# Investigating Exponential Growth and Decay: f(x) = a(b^x)

- Author:
- Mr. Bokar

This application will help us explore exponential functions. We will look at functions of the form f(x) = a(b^x), where a is a positive real number and b > 0 but b is not 1. This applet will help you understand how changing the values of a and b affect the shape of the graph.

Questions:
1. Pick a value for a. Then, slide the value for b. For five different values for b, write the function's equation and sketch a graph of the function with the y-intercept labeled. For example, if I pick a = 2 and b = 3, then the equation is f(x) = 2(3^x). Don't be afraid to use decimals!!
2. Describe how changing the value of b changes the graph of the function.
3. Of your five selected functions, which graphs represented growth and which represented decay? Explain how you can determine if a function is exponential growth or decay based on the equation of the function, not the graph.
4. Pick a value for b. Then, slide the value for a. For three different values of a, write the function's equation and sketch a graph of the function with the y-intercept labeled. See the example in 1.
5. Explain how the value of a changes the graph of the function.
6. Based on your observations, how do a and b affect whether or not the graph is exponential growth or exponential decay?
7. Based on your observations, how do a and b affect the rate of increase or decrease in an exponential function?