Exponential functions for the LCAT: 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 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.
Is exponential functions tested on the LCAT?
Yes. It falls under Advanced Math, one of the content domains LUMS lists for the LCAT in its official sample questions.
Can I practice this without a calculator?
Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.