1. Lesson 8.2 Exponential Decay

2. Lesson 8.2 Exponential Decay

3. Lesson 8.2 Exponential Decay

4. Lesson 8.2 Exponential Decay

5. Lesson 8.2 Exponential Decay

6. Lesson 8.2 Exponential Decay

7. Warm-Up Exercises Evaluate the expression without using a calculator. ANSWER 27 ANSWER – 1 1. – 3 3 1 2. 5 2 – 0 3. 4 1 3 2 – 2 + ANSWER 10

8. Warm-Up Exercises ANSWER about 38,238 The population of a town is modeled by where x is the number of years after 1990. What was the population in 1998 ? 4. 12,500 1.15 x = P

9. Example 1 Graph when Graph the function. b x y = x y = 1b < 0 < 2 1 Make a table of values for the function. SOLUTION y 0 1 12 x 3 – 2 – 2 1 – 8 4 1 4 2 1

10. Example 1 Plot the points from the table. Draw a curve that passes through the plotted points, as shown. Graph when b x y = 1b < 0 <

11. Example 2 Graph when ab x y = 1b < 0 < Graph the function. a. x y = 2 1 3 b. x y = 3 2 4 1 a. Make a table of values. SOLUTION y 23 3 x 1 – 2 3 01 6 8 3 4 3

12. Example 2 Plot the points. Graph when ab x y = 1b < 0 < Draw a curve that passes through the plotted points, as shown at the right. b. Make a table of values. y 12 x 2 – 1 – 8 3 16 9 4 1 0 6 1 9 1

13. Example 2 Graph when ab x y = 1b < 0 < Plot the points. Draw a curve that passes through the plotted points, as shown at the right.

14. Checkpoint 1. Graph the function. Graph when ab x y = 1b < 0 < x y = 4 1 2. x y = 5 2 ANSWER

15. Checkpoint Graph when ab x y = 1b < 0 < 3. x y = 5 1 2 4. x y = 2 1 10 ANSWER Graph the function. ANSWER

16. Checkpoint Graph when ab x y = 1b < 0 < x y = 3 1 2 1 6. x y = 5 4 5 2 ANSWER Graph the function. ANSWER 5.

17. Graph. Describe the horizontal asymptote. State the domain and range. 1 x 2 Example 3 Graph an Exponential Function y = 2 – Translate the graph 2 units to the right. SOLUTION Sketch the graph of the function. It passes through and x y = 2 1 () 0, 1 ( ).). 2, 4 – The graph ’ s asymptote is the x -axis.

18. Example 3 Graph an Exponential Function The domain is all real numbers, and the range is 0.0. y >

19. Example 4 Graph an Exponential Function It passes through and. () 0, 1 4 1 2, 2 – Translate the graph 3 units up. It passes through and. () 0, 4 4 1 2, 5 – Sketch the graph of the function. SOLUTION x y = 3 2 Graph Describe the horizontal asymptote. State the domain and range. x y = 3 2 + 3.

20. Example 4 The graph ’ s asymptote is the line y = 3.3. The domain is all real numbers, and the range is 3.3. y > Graph an Exponential Function

21. Checkpoint 7. Graph the function. Describe the horizontal asymptote. State the domain and range. Graph an Exponential Function y = 2 1 x 3 + ANSWER domain: all real numbers, range: 0y > 0;0; y =

22. Checkpoint Graph the function. Describe the horizontal asymptote. State the domain and range. Graph an Exponential Function y = 4 3 x 5 – ANSWER domain: all real numbers, range: 0y > 0;0; y = 8.

23. Checkpoint Graph the function. Describe the horizontal asymptote. State the domain and range. 9. Graph an Exponential Function x y = 4 1 2 – domain: all real numbers, range: ANSWER 2;2; y = – y > – 2

24. Checkpoint Graph the function. Describe the horizontal asymptote. State the domain and range. 10. Graph an Exponential Function x y = 3 1 + 3 ANSWER domain: all real numbers, range: 3y > 3;3; y =

25. Example 5 Model Exponential Decay Depreciation The value of a used car y (in thousands of dollars) can be approximated by the model where x is the number of years since the car was new. Graph the model. Then use the graph to estimate when the value of the car will be $4000. 20y = ()x)x 0.85 Choose a suitable viewing window. For example, use and 15x ≤≤ 0 25. y ≤≤ 0 You can use a graphing calculator to make a graph. Enter the equation. SOLUTION 20y = ()x)x 0.85

26. Example 5 ANSWER The value of the car will be $4000 after nearly 10 years. Model Exponential Decay The graph passes through,,, and so on. Use the Trace feature to determine that when x is about 10. () 0, 20 () 1, 17 () 2, 14.45 4y =

27. Checkpoint 11. The value of a van y (in thousands of dollars) can be approximated by the model where x represents the number of years since the van was new. Graph the model. Then use the graph to estimate when the value of the van will be $10,000. Model Exponential Decay 24y = ()x)x 0.9 ANSWER after 8 yr.