Exponential functions on the SAT: lesson and practice
An exponential function has the form f(t) = a(b)t. a is the starting value and b is the growth factor: a p% increase each period means b = 1 + p/100, and a p% decrease means b = 1 − p/100. Linear patterns add the same amount each step; exponential patterns multiply by the same factor.
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| Situation | Model |
|---|---|
| Starts at 2,000, grows 5% per year | 2,000(1.05)t |
| Starts at 800, loses 12% per year | 800(0.88)t |
| Starts at 300, doubles every 4 hours | 300(2)t/4 |
Linear or exponential?
Check the steps in a table. Equal differences (3, 6, 9, 12) mean linear. Equal ratios (3, 6, 12, 24) mean exponential.
Common mistakes
Sample questions with answers
1. f(t) = 5,000(1.05)^t models subscribers after t months. What does 1.05 mean?
- Increase of 105% per month
- Increase of 5% per month (correct)
- Increase of 5,000 per month
- Decrease of 5% per month
1.05 = 1 + 0.05, a 5% increase each month.
2. A town of 2,000 grows 3% a year. Which function gives the population after t years?
- 2,000(0.03)^t
- 2,000(1.3)^t
- 2,000(1.03)^t (correct)
- 2,000 + 0.03t
A 3% increase multiplies by 1.03 each year.
3. An investment grows 20% a year. What is the total growth over 2 years?
- 40%
- 44% (correct)
- 20%
- 400%
1.2 × 1.2 = 1.44, a 44% increase.
Frequently asked questions
How many exponential questions are on the SAT?
Exponential functions are part of Advanced Math. Advanced Math makes up about 35% of SAT Math, roughly 13 to 15 questions.
How do I tell growth from decay?
In a(b)^t, b > 1 means growth and 0 < b < 1 means decay.
How do I write a model for doubling every k hours?
Use a(2)^(t/k), where a is the starting amount.