Unit 9, Lesson 9
Interpreting Exponential Functions
Integrated Math 1
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The cost, in dollars, to produce 1 Watt of solar power is a function of the number of years since 1977, .
From 1977 to 1987, the cost could be modeled by an exponential function, . Here is the graph of the function.
Graph of a function on grid, origin O. Horizontal axis, years since 1977, from 0 to 11 by 1's. Vertical axis, dollars per watt, from 0 to 80 by 5's. Plotted coordinates as follows: 0 comma 80, 1 comma 60, 2 comma 45. Approximate coordinates as follows: 3 comma 34, 4 comma 25, 5 comma 19, 6 comma 14, 7 comma 11, 8 comma 8, 9 comma 6, 10 comma 4 point 5, 11 comma 3.
An exponential, decreasing line drawn through points.
Earlier, we used equations to represent situations characterized by exponential change. For example, to describe the amount of caffeine, , in a person’s body hours after an initial measurement of 100 mg, we used the equation .
Notice that the amount of caffeine is a function of time, so another way to express this relationship is , where is the function given by .
We can use this function to analyze the amount of caffeine. For example, when is 3, the amount of caffeine in the body is or , which is 72.9. The statement means that 72.9 mg of caffeine are present 3 hours after the initial measurement.
We can also graph the function to better understand what is happening. The point labeled , for example, has the approximate coordinates so it takes about 10 hours after the initial measurement for the caffeine level to decrease to 35 mg.
Graph of function, origin O. Horizontal axis, from 0 to 10, by 1's, labeled time, hours. Vertical axis, from 0 to 100, by 20's, labeled caffeine, in milligrams. Line starts at 0 comma 100, passes through approximate points 2 comma 80, 5 comma 60, 9 comma 40, and Point P, 10 comma 35.
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 is 3.5). In this situation, negative values for the domain would represent the time before the initial measurement. For example 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.