Modeling Prompt

Ask students to give examples of situations where there is exponential change (population growth, compounding interest, radioactive decay). Write down students’ ideas for all to see.

Then ask students for an example of an exponential function (\(y=2e^{\text-t}\)). Write it for all to see. Here are some possible questions for discussion:

• “What situation could this function model?” (Since the exponent is negative, this models decay. The coefficient of the exponential part is 2, so the initial value is 2. The amount is divided by \(e\) every time \(t\) increases by 1. So it’s a small amount of something and it’s decreasing. How fast it decreases depends on what \(t\) represents.)
• “What questions about a situation could this function help you answer?” (I could find out what \(t\) is when the value of the function is a specific number, like 0.01. If \(t\) represents time, I could find out how often the value of the function doubles.)

Tell students that in this task they will ask and answer their own questions about a situation that can be modeled with an exponential function.