Exploring Exponential Functions

1. Exploring Exponential Functions You should be able to recognize all exponential functions from an equation, graph, or a table of values and determine if it is an exponential growth or decay model.

2. Exponential Functions What do equations of exponential functions look like? Exponential functions have the following forms: Exponential Growth: y = abx, b > 1 Exponential Decay: y = abx, 0 < b < 1 b: Represents the growth or decay factor b - 1: Represents the growth or decay rate (percentage)

3. Determine if the exponential function is a growth or decay model and then identify the initial value, growth or decay factor and the growth or decay rate. Example 2: f(x) = 24.5(0.78)x Example 1: f(x) = 3.42(1.78)x Initial Value: 24.5 Initial Value: 3.42 Growth/Decay Factor: 0.78 Growth/Decay Factor: 1.78 Growth/Decay Rate: Decay 22% Growth/Decay Rate: Growth 78% Example 4: f(x) = 135.97(2.12)x Example 3: f(x) = 5.98(0.28)x Initial Value: 135.97 Initial Value: 5.98 Growth/Decay Factor: 2.12 Growth/Decay Factor: 0.28 Growth/Decay Rate: Growth 112% Growth/Decay Rate: Decay 72%

4. How can you determine if a function is an exponential growth or exponential decay by examining a table of values? Example 5: Example 6: Exponential Growth Exponential Decay

5. Example 7: Determine if a function is an exponential growth or exponential decay by examining a table of values then write its equation. Exponential Growth Growth Factor: 1.24 Growth Rate: 24% Initial Value: 5.68 Equation: f(x) = 5.68(1.24)x

6. Example 8: Determine if a function is an exponential growth or exponential decay by examining a table of values then write its equation. Exponential Decay Decay Factor: 0.82 Decay Rate: 18% Initial Value: 12.8899 Equation: f(x) = 12.8899(0.82)x

7. Writing and Predicting with Exponential Functions The population of Johnson City found during the 2000 census was 25,876. Since then the city has been growing at a rate of 3.2% per year. Write an equation to represent the population of Johnson City since 2000. Exponential Growth Initial Value: 25,876 Growth Rate: 3.2% Equation: f(x) = 25,876(1.032)x Growth Factor: 1.032 Use this equation to predict the current population of Johnson City. f(x) = 25,876(1.032)12 37,761 people f(x) = 37,761.87149

8. Writing and Predicting with Exponential Functions The Garcias have $12,000 in a savings account. The account is increasing by an average rate of 3.76% each year. Write an equation to represent the amount in the savings account since the initial deposit. Exponential Growth Initial Value: $12,000 Growth Rate: 3.76% Equation: f(x) = 12,000(1.0376)x Growth Factor: 1.0376 • Use this equation to predict the amount in the savings account 8 years after the initial deposit. f(x) = 12,000(1.0376)8 f(x) = $16,122.08

9. Writing and Predicting with Exponential Functions A new car costs $32,000.. It is expected to depreciate 12% each year for 4 years and then depreciate 8% each year thereafter. • Find the value of the car 7 years after it was purchased. • First 4 years: Exponential Decay Initial Value: 32,000 Decay Rate: 12% Equation: f(x) = 32,000(.88)x Decay Factor: .88 f(x) = 32,000(.88)4 f(x) = $19,190.25 • Next 3 years: Exponential Decay Initial Value: 19,190.25 Decay Rate: 8% Equation: f(x) = 19,190.25(.92)x Decay Factor: .92 f(x) = 19,190.25(.92)3 f(x) = $14,943.22