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

8.6 Warm-Up ANSWER 16 ANSWER Evaluate Evaluate. 1 4 –2 3. The table shows how much money Tess owes after w weeks. Write a rule for the function. ANSWER m = 50 – 5w Week, w 0123 Owes, m

8.6 Example 1 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 1 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 = The table represents the exponential function y = 3 x. 1 3

8.6 Example The y- values are multiplied by for each increase of 1 in x, so the table represents an exponential function of the form y = ab x with b =. 1 4 b. The value of y when x = 0 is 1, so a = The table represents the exponential function y =. x

8.6 Guided Practice Tell whether the table represents an exponential function. If so, write a rule for the function. ANSWERyes; y = ( ) x 1 5

8.6 Example 2 Graph the function y = x and identify its domain and range. 1 2 SOLUTION STEP 1 Make a table of values. The domain is all real numbers.

8.6 Example 2 STEP 2 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.

8.6 Guided Practice Graph the function y = (0.4) x and identify its domain and range. 2. ANSWER Domain: all real numbers Range: all positive real numbers

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

8.6 Example 3 SOLUTION

8.6 Example – – 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 corresponding y- values for y = x, the graph of y = x is a vertical shrink with reflection in the x- axis of the graph of y = x –

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

8.6 Example 4 SOLUTION The graph represents exponential growth (y = ab x where b > 1). The y- intercept is 10, so a = 10. Find the value of b by using the point (1, 12) and a = 10. y = ab x Write function. 12 = 10  b 1 Substitute. 1.2 = b Solve. A function rule is y = 10(1.2) x. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function. a.

8.6 Example 4 The graph represents exponential decay (y = ab x where 0 < b < 1). The y- intercept is 8, so a = 8. Find the value of b by using the point (1, 4) and a = 8. y = ab x Write function. 4 = 8  b 1 Substitute. 0.5 = b Solve. A function rule is y = 8(0.5) x. SOLUTION Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function. b.

8.6 Guided Practice 4. The graph of an exponential function passes through the points (0, 10) and (1, 8). Graph the function. Tell whether the graph represents exponential growth or exponential decay. Write a rule for the function. ANSWER exponential decay; y = 10  (0.8) x

8.6 Example 5 FORESTRY The number of acres of Ponderosa pine forests decreased in the western United States from 1963 to 2002 by 0.5% annually. In 1963 there were about 41 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 2002?

8.6 Example 5 SOLUTION P = a(1 – r) t Write exponential decay model. = 41(1 – 0.005) t Substitute 41 for a and for r. = 41(0.995) t Simplify. a.a. Let P be the number of acres (in millions), and let t be the time (in years) since The initial value is 41, and the decay rate is

8.6 Example 5 P = 41(0.995) Substitute 39 for t. Use a calculator. ANSWER There were about 33.7 million acres of Ponderosa pine forests in b. To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 2002?

8.6 Guided Practice WHAT IF? In Example 5, suppose the decay rate of the forests remains the same beyond About how many acres will be left in 2010? 5. There will be about 32.4 million acres of Ponderosa pine forest in ANSWER

8.6 Lesson Quiz 2. The population in a town has been declining at a rate of 2% per year since The population was 84,223 in What was the population in 2006? ANSWER about 76, Graph y = ( ) x