SAT Math · Advanced Math

Exponential growth and decay

Q: What is the formula for exponential decay?

Exponential decay is y = a(1 − r)ˣ, where r is the rate of decrease as a decimal. Losing 15% a year from $20,000 gives y = 20,000(0.85)ˣ.

Q: How do you tell if a function is exponential or linear?

Look at equal steps in x. If the outputs change by the same amount each time, it is linear. If they change by the same factor, so the ratio of one output to the previous one is constant, it is exponential.

Q: What is a growth factor?

The growth factor is the base of the exponential, the number you multiply by each step. A factor of 1.25 means 25% growth, and a factor of 0.8 means 20% decay.

Q: How do you solve half-life problems?

Divide the time by the half-life to count how many halvings happen, then multiply the starting amount by ½ that many times. 80 grams after three half-lives is 80 × (½)³ = 10 grams.

Q: How can Aristotle help with SAT advanced math?

Aristotle is a voice AI SAT tutor. Your child talks through an exponential model out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including nonlinear functions.

Read the starting value and the percent rate from y = a(1 ± r)ˣ, and tell exponential from linear.

Try Aristotle free

Updated October 30, 2026

$y = a(1 \pm r)^x$

Both start at 10. The line adds 5 each step; the curve multiplies by 1.25 and pulls ahead by step 7.

What are exponential growth and decay?

y = a\,(1 \pm r)^x

5 \to 10 \to 20 \to 40 \to 80

A quantity is exponential when it is multiplied by the same factor every step: times 2 for doubling, times 1.25 for 25% growth, times 0.7 for 30% decay. The exponent $x$ counts the steps, because it counts how many times you multiplied.

y = 5 · 2ˣ: each step doubles the last.

Linear adds, exponential multiplies

\text{linear: } +5,\ +5,\ +5

\text{exponential: } \times 1.25,\ \times 1.25

A linear model adds the same amount each step; an exponential one multiplies by the same factor. Test a table both ways: constant differences mean linear, constant ratios mean exponential. Multiplying by anything above 1 eventually outruns adding.

Same start; the multiplier passes the adder by step 7 and never looks back.

Decay multiplies by what is left

\text{lose } 30\% \;\Rightarrow\; \times\,0.7

y = 100\,(0.7)^x

Decay is the same machine with a factor below 1. Lose 30% a step and you keep 70%, so multiply by 0.7, never by 0.3. Each step removes a share of what remains, so the drop is steep at first and then flattens toward zero.

100, 70, 49: each value is 0.7 of the one before.

Why it is multiplying, not adding

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Something grows 25%. What do you multiply by?
Answer
1.25: all of it, plus a quarter more.
Question 2
It grows 25% again. What is the total change from the start?
Answer
$1.25 \times 1.25 = 1.5625$ , a 56.25% increase, not 50%. That repeated multiplying is the exponent.
Question 3
Losing 20% a step. What is the factor?
Answer
0.8, because 80% is left each time.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

A population is modeled by $P = 500(1.06)^t$ , where $t$ is years. What do 500 and 1.06 mean?

Question 2

A car costs $20,000 and loses 15% of its value each year. Which function gives its value after $t$ years?

Question 3

The table shows $f(0) = 6$ , $f(1) = 12$ , $f(2) = 24$ , $f(3) = 48$ . Is $f$ linear or exponential, and what is $f(x)$ ?

Question 4

A substance has a half-life of 5 hours. A sample starts at 80 grams. How much is left after 15 hours?

Stuck on a question like this?

Talk it through with Aristotle. It asks guiding questions instead of giving the answer. Intro is free with three sessions.

Try Aristotle free

The formulas

Exponential growth

$y = a(1 + r)^x$

$a$ is the start. 6% growth is a base of 1.06.

Exponential decay

$y = a(1 - r)^x$

A 20% decrease each step is a base of 0.8. The amount shrinks but never reaches zero.

Doubling or halving

$y = a \cdot 2^{t/d} \quad\text{or}\quad y = a\left(\tfrac{1}{2}\right)^{t/h}$

$d$ is the doubling time and $h$ is the half-life. After one $d$ , the amount has doubled.

Linear, for comparison

$y = mx + b$

Linear models add the same amount each step. Exponential models multiply by the same factor.

Reading the growth factor
Base	Rate	Meaning
1.06	+6%	grows 6% per step
2	+100%	doubles each step
1	0%	no change
0.8	−20%	loses 20% per step
0.5	−50%	halves each step

Try it

Set the rate and watch it compound

Pick a start and a rate. Positive rates grow, negative rates decay.

Starting value a10

Rate per step+25%

Equationy = 10(1.25)ˣ

Growth factor1 + 0.25 = 1.25, so growth of 25% each step

After 5 steps10 × 1.25⁵ ≈ 30.5

Common mistakes

Using the rate as the base

6% growth is a base of 1.06, not 0.06 or 6.

Writing decay with a plus

Decreasing by 15% is a factor of 0.85. A factor of 1.15 would be growth.

Treating it as linear

Growing 10% a year for 3 years is not 30% growth. It is $1.1^3 = 1.331$ , a 33.1% increase.

Ignoring the time units

Monthly rate, $t$ in years: the exponent is $12t$ . Half-life of 5 hours: $t \div 5$ .

How it shows up on the SAT

Advanced Math is about 35% of SAT Math. Expect to read the start and rate from an equation, pick linear or exponential, and write a percent model.

What Aristotle tracks in nonlinear functions

16 SAT Math skills, including:

Solving problems with quadratic and exponential functions
Writing quadratic or exponential functions to model relationships
Explaining inputs, outputs, intercepts, and initial values in context
Explaining parts of quadratic and exponential functions in context

What is the formula for exponential growth?

Exponential growth is y = a(1 + r)ˣ, where a is the starting value, r is the growth rate as a decimal, and x is the number of time steps. For 6% yearly growth from 500, it is y = 500(1.06)ˣ.

What is the formula for exponential decay?

How do you tell if a function is exponential or linear?

What is a growth factor?

How do you solve half-life problems?

How can Aristotle help with SAT advanced math?

More SAT concepts

Percent change

Percent increase, decrease, and back-to-back changes, done with one multiplier instead of guesswork.

The quadratic formula

Solve any ax² + bx + c = 0, and use the discriminant to count solutions without solving.

Slope-intercept form

Read a line's slope and y-intercept straight from y = mx + b, and write one from a graph or two points.

Tutoring for this topic

SAT tutor

SAT Math and Reading and Writing, organized by official domain.

Algebra tutor

Algebra 1 and Algebra 2, from linear equations to logarithms.

Precalculus tutor

Precalculus and AP Precalculus, the runway into calculus.

A learner having an idea, surrounded by maps and equations

Practice it out loud

Talk it through with Aristotle

Work a question like these out loud. Aristotle guides you one step at a time, so you can do the next one on your own.

Try Aristotle today