1. 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 Presentation Holt McDougal Algebra 1

2. 9-4-EXT Linear and Nonlinear Rates of Change Identify linear and nonlinear rates of change. Compare rates of change. Objectives

3. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change 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 variable The 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 - 513 2008 - 2005 = 357 3 =119, Or $119 per year.

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

5. Holt McDougal Algebra 1 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.

6. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 1: Continued The rates of change are 1 2, 2 4 = 1 2, 1 2,and 2 4 = 1 2. The function has a constant rate of change. B. 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.

7. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change The rates of change are 2 1, 4 2 = 2, 4 1, and 3 3 =. = 4 1 The function has a variable rate of change.

8. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates 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)} Find the ratio of the amount of change in the dependent variable y to the corresponding amount of change in the independent variable x.

9. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 1 The rates of change are -3 3 1 -3 3 = -1, = = -3 3 = and. The function has a constant rate of change. B. {(–2, –3), (2, 5), (3, 7), (5, 9), (8, 12)} Find the ratio of the amount of change in the dependent variable y to the corresponding amount of change in the independent variable x.

10. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change The rates of change are 2, 2 1 = 2, 2 2, and 3 3 =. = 1 1 8 4 = The function has a variable rate of change.

11. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 2: Linear and Nonlinear Functions A. Use rates of change to determine whether each function is linear or nonlinear.

12. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 2: Continued Find the rates of change. 0 2 = 0, -3 1 = -3, -12 2 = -6, and -9 1 = -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.

13. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 2: Continued Find the rates of change.

14. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 2: Continued -9 3 = -3, -3 1 = -3, -12 4 = -3, and -15 5 = -3. The rates of change not constant, so this function is linear.

15. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 2 A. Use rates of change to determine whether each function is linear or nonlinear.

16. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 2 Continued Find the rates of change. The rates of change are not constant, so this function is nonlinear.

17. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 2 Continued B. Use rates of change to determine whether each function is linear or nonlinear.

18. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 2 Continued Find the rates of change. 0 4 = 0, 0 2 0 2 = 0, and 0 4 = 0. The rates of change are constant, so this function is linear

19. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change 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.

20. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Example 3 : Continued Time (month) 01234 Members of Site A 1002004008001600 Time (month) 01234 Members of Site B 100600 1,1001,6002,100

21. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change 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.

22. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 3 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.

23. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 3 Continued Time (month) 01234 Charlotte $ 500525550575600 Time (month) 01234 Reka $ 500525551.25578.82607.75

24. Holt McDougal Algebra 1 9-4-EXT Linear and Nonlinear Rates of Change Check It Out! Example 3 Continued 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