8.6. Write and Graph Exponential Decay Functions. Warm Up Lesson Presentation Lesson Quiz

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1 8.6 Write and Graph Exponential Decay Functions Warm Up Lesson Presentation Lesson Quiz

2 8.6 Warm-Up. Evaluate. 3 ANSWER 8. Evaluate. 4 ANSWER 6 3. The table shows how much money Tess owes after w weeks. Write a rule for the function. Week, w 0 3 Owes, m ANSWER m = 50 5w

3 8.6 Example Tell whether the table represents an exponential function. If so, write a rule for the function. a. The y-values are multiplied by 3 for each increase of in x, so the table represents an exponential function of the form y = ab x with b = 3. The value of y when x = 0 is, so a =. 3 3 The table represents the exponential function y = 3 x. 3

4 8.6 Example b. The y-values are multiplied by for each increase 4 of in x, so the table represents an exponential function of the form y = ab x with b =. 4 The value of y when x = 0 is, so a =. x The table represents the exponential function y =. 4

5 8.6 Guided Practice Tell whether the table represents an exponential function. If so, write a rule for the function. ANSWER yes; y = ( ) x 5

6 8.6 Example Graph the function y = range. SOLUTION x and identify its domain and STEP Make a table of values. The domain is all real numbers.

7 8.6 Example STEP Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range is all positive real numbers.

8 8.6 Guided Practice. Graph the function y = (0.4) x and identify its domain and range. ANSWER Domain: all real numbers Range: all positive real numbers

9 8.6 Example 3 Graph the functions y = 3 and y = Compare each graph with the graph of y = x 3 x x

10 8.6 Example 3 SOLUTION

11 8.6 Example 3 Because the y-values for y = 3 x are 3 times the corresponding y-values for y = x, the graph of y = 3 x is a vertical stretch of the graph of y = x. Because the y-values for y = x are times the 3 3 corresponding y-values for y = x, the graph of y = x 3 is a vertical shrink with reflection in the x-axis of the graph of y = x.

12 8.6 Guided Practice 3. Graph the functions y = 5 (0.4) x. Compare graph with the graph of y = (0.4) x. ANSWER The graph is a vertical stretch of y = (0.4) x.

13 8.6 Example 4 a. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function. SOLUTION The graph represents exponential growth (y = ab x where b > ). The y-intercept is 0, so a = 0. Find the value of b by using the point (, ) and a = 0. y = ab x Write function. = 0 b Substitute.. = b Solve. A function rule is y = 0(.) x.

14 8.6 Example 4 b. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function. SOLUTION The graph represents exponential decay (y = ab x where 0 < b < ).The y-intercept is 8, so a = 8. Find the value of b by using the point (, 4) and a = 8. y = ab x Write function. 4 = 8 b Substitute. 0.5 = b Solve. A function rule is y = 8(0.5) x.

15 8.6 Guided Practice 4. The graph of an exponential function passes through the points (0, 0) and (, 8). Graph the function. Tell whether the graph represents exponential growth or exponential decay. Write a rule for the function. ANSWER exponential decay; y = 0 (0.8) x

16 8.6 Example 5 FORESTRY The number of acres of Ponderosa pine forests decreased in the western United States from 963 to 00 by 0.5% annually. In 963 there were about 4 million acres of Ponderosa pine forests. a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time. b. To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 00?

17 8.6 Example 5 SOLUTION a. Let P be the number of acres (in millions), and let t be the time (in years) since 963. The initial value is 4, and the decay rate is P = a( r) t Write exponential decay model. = 4( 0.005) t Substitute 4 for a and for r. = 4(0.995) t Simplify.

18 8.6 Example 5 b. To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 00? P = 4(0.995) Substitute 39 for t. Use a calculator. ANSWER There were about 33.7 million acres of Ponderosa pine forests in 00.

19 8.6 Guided Practice 5. WHAT IF? In Example 5, suppose the decay rate of the forests remains the same beyond 00. About how many acres will be left in 00? ANSWER There will be about 3.4 million acres of Ponderosa pine forest in 00.

20 8.6 Lesson Quiz x. Graph y = ( ) 3 ANSWER. The population in a town has been declining at a rate of % per year since 00. The population was 84,3 in 00. What was the population in 006? ANSWER about 76,30

Section 4.3. Graphing Exponential Functions

Section 4.3. Graphing Exponential Functions Graphing Exponential Functions Graphing Exponential Functions with b > 1 Graph f x = ( ) 2 x Graphing Exponential Functions by hand. List input output pairs (see table) Input increases by 1 and output