The purpose of this lesson is for students to revisit the idea of average rate of change and apply it to exponential functions. While rate of change has an unambiguous meaning for linear functions, for nonlinear functions rates of change are not constant so an interval must be specified.
In this lesson, students first are invited to recall how to calculate an average rate of change for a specific interval for a function represented by a data table. Students then consider the average rate of change for an exponential function describing the increase in the number of coffee shops for a business. By comparing the average rate of change of different intervals that all start at the same point, students can observe that while the average rate of change can describe how the data is changing with reasonable accuracy over some intervals, it is not a good predictor over larger intervals since exponential functions do not have a constant rate of change. In the last activity, students use the average rate of change to describe how the cost of producing 1 Watt from a solar cell decreases in two separate intervals, which illuminates the nature of exponential decay to decrease at a slower rate. Contexts are used throughout this lesson to give students an opportunity to reason abstractly and quantitatively (MP2).
In future lessons, students will analyze multiple exponential functions at the same time and will be able to use average rate of change to make comparisons between the functions over the same intervals.
- Calculate the average rate of change of a function over a specified interval.
- Explain (in writing) how well given rates of change reflect the changes in an exponential function.
- Understand that an exponential function, unlike a linear function, has different average rate of change values for different intervals.
Let's calculate average rates of change for exponential functions.
- I can calculate the average rate of change of a function over a specified period of time.
- I know how the average rate of change of an exponential function differs from that of a linear function.
An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3.