# Exponential growth and decay on the SAT

Canonical: https://duckyhelper.com/learn/sat-math/exponential-functions/
Updated: 2026-10-01

An exponential function multiplies by the same factor each step: \(f(t) = a \cdot b^t\). Here a is the starting amount and b is the growth factor. If b is greater than 1 the amount grows, and if b is between 0 and 1 it decays. A growth rate of r percent gives \(b = 1 + \frac{r}{100}\); a decay rate gives \(b = 1 - \frac{r}{100}\). Linear functions add the same amount; exponential functions multiply.

## The key idea

$$
f(t) = a(1 + r)^t \qquad f(t) = a(1 - r)^t \qquad f(t) = a \cdot 2^{\frac{t}{d}}
$$

The first is growth by rate r per step, the second is decay by rate r per step (r as a decimal), and the third doubles every d units of time. Dividing t by d counts how many doubling periods have passed.

**Linear or exponential?**

|  | Linear | Exponential |
| --- | --- | --- |
| Each step | Adds the same amount | Multiplies by the same factor |
| Table test | Equal differences | Equal ratios |
| Example | \(5, 8, 11, 14\) (add 3) | \(5, 10, 20, 40\) (times 2) |

## Worked examples

**Example 1: growth by a percent**

Problem: A savings account starts with $2,000 and grows 5% per year. How much is in the account after 3 years?

1. Growth of 5% means a factor of 1.05 each year.

   $$
   A = 2000(1.05)^3
   $$
2. \(1.05^3 = 1.157625\).

   $$
   2000 \times 1.157625 = 2315.25
   $$

Answer: $2,315.25

**Example 2: build the model from a table**

Problem: A table shows \(f(0) = 400\), \(f(1) = 300\), \(f(2) = 225\), \(f(3) = 168.75\). Write f and describe the change.

1. Check ratios, not differences. Each output divided by the one before:

   $$
   \frac{300}{400} = \frac{225}{300} = 0.75
   $$
2. The ratio is constant, so f is exponential with factor 0.75 and start 400.

   $$
   f(x) = 400(0.75)^x
   $$
3. A factor of 0.75 keeps 75% each step, which is a 25% decrease.

Answer: \(f(x) = 400(0.75)^x\): it decreases by 25% each step.

**Example 3: doubling time**

Problem: A bacteria population starts at 500 and doubles every 4 hours. What is the population after 1 day?

1. One day is 24 hours, which is 6 doubling periods.

   $$
   \frac{24}{4} = 6
   $$
2. Multiply by 2 six times.

   $$
   500 \cdot 2^6 = 32000
   $$

Answer: 32,000

**Example 4 (SAT-hard): change the time unit**

Problem: A population is modeled by \(P(t) = 1200(1.21)^{\frac{t}{2}}\), where t is in years. By what percent does the population grow each year?

1. Rewrite the power: \((1.21)^{\frac{t}{2}} = \left((1.21)^{\frac{1}{2}}\right)^t\).

   $$
   (1.21)^{\frac{1}{2}} = \sqrt{1.21} = 1.1
   $$
2. So \(P(t) = 1200(1.1)^t\). A yearly factor of 1.1 means 10% growth per year.
3. The trap answer is 21%, which is the growth every 2 years. Over 2 years: \(1.1 \times 1.1 = 1.21\), so it matches.

Answer: 10% per year

## Common mistakes

- **Using the rate as the factor.** A 15% decay is a factor of 0.85, not 0.15. Fix: factor equals 1 plus or minus the rate.
- **Calling a constant difference exponential.** If a table adds the same number each step, it is linear. Fix: check ratios for exponential, differences for linear.
- **Forgetting to divide the time.** "Doubles every 4 hours" uses \(2^{\frac{t}{4}}\), not \(2^{4t}\). Fix: ask how many periods fit in the time given.
- **Multiplying the percent by the number of years.** 5% a year for 3 years is not 15%: \(1.05^3 \approx 1.158\), so it is about 15.8%. Fix: use the exponent.
- **Reading 21% as the yearly rate in a model with \(\frac{t}{2}\).** The factor 1.21 applies every 2 years. Fix: rewrite the base so the exponent is t.

## Quick method

> **Tip: Desmos regression for tables**
>
> Enter a table in Desmos and type \(y_1 \sim a b^{x_1}\). Desmos finds a and b. For exactly exponential data it gives the exact values; for real-world data it gives the best fit. For clean tables, dividing two outputs by hand is just as quick.

## Practice

**5 SAT-style questions**

1. Which statement describes \(f(x) = 80(0.92)^x\)?
   A. It increases by 92% each time x increases by 1.
   B. It decreases by 8% each time x increases by 1.
   C. It decreases by 92% each time x increases by 1.
   D. It decreases by 0.08 each time x increases by 1.

   Answer: It decreases by 8% each time x increases by 1.. The factor 0.92 keeps 92% of the value each step, so 8% is lost. 0.08 is a percent of the value, not a fixed amount, so the last choice describes a linear function.

2. A car worth $24,000 loses 15% of its value each year. Which expression gives its value after t years?
   A. \(24000(0.15)^t\)
   B. \(24000(0.85)^t\)
   C. \(24000(1.15)^t\)
   D. \(24000 - 0.15t\)

   Answer: \(24000(0.85)^t\). Losing 15% keeps 85%, so the factor is 0.85. Using 0.15 would keep only 15% each year. 1.15 is growth. The last choice subtracts 15 cents a year.

3. A sample of 3,000 cells triples every 5 days. How many cells are there after 15 days?
   A. 9,000
   B. 27,000
   C. 45,000
   D. 81,000

   Answer: 81,000. 15 days is 3 tripling periods: \(3000 \cdot 3^3 = 81{,}000\). 9,000 triples once, and 45,000 multiplies 3,000 by 15.

4. The value of an account t years from now is \(V(t) = 5000(1.04)^t\). Which expression gives the value m months from now?
   A. \(5000(1.04)^{12m}\)
   B. \(5000(1.04)^{\frac{m}{12}}\)
   C. \(5000\left(\frac{1.04}{12}\right)^m\)
   D. \(5000(1.0033)^m\)

   Answer: \(5000(1.04)^{\frac{m}{12}}\). m months is \(\frac{m}{12}\) years, so replace t with \(\frac{m}{12}\). 1.0033 is a rounded guess, and \(1.04^{\frac{1}{12}}\) is about 1.00327, so it is not equal.

5. Student-produced response: \(f(x) = a \cdot b^x\), where a and b are positive constants. If \(f(0) = 6\) and \(f(2) = 54\), what is \(f(3)\)?

   Answer: 162. \(f(0) = a = 6\). Then \(6b^2 = 54\), so \(b^2 = 9\) and \(b = 3\) (b is positive). \(f(3) = 6 \cdot 27 = 162\).

## Frequently asked questions

### How do I tell exponential from linear in a table?

Look at how the outputs change when x goes up by 1. Equal differences (add 3 each time) mean linear. Equal ratios (multiply by 2 each time) mean exponential. Make sure the x values are evenly spaced before you compare.

### What is the difference between growth rate and growth factor?

The rate is the percent change, like 5%. The factor is what you multiply by each step, like 1.05. For decay, a rate of 15% gives a factor of 0.85. The factor goes in the base of the exponential.

### Are exponential questions hard on the SAT?

Basic ones are quick. The harder ones change the time unit, like a model in years asked about months, or hide the rate inside an exponent like \(\frac{t}{2}\). Rewriting the base so the exponent is plain t solves most of them.

## Sources

- [College Board: SAT Math, Advanced Math skills (nonlinear functions)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced), accessed 2026-10-01

## Related

- [Percentages on the SAT](https://duckyhelper.com/learn/sat-math/percentages/)
- [Functions and function notation on the SAT](https://duckyhelper.com/learn/sat-math/functions/)
- [Logarithms](https://duckyhelper.com/learn/algebra-2/logarithms/)
- [SAT Math study guides](https://duckyhelper.com/learn/sat-math/)

## Try asking Ducky

- "Why is a 15% loss a factor of 0.85?"
- "How do I know if this table is linear or exponential?"
- "Can you explain the t over 2 exponent again, slower?"

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