Lesson 8.2 Exponential Decay

Lesson 8.2 Exponential Decay

Lesson 8.2 Exponential Decay

Lesson 8.2 Exponential Decay

Lesson 8.2 Exponential Decay

Lesson 8.2 Exponential Decay

Warm-Up Exercises Evaluate the expression without using a calculator. ANSWER 27 ANSWER – 1 1. – – – 2 + ANSWER 10

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

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 x 3 – 2 – 2 1 –

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 <

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

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 –

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.

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

Checkpoint Graph when ab x y = 1b < 0 < 3. x y = x y = ANSWER Graph the function. ANSWER

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

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.

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

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

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

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 =

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.

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

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

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 $ y = ()x)x 0.85 Choose a suitable viewing window. For example, use and 15x ≤≤ y ≤≤ 0 You can use a graphing calculator to make a graph. Enter the equation. SOLUTION 20y = ()x)x 0.85

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, y =

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.