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Algebra 1

5.5.4 Justifying Exponential Models

Algebra 15.5.4 Justifying Exponential Models

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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.

A scatter plot shows insulin levels (micrograms) decreasing over 20 minutes. Points (1,90), (2,81), and (3,72.9) are labeled along the curve; insulin drops as time increases.

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?

2. Answer the following questions:

a. How much insulin broke down in the first minute?

b. What fraction of the original insulin is that?

3. Answer the following questions. 

a. How much insulin broke down in the second minute?

b. What fraction is that 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.

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    

6. Answer the following questions.

a. Describe how you would find how many micrograms of insulin remain in his bloodstream after 10 minutes. 

b. After mm minutes?

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

Which of the following is the equation of the graph provided?

GRAPH OF A DECREASING EXPONENTIAL FUNCTION WITH \(y\)-intercepts OF 1 AND PASSING THROUGH THE POINT (1, 0.333). THE POINT (1, 0.333) IS LABELED.

  1. y = ( 1 3 ) x
  2. y = 1 3 x
  3. y = 3 x + 1
  4. y = 3 x

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.

A graph showing acetaminophen levels (mg) decreasing over 8 hours; starting at 1000 mg at 0 hours, dropping to 750 mg at 1 hour, 562.5 mg at 2 hours, and continuing downward.

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 6001000=360600=6106001000=360600=610, the growth factor is 610610. Therefore, it is decaying exponentially, and the equation is y=1000(610)xy=1000(610)x, where yy is the amount of chemical in mg in the patient and xx 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 xx-axis where y=0y=0. The equation of the line that the data points approach is called the asymptote. In this example it is y=0y=0.

Try it

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

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