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.

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)

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 1: f(x) = 3.42(1.78)x Example 2: f(x) = 24.5(0.78)x Initial Value: 3.42 Initial Value: 24.5 Growth/Decay Factor: 1.78 Growth/Decay Factor: 0.78 Growth/Decay Rate: Growth 78% Growth/Decay Rate: Decay 22% Example 3: f(x) = 5.98(0.28)x Example 4: f(x) = 135.97(2.12)x Initial Value: 5.98 Initial Value: 135.97 Growth/Decay Factor: 0.28 Growth/Decay Factor: 2.12 Growth/Decay Rate: Decay 72% Growth/Decay Rate: Growth 112%

How can you determine if a function is an exponential growth or exponential decay by examining a table of values? Example 5: Example 6: X Y 2 8.6672 3 7.1071 4 5.8278 5 4.7788 6 3.9186 x -2 -1 1 2 y .25 .5 4 Exponential Growth Exponential Decay

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 X Y 4 13.429 5 16.652 6 20.648 7 25.604 8 31.748 Growth Factor: 1.24 Growth Rate: 24% Initial Value: 5.68 Equation: f(x) = 5.68(1.24)x

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 X Y 2 8.6672 3 7.1071 4 5.8278 5 4.7788 6 3.9186 Decay Factor: 0.82 Decay Rate: 18% Initial Value: 12.8899 Equation: f(x) = 12.8899(0.82)x

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

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

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