understand linear functions name · 2016. 4. 8. · 88 esson 8 understand linear functions...
TRANSCRIPT
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©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
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©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
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©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
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©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
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©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.
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©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?