Advanced Math · SAT Math

Exponential Functions on the Digital SAT

Exponential functions model the second-most-common SAT word-problem scenario after linear functions — population growth, bank interest, half-life, depreciation, viral spread. This lesson covers the form y = ab^x, identifying growth vs decay, the doubling and half-life patterns, compound interest, and how to tell exponential from linear. Plus 8 practice problems.
By the Prepiii Editorial TeamUpdated 2026-05-23~11 min read

The form: y = ab^x

An exponential function has the form y = a · b^x, where:

a is the starting value — the output when x = 0. (Anything to the 0 power is 1, so a · b^0 = a · 1 = a.)

b is the base (also called the growth factor or multiplier). It controls how fast the function grows or shrinks.

Two cases based on the base:

  • If b > 1: exponential growth. Each step multiplies by b. Example: y = 100 · 2^x doubles every step.
  • If 0 < b < 1: exponential decay. Each step shrinks by a factor of b. Example: y = 100 · 0.5^x halves every step.

Translating percent change into the base

Most SAT exponential word problems give a percent change per period (year, hour, etc.) and you need to write the function.

X% growth per period → base b = 1 + X/100.

X% decay per period → base b = 1 - X/100.

Example. A population starts at 5,000 and grows by 4% per year. The function is P(t) = 5000 · 1.04^t where t is years.

Example. A car worth $20,000 loses 12% of its value each year. The function is V(t) = 20000 · 0.88^t.

Doubling and half-life patterns

When the SAT gives you a doubling time or half-life, it's cleaner to write the function around that period than to calculate a base.

Doubling every k periods: y = a · 2^(t/k).

Halving every k periods: y = a · (1/2)^(t/k).

Tripling every k periods: y = a · 3^(t/k).

Example. A bacterial culture starts at 200 cells and doubles every 3 hours. After t hours: P(t) = 200 · 2^(t/3).

Compound interest

Compound interest is exponential growth applied to money. The standard formula:

A = P(1 + r/n)^(nt)

P = principal · r = annual rate (decimal) · n = compounding periods per year · t = years

Example. $1,000 invested at 6% annual interest compounded monthly for 5 years: A = 1000(1 + 0.06/12)^(12·5) ≈ $1,348.85.

Annual compounding shortcut: when n = 1, the formula simplifies to A = P(1 + r)^t — exactly the percent-growth form.

Exponential vs linear: how to tell them apart

The SAT frequently asks "which model best fits this data?" The trick is to check the pattern of change:

  • Linear: the output changes by the same amount each step. 100, 110, 120, 130 → adds 10 each step.
  • Exponential: the output changes by the same factor (ratio) each step. 100, 110, 121, 133.1 → each term is 1.10× the previous (10% growth).

Quick test: compute consecutive differences. If constant → linear. If consecutive ratios are constant → exponential.

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Solving with Desmos

Desmos plots exponentials cleanly and lets you sanity-check models instantly. Three techniques:

1. Plot to verify a percent-change model

For "a population grows by 4% per year starting at 5,000," type y = 5000 · 1.04^x. Check that y(0) = 5000 and y(1) ≈ 5200 — confirms the model.

2. Use a slider to find a base or starting value

If a problem gives a data point (e.g., 'after 5 years the value is 800'), type y = a · b^x with sliders for a and b. Drag until the curve passes through the given points.

3. Tilde regression for compound problems

For "starting at 1000, the value doubled in 5 years — find the annual rate": type 2000 ~ 1000 · (1 + r₁)^5. Desmos solves for r₁ ≈ 0.1487, i.e., ~14.87% annual rate.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Confusing the starting value with the base

In y = 200 · 1.5^x, the starting value is 200 (a) and the growth factor is 1.5 (b). Swapping them produces a wildly wrong model.

Using 1 - r instead of 1 + r for growth

For 4% growth, the base is 1.04 (not 0.96). For 4% decay, the base is 0.96. Read the verb: 'grows by,' 'increases by' → 1 + r. 'Decays,' 'depreciates,' 'loses' → 1 - r.

Adding percentages instead of compounding

A 10% increase followed by another 10% increase is NOT 20%. It's 1.10 × 1.10 = 1.21, or 21%. Exponentials compound — they don't add.

Misreading the time exponent

If something doubles every 3 hours and you want the value after t hours, the exponent is t/3, not 3t. The exponent is 'how many doubling periods have passed' — total time divided by period length.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

A bacterial colony starts at 500 cells and doubles every hour. How many cells are there after 4 hours?

  1. 2,000
  2. 4,000
  3. 8,000
  4. 16,000

Click to reveal solution →

Answer: (C) 8,000

Doubling 4 times: 500 · 2^4 = 500 · 16 = 8000.

2

A car's value depreciates by 15% per year. If it's worth $24,000 today, what is its approximate value in 3 years?

  1. $14,500
  2. $15,300
  3. $17,300
  4. $20,400

Click to reveal solution →

Answer: (B) $15,300

Multiplier per year: 0.85. After 3 years: 24000 · 0.85^3 = 24000 · 0.6141 ≈ 14,738.

Rounded: ~$14,700–15,300 (depending on rounding). Choice (B) is closest.

3

Which function models a quantity that starts at 200 and grows by 7% per year?

  1. y = 200 · 0.93^t
  2. y = 200 · 1.07^t
  3. y = 200 · 7^t
  4. y = 7 · 200^t

Click to reveal solution →

Answer: (B) y = 200 · 1.07^t

Starting value 200 → a = 200. 7% growth → base b = 1 + 0.07 = 1.07.

4

A radioactive isotope has a half-life of 8 years. If a sample starts at 40 grams, how many grams remain after 24 years?

  1. 2.5 grams
  2. 5 grams
  3. 10 grams
  4. 20 grams

Click to reveal solution →

Answer: (B) 5 grams

24 years is 3 half-lives. Halve three times: 40 → 20 → 10 → 5.

Or: P(24) = 40 · (1/2)^(24/8) = 40 · (1/2)^3 = 40 · 1/8 = 5.

5

Which best describes the data: 100, 150, 225, 337.5, ...?

  1. Linear with slope 50
  2. Linear with slope 75
  3. Exponential with base 1.5
  4. Exponential with base 2

Click to reveal solution →

Answer: (C) Exponential with base 1.5

Ratios of consecutive terms: 150/100 = 1.5, 225/150 = 1.5, 337.5/225 = 1.5. Constant ratio → exponential. Base = 1.5.

6

If $5,000 is invested at 4% annual interest, compounded annually, what will the balance be after 10 years?

  1. $5,400
  2. $6,500
  3. $7,401
  4. $8,236

Click to reveal solution →

Answer: (C) $7,401

A = 5000(1.04)^10 ≈ 5000 · 1.4802 ≈ 7401.

7

The function f(t) = 2000 · 1.05^t models the population of a town t years after 2020. What was the population in 2020?

  1. 1.05
  2. 1,000
  3. 2,000
  4. 2,100

Click to reveal solution →

Answer: (C) 2,000

When t = 0: f(0) = 2000 · 1 = 2000. The starting value is the coefficient a.

8

A virus spreads such that the number of infected people triples every 4 days. If 50 people are infected today, how many will be infected in 12 days?

  1. 150
  2. 450
  3. 1,350
  4. 1,500

Click to reveal solution →

Answer: (C) 1,350

12 days = 3 tripling periods. Triple three times: 50 · 3^3 = 50 · 27 = 1350.

Frequently asked questions

What is an exponential function?

+
A function of the form y = a · b^x where a is the starting value and b is the base (growth factor). The graph either grows rapidly (b > 1) or decays toward zero (0 < b < 1). Real-world examples: population growth, compound interest, radioactive decay.

What's the difference between linear and exponential growth?

+
Linear growth adds the same amount each step (100, 110, 120, 130 — adding 10). Exponential growth multiplies by the same factor each step (100, 110, 121, 133.1 — multiplying by 1.10). Over time, exponential outpaces linear dramatically.

How do I convert a percent growth rate into the base?

+
For X% growth, the base is 1 + X/100 (so 4% growth means base 1.04). For X% decay, the base is 1 − X/100 (so 12% depreciation means base 0.88).

What is half-life?

+
Half-life is the time it takes for a quantity to be reduced to half its starting value. A function with half-life k is y = a · (1/2)^(t/k), where t is total time elapsed.

What's the compound interest formula?

+
A = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the time in years. For annual compounding (n = 1), it simplifies to A = P(1 + r)^t.

Keep going

Linear functions

How exponential growth differs from linear

Percent problems

Foundation for percent-growth modeling

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