Lesson 9
Interpreting Exponential Functions
Let’s find some meaningful ways to represent exponential functions.
Problem 1
The number of people with the flu during an epidemic is a function, \(f\), of the number of days, \(d\), since the epidemic began. The equation \(f(d) = 50 \boldcdot \left(\frac{3}{2}\right)^d\) defines \(f\).
- How many people had the flu at the beginning of the epidemic? Explain how you know.
- How quickly is the flu spreading? Explain how you can tell from the equation.
- What does \(f(1)\) mean in this situation?
- Does \(f(3.5)\) make sense in this situation?
Problem 2
The function, \(f\), gives the dollar value of a bond \(t\) years after the bond was purchased. The graph of \(f\) is shown.
- What is \(f(0)\)? What does it mean in this situation?
- What is \(f(4.5)\)? What does it mean in this situation?
- When is \(f(t) = 1500\)? What does this mean in this situation?
Problem 3
Technology required. A function \(f\) gives the number of stray cats in a town \(t\) years since the town started an animal control program. The program includes both sterilizing stray cats and finding homes to adopt them. An equation representing \(f\) is \(f(t) = 243 \left( \frac13 \right)^t\).
- What is the value of \(f(t)\) when \(t\) is 0? Explain what this value means in this situation.
- What is the approximate value of \(f(t)\) when \(t\) is \(\frac{1}{2}\)? Explain what this value means in this situation.
- What does the number \(\frac13\) tell you about the stray cat population?
- Use technology to graph \(f\) for values of \(t\) between 0 and 4. What graphing window allows you to see values of \(f(t)\) that correspond to these values of \(t\)?
Problem 4
Function \(g\) gives the amount of a chemical in a person's body, in milligrams, \(t\) hours since the patient took the drug. The equation \(g(t) = 600 \boldcdot \left(\frac{3}{5}\right)^t\) defines this function.
- What does the fraction \(\frac{3}{5}\) mean in this situation?
- Sketch a graph of \(g\).
- What are the domain and range of \(g\)? Explain what they mean in this situation.
Problem 5
The dollar value of a moped is a function of the number of years, \(t\), since the moped was purchased. The function, \(f\), is defined by the equation \(f(t) = 2,\!500 \boldcdot \left(\frac{1}{2}\right)^t\) .
What is the best choice of domain for the function \(f\)?
\(\text-10 \leq t \leq 10\)
\(\text-10 \leq t \leq 0\)
\(0 \leq t \leq 10\)
\(0 \leq t \leq 100\)
Problem 6
A patient receives 1,000 mg of a medicine. Each hour, \(\frac{1}{5}\) of the medicine in the patient's body decays.
- Complete the table with the amount of medicine in the patient's body.
- Write an equation representing the number of mg of the medicine, \(m\), in the patient's body \(h\) hours after receiving the medicine.
- Use your equation to find \(m\) when \(h = 10\). What does this mean in terms of the medicine?
| hours since receiving medicine |
mg of medicine left in body |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| \(h\) |
Problem 7
The trees in a forest are suffering from a disease. The population of trees, \(p\), in thousands, is modeled by the equation \(p = 90 \boldcdot \left(\frac{3}{4}\right)^t\), where \(t\) is the number of years since 2000.
- What was the tree population in 2001? What about in 1999?
- What does the number \(\frac{3}{4}\) in the equation for \(p\) tell you about the population?
- What is the last year when the population was more than 250,000? Explain how you know.
Problem 8
All of the students in a classroom list their birthdays.
- Is the birthdate, \(b\), a function of the student, \(s\)?
- Is the student, \(s\), a function of the birthdate, \(b\)?
Problem 9
Mai wants to graph the solution to the inequality \(5x - 4 > 2x - 19\) on a number line. She solves the equation \(5x - 4 = 2x - 19\) for \(x\) and gets \(x = \text -5\).
Which graph shows the solution to the inequality?
