5.5.4 Justifying Exponential Models
5.5.4 • Justifying Exponential Models
Activity
A patient who is diabetic receives 100 micrograms of insulin. The graph shows the amount of insulin, in micrograms, remaining in his bloodstream over time, in minutes.
1. Scientists have found that the amount of insulin in a patient’s body changes exponentially. How can you check if the graph supports the scientists’ claim?
Compare your answer:
If the insulin decays exponentially, that means that after each minute, the same fraction of insulin will decay and remain. So one way to check if this prediction matches the data is to see what fraction of insulin decays (or remains) after the first minute, then after the second minute, and so on. If these fractions are the same, then the graph supports the scientists’ claim.
2. Answer the following questions:
a. How much insulin broke down in the first minute?
Compare your answer:
10 micrograms broke down.
b. What fraction of the original insulin is that?
Compare your answer:
of the initial injection
3. Answer the following questions.
a. How much insulin broke down in the second minute?
Compare your answer:
9 micrograms broke down.
b. What fraction is that of the amount one minute earlier?
Compare your answer:
of the amount one minute earlier
4. What fraction of insulin remains in the bloodstream for each minute that passes? Be prepared to show your reasoning.
Compare your answer:
. For example: In the next minute, another of insulin broke down. Each time, of the amount remains.
5. Complete the table to show the predicted amount of insulin 4 and 5 minutes after injection.
| Time after Injection (Minutes) | 0 | 1 | 2 | 3 | 4 | 5 |
| Insulin in the Bloodstream (Micrograms) | 100 | 90 | 81 | 72.9 |
Compare your answers:
| Time after Injection (Minutes) | 0 | 1 | 2 | 3 | 4 | 5 |
| Insulin in the Bloodstream (Micrograms) | 100 | 90 | 81 | 72.9 | 65.61 | 59.05 |
6. Answer the following questions.
a. Describe how you would find how many micrograms of insulin remain in his bloodstream after 10 minutes.
Compare your answer:
Multiply 100 by ten times: .
b. After minutes?
Compare your answer:
For example: Multiply 100 by times: .
Video: Analyzing Exponential Graphs and Writing Equations
Watch the following video to learn more about analyzing exponential graphs and using the graphs to write an equation.
Self Check
Additional Resources
More Exponential Models
The graph shows the amount of a chemical in a patient’s body at different times, measured in hours since the levels were first checked.
Could the amount of this chemical in the patient be decaying exponentially? Explain how you know.
The graph does seem to have the shape of exponential decay. To confirm if the chemical in the patient is decaying exponentially, you can find the growth factor. Since terms are usually multiplied by the growth factor, you can divide consecutive terms to see if there is the same growth factor each time.
Since , the growth factor is . Therefore, it is decaying exponentially, and the equation is , where is the amount of chemical in mg in the patient and is the time in hours.
As the time continues, the amount of chemical in the patient is approaching 0 mg. We can see this in the graph as the data points approach the -axis where . The equation of the line that the data points approach is called the asymptote. In this example it is .
Try it
More Exponential Models
The following table represents the number of people who have read a book. Does the table represent exponential growth? Be prepared to show your reasoning.
| Number of Months | Number of People who have Read a Book |
|---|---|
| 0 | 300 |
| 1 | 600 |
| 2 | 1200 |
| 3 | 2400 |
Compare your answer:
Here is how to check the table to see if it represents an exponential relationship:
Just like the values of a graph, you can divide consecutive terms to find the growth factor. If each set of consecutive terms has the same ratio, the relationship is exponential. In this situation, dividing consecutive terms results in . Therefore, the situation represents exponential growth and can also be represented by the equation , where is the number of people who have read a book and is the number of months.