a

Create a strategy We can work backwards from solving to ﬁnd the equation that had the given solutions.

Apply the idea Given solutions

Add 1 to both sides

Square both sides

Reﬂect and check This is one equation, but we could also subtract 7 from both sides to get an equivalent equation with the same solutions. (x + 1)2 − 7 = 0

Create a strategy We can use a similar process as the previous problem, but this time we need to multiply both sides by 3 ﬁrst.

Apply the idea Given solutions

Multiply 3 to both sides

Subtract 5 from both sides

Square both sides

Reﬂect and check When completing the square, fractional solutions come from equations where a ≠ 1. The 3 in the denominator came from the coeﬃcient of x inside the parentheses.

8.03 Solve quadratics using square roots mathspace.co

471

Example 3 A square ﬁeld has perpendicular lines drawn across it dividing it into 36 equal sized smaller squares. If the total area of the ﬁeld is 225 square feet, determine the side length of one of the smaller squares.

Create a strategy We know that there are 36 smaller squares in total on a larger square grid, so there must be 6 by 6 smaller squares on the grid. If we let the side of a smaller square be x, then the side of the larger square can be 6x. This gives us the quadratic equation (6x)2 = 225. We can then solve this equation by taking square roots.

Apply the idea Write the equation

Take the square root of both sides

Divide both sides by 6

Simplifying the expression gives us the solutions x = ±2.5. We can exclude the negative solution as the length of the square must be positive. So, the smaller square has a side length of 2.5 feet.

Reﬂect and check In most real-life applications, we will exclude the negative solution as it will be non-viable for the context.

Idea summary When we use the square root property, we always include the ± symbol to denote the positive and negative root. We can use the following facts to simplify radical expressions, for a, b ≥ 0: Multiplication property of radicals

Division property of radicals

Completing the square Completing the square is a method we use to rewrite a quadratic expression so that it contains a perfect square trinomial which can be factored as A2 + 2AB + B2 = ( A + B)2. We used this method in a previous lesson to convert a quadratic equation from standard form to vertex form. We will now learn to use the completing the square method combined with the square root property to solve quadratic equations.

Exploration Consider the equation x2 + 6x = 11.

472

Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial.

Remember that we need to keep both sides of the equation balanced. After making the perfect square trinomial, check that your equation is still balanced.

How could we solve the equation in this form?

How could you apply your method to x2 − 10x − 5 = 0?

Mathspace Virginia SOL Algebra 1 mathspace.co

For quadratic equations where a = 1, we can write them in perfect square form by following these steps: 1 2

Subtract c from both sides

Rewrite the x term

Add

to both sides

Factor the perfect square trinomial

If a ≠ 1, we can ﬁrst divide through by a to factor it out. Note that when we were using completing the square to write an equation in vertex form, we keep the constant term on the same side of the equation as the variable terms. Then, to maintain equivalency and complete square, we term. This results in all the terms being on the same side of the equation so we can

add and subtract the same

identify the vertex of the parabola. But, if we want to solve the equation, we keep the x terms together and move the constant term to the other side of the equation. Then the

term is added to both sides of the equation to maintain equivalency and create a perfect

trinomial. This gives us a squared factor on one side and a constant term on the other side of the equation, allowing us to use the square root property to solve for x. If we can rewrite an equation by completing the square, then we can solve it using square roots.

Example 4 Solve the following quadratic equations by completing the square. a x2 + 18x + 32 = 0

Create a strategy To solve an equation by completing the square, start by moving the constant term to the other side of the equation. We will complete the square by ﬁnding half the coeﬃcient of the x term, squaring it, and adding it to both sides of the equation. Once we’ve completed the square, we can solve.

Apply the idea x2 + 18x + 32 = 0

Given equation

x + 18x = −32

Subtract 32 from both sides

Since the coeﬃcient of the x-term is 18, we will need to add

to both sides of our equation.

Complete the square

Factor the perfect square trinomial

Take the square root of both sides

This leaves us with two equations x + 9 = 7 and x + 9 = −7. We can solve both equations by subtracting 9, so we get the solutions x = −2 and x = −16.

8.03 Solve quadratics using square roots mathspace.co

473

b 2x2 − 10x + 7 = 0

Create a strategy In this example, the coeﬃcient of x2 is 2, so we will need to divide this coeﬃcient out before completing the square. We can then perform steps similar to the previous example.

Apply the idea

Given equation Divide both sides by 2

Subtract The coeﬃcient of x is

from both sides

Taking half of −5 and squaring it gives us

so this is the value that

completes the square.

Complete the square

Factor the left side, evaluate the addition on the right side

Square root property

Division property of radicals

Multiplication property of radicals

Evaluate the radicals

This leaves us with two equations: Next, we add

to solve both equations, and we ﬁnd that the solutions are

and

Reﬂect and check These can also be combined into one fraction:

Idea summary Completing the square can be used to solve any quadratic in the form ax2 + bx + c = 0, but it is easiest to use when a = 1 and b is even.

474

Mathspace Virginia SOL Algebra 1 mathspace.co

Practice What do you remember? 1

Fully simplify each square root. a

x2 = 25

x = 49

x2 = 81

x2 = 100

x2 = 1

x2 + 5x + ⬚

x − 25 = 0

State the quadratic equation that has the given solutions: x = ±5

Complete the following expressions so they form a perfect square trinomial. a

Solve the following equations by using square roots:

a 4

x2 − ⬚ x + 16

x2 − ⬚ x + 1

x2 − x + ⬚

x2 − 5x + ⬚ = (x − ⬚)2

Complete the square by ﬁnding the missing values. a

x2−

+ ⬚ = (x − ⬚)2

x2 + 4x + ⬚ = (x + ⬚)2

Solve the following quadratic equations by completing the square: a

x2 + 18x + 32 = 0

x2 − 6x + 8 = 0

x2 − 9x + 8 = 0

x2 − 2x − 32 = 0

25y2 = 36

5x2 − 45 = 0

(4x + 3)2 = 64

5( p2 − 3) = 705

(x + 2)2 = 20

2(x − 3)2 = 8

x2 − 8x − 9 = 0

x2 + 14x − 51 = 0

Let’s practice 7

Solve the following equations by ﬁnding square roots: a

x2 − 5 = 31

(m − 7)2 = 81

(x − 10)2 = 26

Solve the following equations by completing the square: a

x2 + 2x − 8 = 0

x2 − 6x + 5 = 0

2x2 − 12x − 32 = 0

4x2 + 11x + 7 = 0

Solve the following quadratic equations by completing the square. Express your answers in simplest form. a e

x2 + 11x + 5 = 0 2

6x + 48x + 24 = 0

x2 − 7x + 8 = 0

c g

x2 + 22x + 9 = 0 2

5x + 55x + 3 = 0

x2 + 24x + 5 = 0

2x2 + 5x + 1 = 0

State the quadratic equation that has the given solutions: a

(4 − d)2 = 9

Solve each quadratic equation below using either the square root property or by completing the square. Justify each step work. a

x2 + 5 = 30

x2 + 12x + 32 = 0

x2 + 8x + 5 = 0

(8x + 9)2 = 256

8.03 Solve quadratics using square roots mathspace.co

475

Solve each quadratic equation below using either the square root property or by completing the square. Verify your work by graphing or with substitution. a

(x − 5)2 = 36

x2 + 13x + 36 = 0

2x2 − 12x − 54 = 0

(5x − 3)2 = 49

Harry is using a diving board to dive into a swimming pool. The distance from his head to the surface of the water can be represented as (x − 7) (x + 7) = 147. Find the viable solution to the quadratic equation.

Consider the equation x2 + 24x = 10. Janessa tried to solve the equation by completing the square. 1 2 3 4 a

Identify the mistakes she made.

Solve the equation correctly.

On the graph of y = x2 − 4, there are two points where y = 12. Without drawing the graph, find the x-coordinates of these two points.

On Earth, the equation d = 4.9t2 is used to ﬁnd the distance (in meters) an object has fallen through the air after t seconds. Willow is sky diving and wants to release her parachute once she has fallen 400 m. Determine the time it will take her to fall 400 m, rounding your answer to the nearest second.

The kinetic energy E of a moving object is given by speed in meters/second.

where m is its mass in kilograms and v is its

If a vehicle weighing 1600 kilograms has kinetic energy E = 204 800, determine what speed it is moving. 18

Eduardo is trying to solve the equation (x + 9)2 = 25. He thinks that the equation is equivalent to x + 9 = 5. Nicolette, however, thinks he has performed the operations incorrectly, and that he should have subtracted 9 from both sides of the equation ﬁrst giving an equivalent equation of x2 = 16. Describe any errors Eduardo and/or Nicolette have made, and ﬁnd the correct solution to the equation.

Let’s extend our thinking 19

The revenue y (in millions of dollars) of a company x years after it ﬁrst started is modeled by y = 12.5x2 − 64x + 135

476

Use this equation to predict the number of years it will take for the revenue to reach $1 167 million.

Describe another method you could use to calculate this.

Mathspace Virginia SOL Algebra 1 mathspace.co

Sauya’s teacher gave her a square and a rectangle and asked her to use them to create a larger square. x

Sauya cut the rectangle in half, and placed the two pieces on either side of the square as shown below.

Determine the area of the smaller square that, when added, completes the larger square.

Find the total area of the larger square.

Complete the square of the quadratic equation, ax2 + bx + c = 0, to ﬁll in the blanks: (x + ⬚ )2 = ⬚

8.03 Solve quadratics using square roots mathspace.co

477

Example 1 The standard form of a quadratic equation is ax2 + bx + c = 0. a Derive the quadratic formula by solving this equation for x.

Create a strategy To solve for x, we can complete the square.

Apply the idea

Division property of equality

Subtraction property of equality

Complete the square

Factor the left side

Evaluate the exponent

Evaluate the subtraction

Square root property

Evaluate the square root

Subtraction property of equality

Evaluate the addition

Reﬂect and check The standard form of a quadratic equation represents any quadratic equation. Since we solved this equation for x, this formula can be used to ﬁnd the solution to any quadratic equation.

b Use the quadratic formula to solve the equation −x2 + 6x − 8 = 0

Create a strategy First, we need to make sure the equation is in standard form and equal to zero. This one already is, so we can see that a = −1, b = 6, and c = −8. We can substitute these values into the quadratic formula to solve for x.

8.04 Solve quadratics using the quadratic formula mathspace.co

479

Apply the idea Quadratic formula with a = −1, b = 6, and c = −8

Evaluate the exponent and multiplication

Evaluate the subtraction

Evaluate the square root

Now, we can separate this into the two answers:

When we simplify, we ﬁnd the answers to be x = 2 and x = 4.

Reﬂect and check Since the answers are rational, we could have solved the quadratic equation by factoring. −x2 + 6x − 8 = 0 2

Given equation

−(x − 6x + 8) = 0

Factor out −1

−(x − 4) (x − 2) = 0

Factor the trinomial

Using the zero product property, we get the answers x = 4 and x = 2.

c Use the quadratic formula to solve the equation 5x2 = 8x + 1.

Create a strategy Before using the quadratic formula, we need to get the equation in the form ax2 + bx + c = 0. Then, we can correctly identify the values of a, b, and c.

Apply the idea 5x2 = 8x + 1 2

5x − 8x − 1 = 0

Given equation Subtraction property of equality

Now we can see a = 5, b = −8, c = −1. Quadratic formula with a = 5, b = −8, and c = −1 Evaluate the exponent and multiplication Evaluate the addition Product of radicals Evaluate the square root Simplify by a factor of 2 The answers are

480

Mathspace Virginia SOL Algebra 1 mathspace.co

Reﬂect and check When the answer is irrational, then the quadratic formula or completing the square are the only methods we could use to solve the quadratic equation.

Example 2 Solve: a 5x2 − 15x + 2 = 0

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula.

Apply the idea The equation 5x2 − 15x + 2 = 0 is already in standard form so we can identify a, b, and c. We can see that a = 5, b = −15, and c = 2 and substitute them into the quadratic formula.

Quadratic formula

Substitute a = 5, b = −15, c = 2

Simplify the adjacent signs

Evaluate the multiplication

Evaluate the exponent

Evaluate the subtraction

and

So the solutions are

b 10 − 6m + 2m2 = m2 + 8m + 9

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, and use the quadratic formula.

Apply the idea 10 − 6m + 2m2 = m2 + 8m + 9 2

Original equation

10 − 6m + m = 8m + 9

Subtract m2 from both sides

10 − 14m + m2 = 9

Subtract 8m from both sides

1 − 14m + m = 0 2

m − 14m + 1 = 0

Subtract 9 from both sides Write in descending order

8.04 Solve quadratics using the quadratic formula mathspace.co

481

Now we can see that a = 1, b = −14, and c = 1 so we can substitute these values into the quadratic formula.

Quadratic formula

Substitute a = 1, b = −14, c = 1

Simplify the adjacent signs

Evaluate the multiplication

Evaluate the exponent

Simplify the expression inside the square root

Simplify the square root

Divide out the common factor of 2

So the solutions are

and

Example 3 A ball is launched from a height of 80 ft with an initial velocity of 107 ft per second. Its height, h feet, after x seconds is given by h = −16x2 + 107x + 80 Determine the number of seconds it will take the ball to reach the ground. Explain your reasoning.

Create a strategy On the ground the ball will have a height of 0 ft, so h = 0. This means we want to solve the equation 0 = −16x2 + 107x + 80. Since the numbers are large, the quadratic formula is an appropriate method for solving.

Apply the idea For this equation, a = −16, b = 107, and c = 80. Substituting into the quadratic equation, we get

Quadratic formula with a = −16, b = 107, and c = 80

Evaluate the exponent and multiplication

Evaluate the addition

Product of radicals

Evaluate the square root

Therefore,

482

Mathspace Virginia SOL Algebra 1 mathspace.co

The two solutions, rounded to two decimal places, are x = −0.68 and x = 7.37. Remember that x represents seconds. We can exclude the negative solution as it is outside the domain, which is x ≥ 0, since time cannot be negative. The ball will reach the ground after 7.37 seconds.

Reﬂect and check In many real-world situations, negative numbers do not make sense. Always check that your answers satisfy the constraints of the variables.

Example 4 The amount of litter in a park at the end of the day can be modeled against the number of people who visited the park that day by the equation:

where L is the number of pieces of litter and P is the number of people. Determine the number of people who visited the park if there are 20 pieces of litter at the end of the day.

Create a strategy We want to ﬁnd the number of people, P, when there are 20 pieces of litter at the end of the day, L = 20. We can do this by substituting L = 20 into the equation, rearranging the equation into quadratic standard form, and then using the quadratic formula to solve for P.

Apply the idea

Model equation

Substitute in L = 20

Multiply both sides by −50

Add 1000 to both sides

Quadratic formula with a = 1, b = −73, and c = 850

Evaluate the operations

The two solutions (rounded to two decimal places) are P = 58.46 and P = 14.54. Since we are counting the number of people who visited the park, we want to round to the nearest whole number. If there are 20 pieces of litter in the park at the end of the day, then either 58 or 15 people visited the park that day.

Reﬂect and check In real life applications where we are counting whole objects, we want to round to the nearest integer, so our solution makes sense in context.

8.04 Solve quadratics using the quadratic formula mathspace.co

483

Idea summary For any quadratic equation of the form 0 = ax2 + bx + c where a ≠ 0 and a, b, and c are real numbers, the quadratic formula can be used to solve for x.

The discriminant The radicand in the quadratic formula is called the discriminant.

b2 – 4ac

discriminant

The discriminant can be used to determine the number and type of solutions to any quadratic equation.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.04 to answer these questions. 1.

What do each of the sliders represent?

How many types of roots can there be?

How do the types of roots relate to the value of the discriminant, D?

How do the x-intercepts relate to the value of the discriminant, D?

The values of a, b, and c affect the value of the discriminant. The type of number the discriminant is affects the number and type of x-intercepts on the graph. Quadratic equations can have 3 types of solutions: 2 real solutions, 1 real solution, or no real solutions. The value of the discriminant quickly reveals which type of solution a quadratic equation has. 13 12 11 10 9 8 7 6 5 4 3 2 1 −3 −2 −1

484

Discriminant (> 0):

b2 − 4ac = (2)2 − 4(−1) (12) = 52 Solutions:

x-intercepts:

x 1

Mathspace Virginia SOL Algebra 1 mathspace.co

The square root of a positive number is a real number, so the plus or minus sign ensures there will always be 2 real solutions when the discriminant is positive. Discriminant (= 0):

b2 − 4ac = (−2)2 − 4(1) (1) = 0

Solution:

x2 − 2 x + 1 = 0

3 2

x-intercept: (1, 0)

1 x −1

The square root of zero is zero. This eliminates the radical part of the quadratic equation, leaving only

which

will result in a single value. So, when the discriminant is zero, there will be one real solution. Discriminant (< 0):

b2 − 4ac = (−2)2 − 4(1) (2) = −4

Solutions: 3 2

x-intercepts: None

1 x −1

No real number gives us a negative number when squared. Therefore, there are no real solutions when the discriminant is negative.

Example 4 Use the discriminant to determine the number and nature of the solutions of the following quadratic equations: a 2x2 − 8x + 3 = 0

Create a strategy For this equation, we have a = 2, b = −8, c = 3.

Apply the idea The discriminant is (−8)2 − 4(2) (3) = 40. Since it is positive, the equation has two real solutions.

Reﬂect and check Two real solutions means the function has two x-intercepts.

8.04 Solve quadratics using the quadratic formula mathspace.co

485

b −5x2 + 6x − 2 = 0

Create a strategy For this equation, we have a = −5, b = 6, c = −2.

Apply the idea The discriminant has a value of (6)2 − 4(−5) (−2) = −4. Since it is negative, the equation has no real solutions.

Reﬂect and check If the discriminant is negative, then the formula will involve taking the square root of a negative number which will result in no real solutions. This quadratic will not intercept the x-axis. We can also see that if 4ac > b2 the discriminant will be negative, and the corresponding equation will have no real solutions.

c x2 − 3x + 9 = 3x

Create a strategy Before identifying our variables, the equation must be in the form ax2 + bx + c = 0, so we need to begin by subtracting 3x from both sides. x2 − 6x + 9 = 0 For this equation, we have a = 1, b = −6, c = 9.

Apply the idea

Reﬂect and check 2

The discriminant has a value of (−6) − 4(1) (9) = 0, so the equation has one real solution.

When the discriminant is zero, the quadratic equation simpliﬁes to be

which is equal to the x-value of

the vertex. This means the vertex lies on the x-axis and the x-coordinate of the vertex is the only solution to the quadratic equation.

Idea summary The discriminant, b2 − 4ac, can help us determine the type and number of solutions to a quadratic equation without needing to solve the equation fully. • • •

486

b2 − 4ac > 0 two real solutions b2 − 4ac = 0 one real solution b2 − 4ac < 0 no real solutions

Mathspace Virginia SOL Algebra 1 mathspace.co

Practice What do you remember? 1

Is each statement true or false? a

Any quadratic equation that can be solved by completing the square can also be solved by the quadratic formula.

Any quadratic equation that can be solved by factoring can also be solved by the quadratic formula.

The equation 3x2 + 3x − 7 = 0 can be solved using the quadratic formula.

The quadratic formula will always give 2 unique solutions.

For the equation 5x2 − x − 2 = 0, we would set a = 5, b = 1,and c = −2.

The standard form of a quadratic equation is ax2 + bx + c = 0. Find the values of a, b and c in the following quadratic equations: a

x2 − 6x + 5 = 0

−4x2 + 15x − 8 = 0

2x2 + 9x = 0

−5(x − 3)2 + 4 = 0

x2 + 7x = 10

Jeremy is using the quadratic formula for the equation 9y2 = 8y. He has correctly identified that b = 8, what are the values of a and c?

Given the quadratic equation x2 − 4x + k = 0, where k is a constant, if one of the roots is other root?

The solutions of a quadratic equation are 9 and −9. What can be said about the value of b2 − 4ac?

Consider the quadratic equation:

, what is the

x2 + 3x − 5 = 0

Select the solution to the quadratic equation.

Consider the quadratic equation:

x2 − 16x = 5x − 9

Select the solution to the quadratic equation. A

Let’s practice 8

For the following equations: i

Find the value of the discriminant.

State the number of real solutions for the equation.

3x2 − 5x + 7 = 0

x2 − 4 = 0

2x2 − 2x = x − 1

4x2 − x = x2 − 5

−x2 − 8x − 16 = 0

+ 3x + 9 = 0

8.04 Solve quadratics using the quadratic formula mathspace.co

487

For the following equations: i

Determine the number of real solutions. Explain your reasoning.

Find the real solution(s) of the equation.

x2 − 8x − 48 = 0

4x2 − 4x + 1 = 0

x2 + 5x +

x2 + 12x + 36 = 0

Find the value of the discriminant.

State the number and nature of the solutions to the equation.

x2 + 6x = −90

4x2 = 6x − 7

2x2 − 2x = x − 1

13 + x2 = −7x 2

−x2 − 5x = 8

5x + 13x + 10 = x − 7x − 15

−7 + 3x = 2x2 − 5

1.8x2 + 5.2x − 2.3 = 0

3x(x + 4) = −3x + 4

x2 + 11x + 28 = 0

6 − 9x = −2x2 − 4x

Solve each equation below using the quadratic formula. Justify your work. Leave your answer(s) in exact, simplified form. a

x2 − 7x + 9 = 0

x2 − 5x − 2 = 0

−2x2 − 15x − 4 = 0

3x2 + 9x − 4 = 0

5x = (x − 5) (3x + 3)

12 − 8m + 2m2 = m2 + 12m + 15

Solve the following equations using the quadratic formula. Round your answers to two decimal places.

x2 − 4x + 7 = 0

For the following equations:

−5x − 15x + 3 = 0 2

3n = 2n − 2n + 7

Solve each equation below using the quadratic formula. Verify your solution(s) by graphing. a

x2 + 5x + 6 = 0

x2 − 5x + 6 = 0

2x2 + 7x + 3 = 0

4x2 − 17x − 15 = 0

2x2 + 6x − 8 = 0

4x2 − 10x + 4 = 0

Yuri is playing baseball and hits a homerun with an initial velocity of 101 ft/s, from a height of 3 ft. After x seconds, its height (in feet) is given by h = −16x2 + 101x + 3 LaDeana is in the crowd and catches the homerun ball in the stands from a height of 15 ft above the ground. Determine the number of seconds after Yuri hits the ball that LaDeana catches it. Round your answer to two decimal places.

The stopping distance of a car when the brakes are applied can be modeled by the equation where s is the stopping distance in feet and u is the initial speed in miles per hour.

Determine the speed the car was travelling if its stopping distance was 100 ft, rounding your answer to the nearest hundredth of a mile.

It takes 399 ft to stop when travelling at the speed limit of 70 mph. If it takes Ray 496 ft to stop, determine how fast over the speed limit Ray was driving.

The number of customers at a restaurant can be estimated by the equation y = −x2 + 28x − 159 where y is the number of people and x is the hour of the day (in 24-hour time). Determine the opening hours of the restaurant. Explain your reasoning.

488

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 17

For each of the given solutions to a quadratic equation: i

Find the values of a, b and c.

Write down the quadratic equation that has these solutions. b

a 18

Use the discriminant to match each quadratic equation below with the correct graph. Explain your reasoning a

5x2 − 10x − 35 = 0

2x2 + 8x + 8 = 0

−1

x2 − x + 1 = 0

9 8 7 6 5 4 3 2 1

−3 −2

y x −3 −2 −1

−10 −20 −30 x 1

−40

8 7 6 5 4 3 2 1 −5 −4 −3 −2

−1

x 1

Consider the equation in terms of x: mx2 − 3x − 5 = 0 a

Given that it has two unique solutions, determine the possible values of m.

There is one value of m that must be eliminated from the range of solutions found in the previous part. Determine the solution and explain why it must be eliminated.

Find the values of n for which x2 − 8nx + 1296 = 0 has one solution.

With reference to the discriminant, explain what determines the nature of the solutions to a quadratic equation.

Determine the range of values of the constant k such that the equation 3x2 + kx + 12 = 0 has no real solutions. Justify your answer.

Use an algebraic method to show that the graphs of the functions f (x) = 3x2 − x + 8 and g (x) = −x2 + 2x − 4 do not intersect.

A quadratic equation has two real solutions whose difference is k, for some positive value k. Determine algebraically a simpliﬁed expression for the discriminant of this equation. 8.04 Solve quadratics using the quadratic formula mathspace.co

489

Consider a right-angled triangle with side lengths x units, x + p units and x + q units, ordered from shortest to longest. No two sides of this triangle have the same length. a

Complete the statement:

p and q have lengths such that 0 < ⬚ < ⬚.

Find the discriminant of this quadratic equation.

Determine the number of real solutions.

Find the value of x when q = 2p. Give your answer in terms of p.

Write a quadratic equation in standard form that describes the relationship between the sides of the triangle in terms of x.

x+p

x+q

490

Mathspace Virginia SOL Algebra 1 mathspace.co

8.05 Solve quadratics using appropriate methods After this lesson, you will be able to… • use the structure of a quadratic expression to identify ways to rewrite it. • choose appropriate methods for solving quadratic equations based on the structure of the quadratic expression. • create and solve quadratic equations for real-world contexts.

Solving quadratic equations using appropriate methods We have several methods we can use to solve quadratic equations. To determine which method is the most suitable we need to look at the form of the quadratic equation. Graphing

Advantages: Helps us visualize the quadratic and its key features

Disadvantages: Only best when intercepts are integers, in which case it could have been factored instead x

Solution

Factoring

Equation form: Any form is ﬁne if using technology, otherwise it is best in a form that is equal to 0

Advantages: This is usually the fastest method Disadvantages: Not all polynomials are factorable, some factorable polynomials are diﬃcult to factor Equation form: ax2 + bx + c = 0 where a, b, c are small

Square root property

Advantages: Simplest method for solving equations in vertex form or equations missing an x-term Disadvantages: Few equations are given in this form Equation form: x2 = k or a(x − h)2 = k

Completing the square

Advantages: Can be used to solve any quadratic equation Disadvantages: Requires more steps than other methods, fractions make it diﬃcult Equation form: x2 + bx + c = 0 where b is even

8.05 Solve quadratics using appropriate methods mathspace.co

491

Advantages: Can be used to solve any quadratic equation

Quadratic formula

Disadvantages: Can be time-consuming, many opportunities to make miscalculations, although calculators simplify its use Equation form: ax2 + bx + c = 0 where a, b, c are large There is not one correct method for solving a quadratic equation. You would not be wrong by using one method over another; it is just easier, sometimes more practical, to use some methods over others.

Example 1 For the following quadratic equations, ﬁnd the solution using an eﬃcient method. Justify which method you used. a x2 − 7x + 12 = 0

Create a strategy The leading coeﬃcient of x2 is 1, so we can check if this can be easily factored. The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and we want to ﬁnd two factors that have a product of 12 and sum to −7. As the product is positive but the sum is negative, we know both factors must be negative.

Apply the idea Since the equation can be factored by grouping, we will factor the equation and solve it. The two factors that have a product of 12 and a sum of −7 are −3 and −4. We can write the equation in factored form as (x − 4) (x − 3) = 0, which gives us two solutions x = 3 and x = 4.

Reﬂect and check In general, if the coeﬃcients are small, and especially if a = 1, it is worth checking to see if we can easily factor the equation to solve.

b x2 − 11 = 21

Create a strategy Here we have b = 0, and can easily isolate the x2, which means we can solve this by using square roots.

Apply the idea Since we can easily isolate x2, we will solve the equation using square roots as follows: Given equation

Add 11 to both sides

Evaluate the square root of both sides

Factor 32

Multiplication property of radicals

Evaluate the radical

giving us two solutions:

492

Mathspace Virginia SOL Algebra 1 mathspace.co

and

Reﬂect and check In general, if we can easily rearrange the equation into the form (x − h)2 = k for some positive value of k then solving using square roots is a suitable method.

c 3x2 − 24x + 20 = 5

Create a strategy For most of the methods we know, the quadratic needs to be equal to zero ﬁrst. We can subtract 5 from both sides, then check see if factoring can be used.

Apply the idea Since the trinomial is equal to a constant, we will ﬁrst set the equation equal to zero and attempt to factor the trinomial. Then, we can determine an approach that is appropriate for solving this equation. 3x2 − 24x + 20 = 5 2

3x − 24x + 15 = 0

Given equation Subtract 5 from both sides

Factor the GCF of 3

Divide by 3 on both sides

3(x − 8x + 5) = 0 x − 8x + 5 = 0

From here, we can see that the equation cannot be factored further. Since a = 1 and b is even, we can use completing the square to solve. Subtraction property of equality

Complete the square

Factor the left side, evaluate the right side

Evaluate the square root of both sides

Add 4 to both sides

Reﬂect and check The quadratic formula could have been used, but it may have been more time-consuming, especially if we didn’t factor out the GCF ﬁrst. Original equation set equal to 0

Substitute a, b, c into quadratic formula

Evaluate the division

Multiplication property of radicals

Evaluate the radical

Evaluate the division

8.05 Solve quadratics using appropriate methods mathspace.co

493

Example 2 A rectangular enclosure is to be constructed from 100 meters of wooden fencing. The area of the enclosure is given by A = 50x − x2, where x is the length of one side of the rectangle. If the area is 525 m2, determine the side lengths.

Create a strategy We can set up and solve a quadratic equation, 50x − x2 = 525. Since the values are large we will try solving this problem with the quadratic formula. The two solutions will be the side lengths of the enclosure.

Apply the idea Rearranging the equation into standard form we get x2 − 50x + 525 = 0. We can solve this using the quadratic equation: Quadratic formula

Substitute a = 1, b = −50, c = 525

Evaluate the operations

Evaluate the square root

This leaves us with two values,

and

Evaluating each expression for x we

get x = 35 and x = 15 as the side lengths of the rectangular enclosure.

Reﬂect and check We can conﬁrm our answer is correct by checking the conditions of the problem. We had 100 meters of fencing and 2(35 + 15) = 100. We needed the area to be 525 m2 and 35(15) = 525 as required. Since there are two rational solutions, the quadratic equation was also factorable: x2 − 50x + 525 = (x − 35) (x − 15), but these factors are not immediately obvious.

Idea summary Below is a list of the easiest method to use and the form of the quadratic equation for which we should use it:

Graphing Factoring Square root property Completing the square Quadratic formula

494

Mathspace Virginia SOL Algebra 1 mathspace.co

Easiest equation form Any form is fine when using technology ax2 + bx + c = 0 where a, b, c are small x2 = k or a(x − h)2 = k x2 + bx + c = 0 where b is even ax2 + bx + c = 0 where a, b, c are large

Practice What do you remember? 1

Solve the following equations: a

(x − 3)2 = 64

(2 − x)2 = 81

x (x + 7) = 0

(10x − 9)2 = 0

(x − 6) (x + 7) = 0

x2 − 8x + 15 = 0

x2 − 4x − 22 = 0

The formula for the surface area of a sphere is S = 4π r2, where r is the radius. Determine the radius of a sphere that has a surface area of 804 in2, rounding your answer to two decimal places.

Consider the quadratic equation: x2 − 3x − 108 = 0 Select the solution to the quadratic equation. A

x = 9, x = −12

x = −9, x = 12

x = −9, x = −12

x = 9, x = 12

Consider the quadratic equation: x2 − 200 = −79 Select the solution to the quadratic equation.

A square lot has an area of 289 m2. Find the length of one side of the square lot if A = s2. A

s = 144.5 m

s = 72.25 m

s = 17 m

Let’s practice 6

For the following quadratic equations, ﬁnd the solution using an eﬃcient method. Justify which method you used. a

x2 − 10 = 15

x2 − 7x + 6 = 0

25y2 = 36

4x2 + 5x + 1 = 0

x2 + 24x + 63 = 0

x2 − 7x = 0

5k2 − 17k + 13 = 0

x2 + 9x + 20 = 0

x2 − 4x = −1

3x2 − 2x + 5 = 10x + 1

Solve the following equations: a

(x − 6)2 − 2 = 0

24x2 = 71x − 35

x + 27x + 23 = 3x − 40

3x2 − 36x + 33 = 0

3x2 − 12x − 36 = 0

4x2 − 13x + 2 = 0

x + 18x + 32 = 0

(x + 5)2 − 2 = 15

11x + x2 + 5 = 0

−4 + 2x2 − 5x = 0

x2 − 2x − 15 = 0

Solve the following equations. Justify your work. a

4x2 = 2 + 8x

x2 − 18x = −6

Solve the following equations. Verify your solution(s) by graphing or substitution. a

x2 − 4x = 32

2x2 + 3x − 5 = 0

x2 = −2x + 24

8.05 Solve quadratics using appropriate methods mathspace.co

495

The following rectangle has a length of L = 56y + 11 and a width of W = 5y2: a

Write the perimeter of the ﬁgure in terms of y.

If the perimeter is equal to 630, ﬁnd y.

At time t seconds, the distance, s, traveled by an object moving in a straight line is given by

where u is its starting speed and a is its acceleration. When u = 16 and a = 8, ﬁnd how long it would take for the object to travel 128 m. 12

The base of a triangle is 3 m more than twice its height. The area of the triangle is 115 m2. Let x be the height of the triangle. a

Find the height by solving for x.

Find the length of the base.

A rectangular swimming pool is 16 m long and 6 m wide. It is surrounded by a pebble path of uniform width x m. The area of the path is 104 m2. a

Find an expression for the area of the path in terms of x.

Write an equation and solve for x, the width of the path.

16 m x

14 Harry is using a diving board to dive into a swimming pool. The distance from his head to the surface of the water can be represented as (x − 7) (x + 7) = 147. Select the viable x-value to the quadratic equation. A

x = 14

x = −14

x = 49

x=7

Let’s extend our thinking 15

For each of the following equations determine, without solving them, the most eﬃcient method for solving them. Explain your thinking. a

x2 − 3x + 2 = 0

16x2 − 81 = 0

8x = x2

Executives at the Widget Emporium are discussing whether to merge their company with the Trinket Bazaar, a large competitor. Market analysis shows that the extra revenue the company will receive can be modelled by the equation R = 0.25t2, and the extra costs by the equation C = 3.5t. R and C are measured in thousands of dollars and t is measured in months after the merger.

496

Find the times at which the extra revenue R will match the extra cost C.

The executives decide that they can only afford to operate at a loss for one year. Based on this requirement, state whether you would advise that the Widget Emporium merge with the Trinket Bazaar.

Mathspace Virginia SOL Algebra 1 mathspace.co

9 Data Analysis Big ideas • Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. • Different representations of data highlight different characteristics of the data. • Many sets of bivariate data can be modeled using familiar functions.

Chapter outline 9.01 9.02 9.03 9.04 9.05

Data and sampling Scatterplots Linear regression Quadratic regression Analyze bivariate data

500 514 529 546 562

Variables

Bivariate data

Variables are quantities or qualities that can be measured or classified.

Bivariate data is data that is collected from two different variables and compared against each other. This data is typically numerical.

Independent variable

Example: Age versus height, or 1-mile time versus 5-mile time

The variable that is varied or controlled to explore the effect it has on the dependent variable. Dependent variable The variable that depends on the independent variable. We typically want to explore the effect that the independent variable has on the dependent variable.

Person

Age (years)

Art Kumi Isla Daria Xia

30 40 50 60 70

Systolic blood pressure (mmHg) 121 140 134 154 146

For a study about heart health, a person’s systolic blood pressure is measured against their age. Their age is the independent variable and can be any value. Their systolic blood pressure is the dependent variable that is recorded against their age. Notice that each person’s age and systolic blood pressure make a pair of values in the bivariate data set.

To start working with bivariate data, we need to formulate a statistical question. Statistical question A statistical question that can be answered by collecting data and whose answer may vary depending on the sample the data is collected from. Also called an investigative question. Statistical question Is there a relationship between age and systolic blood pressure? Do test scores increase as the amount of time studying increases? Does how long a pen lasts impact the cost of the pen?

Not statistical question What is your blood pressure? How long did you study and what was your grade? Are there pens under five dollars that will last all year?

A statistical question is different from a survey question that is asked to those in the people in a study. We need to make sure that questions are not leading people to answer a particular way. This means not using emotive language or suggesting a particular answer. Good survey question Do you watch soccer? How would you rate your meal? What was your average speed driving here?

Leading question Do you watch the most popular sport in the world, soccer? What did you think of the meal from the outstanding chef? Did you do the wrong thing and go over the speed limit to get here?

Notice how the good questions are very neutral and the leading questions may encourage people to respond in a particular way.

9.01 Data and sampling mathspace.co

501

Each of these questions is clear and concise, focusing specifically on the relationship between temperature and ice cream sales. They each propose a different aspect of the relationship to investigate, making them effective statistical questions.

Reﬂect and check After one round of the data cycle we may formulate a new question to further explore the topic.

Example 3 In a study conducted at a high school, students’ study times (in hours) and their corresponding test scores (out of 100) were recorded for one particular examination. The data is to be analyzed to understand the relationship between study time and test scores. a Identify the variables involved in this scenario.

Create a strategy A variable is any characteristic, number, or quantity that can be measured or counted. There are two types of variables: dependent and independent. The dependent variable is what is being measured or observed (the outcome), while the independent variable is what is being manipulated or changed (the likely cause).

Apply the idea In this case, the two variables are study time and test scores. The independent variable is the study time. This is because it is the variable that we think will cause changes in the test scores. The dependent variable is the test score. This is because it may change in response to changes in the study time. Test scores are what we are interested in predicting or explaining.

Reflect and check It’s important to consider potential confounding variables in any study. A confounding variable is an outside influence that may impact one or both of the variables. These may lead to a false conclusion. For example, the difficulty of the test, the student’s previous knowledge, and other external factors (like health, sleep, etc.) could all potentially impact a student’s test score.

b Rewrite this question so it is not leading and it could be used to accurately collect data. “We believe students who study more do better on tests. How much time did you spend studying last night? Did you do well on the test?”

Create a strategy We need to make sure that the question is not leading people to answer a particular way. This means not using emotive language or suggesting a particular answer.

Apply the idea “How much time did you spend studying last night? Did you do well on the test?”

9.01 Data and sampling mathspace.co

503

c Explain why doing a sample of the children at a park one Monday morning would not be a good sample.

Create a strategy For the sample to be representative of the population, different children of varying ages with various playground habits should be included.

Apply the idea This sample would be convenient, so is a convenience sample which is not representative. This sample likely wouldn’t include school age children who would be at school, not at the playground on a Monday morning. It also wouldn’t include children who don’t regularly get to go to the playground. Finally, there may not be many children at the park, but the population of children might be large in comparison.

Example 6 Dr. Jane is a health researcher and she formulated the question “Is there a relationship between the frequency of exercise and overall health among working adults in Washington, DC?” She wants to collect data for her research. Choose an appropriate sampling method.

Create a strategy When selecting a sampling method, Dr. Jane needs to consider several factors such as the size of her target population, the resources she has available, and potential biases that could influence the results.

Apply the idea An appropriate sampling method for this study could be stratified sampling. Considering Washington, DC’s large and diverse population, stratified sampling would ensure that all segments of the population are represented in the sample. Dr. Jane could divide the population into different strata based on factors like age, occupation, or zipcode, and then randomly select participants from each stratum.

Reﬂect and check While stratified sampling can provide a representative sample, it can be more complex and time-consuming to implement, and it might not be feasible if information about the different strata is not readily available. An alternative method might be simple random sampling, where every individual in the population has an equal chance of being selected. However, this method might not guarantee that all segments of the population are adequately represented, especially for a diverse population.

508

Mathspace Virginia SOL Algebra 1 mathspace.co

Idea summary After we formulate a clear statistical question, we use the data cycle to collect, show, and explain information. To get data, we can use methods like: • • • • •

Watching (Observation) Measuring Asking questions (Survey) Doing experiments Acquiring existing secondary data

Sampling methods are techniques to collect data from a representative subset of the population, known as a sample. • •

Population: every member of a group. Sample: a subset of the population.

Types of sampling methods include: • • • •

Simple Random Sampling: every member of the population has an equal chance of being selected. Systematic Sampling: involves selecting every nth member of the population. Stratified Sampling: dividing the population into subgroups, and then selecting a separate random sample from each subgroup. Cluster Sampling: the population is divided into groups, or clusters. Then, a random sample of clusters is selected, and all members within selected clusters are included in the sample.

The type of sampling method chosen can greatly influence the quality of data collected and the conclusions drawn from it.

Practice What do you remember? 1

What is the difference between bivariate data and univariate data?

Give an example of a real-world situation where the relationship between two variables can be investigated using bivariate data.

State whether each statistical question could be answered by collecting univariate data or bivariate data:

How are the weights of students on the wrestling team distributed?

Is there a relationships between iron levels in soil and weed growth?

What shoe sizes are the most common at each of my local schools?

Is taxable income related to latitude of home address?

Does the number of days children spend in daycare affect the number of days spent home sick?

Is there a relationship between the amount of natural sunlight in a classroom and students’ exam results?

Typically how old are people when they learn to skate on ice? Does it vary by country?

State whether each statement about statistical questions is true or false. a

The question must have a yes or no answer.

The question allows for surveys to be conducted.

The answers to question may vary from one person to another.

The answers to question requires only numerical values.

They are only used for bivariate data

They should include your hypothesis for what you think the answer will be. 9.01 Data and sampling mathspace.co

509

510

State whether each of the following questions are statistical questions: a

Which city is the capital of France?

How far away is the moon from the Earth, right now?

Which is a typical maximum temperature during the summer in Chesterfield, VA?

How much do kittens weigh?

How far do you have to travel from home to school each day?

How old are Olympic gold medal winners when they win their medal?

How far do students have to travel to get from their home to school each day?

How far do you have to travel from home to school each day?

How many calories do people burn per day?

Determine which data collection method best describes each of these scenarios. a

Asking people on the street about the age of their oldest living relative and recording their answers.

Using a batter bowl and a scale to determine the density of dough before and after it rises to see if there is a relationship.

A long term study that provides different levels of subsidies for childcare and looks to see if this affects the income and mental well-being of those children when they reach adulthood.

Watching a variety of gardens to see how many pollinators visit per hour.

Is each question leading or not? a

Do you take a multi-vitamin?

How much time do you waste on social media per day?

How much time do you spend reading every week?

Do you think the government should be allowed to cut down some of the oldest trees in the area to construct a metro railway line in the city?

Do you think bike helmets should be mandatory for all bike riders?

Do you eat at least the recommended number of servings of fruits and vegetables to ensure a healthy and long life?

How much time do you spend sitting every day?

Determine whether the scenario represents collecting data from a population or sample: a

Oscar has determined the cost of 5% of houses from each suburb in Richmond.

Ainsley tests every lamp that the factory produces.

Habib scans every carry-on bag for a flight Roanoke.

Drake does a checkup on all children brought to his doctor’s office to assess the health of all children in the city.

In which type of sampling does every individual in the population have an equal chance of being selected? A

Random sampling

Systematic sampling

Stratified sampling

What is the main difference between random and systematic sampling? A

Random sampling involves selecting individuals at regular intervals, while systematic sampling involves selecting individuals randomly.

Random sampling involves selecting individuals randomly, while systematic sampling involves selecting individuals at regular intervals.

There is no difference between random and systematic sampling.

Mathspace Virginia SOL Algebra 1 mathspace.co

John is conducting a research study on the satisfaction levels of employees in a company. He decides to use random sampling. a

Which option best describes the process of random sampling? A Random sampling involves dividing the population into distinct subgroups based on specific characteristics and then randomly selecting participants from each subgroup. B Random sampling involves selecting participants from a population in a completely random manner, ensuring each individual has an equal chance of being chosen. C Random sampling involves selecting participants from a population at regular intervals, using a predetermined starting point.

Which option best describes one advantage of using this random sampling? A One advantage of using this method is that it reduces the chances of bias and provides a representative sample of the population. B One advantage of using this method is that it ensures representation from each subgroup, allowing for more accurate estimates for each group.

A school principal wants to estimate the number of students who ride a bicycle to school. Is each sample biased? a

All students who are in the school band.

Eight students in the hallway after school.

Ten students from each grade, chosen at random.

130 randomly selected students during the lunch periods.

For each statistical question, identify the independent and dependent variables. a

Is there a relationship between the number of website visitors and the percentage of visitors who make a purchase?

Is there a relationship between the number of words in a child’s vocabulary and the number of books that are read to them per week?”

Is there likely to be a relationship between each pair of variables? a

Time spent on phone per day and number of apps on phone

Amount spent on pet food and amount spent on candy

Price of lemons and average SAT scores

Car speed and gas mileage

Let’s practice 15

For each scenario: i

Identify possible independent and dependent variables.

Write a statistical question related to the scenario.

Gertrude notices the eggs with more Omega-3 tend to cost more and wonders if these are related.

Latisha has started baking bread from scratch. As she gets more experienced she finds it takes less time. She wonders if there is a link between experience and time it takes to bake bread.

Theo likes running and is working on his hill sprints on different hills. He is curious about his maximum speed on different slopes.

Change the following questions to make them statistical questions. a

How many books does your teacher have?

How many points did the grade school basketball team score in its last game?

What is your grade in Algebra 1 during the first term?

9.01 Data and sampling mathspace.co

511

Jeremiah formulated the statistical question “Is there a relationship between spending on advertisement and business revenue?” Design a simple study that could be used to collect data to answer her question.

Natalie notices that finding coats that are long enough for her is difficult. She is curious what other body measurements are the best predictor of torso length. She formulates the question “Is there a relationship between arm length and torso length in teenagers?”

Should she use measurement, observation, or acquire data to collect the data?

What type of data would she be collecting?

What type of sample should she use to ensure it is representative of the population?

Rewrite each question so it is not leading and it could be used to accurately collect data using a sample survey. a

A new study said that using more than two bottles of shampoo per year is wasteful. How many bottles of shampoo do you use per year?

Most people with nice hair use some kind of oil product, what is your favorite hair product?

Would you describe your hair as beautiful?

How strong is the relationship between hair length and intelligence?

The owner of a movie theater wants to use stratified sampling in their survey of people who come to their theater. Are these methods considered to be stratified sampling?

Interview 10% of the people who used the concessions and 10% of people who didn’t.

Interview every person that sees a romantic movie.

Interview 10% of the people from each movie.

Interview every 10th person that purchases a ticket.

For each scenario, determine the type of sampling method used: a

Drawing out the winning ticket in a lottery

Choosing every 50th person on the class roll to take part in a survey

Choosing 5% of the of the students in each grade for grades 7−12

David is conducting a research study to investigate the eating habits of people in a particular city. He opts for stratified sampling. What is an advantage of using this method?

Sarah is conducting a survey to gather data about the shopping preferences of customers in a large retail store. She chooses to use systematic sampling. What is a disadvantage of using this method?

Explain why the following samples are biased:

512

Hannah is surveying customers at a shopping mall. 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 favorite 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.

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 25

Imagine you are investigating the relationship between a student’s SAT score and their college GPA. Identify potential confounding variables that could influence this relationship and how you might account for them in your investigation.

Patricia surveys her class about their favorite music. Her results are shown in the table: Genre No. students

Country 15

Pop 2

According to the survey, which genre is most popular among her class?

This was her survey question: “The coolest kids like country music, and nobody likes pop. Do you like country music or pop music?” Do you trust the results of Patricia’s survey? Explain your answer.

You want to survey a group of n students about their favorite sports, but you only have time to survey 20 students. How would you use systematic sampling to select the 20 students for the survey?

You want to conduct a survey of the reading habits of students at your school. How would you use stratified sampling to ensure a representative sample?

Discuss the importance of formulating appropriate statistical questions in the data cycle. How does this impact the subsequent steps in the cycle such as data collection, analysis, and interpretation?

For a topic that interests you, consider a relationship that might exist. a

Formulate a statistical question that could be used to explore the relationship.

Describe a sampling and data collection method that could be used.

9.01 Data and sampling mathspace.co

513

7 6

5 2

3 1

x 1

10 20 30 40 50 60 70 80 90

Strong negative relationship since points are tightly clustered along the model

2 3 4 5 6 7 8 9

Moderate negative relationship since points are relatively clustered along the model

260

259

258

257

5 4

256

255

254

x 0.2

x 1

0.3

Weak positive relationship since points are loosely clustered along the model

2 3 4 5 6 7 8 9

No relationship since there is no evident clustering of the data

When comparing bivariate data, it may be necessary to separate the data into categories. For example, when comparing the weights of dogs during their first year after birth, the data might not show a relationship because large dogs (like Boxers) will grow much more than small dogs (like Yorkies). We can compare categorical variables in scatterplots by using different colors or symbols. The weights of small, medium, and large dogs over time are shown in the scatterplot. Weight of dogs over time small dogs

weight (lb)

60 50

medium dogs

large dogs

30 20 10 0

9 10 11 12

Age (in months)

Different colored dots represent the different categories or sizes of dogs. For each category, there is a strong, positive linear relationship between the dogs’ age and weight. It is important to note that the existence of a relationship between two variables in a scatterplot, regardless of strength, does not necessarily imply that one causes the other. Causation can only be determined from an appropriately designed statistical experiment. 9.02 Scatterplots mathspace.co

515

Example 1 For each scatterplot, determine whether the variables have a linear relationship, a nonlinear relationship, or no relationship. If there is a relationship, describe its strength. If the relationship is linear, describe the direction as positive or negative. a

55 50 45 40 35 30 25 20 15 10 5

x 1

Create a strategy A relationship between two variables exists if the points follow a similar trend. The points will roughly form a line (linear) or a curve (nonlinear) if there is a relationship. To describe the strength of the relationship, we can analyze how tightly the data points are clustered or grouped together.

Reﬂect and check The y-values are decreasing at a slower and slower rate, causing the point to form a curve. This shows there is a nonlinear relationship between the variables. Because the points are tightly clustered, the relationship is strong.

b 90

80 70 60 50 40 30 20 10

x 1

8 9

Create a strategy A relationship between two variables exists if the points follow a similar trend. If there is no trend or no shape to the data, then there is no relationship between the variables.

Apply the idea There is no trend in this data, meaning there is no relationship between the variables.

516

Mathspace Virginia SOL Algebra 1 mathspace.co

Reflect and check We could try to sketch a line of fit for the data, like the one shown, but the points are far from the line. A negative, linear trend would suggest that y decreases as x increases, which we cannot conclude for this data set. 90

80 70 60 50 40 30 20 10

x 1

50 45 40 35 30 25 20 15 10 5

2 3

4 5

7 8

x 2 4 6 8 10 12 14 16 18

Create a strategy First, we must determine if a relationship between the variables exists. If a relationship exists, we can describe the strength by analyzing how tightly the data points are clustered. If the relationship between the variables is linear, the direction of the relationship can be described as positive or negative. • Positive relationship: as the independent variable increases, the dependent variable increases • Negative relationship: as the independent variable increases, the dependent variable decreases

Apply the idea As the x-values increase, the y-values also increase. This indicates there is a positive, linear relationship between the variables. However, the points are not tightly clustered, so the relationship between the variables is moderate.

9.02 Scatterplots mathspace.co

517

The data should be separated into two categories: • Patients that take medication only • Patients that take medication and attend physical therapy sessions

Reflect and check Since the data is bivariate and numerical, it can be represented by a scatterplot. The independent variable is time, and the dependent variable is the patients’ pain level. An example scatterplot is shown: Pain level changes over time

Pain level

8 7

Group A (Physical therapy and medication)

Group B (Medication only)

5 4 3 2 1 0

3 4 5 6 Time (weeks)

Example 3 A surfing company is located in various coastal states across the U.S. When analyzing their data, they separate the store locations into two regions: the Western region and the Eastern region. The scatterplot shows data collected to answer the question, “How have the sales of our product changed over time in each of the sales regions?” Product sales over time 720 640

Western region

Sales ($)

560

Eastern region

480 400 320 240 160 80 0

0 2

8 10 12 14 16 18 20 Time (months)

a Identify the independent and dependent variables in this context.

Create a strategy Recall that the independent variable is not affected by the other variable, while the dependent variable may be affected or changed by the other variable. On a scatterplot, the independent variable is placed on the horizontal axis, and the dependent variable is placed on the vertical axis.

9.02 Scatterplots mathspace.co

519

Apply the idea The independent variable is time (measured in months), and the dependent variable is the amount of sales (measured in dollars).

b The owner of the company makes this conclusion: “The sales of the product are improving with time.” Which sales region was the owner analyzing?

Create a strategy In part (a), we found that the amount of sales is the dependent variable, and time is the independent variable. This means the owner concluded that the dependent variable increases as the independent variable increases.

Apply the idea

Reflect and check

According to the owner’s statement, both variables are increasing which indicates a positive relationship. Both sets of data values show a linear relationship, but only the blue dots show a positive relationship. According to the key (or legend), the blue points represent data from the Western region.

If the owner was analyzing the Eastern region (the black points), the conclusion would have been, “The sales of the product are decreasing over time.”

Example 4 Adria heard that children who learn to speak at a young age are more likely to be gifted and talented in later stages of life. She decides to investigate this using the data cycle. a Formulate a statistical question for Adria that would lead to the collection of data that can be represented in a scatterplot.

Create a strategy First, we need to identify the variables of interest. Then, we need to write a question such that the answer to the question addresses both variables.

Apply the idea From the given information, we gather that Adria is interested in two variables: 1. The age when a child first spoke 2. Their intelligence level later in life The information is not specific about the later stages of life. We can choose any stage of life after birth, such as the teenage years. One possible statistical question is, “What is the relationship between the age at which a child first spoke and their level of intelligence as teenagers?”

Reflect and check Other possible questions are: • How does the age at which a child first spoke influence their level of intelligence as adults? • If a child first spoke at 6 months old, what level of intelligence are they expected to have as a teenager? • Which range of ages for when a child first spoke correspond to the highest levels of intelligence? This could also be separated into multiple categories: age when a child first spoke versus intelligence level after middle school, after high school, and after university.

520

Mathspace Virginia SOL Algebra 1 mathspace.co

b The table shows the ages of some teenagers when they first spoke and their results in an aptitude test: Age when ﬁrst spoke (months) Aptitude test results

14 96

27 69

9 93

16 101

21 87

17 92

10 99

7 104

19 93

24 97

Create a scatterplot to model the data.

Create a strategy Let x represent the age when the child first spoke and y represent the aptitude test results as a teenager. The minimum value for x is 7 and the maximum is 27, so we can use a scale of 5 to label the x-axis. The minimum value for y is 69 and the maximum is 104, so we can use a scale of 20 to label the x-axis.

Apply the idea Aptitude score 100 80 60 40 20 Age (months) 5

c Draw a conclusion about the data by answering the statistical question from part (a).

Create a strategy To describe the relationship between the age at which a child first spoke and their level of intelligence as teenagers, we can analyze the following features of the data: • Form: linear or nonlinear • Strength: strong or weak If the data follows a linear trend, we can describe the direction as positive or negative.

Apply the idea

Reﬂect and check

The points are relatively close together, indicating a strong relationship. As the age increases, the aptitude score decreases slightly, indicating a negative, linear relationship.

The closer the points are to forming a line or curve, the stronger their relationship will be. A strong relationship between two quantities suggests that the value of one quantity can be predicted with some accuracy given the other quantity, but is not enough evidence to suggest that changes in one quantity directly cause changes in the other.

The relationship between the age when a child first spoke and their aptitude test score as a teenager has a strong, negative, linear relationship. This suggests that as the age at which a child first spoke increases, their intelligence level as a teenager tends to decrease.

9.02 Scatterplots mathspace.co

521

Idea summary The analysis of bivariate data should include: • •

Form, usually described as a linear relationship or a nonlinear relationship Strength, describing how closely the data points match the model line or curve

If the relationship between the variables is linear, the direction of the relationship can be described as positive or negative. • •

Positive relationship: as the independent variable increases, the dependent variable increases Negative relationship: as the independent variable increases, the dependent variable decreases

Practice What do you remember? 1

Create a scatterplot that models each set of data. Include labels and scales on each axis. a

The heights and weights of the female Olympic “All around champions” in gymnastics.

Suni Lee Simone Biles Gabby Douglas Nastia Liukin Carly Patterson Simona Amanar b

Weight (lbs) 112 104 90 99 97 97

The test scores on the midterm and ﬁnal exam for a sample of students. Midterm Exam Final Exam

Height (inches) 60 57 59 62 59 62

95 90

90 92

88 83

84 80

75 62

77 80

65 60

70 74

99 100

85 85

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 relationship be positive or negative?

522

For each pair, identify the independent and dependent variable: a

Amount of fertilizer and plant height

Length of stride and height

Number of family members and time (in hours) spent cooking

Time spent traveling and distance to destination

Time spent practicing and performance in piano lessons

Mathspace Virginia SOL Algebra 1 mathspace.co

Which question would lead to data that could be represented by a scatterplot? A

On which days of the week do most teenagers play videos?

How many hours does an average teenager spend playing video games each day?

What is the difference in the number of hours teenagers play video games?

How does the amount of time spent outside impact the amount of time spent playing video games?

Does each scatterplot show a linear or nonlinear relationship? a

5 x 5

9.02 Scatterplots mathspace.co

523

Let’s practice 6

Describe the strength and direction for each linear association. a

5 x 5

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 grade? a

b 100

Final exam grade

100

x 0

Number of class attended

524

Mathspace Virginia SOL Algebra 1 mathspace.co

x 0

Number of class attended

d 100

Final exam grade

100

x 10

Number of class attended

This scatterplot shows the relationship between air and sea temperature. a

Which is the best description of the relationship between the variables?

35 Sea temperature

A Strong, positive, linear B Moderate, negative, linear C Weak, nonlinear b

Describe the relationship between the variables in context.

30 25

Air temperature

This table shows the scores of 12 students in math and P.E. class. a

Select the question that could be answered by the data. A Do students prefer Math or P.E.? B If a student does well in Math, do they also do well in P.E.? C Does a student’s Math grade impact their P.E. grade? D How many students are enrolled in Math and P.E.?

Construct a scatterplot for the students’ scores in math versus their scores in P.E. class.

Is the relationship between students’ grades in math and P.E. linear or nonlinear? If it is linear, describe the direction as positive or negative.

Describe the strength of the relationship between students’ grades in math and P.E.

Student 1 2 3 4 5 6 7 8 9 10 11 12

Maths 63 82 60 79 88 81 61 91 72 62 66 92

P.E. 44 94 52 70 67 60 73 86 84 93 57 92

A shop owner in Morocco collected data to answer the statistical question, “How does the temperature outside impact the number of fans sold?” The data is shown in the table: Temperature (°C) Number of fans sold

6 12

8 13

10 14

12 17

14 18

Identify the independent and dependent variables.

Which method was most likely used to collect the data? A Measurement

Observation

16 19

18 21

20 23

Survey

Construct a scatterplot using the data from the table.

Describe the relationship between the temperature and the number of fans sold.

Experiment

9.02 Scatterplots mathspace.co

525

Data was collected to answer the statistical question, “What is the relationship between the age at which a child first walked and the age at which they first spoke?” The data collected is shown in the table: Age first walked Age first spoke

12 12

18 21

15 16

13 20

17 19

9 13

16 22

20 20

What is the relationship between the age at which a child first walked and the age at which they first spoke?

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. 1 8

2 19

3 26

4 37

5 51

6 65

Was the data most likely collected through measurement, observation, a survey or an experiment?

Construct a scatterplot using the data from the table.

Describe the relationship between the number of passers-by and the time until assistance is offered.

As more passers-by are present, what happens to the time taken until help is offered?

Each point on the scatterplot shows the time (in weeks) Sumon spent training for a half marathon and the corresponding miles they were able to run. a

The number of weeks that Sumon trained for the half marathon is the independent variable.

The y-coordinates of the points represent the time spent by Sumon training.

There is evidence to suggest that the longer Sumon trains, the further they can run.

526

11 15

Construct a scatterplot for the data.

Using the scatterplot, are these statements true or false?

14 17

Number of passers-by (n) Time until help is offered (t)

10 12

The relationship between the number of weeks training and the number of miles Sumon is able to run is positive.

As preparation for a science test, a group of 10 students was given a practice worksheet containing 60 questions. The table shows the number of questions from the worksheet successfully completed by each student and the score out of 100 of that student on the test. a

Formulate a question that could be answered by the data.

Was the data most likely collected through measurement, observation, a survey or an experiment?

Which variable is independent and which variable is dependent?

Construct a scatterplot of the data.

Draw a conclusion about the data by answering the statistical question from part (a).

Mathspace Virginia SOL Algebra 1 mathspace.co

Running distance (mi) 20 15 10 5

Number of questions 11 14 36 60 57 42 20 27 50 59

No. of weeks 15 20

Test result 20 23 62 97 100 66 35 52 87 99

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)

D 100 Snow remaining (%)

100

Snow remaining (%)

Temperature outside (°C)

x 0

Temperature outside (°C)

A snow sculpture company wants a chemical that will keep snow from melting. Which chemical should they choose to ensure that at least half of the snow is still there after 3 hours?

If the company operates at a temperature of 3 °C to 4 °C, which chemical should they choose?

Mona has a checking account and a savings account with her bank. Her savings accout accrues interest from the bank, and she tries to deposit and withdraw from both accounts equally. She has been tracking the balance of each account over the past year.

Balance ($)

1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

Checking account Savings account

Formulate a question that could be answered by the scatterplot.

Describe a method that Mona could have used to collect the data.

Identify the independent and dependent variables.

Which account had a higher balance at the beginning of the year?

Which account had a higher balance at the end of the year?

Draw a conclusion about the data by answering the statistical question from part (a).

0 1 2 3 4 5 6 7 8 9 10 11 12 Time (months)

9.02 Scatterplots mathspace.co

527

Let’s extend our thinking 17

Determine whether each statement is true or false. Provide an example to support your claim. a

If the value of variable A increases as the value of variable B increases, there is a relationship between variable A and variable B.

If there is a relationship between variable A and variable B, then changes in variable A directly cause changes in variable B.

If variable A and variable B move in opposite directions (as one increases, the other decreases), then they have a negative relationship.

Consider the scatterplot: a

Explain why the relationship between the variables is weak.

Determine whether the following pairs of variables could be represented by the data set:

Scores in an English test and distance traveled from home to school.

Cost of cars and cost of gasoline.

iii Distance traveled in a car and the cost of a driver’s license. x

Brody wants to take his dog on a hike, then stop to pick up groceries on the way home from the hike. However, he is worried about leaving his dog in the car for half an hour while he is inside the store because he has heard it is unsafe. Temperature over time in a closed car Brody wants to use the data cycle to investigate the temperature inside the car over time a

Formulate a question which could be investigated using a scatterplot.

Determine what variables could be used to answer your investigative question.

The current temperatures where Brody lives are between 70–80 °F. He acquired data on the temperature inside a car on a 70 °F day and an 80 °F day, shown in the table. Plot each set of data on the same scatterplot.

Draw a conclusion about the data by answering the statistical question from part (a).

Brody learns that at 103 °F, dogs lose their ability to regulate their body temperature. Determine an approximate range of time that it takes for the inside of a car to reach 103°F when outside temperatures are between 70°F and 80°F.

Time in minutes 0 5 10 15 20 25 30 35 40 45 50 55 60

Temperature in car on 70°F day 70°F 83°F 89°F 94°F 99°F 102°F 104°F 106°F 108°F 110°F 111°F 112°F 113°F

Temperature in car on 80°F day 80°F 94°F 99°F 105°F 109°F 111°F 114°F 117°F 118°F 119°F 121°F 122°F 123°F

When Sherrie was trying on shoes, the sales attendant told her to always try on both shoes because, for most people, one foot is longer than the other. Go through the whole data cycle at least once to investigate tge relationship between the lengths of a person’s feet.

528

Mathspace Virginia SOL Algebra 1 mathspace.co

A line of best fit (or trend line) is a straight line that best represents the data on a scatterplot. We can use lines of best fit to help us make predictions or conclusions about the data. We previously approximated a line of best fit by trying to balance the number of points above the line with the number of points below the line. This can result in multiple different models. Height (cm)

Height (cm)

8 y = 1.4x + 1.6

6 4

Weekly growth 1

Weekly growth

8 9

3 points above, 3 points below

y=x+3

8 9

5 points above, 4 points below We get a more accurate line of best fit when we use technology, referred to as linear regression analysis.

Height (cm)

Once we have found the line of best fit for a scatterplot, we can interpret the key features and use the line to predict values that don’t appear in the data set.

10 8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

In the context of a line of best fit, the slope-intercept form represents

y = mx + b m

the rate of change for y with respect to x

the starting value of y when x is 0

For example, this graph models a plant’s growth over several weeks. 14

The slope of the line y = 1.21x + 2.14 means that the plant is growing at a rate of 1.21 centimeters per week.

Height (cm)

The y-intercept of 2.14 means the plant was 2.14 centimeters tall at week 0. This is feasible if the plant was not a seed when measurements began.

12 10 8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

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

530

Mathspace Virginia SOL Algebra 1 mathspace.co

Interpolations

Extrapolations

Using the previous example of the plant height over time: 14

Interpolating which week the plant was 9 centimeters tall, we will solve 9 = 1.21x + 2.14. The plant was 9 centimeters tall at 5.67 weeks.

Height (cm)

Extrapolating the plant’s height at 10 weeks, we will evaluate y = 1.21(10) + 2.14. The plant will be 14.24 centimeters tall at 10 weeks.

10 8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set. • A larger sample size increases reliability. • Interpolation with a strong correlation implies a reliable prediction. • Interpolation with a moderate or weak correlation leads to a less reliable prediction. • Extrapolation generally leads to an unreliable prediction. The further outside the range of known values, the less reliable it is.

Example 1 Natalia collected data to answer the question, “What is the relationship between the years since purchasing a car and its value?” Her data is shown in the table. Time since purchase (years) Value (thousands of dollars)

0.5 29

0.8 1.2 28.5 28.5

1.3 27.4

1.5 28.5

1.7 27

1.8 2.1 25.9 25.9

2 24.7

Time since purchase (years) Value (thousands of dollars)

2.6 24.6

2.8 23.5

3.4 23.3

3.6 21

3.9 21

4.05 22

4.8 20.1

3.1 24.6

4.6 21

2.5 26.4

a Find the equation of the line of best fit.

Create a strategy To find the equation using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Highlight the data and select Two Variable Regression Analysis. 3. Under the Regression Model drop down menu, choose Linear.

9.03 Linear regression mathspace.co

531

Apply the idea Enter the x-values and y-values in two separate columns. 1. Enter the x-values and y-values in two separate columns.

2. Highlight the data and select Two Variable Regression Analysis.

532

Mathspace Virginia SOL Algebra 1 mathspace.co

3. Under the Regression Model drop down menu, choose Linear.

If we round the coefficients to two decimal places, the equation of the line of best fit is y = −2.2x + 30.46.

Reflect and check The points are tightly clustered around the line, indicating that the relationship between the years since the car was purchased and the value of the car is strong. This means the line of best fit can be used to make relatively reliable predictions. Remember, a strong relationship does not imply that one variable causes changes in the other. We cannot say that the year since the car was purchased causes the value of the car to decrease, as there may be other factors that affect the value of the car.

b Interpret the slope and y-intercept of the line.

Create a strategy Use the independent and dependent variables to determine the units of the slope and y-intercept. Car value over time Value (thousands of dollars) 30

To help us visualize the relationship better, we can sketch the scatterplot and line of best fit, and add labels on the axes of the graph. Remember that the y-values are in thousands of dollars. This means we will need to multiply the y-value of the slope and y-intercept by 1000 when interpreting them in context.

25 20 15 10 5 Time since purchase (years) 1

9.03 Linear regression mathspace.co

533

Apply the idea The y-intercept of (0, 30.46) means that at the time of purchasing the car, it would have a value of $30460. The slope of −2.2 means that each year, the car’s value would decrease by $2200.

c Make a prediction about the value of a car after 3 years.

Create a strategy We are given the years since the car was purchased, which is the indpendent variable (x), and we are looking for the value of the car, which is the dependent variable (y). We can use the graph to estimate the y-value at x = 3 or use the line of best fit to get a more accurate prediction.

Apply the idea When we substitute x = 3 into the equation, we get y = −2.2(3) + 30.46 = 23.86 Based on the equation of the line of best fit, a car that is initially valued at $30460 will be worth $23860 three years after it was purchased.

Reflect and check When using technology to evaluate x = 3, we will get a slightly different answer. This is because the coefficients were rounded in our line of best fit. The calculator does not round the coefficients, making its result more accurate.

534

Mathspace Virginia SOL Algebra 1 mathspace.co

d Make a prediction about the value of a car after 10 years.

Create a strategy Since 10 years after purchase is not shown on the graph, we can use the equation of the line of best fit to determine the value of a car at that time.

Apply the idea We can use technology to find the value of y when x = 10.

A car that is initially valued at $30 460 will be worth $8513 ten years after it was purchased.

e Is the prediction for the car’s value after 3 years or after 10 years more reliable?

Create a strategy To determine the reliability of the predictions, consider whether interpolation or extrapolation was used to make the prediction. Interpolation leads to a more reliable outcome than extrapolation.

Apply the idea The given data ranges between x = 0.5 and x = 4.8. This means the prediction of the car’s value after 3 years falls within the range of known data, while the prediction after 10 years falls outside of that range. The prediction of the car’s value after 3 years is more reliable.

Reflect and check Interpolation is more reliable than extrapolation because the predictions follows the same pattern as the known data values. With extrapolation, we assume that the trend continues beyond the known data values. Realistically, the trend may not continue which makes extrapolation less reliable.

9.03 Linear regression mathspace.co

535

Example 2 A teacher recorded the number of days since a student last studied for an exam and their score out of a possible 80 points on the exam. Days since studying Exam score

3 64

2 59

6 42

4 57

4 58

1 72

6 33

3 63

4 55

2 62

a Formulate an investigative question that can be answered by the data.

Create a strategy The question should be focused on the relationship between the variables represented by the data. The independent variable is the number of days since studying, and the dependent variable is the score on the exam.

Apply the idea One possible question is, “How does the number of days since a student last studied impact their exam score?”

Reflect and check Other possible questions are: • What is the relationship between the number of days since a student last and their exam score? • How many days prior to the exam should a student study to increase their exam score? • If a student studies on the same day as the exam, what is their expected score on the exam?

b Was the data most likely collected through measurement, observation, a survey or an experiment?

Apply the idea The teacher did not measure, observe, or control the time since a student studied. Instead, it is more likely that the teacher asked the students how many days it has been since they last studied. The data was most likely collected through a survey.

Reflect and check Although the teacher may have had access to the students’ exam scores (assuming the teacher was the one that assigned the exam), they could have still included a survey question about the exam score to keep the data organized. For example, their survey questions could have been, “How many days has it been since you last studied for this subject?” and “What was your score on the exam?”

c Describe the relationship between the number of days since studying and the exam score.

Create a strategy To describe the relationship, we should construct a scatterplot to get a visual of the data. Then, we will consider the form (linear or nonlinear), strength (strong, moderate or weak), and direction (positive or negative).

536

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea 80

Score

70 60 50 40 30 20 10

Days since studying 1

The data appears to have a strong, negative, linear relationship. Relating this back to the context, we can say that as the number of days since a student last studied increases, and their score on the exam tends to decrease.

d Calculate the line of best ﬁt using technology.

Create a strategy To find the equation using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Hightlight the data and select Two Variable Regression Analysis. 3. Under the Regression Model drop down menu, choose Linear

Apply the idea 1. Enter the x- and y-values in two separate columns:

9.03 Linear regression mathspace.co

537

2. Highlight the data and select Two Variable Regression Analysis:

3. Choose Linear under the Regression Model drop down menu to ﬁnd the line of best ﬁt:

Apply the idea The equation of the line of best ﬁt is y = −6.22x + 78

Reflect and check If the instructions do not specify to round the coefficients, it is best to include all the digits given by the calculator. This increases the accuracy of the model and the predictions.

538

Mathspace Virginia SOL Algebra 1 mathspace.co

e Answer the question formulated in part (a).

Create a strategy To answer the question, “How does the number of days since a student last studied impact their exam score?”, we can describe the direction of the linear relationship. To be more specific, we can interpret the slope of the line in context. In the previous part, we found the equation of the line of best fit to be y = −6.2245x + 78.2857, which tells us the slope is −6.2245.

Apply the idea As the number of days since a student last studied increases, their exam score decreases. More specifically, for each additional day since a student last studied, their exam score is expected to decrease by about 6 points.

Reflect and check Matching the rise and run of the slope to their respective units can help us interpret its meaning in context.

The y-values represent the exam score, which is the “rise” of the slope. The x-values represent the number of days since studying, which is the “run” of the slope.

Score

70 60

Since the slope is negative, it represents a decrease of 6.2245 in the exam score for every 1 day since studying.

50 40 30 20 10

Days since studying 1

If a student studied the same day as the exam, what would we expect their score to be?

Create a strategy If the number of days since a student last studied is 0, then their exam score is the y-value of the y-intercept.

Apply the idea The y-intercept tells us that a student who has studied on the day of the exam has a predicted score of 78.2857, according to the linear model.

Reflect and check Although this value was found through extrapolation, x = 0 is not very far outside of the range of known values. Since the relationship is strong, this prediction is relatively reliable.

9.03 Linear regression mathspace.co

539

Idea summary A line of best fit for a set of data can be used to interpret a given situation and make predictions about values not represented by the data. A line of best ﬁt has an equation of the form y = mx + b. We can use technology to perform the linear regression analysis. In the context of a line of best ﬁt, the slope-intercept form represents

y = mx + b m

the rate of change for y with respect to x

The starting value of y when x is 0

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

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set. In general, interpolation is more reliable than extrapolation.

Practice What do you remember? 1

Choose the line of best ﬁt for this scatterplot.

8 7 Line 3

Line 2

5 4 3

Line 1

2 1

x 1

2 3 4 5 6 7 8 9

Sketch the line of best fit for each scatterplot: a

5 x 5

540

Mathspace Virginia SOL Algebra 1 mathspace.co

x 5

5 x 5

The scatterplot represents the given data set: {(36, 114), (20, 164), (22, 154), (24, 150), (26, 140), (28,138), (30,134), (32, 122), (34,118)} a

Create a table of values for the data set.

Find the equation of the line of best fit.

180

170 160 150 140 130 120 110 100

x 20 22 24 26 28 30 32 34 36

Does each graph show interpolation or extrapolation? a

10 20 30 40 50 60 70 80 90

x 10 20 30 40 50 60 70 80 90

9.03 Linear regression mathspace.co

541

Is each statement true or false? a

The line of best fit can only be used for interpolation, not extrapolation.

Using the line of best fit for extrapolation is generally more reliable than interpolation.

Interpolation is more accurate when the data points are tightly clustered around the line of best fit.

A line of best fit may not accurately represent the relationship between variables if the relationship is non-linear.

It is not important to consider the validity of predictions when using interpolation or extrapolation.

Considering the strengths and weaknesses of a regression model helps ensure that the conclusions are accurate and reliable.

Let’s practice 6

Find the equation of the line of best fit for each data set. a

x y

28 3926

x y

17 16

Speed Time

Age Accidents

30 6482 19 19

20 85

32 10 589

21 21 25 87

20 41

23 16

25 17

30 75 25 44

34 17 098

35 82 30 39

36 28 236

27 27 40 69

35 34

38 46 985

29 19 45 73

40 30

50 60 45 25

55 57 50 22

60 45 55 18

65 49 60 19

65 17

For the data set shown: {(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)}

Find the equation of the line of best fit.

Predict the value of y when x = 3.49. Round your answer to two decimal places.

Is the prediction in part (b) an example of interpolation or extrapolation?

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 °F, t Number of hot cocoas sold, n

0 20

2 16

5 3

2 16

1 18

3 17

6 8

4 12

Is the relationship between the outside temperature and the number of hot cocoa sold linear or nonlinear?

Calculate the regression model for this data set. Round all values to the nearest tenth.

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.

542

Explain the meaning of the y-intercept.

State the slope 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?

Mathspace Virginia SOL Algebra 1 mathspace.co

The life expectancy (E), in years, of individuals at different annual incomes (I), per $1000, is shown:

E 100

The equation of the line of best ﬁt is E = 0.09I + 72.55. a

By how much does average life expectancy change for each $1000 of annual income?

Find the average life expectancy of someone who earns no income.

75 50 25 I 50

Scientists collect data to answer the statistical question, “What is the relationship between the number of aphids and the number of ladybugs in various areas?” The data is shown in the scatterplot.

100

150

A 3200

A = −3.82L + 3865.21 represents the line of best ﬁt.

Describe the variables the scientists used to answer their statistical question.

How much does the average aphid population change by with each extra ladybug? Round your answer to the nearest aphid.

Find the average aphid population of a region with no ladybugs. Round your answer to the nearest aphid.

2400 1600 800 L 200 400 600 800 1000

The average monthly temperature and the average wind speed in a particular location was plotted over several months. The graph shows the points for each month’s data and their line of best fit. a

Identify the independent and dependent variables used in this study.

Use the line of best fit to approximate the wind speed on a day when the temperature is 41 °F.

How reliable is this prediction? Explain your answer.

Wind speed (knots)

6 5 4 3 2 1

Temperature (°F ) 39

The scatterplot shows data collected on the amount of caffeine consumed, in milligrams, in a day and the number of hours of sleep for 30 adults.

Which equation is most likely the line of best fit?

B y = 0.015x − 9.58

C y = −0.015x + 9.58

Describe the strength of the relationship. Explain what this relationship implies in context.

Formulate a question that can be answered by the scatterplot.

D y = −0.015x − 9.58

Hours of sleep

A y = 0.015x + 9.58

4 Caffeine (mg) 50

100 150 200 250 300

9.03 Linear regression mathspace.co

543

Scientists conducted a study to analyze the time it took people to perform a simple matching activity after they’ve had different amounts of sleep. The participants were placed in similar rooms under the same conditions, and they were all given the same matching activity. The data is shown in the table and scatterplot, along with the line of best fit. Number of hours sleep (x) Completion time in seconds (y)

3.5 3.6

Which data collection method was used? Observation

4 3.4

Survey

Experiment

Use technology to find the equation for the line of best fit.

Use the line of best fit to predict the task completion time for someone who has slept 5 hours.

Predict the number of hours someone has slept if they complete the matching task in 4 seconds.

A student collected data to answer the statistical question, “How does the temperature outside impact the number of people at the beach?” Their data is shown in the table. 69 80

72 88

74 120

77 134

79 162

80 177

82 180

83 188

85 220

87 230

Did the student collect the data through measurement, observation, a survey or an experiment?

About how many people might be at the beach when the temperature is 80°F?

What is the temperature when there are 81 people at the beach?

About how many people might be at the beach when the temperature is 90°F?

Is the prediction for 80°F or 90°F more reliable?

One liter of gas is raised to various temperatures, and its pressure is measured. The results are shown in the table. 300 2400

302 2416

304 2434

308 2462

310 2478

312 2496

314 2512

316 2526

318 2546

Use technology to calculate the equation for the line of best fit.

Use the line of best 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?

Will using the line of best fit to predict pressure within each of these ranges of temperatures result in a reliable prediction? 300 ≤ Temp ≤ 320

300 ≤ Temp ≤ 600

iii 0 ≤ Temp ≤ 320

280 ≤ Temp ≤ 340

Concern over student use of the social media app SnappyChatty leads to a study of student grades in Mathematics versus minutes spent using the app. The results are shown in the table. Minutes, M Grade, P %

544

2.5 3.7

Formulate a question that could be answered by the data.

Temperature (K) Pressure (Pa)

2.1 4.66

Temperature (°F) Number of people

1.5 4.1

A Measurement

1.1 4.66

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 SnappyChatty app. Use a model to justify your response.

Use your model to explain and interpret the relationship between minutes spent on the SnappyChatty app and mathematics grades.

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 18

Based on the given scatterplot and line of best fit, could the model be used to make reliable predictions? Explain.

8 7 6 5 4 3 2 1

x 1

Lorena and Frasier each draw a possible line of best ﬁt. a

2 3 4 5 6 7 8 9

y Frasier

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 represent a line of best ﬁt? Explain.

6 5

Lorena said she noticed that there was a point that was far away, but decided to ignore it. Would Lorena’s line better represent a line of best ﬁt? Explain.

4 Lorena

3 2 1

x 2 4 6 8 10 12 14 16 18

Each week, a school counselor helps students who are struggling in Math and English organize tutoring sessions. He hopes that the tutoring sessions will have a positive effect on students’ grades. a

Formulate a question which could be investigated using a scatterplot.

Determine what variables could be used to answer the statistical question from part (a).

Describe a method the school counselor could use to collect the data.

The data the school counselor collected on the students who receive Math and English tutoring is shown in the tables. Create a scatterplot of the data. Math students Hours of tutoring per week Math grade

1.5 63

3.5 75

1 60

4 79

2 68

2.5 69

3 71

Hours of tutoring per week Math grade

3.25 72

1.75 66

2.25 68

3.75 74

1.25 62

2.75 70

4.5 79

English students

Hours of tutoring per week English grade

1 69

2.5 80

0.75 64

1.5 72

4 91

1.25 71

2 77

Hours of tutoring per week English grade

3.5 89

1.25 67

2.75 83

1.75 73

3 85

2.25 79

3.75 87

Find the regression model for each set of data.

Draw a conclusion about the data by answering the statistical question from part (a).

Use the regression models to predict the grades of a math student and an English student who each receive 3 hours and 15 minutes of tutoring each week. Explain whether these predictions are reliable.

Go through the whole data cycle at least once to investigate whether a relationship between a person’s height and the length of their stride exists. 9.03 Linear regression mathspace.co

545

9.04 Quadratic regression After this lesson, you will be able to… • determine if a table of values represents a quadratic model. • determine the equation of the curve of best fit given a table of values. • predict values using a quadratic model.

Quadratic regression Functions can be used to model real-world events and interpret data from those events. Data that measures or compares two characteristics of a population is known as bivariate data. When analyzing data, we previously described the relationship between two variables as linear or nonlinear. In this lesson, we will focus on nonlinear relationships that can be modeled by a quadratic function.

Exploration Each table shown represents a different set of data. Table 1 x y

0 13

0.2 7

0.4 4

0.6 3

0.8 1

1 0

1.2 2

1.4 3

1.6 6

1.8 9

6 112

6.5 120

7 114

7.5 127

8 127

6 54

7 55

8 53

9 52

10 50

6.3 7

6.8 6

7.2 5.5

7.4 4

8 2

Table 2 x y

3 63

3.5 68

4 77

4.5 90

5 104

5.5 100

Table 3 x y

0 63

1 65

2 61

3 59

4 58

5 59

Table 4 x y

1 1.5

2 3

2.5 4.8

3 5

4 7.4

5 8

Without creating a scatterplot:

546

Does the data in Table 1 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 2 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 3 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 4 have a linear or quadratic relationship? Explain your answer.

Mathspace Virginia SOL Algebra 1 mathspace.co

To more easily analyze a set of data and determine if there is a quadratic relationship between the variables, we often construct a scatterplot.

8 7

Data presents a quadratic relationship if it forms a symmetric curve or parabolic shape.

6 5 4

The quadratic curve of best fit that approximately models the data can be calculated using technology. Most calculators will write the model in standard form (of a quadratic function), y = ax2 + bx + c.

3 2 1 1

2 3 4 5 6 7 8 9

If points are more tightly clustered along the model, it represents a stronger relationship between the variables. The curve of best fit can help us make predictions or conclusions about the data. If we are given an x-value, we can predict the y-value by substituting x into the equation and solving for y. We can also use the graph of the model to approximate x and y-values. 9

x 1

2 3 4 5 6 7 8 9

When x = 8, y ≈ 3

When y = 7, x ≈ 4 and 6

When anayzing the data, it is often helpful to interpret the x-intercepts or the vertex in context. For example, if the equation models a company’s sales over time, the x-intercepts represent the times the company made no sales, and the vertex represents the time the highest amount of sales were made. It is important to consider the context of the data when communicating results as the model may only be appropriate over a part of the domain. Domain constraint A limitation or restriction of the possible x-values, usually written as an equation, inequality, or in set-builder notation

9.04 Quadratic regression mathspace.co

547

Example 1 For each scatterplot, determine whether the variables have a linear relationship or a quadratic relationship. If there is a relationship, describe its strength. a

y 35 30 25 20 15 10 5

x 1

Create a strategy A relationship between two variables exists if the points follow a similar trend. The points will roughly form a line if there is a linear relationship or a parabola if there is a quadratic relationship. To describe the strength of the relationship, we can analyze how tightly the data points are clustered or grouped together.

Apply the idea As the x-values increase, the y-values decrease then increase, causing the points to form a U-shaped curve. This shows there is a quadratic relationship between the variables. Because the points are tightly clustered, the relationship is strong.

b 90

80 70 60 50 40 30 20 10

x 1

Apply the idea As the x-values increase, the y-values decrease. This indicates there is a linear relationship between the variables. However, the points are not tightly clustered, so the relationship between the variables is moderate.

Reflect and check Recall that we can describe a linear relationship as positive or negative. For this data set, the relationship is negative since one variable increases and the other decreases. This implies that the equation of the line of best fit would have a negative slope.

548

Mathspace Virginia SOL Algebra 1 mathspace.co

50 45 40 35 30 25 20 15 10 5

x 2 4 6 8 10 12 14 16 18

Apply the idea A relationship between two variables exists if the points follow a similar trend. If the y-values increase and decrease over the domain, the relationship can be modeled by a quadratic function.

Reflect and check As the x-values increase, the y-values increase then decrease, causing the points to form an upside down, U-shaped curve. This shows there is a quadratic relationship between the variables. Because the points are not tightly clustered, the relationship is moderate.

Example 2 A conservationist tracks the population, y, of manatees that regularly visit a river over a number of years, x, (starting at zero). The data is displayed in the table: x y

0 65

1 61

2 58

3 60

4 66

5 74

6 90

a Was the data most likely collected through measurement, observation, a survey or an experiment?

Create a strategy Consider whether the population was measured (with a measurement tool such as a rule or protactor) or observed. Also consider whether anyone was surveyed or whether any variables were controlled.

Apply the idea The population of manatees was not measured, and the conservationist did not survey anyone to collect the data. The information does not specify whether any other variables were controlled, so we can assume that an experiment was not used. The data was most likely collected by observation.

Reﬂect and check Many times, populations of species are tracked using tracking devices. It is possible that the manatees each have a tracking device, and a conservationist collects data from those devices each year.

9.04 Quadratic regression mathspace.co

549

b Determine if the manatee population over time has a quadratic relationship.

Create a strategy Construct a scatterplot to visually determine if a linear or quadratic model is a better ﬁt.

Apply the idea After plotting the data on a graph, we get the following scatterplot: y 90 85 80 75 70 65 60

x 1

There is a clear curve in the pattern of the data, so a quadratic function would better ﬁt the data.

c Using technology, determine an appropriate equation to model the data set. Round all values to two decimal places.

Create a strategy We can use technology to calculate the quadratic regression equation. Remember that a quadratic function is a polynomial of degree 2. To ﬁnd the equation using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Highlight the data and select Two Variable Regression Analysis. 3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

Apply the idea 1. Enter the x-values and y-values in two separate columns.

550

Mathspace Virginia SOL Algebra 1 mathspace.co

2. Highlight the data and select Two Variable Regression Analysis.

3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

Rounding the values to two decimal places, we ﬁnd the approximate curve of best ﬁt is y = 1.94x2 − 7.75x + 65.74.

9.04 Quadratic regression mathspace.co

551

d Using the model in part (b), determine the population 10 years afer the numbers were ﬁrst recorded.

Create a strategy We can ﬁnd the population, y, after 10 years by substituing x = 10 into the equation of the curve of best ﬁt.

Apply the idea y = 1.94x2 − 7.74x + 65.74 2

State the equation

y = 1.94 (10) − 7.74 (10) + 65.74

Substitute x = 10

y = 182.34

Evaluate

We can see that after 10 years, the population will have grown to about 182 manatees.

Reflect and check Remember that the coefficients in the equation for the curve of best fit have been rounded. Rounding values reduces the accuracy of the prediction. If we had used technology to make this prediction, we would have gotten a slightly different answer.

The calculator’s answer is more accurate because it includes more decimal values in the coeﬃcients and does not round them to only four place values. However, the differences between these values is small and does not change our ﬁnal, rounded answer.

552

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 3 Carlos is a goalie on the school soccer team. When he kicks a soccer ball dropped from his hands, he notices that the angle of trajectory for each kick is different. He also notices that there are times when the ball does not travel as far as other times. He wants to investigate this further using the data cycle. a Formulate a statistical question that Carlos can use for his investigation.

Create a strategy We can assume that Carlos is interested in determining the optimum angle at which he should kick a soccer ball dropped from his hands to achieve the maximum distance. There are many statistical questions we can ask, but we should focus the question around the purpose of the investigation.

Apply the idea One possible statistical question is, “At what angle should Carlos kick the soccer ball for it to travel farthest?”

Reﬂect and check Other possible questions are: • How does the distance the ball travels change with the angle of trajectory? • If Carlos kicked the ball and it traveled 130 feet, what was the ball’s angle of trajectory? • If the ball is kicked at the optimum angle, what is the farthest distance the ball will travel?

b Determine what variables could be used to answer the statistical question formulated in part (a).

Apply the idea The two things that Carlos would need to collect data on to answer the question are the angle of trajectory for each kick and the distance the ball travels.

Reﬂect and check The angle of trajectory can impact the distance the ball travels, but the distance the ball travels cannot impact the angle of trajectory. This means the angle of trajectory is the independent variable, and the distance the ball travels is the dependent variable.

c Carlos records 10 kicks and analyzes them to determine the angle of trajectory and also the distance traveled. His results are recorded in the table: Angle (degrees) Distance (feet)

24 112

30 129

33 138

37 155

43 161

48 164

51 158

56 148

60 134

64 124

Determine if the data suggests a linear or quadratic relationship. Explain your answer.

Create a strategy We can determine if the data suggests a linear or quadratic relationship by plotting the points on a coordinate plane and determining if the data resembles a line or a parabola. To do this using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Highlight the data and select Two Variable Regression Analysis.

9.04 Quadratic regression mathspace.co

553

Apply the idea 1. Enter the x-values and y-values in two separate columns.

2. Highlight the data and select Two Variable Regression Analysis.

The data has a parabolic shape which is symmetric. The y-values begin increasing, then reach a maximum value, then decrease after. This means the data has a quadratic relationship.

554

Mathspace Virginia SOL Algebra 1 mathspace.co

d Using technology, determine an appropriate equation to model the data set.

Create a strategy We can use technology to calculate the quadratic regression equation. Remember that a quadratic function is a polynomial of degree 2.

Apply the idea

y = −0.1132x2 + 10.3245x − 74.5885, where x is the angle of trajectory (in degrees) and y is the distance traveled (in feet).

Reﬂect and check If the instructions do not specify to round the coeﬃcients, it is best to include all the digits given by the calculator. This increases the accuracy of the model and the predictions.

e Draw a conclusion about the data by answering the statistical question from part (a).

Create a strategy The statistical question from part (a) is, “At what angle should Carlos kick the soccer ball for it to travel farthest?” When considering the quadratic regression model, the largest y-value represents the farthest distance traveled by the ball. The vertex is the maximum point of the parabola and represents the angle of trajectory (x) that Carlos should kick the ball for it to travel farthest (y). We can ﬁnd this angle (x-value) using the equation

Apply the idea The vertex represents the optimum angle to kick the ball to achieve the maximum distance traveled.

9.04 Quadratic regression mathspace.co

555

The equation of the curve of best ﬁt is y = −0.1132x2 + 10.3245x − 74.5885, where a = −0.1132 and b = 10.3245. Equation of the x-value of the vertex

Substitute a = −0.1132, b = 10.3245

Simplify

For the ball to travel farthest, Carlos would need to kick the ball at an angle of about 45.6°.

Reﬂect and check To ﬁnd the farthest distance the ball is expected to travel, we can substitute x = 45.6 into the equation and solve for y. y = −0.1132x2 + 10.3245x − 74.5885 2

State the equation

y = −0.1132(45.6) + 10.3245(45.6) − 74.5885

Substitute x = 45.6

y ≈ 160.8

Simplify

The vertex occurs at about (45.6, 160.8) which means that the maximum distance of 160.8 feet is achieved by kicking the ball at an angle of 45.6°. When looking at the raw data, we see that Carlos actually kicked the ball farther than this. One of his kicks traveled 164 feet when it was kicked at an angle of 48°. This implies that there are other factors that affect the distance the ball travels, such as the force Carlos uses to kick the ball.

Idea summary Data presents a quadratic relationship if it forms a symmetric curve or parabolic shape. If points are more tightly clustered along the model, it represents a stronger relationship between the variables.

Practice What do you remember? 1

Masturah is using an app on her phone to learn French. She uses the app to learn and practice her French each day, and the following day, the app quizzes her on how much she remembered from the previous day. a

Which statistical question would lead to data that can be represented by a scatterplot? A What is the average amount of time Masturah spends learning French each day? B What day of the week does Masturah practice French the longest? C How many times does Masturah practice French in a week? D What amount of time should Masturah practice French each day to maximize the amount she remembers for the following day?

b 2

556

Determine the variables that could be used to answer the statistical question.

Determine whether or not the following are quadratic functions: a

y = 4x2 + 5

y2 = x2 − 5x + 6

y = 10x + 9

y = 4(x − 7)2 + 8

Mathspace Virginia SOL Algebra 1 mathspace.co

y=x+2

y = (x − 5) (x − 8)

Determine whether or not the following graphs could represent a quadratic relation: a

b 9

−3 −2

2 x

−1

1 1

x 2

x 1

Determine whether or not the following tables could represent a quadratic function: a

x −2 −1 0 y 40 24 10

1 8

2 18

x y

1 −6

2 −16

3 −24

4 −29

5 −26

x 12 13 14 15 y 6 2 1 0

16 −6

x y

0 9

1 9

2 9

3 9

4 9

Consider the scatterplot:

1.2

Select a quadratic function that ﬁts the data the best. A

y = 0.3(x − 9)2 + 1.2

y = −0.03(x − 20)2

y = −0.003(x − 19) + 1

y = 3(x − 25)2 + 0.8

1 0.8 0.6 0.4 0.2 0

15 20 25 30 35

Let’s practice 6

Consider the data set shown: {(−8, 15.3), (−6, 25.1), (−4, 31.5), (−2, 35.2), (0, 37.8), (2, 35.6), (4, 30.1), (6, 21.7), (8, 10.4)} Select the equation of the curve of best fit. A C

y = 0.38x2 − 0.3x + 37 2

y = −1.3x − 0.3 + 37.1

y = 1.3x2 + 0.27x + 37

y = −0.38x2 − 0.27x + 37.1 9.04 Quadratic regression mathspace.co

557

Calculate the quadratic curve of best fit for each data set: a

x y

−3 −18

−2 −20

−1 −26

0 −23

1 −24

2 −19

3 −14

x y

0 17

1 10

2 3.5

3 2

4 0

5 2

6 5.5

x y

−3 3.1

−2 8.8

−1 15.9

0 19.5

1 17.2

2 10.7

3 5.1

x y

−7 −12

−6 −4

−5 1

−4 6

−3 5

−2 −2

−1 −9

Match each regression model to the set of data it fits best. i

y = 0.4x2 − 5.67x + 41

y = −0.22x2 + 2.72x + 8.52

iii

y = −0.32x2 + 0.48x + 19.7

y = 0.22x2 − 1.6x + 2.27

x 0.9 1.8 6.2 7.3 y 1 0.1 1 2

8.5 4.7

7.6 2.9

y 8

7 20

2.8 27

16 52

0 42

19 75

13 6.8

15 0.6

x 1

2 4 6 8 10 12 14 16 18 20

x 3.7 8.3 0.5 2.8 y 15 6 19 18

6.1 12

9 3

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

x y 18

4 16

11 12

0.6 9.6

1.7 13

16 14 12 10 8 6 4 x 1 2 3 4 5 6 7 8 9 10

558

13 35

x y

Mathspace Virginia SOL Algebra 1 mathspace.co

x 2 4 6 8 10 12 14 16 18

The creators of the online game Nomad’s Horizon formulated the statistical question, “How has the number of people playing our game changed over time?” They collected data on the number of people, y, playing the game in the years after its release, x. x y

0 419

1 112

2 13

5 855

6 1602

Describe the independent and dependent variables.

Is the relationship between the variables linear or quadratic?

Find the equation of the curve of best fit. Round all values to two decimal places.

The population, y, of a particular species of bird is tracked over a number of years, x, (starting at zero), with the data displayed in the table: 0 64

1 63

2 65

3 75

4 82

5 86

6 96

7 113

8 127

Formulate a question that could be answered by the data.

Which data collection method was most likely used? A Measurement

Observation

9 149

10 161

Survey

11 180

12 208

Experiment

Determine an appropriate equation to model the data. Round all values to two decimal places.

Using the model in part (c), predict what the population will be 20 years after the species was first recorded.

Ten pregnant women at various weeks of pregnancy were asked at their medical appointments to rate their level of discomfort on a scale of 0 to 10 where 0 is completely comfortable and 10 is in severe discomfort or pain. The results are displayed in the given graph table. Week Discomfort Level

4 397

x y

3 148

8 5

12 3

16 2

20 1

22 1

24 2

28 3

32 6

36 7

40 10

Was the data collected through measurement, observation, a survey or an experiment?

Determine if the data suggests a quadratic relationship. Explain your answer.

Determine an appropriate equation to model the data set. Round all values to four decimal places.

Interpret the meaning of the vertex of the model.

Explain the significance of there being no x-intercepts.

Jiang is helping his mom to determine the best price for a dozen eggs for new contracts. Some experimentation and research provided the results shown for different expected profits based on the price. Price per dozen Proﬁt per month($)

1.3 3200

1.35 3230

1.4 3250

1.45 3210

1.5 3100

1.55 3000

1.6 2800

Formulate a question that could be answered by the data.

Determine if the data suggests a quadratic relationship. Explain your answer.

Determine an appropriate equation to model the data set with integer coefficients.

Determine the price which the model predicts would result in the highest profit.

Interpret the y-intercept.

Interpret the x-intercepts.

9.04 Quadratic regression mathspace.co

559

Researchers collected data to answer the statistical question, “How does the electricity consumption (in kilowatt-hours) of households change throughout the afternoon and evening hours?” a

Describe the variables that could be used to answer the statistical question.

The data the researchers collected is shown in the table. Describe the relationship between the variables. Hours after noon Usage (kWh)

0.5 1.2

2.5 2.3

3.5 2.7

4.5 3

5 3.4

6.5 3

7.5 3.2

1 1.5

8.5 2.6

Hours after noon Usage

2 1.9

9.5 2.3

10.5 2

0 1

11.5 1.4

1.5 1.7

3 2

4 2.2

12 0.9

Hours after noon Usage (kWh)

5 2.8

6 3.5

7 3.8

8 3.6

9 3

10 2.5

11 1.8

12 1.2

Calculate the regression model for this data.

Answer the researchers’ statstical question.

Use the curve of best fit to predict a household’s electricity consumption at 5 p.m.

What time(s) of the day is the model’s predictions most reliable? A From noon to 2:30 p.m. and from 10:30 p.m. to midnight B From 2:30 p.m. to 10:30 p.m. C From noon to 7 p.m. D From 7 p.m. to midnight

Let’s extend our thinking 14

At the beginning of the school year, Shirah and her friends decided that they want to take a trip during spring break in March. They researched average prices of flights and found the following data. Months until trip Average price

6 $540

5 $503

4 $461

3 $432

2 $450

1 $520

0 $637

Formulate a question Shirah and her friends can use for their investigation.

Calculate the regression model for this data.

Determine when Shirah should purchase her flight and accommodation. Explain your reasoning.

The table shows the average weekly wage (in dollars) of an American resident from 1996 to 2006, where x is the number of years since 1996. Year x f (x)

560

7 $547

1996 0 800

2000 4 961.76

2003 7 1065.86

2006 10 1155.20

Determine whether a linear or quadratic function would accurately model this situation. Explain your reasoning.

Predict the average weekly wage of an American in 2010.

Would the model from part (a) make sense for long term analysis? Explain your answer.

Write a report about the changes in the average weekly wage of an American resident from 1996–2016.

Mathspace Virginia SOL Algebra 1 mathspace.co

The populations of U.S. cities are constantly changing. Some cities see large increases in population, while others face large decreases in population size. a

Formulate a question about the population of Pittsburgh, Pennsylvania that would require the collection of bivariate data.

Describe the variables that could be used to answer the question from part (a).

Collect the census data on the population of Pittsburgh, Pennsylvania from 1870 to 2000.

Use technology to create a scatterplot and describe the form or shape of the data.

Use technology to find an appropriate equation to model the data set.

Draw a conclusion about the data by answering the question formulated in part (a).

Could the model from part (e) be used to make reasonable predictions after 2000? Explain your answer.

9.04 Quadratic regression mathspace.co

561

9.05 Analyze bivariate data After this lesson, you will be able to… • analyze relationships of variables given a set of bivariate data. • make conclusions given a set of bivariate data.

Analyze bivariate data The process of analyzing bivariate data involves a two-step process. First, we plot the data on a scatterplot. This allows us to visually inspect the relationship between the two variables. Then, we use mathematical models to describe this relationship. Two common models that we have used are the linear regression model and the quadratic regression model.

Interactive exploration Explore online to answer the question

mathspace.co Use the interactive exploration in 9.05 to answer this question. 1.

Which function fits the data better? How do you know?

To determine the curve of best fit for a set of bivariate data, we can use technology such as graphing calculators or software. These tools allow us to perform both linear and quadratic regression on the same set of data and compare the results. To decide which curve best models the data, we can visually assess whether the curves follow the trend in the data and how close the points are to each curve. We can also use the context to determine if a model is a good fit. Distance from ground (ft.)

48 32 16 Time

In the models shown, we can see the data points more closely follow the quadratic curve. Especially upon inspection of x-values closer to 0, the quadratic model more closely aligns with the data in the scatterplot. A better model will have data that is more tightly clustered along the curve.

562

Mathspace Virginia SOL Algebra 1 mathspace.co

Linear model

Quadratic model

A type of relationship between two variables that can be expressed as a straight line on a graph. It is described by the equation y = mx + b, where m is the slope and b is the y-intercept.

A type of relationship between two variables that can be expressed as a curve on a graph. It is described by the equation y = ax2 + bx + c, where a, b, and c are constants.

Example 1 A ball is dropped off of a building that is 25 feet high. The table below shows its distance from the ground over time. Time since being thrown (seconds) Distance from ground (feet)

0 25

1 24.5

2 23

3 20.4

4 17.1

5 11.5

5.5 6.5

6 1

a Describe the relationship between the time since the ball was dropped and its distance from the ground. Is it quadratic or linear?

Create a strategy Construct a scatterplot to get a visual of the data. Distance from the ground (feet) 25 20 15 10 5 Time since being dropped (seconds)

Then consider the form, strength, and direction.

Apply the idea The data appears to ﬁt a strong quadratic model.

b Use technology to create a model and graph the model alongside a scatterplot of the data.

Create a strategy We can use technology to calculate the quadratic regression equation. Remember that a quadratic function is a polynomial of degree 2. To find the equation using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Highlight the data and select Two Variable Regression Analysis. 3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

9.05 Analyze bivariate data mathspace.co

563

Apply the idea 1. Enter the x-values and y-values in two separate columns.

2. Highlight the data and select Two Variable Regression Analysis.

3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

The equation of the curve of best fit is y = −0.8521x2 + 1.4149x + 24.3536.

564

Mathspace Virginia SOL Algebra 1 mathspace.co

c Based off your model, when would you predict the ball would hit the ground?

Create a strategy Looking at the graph, the ball would hit the ground when the distance from the groud is zero feet. Follow the pattern of the scatterplot or look at the model created using technology and predict when that would be. Alternatively, we can verify our solution by ﬁnding the x-intercept for our regression model.

Apply the idea Looking at the model made with technology, the ball would hit the ground after approximately 6.25 seconds, which is the x-intercept when the distance from the ground is 0 feet.

Reﬂect and check Remember, that this prediction is just an educated guess based on our model, and your answer may differ slightly based on the model you chose. According to this one, a more precise answer is about 6.24 seconds.

Example 2 Ronaldo is looking to rent a two-bedroom apartment. He wants something that is spacious, but affordable. He decides to use the data cycle to explore rental options in his area. a Identify the two variables that Ronaldo should collect data for in his investigation of potential apartments and then formulate a statistical question to investigate them.

Create a strategy Consider the factors that Ronaldo is interested in: • A two-bedroom apartment • A spacious apartment • An affordable rental price Then, determine which factors would require the collection of data.

9.05 Analyze bivariate data mathspace.co

565

Apply the idea Ronaldo should collect data that describes the size of the apartment, usually measured by square footage, and the rental price, usually given as a monthly rate. The apartments should all have two bedrooms, since that is the type (category) of apartment he is interested in. One possible question is, “What is the price range of two-bedroom apartments with 1000–1200 square feet?”

Reﬂect and check In this context, the size of the apartment (in square feet) is the independent variable, and the monthly rental price (in dollars) is the dependent variable. Other possible questions are: • How does the monthly rental price of a two-bedroom apartment change with the size of the apartment? • What size apartments are typically $1500–$1700 per month? • How do the prices and sizes of two-bedroom apartments compare to those of two-bedroom houses? b Collect data that could be used to answer the statistical question you formulated.

Create a strategy Previously, we determined that data should be collected on the size of the apartment, usually measured by square footage, and the rental price, usually given as a monthly rate. This information can be acquired online. Typically, rental properties in an area are advertised on websites such as Zillow.com or Apartments.com.

Apply the idea This is an example data set of current rental properties around Norfolk, VA: Square footage Rental price

1000 2049

1400 2179

755 1324

1172 1881

1200 1775

1050 1500

1166 1870

1195 2075

900 1425

900 1700

Square footage Rental price

822 1909

1383 2150

1183 1500

1113 1969

850 1600

783 1350

884 1500

750 1400

1000 1299

1224 1695

Square footage Rental price

980 1200

866 1495

802 1260

904 1750

1250 1700

850 1350

750 1600

1025 1550

1117 2300

1027 1800

Reﬂect and check Remember that the sample should be collected randomly, and there should be a decent amount of two-bedroom apartments in the sample to be representative of the population. c Determine whether a linear or quadratic function would represent the relationship best. Calculate the equation of the curve of best ﬁt.

Create a strategy First, we can use technology to create a scatterplot and examine the shape of the data. After determining which function models the data best, we can find the equation of the curve of best fit with technology. To find the equation using technology, we can follow these steps: 1. Enter the x-values and y-values in two separate columns. 2. Highlight the data and select Two Variable Regression Analysis. This will generate the scatterplot. 3. Under the Regression Model drop down menu, choose Linear or Polynomial, depending on the shape of the data.

566

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea Enter the data into the GeoGebra statistics calculator, and perform the Two Variable Regression Analysis.

The relationship between the variables is not strong, but the y-values tend to increase as the x-values increase. This indicates there is a moderate, linear relationship between the variables. Now, we can ﬁnd the equation of the line of best ﬁt by choosing Linear under the Regression Model drop down menu.

The equation of the line of best ﬁt is y = 1.0159x + 645.7778.

9.05 Analyze bivariate data mathspace.co

567

Reﬂect and check When analyzing the quadratic curve of best ﬁt, we can see that the curve does not model the data better than the linear model. In fact, the section of the parabola shown does not have much curve to it. This means that prediction made with either model would be similar.

d Ideally, Ronaldo would like an apartment that is 1100 ft2. Predict the monthly rental price of an apartment of this size.

Create a strategy In the previous part, we found the equation of the line of best ﬁt to be y = 1.01586x + 645.7778, where x represents the size of an apartment in square feet and y represents the monthly rental price in dollars. We can substitute x = 1100 into the equation to ﬁnd the monthly rental price.

Apply the idea y = 1.0159x + 645.7778

Line of best ﬁt

= 1.0159 (1100) + 645.7778

Substitute x = 1100

= 1763.2678

Evaluate

An 1100 ft apartment will cost about $1763 per month.

Reﬂect and check This prediction was made with interpolation because it falls within the range of the known data values. However, the prediction is not very strong because the points are not tightly clustered around the line.

568

Mathspace Virginia SOL Algebra 1 mathspace.co

e Ronaldo’s budget is $1650. Predict the size of the apartment he can afford.

Create a strategy The monthly rental price is the dependent variable (y), and the size of the apartment is the independent variable (x). We must substitute y = 1650 into the equation of the line of best ﬁt, and solve for the x-value.

Apply the idea y = 1.0159x + 645.7778 1650 = 1.0159x + 645.7778 1004.2222 = 1.0159x 988.505 = x

Line of best ﬁt Substitute y = 1650 Subtract 645.7778 from both sides Divide both sides by 1.0159

$1650 a month can get Ronaldo an apartment with about 988.5 square feet of space.

Draw a conclusion by answering the statistical question from part (b) and summarize the results of the investigation.

Create a strategy The statistical question from part (b) was, “What is the price range of two-bedroom apartments with 1000–1200 square feet?”

Apply the idea If Ronaldo wants a two-bedroom apartment that is 1100 ft2, he should expect to pay about $1763 per month. This is outside of his budget, so he should look for apartments that are around 988 ft2 to stay within his desired price range. However, according to the raw data, the montly rental price of an apartment with 1000–1200 square feet ranges from $1300–$2300. This shows that it is possible to ﬁnd an 1100 ft2 apartment within the $1650 price range. There are most likely other factors, such as the neighborhood or distance from downtown Norfolk, that affect the price of the property that Ronaldo should take into consideration when making his ﬁnal decision.

Reﬂect and check These results could help Ronaldo make a decision about the apartment he would like to rent, or it could lead him to ask another question. For example, Ronaldo might ask the question, “How does the size of an apartment impact the monthly rental price of a one-bedroom or two-bedroom apartment?” He could use the slope of the line of best ﬁt to conclude that for each 1 square foot increase in apartment size he can expect to pay around $1.02 more per month. This might lead Ronolado to explore one-bedroom apartments instead. He could repeat the data cycle, collecting data on one-bedroom apartment sizes and prices. Then, he can plot the data on the same scatterplot in part (d), but use a different color for the points representing one-bedroom apartments.

Idea summary We can use technology to analyze bivariate data by creating and comparing regression models. To choose the model with the best fit, we analyze the visual fit on the scatterplot and the context of the problem. If the points are clustered more closely, the model is the better fit.

9.05 Analyze bivariate data mathspace.co

569

Practice What do you remember? 1

For each of the following scatterplots: i

State the type of function that best models the data.

State whether the slope of the line (for a linear model) or coeﬃcient of x2 (for a quadratic model) is positive or negative.

No. of Restaurants

Average Fuel Economy

Time

Height of an object

Height

Heart rate

Time

Shoe size

Time

Concentration

Speed

No. of Fish

Temp (°F )

Time

Sales of Hot Chocolate

Time

Which one of the following types of functions is an appropriate model for the data shown on each graph? • Linear, f (x) = mx + b • Quadratic, f (x) = ax2 + bx + c, a > 0 • Quadratic, f (x) = ax2 + bx + c, a < 0 Sales (in millions)

Sales (in millions)

Year 4 570

Mathspace Virginia SOL Algebra 1 mathspace.co

Year 4

Hermione has just purchased a new car and wants to know how the speed at which she drives changes the gas consumption of her car. With the help of a friend, she records the gas consumption at several different speeds. Speed, x km/hr 30 40 50 60 70 80

Fuel consumption, y L/100 km 13 7 5 6 14 23 QuadReg y = Ax2 + Bx + C A = 0.020 535 71 B = −2.053 214 C = 56.15

Plot the data points from the table.

Using the data from your plotted graph, what type of model would be most appropriate?

A graphing utility has ﬁtted the data from the table to a quadratic model. The calculator’s output is shown. Use these results to build a quadratic model from the data, giving each of the constants correct to two decimal places.

Let’s practice 4

Nine data points have been plotted with a quadratic curve of best ﬁt: 18 16 14 12 10 8 6 4 2 −2 −2

x 2 4 6 8 10 12 14 16 18

Predict the y-value of a point with an x-value of 2.

Determine whether the following points would be predicted by the quadratic curve of best ﬁt: i

(9, 3)

(3, 4)

iii

(12, 0)

(14, 4)

9.05 Analyze bivariate data mathspace.co

571

Answer the following questions using the scatterplot. 18 16 14 12 10 8 6 4 2 −2

x 2

−2

Using only the scatterplot, decide whether a linear of quadratic regression model would be a better ﬁt. Justify your answer.

Predict the y-value of a point with an x-value of −2. Justify your answer.

Nine data points have been plotted with a quadratic curve of best ﬁt: y

18 16 14 12 10 8 6 4 2

−2 −2

2 4 6 8 10 12 14 16 18

Predict the y-value of a point with an x-value of 13.

Determine whether the following points would be predicted by the quadratic curve of best ﬁt: i

(3, 4)

(14, 10)

iii

(2, 9)

(15, 15)

A scatterplot has been created from a set of data: 14

12 10 8 6 4 2 −2 −2

8 10 12 14

How would you describe the strength and form of the relationship? Justify your answer.

Determine which of the following is the best estimate of y-intercept for the regression model: A 0

572

x 2

Mathspace Virginia SOL Algebra 1 mathspace.co

−4

The distance d in kilometers that Emma runs was measured at different times t minutes after she started. The following quadratic curve of best ﬁt was graphed: 4.5

4 3.5 3 2.5 2 1.5 1 0.5

t 1

Identify the independent and dependent variables.

Formulate a question that could be answered by the scatterplot.

Using the curve of best ﬁt, ﬁnd the predicted distance Emma runs after: i

d 9

2 minutes

8 minutes

Which of the predictions in part (c) is less reliable? Explain your answer.”

A computer program compares and orders the scores of all students who sit an exam. The time taken (T, in milliseconds) for the program to completely order all students is shown in the table for different numbers of students, n: Number of students (n) Time (T )

2 10

4 60

6 150

8 280

10 450

12 660

14 910

16 1200

18 1530

20 1900

Write an equation to model the data. 10

The table shows data collected to answer the question, “What is the relationship between a location’s altitude and its average annual temperature?” The data represents ten randomly selected locations on Earth. Altitude (yd) Temperature °F

2600 −9

2400 −2

1000 28

200 50

600 37

1600 21

2200 7

2800 −13

1200 21

Describe a method that may have been used to collect the data.

What type of function best models the relationship between the altitude of a location and its average annual temperature?

Write and graph a function to model the relationship.

Complete the table by using the model to approximate the average annual temperature of locations at the given altitudes. Altitude (x) 500 1000 2000

Average Annual Temperature ( y)

Dylan starts a mountain hike at an altitude of 940 yd and plans to reach the summit at an elevation of 1850 yd. According to the model, by how much will the temperature decrease?

9.05 Analyze bivariate data mathspace.co

573

A sample of 20 cars were weighed and their average fuel consumption (measured in gallons per mile) measured. The data is shown in the table. a

Formulate a question that can be answered by the data.

What type of function best models the relationship between the weight of a car and its average fuel consumption?

Write and graph a function to model the relationship.

Identify the slope and y-intercept of the function and explain what they mean in terms of the context.

Complete the table by using the model to approximate the average fuel consumption of cars with the given weights. Weight (x) 3200 3100 1500

Average Fuel Consumption ( y)

Bob lives in the city, so he wants to purchase a car that is relatively fuel efficient. Two cars that he is considering weigh 1600 lb and 2900 lb respectively. According to the model, which car should he choose if fuel consumption is the only consideration?

Weight (lb) 3400 3100 3000 2500 2900 2400 2000 2600 3300 3200 2700 2600 3900 2200 3700 3600 2900 2700 2300 3000

Fuel consumption 120 125 110 112 111 93 105 103 128 110 108 110 138 94 131 129 117 115 103 124

A social researcher claims that the longer people stay in their job, the less satisfaction they gain from their work. She asked a sample of people how many years they had been employed in their current job and to rate their level of satisfaction out of 10. The results are presented in the table. Number of years employed 2 3 5 6 8 10 12 13 15

Satisfaction Rating 9 5 4 2 5 7 8 7 9

Was the data collected through measurement, observation, a survey or an experiment?

Create a scatterplot for the data collected.

Does the scatterplot support the social worker’s claims?

Write the equation that would be best suited to model the relationship between the number of years employed and satisfaction with the job.

The social researcher herself has been employed in her current job for 4 years and rates her satisfaction with her work a 10 out of 10. Find the difference between the satisfaction rating approximated by the model and her actual rating.

574

Mathspace Virginia SOL Algebra 1 mathspace.co

Let’s extend our thinking 13

Every year, a popular movie trailer is released and people anticipated the day it will arrive in theaters. It is common for people to try to see the movie soon after it comes out. Typically, the movie’s highest daily box offices sales is the day the movie hits theaters. a

Formulate a question related to this context which could be investigated using a scatterplot.

Describe the variables that could be used to answer the question from part (a).

Collect data on a recent, popular movie that could be used to answer the statistical question from part (a).

Use technology to create a scatterplot and describe the form and strength of the relationship.

Use technology to find an appropriate equation to model the data set.

Draw a conclusion about the data by answering the question formulated in part (a).

Describe the domain over which the curve of best fit found in part (e) could be used to make reasonable predictions. Explain your answer.

People use social media as a way to connect with friends, a way to discover new places to travel, or as a platform for their business, to name a few. In some cases, it is important to track things such as the number of followers you have or the amount of engagement your content receives. Go through the whole data cycle at least once to investigate whether a relationship exists between the amount of time spent on social media and the number of followers someone has.

9.05 Analyze bivariate data mathspace.co

575