05.03 Interpreting Exponential Functions

Due No due date
Points 10
Questions 10
Time Limit None
Allowed Attempts Unlimited

Instructions

unit 5

Interpreting Exponential Functions

Goal

You will identify graphs that represent situations that should be continuous or discrete and graphs of exponential functions written in function notation to answer questions about a context. Then, apply that knowledge by answering questions.

Estimated completion time: 35 minutes

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Earlier, we used equations to represent situations characterized by exponential change. For example, to describe the amount of caffeine c in a person’s body t hours after an initial measurement of 100 mg, we used the equation $LaTeX: c=100\cdot\left(\frac{9}{10}\right)^t$ .

Notice that the amount of caffeine is a function of time, so another way to express this relationship is $LaTeX: c=f\left(t\right)$ where f s the function given by $LaTeX: f\left(t\right)=100\cdot\left(\frac{9}{10}\right)^t$ .

We can use this function to analyze the amount of caffeine. For example, when t is 3, the amount of caffeine in the body is $LaTeX: 100\cdot\left(\frac{9}{10}\right)^3$ or $LaTeX: 100\cdot\left(\frac{729}{1,000}\right)$ , which is 72.9. The statement $LaTeX: f\left(3\right)=72.9$ means that 72.9 mg of caffeine are present 3 hours after the initial measurement.

We can also graph the function f to better understand what is happening. The point labeled P, for example, has coordinates approximately (10,35) so it takes about 10 hours after the initial measurement for the caffeine level to decrease to 35 mg.

Graph with y-axis = caffeine (mg) and x-axis = time (hours).

A graph can also help us think about the values in the domain and range of a function. Because the body breaks down caffeine continuously over time, the domain of the function—the time in hours—can include non-whole numbers (for example, we can find the caffeine level when t is 3.5). In this situation, negative values for the domain would represent the time before the initial measurement. For example, $LaTeX: f\left(-1\right)$ would represent the amount of caffeine in the person's body 1 hour before the initial measurement. The range of this function would not include negative values, as a negative amount of caffeine does not make sense in this situation.

Watch the following video about interpreting exponential functions.

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