©Curriculum Associates, LLC Copying is not permitted. 83Lesson 8 Understand Linear Functions

Name: Understand Linear Functions

Lesson 8

Study the example problem showing how to compare rates of change for two functions. Then solve problems 1–6.

1 What do the rates of change represent?

2 What does it mean in the context of the example to say that the rate of change for Monte’s savings is greater?

3 Find the total savings after working 8 hours for each person .

Example

Monte and Ramon are each saving all of the money they earn . Monte started with $3 and earns $8 an hour at his part-time job . The graph shows Ramon’s total savings . Which function has a greater rate of change?

Make a table of values for Monte’s savings .

Hours Worked 0 1 2 3 4 5

Total Savings ($) 3 11 19 27 35 43

You can use ordered pairs from the table to find Monte’s rate of change .

Monte’s rate of change: vertical change ·············· horizontal change 5 8 ·· 1 5 8, or $8/hr

You can use the graph to find Ramon’s rate of change .

Ramon’s rate of change: vertical change ·············· horizontal change 5 6 ·· 1 5 6, or $6/hr

The rate of change for Monte’s savings is greater .

Vocabularyrate of change the rate at which one quantity

increases or decreases

with respect to a change

in the other quantity . It is

the ratio of the vertical

change to the horizontal

change on a graph .

How can you compare two functions?

Tota

l Mon

ey ($

)

Hours Worked

Ramon’s Savings

2 4 6 8 91 3 5 7Ox

y

126

24

36

18

30

424854

©Curriculum Associates, LLC Copying is not permitted.84 Lesson 8 Understand Linear Functions

Solve.

4 The table shows how much money a concession stand earns selling hamburgers . The graph shows how much money the stand earns selling hot dogs . Find and compare the rates of change for these two functions .

Hamburger SalesNumber of

Hamburgers SoldAmount

Earned ($)

1 8

2 16

3 24

4 32

Am

ount

Ear

ned

($)

Number of Hot Dogs Sold

Hot Dog Sales

2 4 6 8 91 3 5 7Ox

y

42

8

12

6

10

141618

5 Two companies charge differently for canoe rentals, as shown below . What is the rate of change for each function? What is the cost to rent a canoe for 4 hours from each company?

Company A: c 5 8h 1 10, where c 5 total cost (in dollars) and h 5 number of hours

Company B: $15 per hour

6 The graph shows a function . Write an equation with the same initial value and a rate of change that is less than the rate of change of the function shown in the graph .

2 4 6 8 91 3 5 7Ox

y

21

4

6

3

5

789

©Curriculum Associates, LLC Copying is not permitted. 85Lesson 8 Understand Linear Functions

Name: Lesson 8

1 What is the initial value for the function in the example problem? What is the rate of change?

2 Linear functions can be written with equations in the forms y 5 mx and y 5 mx 1 b . In which form is the linear function in the example problem? What are the values of m and b and what do they represent?

3 Do you think that the equation y 5 2x2 is a linear function? Explain why or why not .

Example

Consider the equation y 5 x 1 2 . Use the equation to complete the table and then graph the equation .

Does the equation y 5 x 1 2 represent a linear function?

Complete the table and graph the equation .

x 22 21 0 1 2

y 0

The graph is a straight line, so the equation y 5 x 1 2 represents a linear function .

Identify Linear Functions

Study the example showing how to tell whether a function is linear. Then solve problems 1–6.

Vocabularylinear function a function with a graph

that is a non-vertical

straight line, which can

be represented by a

linear equation in the

form y 5 mx 1 b .

x

y

1

O 1 2 3 4

2

4

3y 5 x 1 1

y 5 x 1 1 is a linear

function .

1

21 3 4

22

2122232421

23

24

3

2

4

x

y

O

©Curriculum Associates, LLC Copying is not permitted.86 Lesson 8 Understand Linear Functions

Solve.

4 Graph each of the equations below on the coordinate grid . Describe each graph and tell whether or not the equation represents a linear function .

a. y 5 2x 1 2

b. y 5 x2 1 2

5 Which of these equations are linear functions that go through the point (0, 6)? Explain your reasoning .

y 5 6x y 5 x 1 6

y 5 x2 1 6 y 5 2x 1 6

6 The graph of an equation is shown at the right . Explain why the equation is a linear function . Then explain how to write an equation for the function .

1

21 3 4

22

22 21232421

23

24

3

2

4

x

y

O

1

21 3 4

22

2122232421

23

24

2

4

3

x

y

O

©Curriculum Associates, LLC Copying is not permitted. 87Lesson 8 Understand Linear Functions

Name: Lesson 8

Example

Safina graphs the functions y 5 x 1 1 and y 5 x ·· 2 on the same

grid . She says that y 5 x 1 1 is linear but y 5 x ·· 2 is not because it cannot be written as an equation in the form y 5 mx 1 b . Describe how you can check Safina’s work and reasoning . Then tell whether Safina is correct or not .

Show your work. Use graphs, words, and numbers to explain your answer .

Reason and Write

Study the example. Underline two parts that you think make it a particularly good answer and a helpful example.

Where does the example . . . • answer both parts

of the problem?• use graphs to

explain?• use words to

explain?• use numbers to

explain?

1

21 3 4

22

22 212324

23

24

2

4

x

y

O

3

21

y 5 x 1 1

y 5 2x

I can graph both functions on the same grid to determine whether each is linear or nonlinear.

My graph shows that both functions are linear.

Then I can find the initial value and rate of change for the second function to write it in the form y 5 mx 1 b.

y 5 x ·· 2 : initial value 5 0 and the rate of

change 5 1 ·· 2 . This function could be written in

y 5 mx 1 b form as y 5 1 ·· 2 x 1 0.

Safina is not correct. Both functions are linear.

©Curriculum Associates, LLC Copying is not permitted.88 Lesson 8 Understand Linear Functions

Solve the problem. Use what you learned from the model.

Graph the functions y 5 2x 1 1 and y 5 2x 1 4 on the same grid to determine whether they are linear or nonlinear . Describe the similarities and differences between the graphs, including their initial values and rates of change .

Then graph the function y 5 2x 2 3 on the same grid . What are its similarities to the other two functions?

Show your work. Use graphs, words, and numbers to explain your answer .

Did you . . . • answer all three

parts of the problem?

• use graphs to explain?

• use words to explain?

• use numbers to explain?