Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change 9-4-EXT Linear and Nonlinear Rates of Change Holt Algebra 1 Lesson Presentation Lesson

Holt McDougal Algebra 1

9-4-EXT Linear and Nonlinear Rates of Change 9-4-EXTLinear and Nonlinear Rates of Change

Holt Algebra 1

Lesson PresentationLesson Presentation



9-4-EXT Linear and Nonlinear Rates of Change

Identify linear and nonlinear rates ofchange.

Compare rates of change.

Objectives



Recall that a rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

Rate of change =change in dependent variable

change in independent variableThe table shows the price of one ounce of gold in 2005 and 2008. The year is the independent variable and the price is the dependent variable. The rate of change is

870 - 5132008 - 2005 = 357

3 =119, Or $119 per year.




9-4-EXT Linear and Nonlinear Rates of Change Example 1: Identifying Constant and Variable Rates of

Change

A. Determine whether each function has a constant or variable rate of change.

Find the ratio of the amount of change in the dependent variable y to the corresponding amount of change in the independent variable x.



Example 1: Continued

The rates of change are

12

, 24

=12

, 12

,and 24

=12

.

The function has a constant rate of change.

B. Determine whether each function has a constant or variable rate of change.





21

, 42

=2, 41

,and 33

= .=4 1

The function has a variable rate of change.



Check It Out! Example 1

A. Determine whether each function has a constant or variable rate of change.

{(–3, 10), (0, 7), (1, 6), (4, 3), (7, 0)}





The rates of change are -33

-11

-33=-1, =-1, =-1, -3

3 =-1and .

The function has a constant rate of change.

B. {(–2, –3), (2, 5), (3, 7), (5, 9), (8, 12)}





2, 21

=2, 22

,and 33

= .=1 184=

The function has a variable rate of change.



Example 2: Linear and Nonlinear Functions

A. Use rates of change to determine whether each function is linear or nonlinear.




Find the rates of change.

02

= 0 , -31

= -3 , -122

= -6 , and -91

= -9 .

The rates of change are not constant, so this function is nonlinear.

B. Use rates of change to determine whether each function is linear or nonlinear.








-93

= -3 , -31

= -3 , -124

= -3 , and -155

= -3.

The rates of change not constant, so this function is linear.




A. Use rates of change to determine whether each function is linear or nonlinear.



Check It Out! Example 2 Continued


1 1 3

1 1, 3, 2 ,

1 17 82 and 8.

2 32

1

The rates of change are not constant, so this function is nonlinear.




B. Use rates of change to determine whether each function is linear or nonlinear.





04

= 0 , 02

= 0, 02

= 0 , and 04

= 0.

The rates of change are constant, so this function is linear



Example 3 : Application

Two auction Web sites start with 100 members each. At site A, the number of members doubles each month. At site B, 500 new members are added each month. Describe the functions that give the number of members for each site as linear or nonlinear. Which web site is growing more quickly from month 3 to month 4?

Use the verbal descriptions to make a table for the number of members each month.



Example 3 : Continued

Time (month)

0 1 2 3 4

Members of Site A

100 200 400 800 1600

Time (month)

0 1 2 3 4

Members of Site B

100 600 1,100 1,600 2,100



Example 3 : Continued

For Web site A, the rates of change are 100, 200, 400, and 800, so the rate of change is variable and the function is nonlinear.

For Web site B, the rates of change are all 500, so the rate of change is constant and the function is linear.

Site A grows more quickly between months 3 and 4.




Reka and Charlotte each invest $500. Rach month, Charlotte’s investment grows by $25, while Reka’s investment grows by 5% of the previous month’s amount. Identify the function that gives the value of each investment as linear or nonlinear. Who is earning money more quickly between month 3 and 4?

Use the verbal descriptions to make a table for the number of members each month.




Time (month)

0 1 2 3 4

Charlotte $ 500 525 550 575 600

Time (month)

0 1 2 3 4

Reka $ 500 525 551.25 578.82 607.75




For Charlotte’s investment, the rates of change are all 25, so the rate of change is constant and the function is linear

For Reka’s investment, the rates of change are 25, 26.25, 27.56, and 28.94 so the rate of change is variable and the function is nonlinear.

Reka is earning more money between months 3 and 4

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Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change 9-4-EXT Linear and Nonlinear Rates of Change Holt Algebra 1 Lesson Presentation Lesson