Free 30-Q SAT Diagnostic →

Free video lesson

How to Solve Exponential Growth and Decay on the SAT

Linear or exponential? The SAT tests this constantly, and there's a 5-second test: linear adds the same amount each step (constant difference); exponential multiplies by the same factor (constant ratio). Here's how to tell them apart — and see them diverge in Desmos.

Math · Nonlinear functions2:10Published July 5, 2026

On YouTube: SAT Growth & Decay: One Multiplier Rule

What this lesson covers

Linear = constant difference (2, 4, 6, 8… +2)
Exponential = constant ratio (2, 4, 8, 16… ×2)
Graph both: same start, wildly different finish
Practice: 5, 10, 20, 40
The three traps (differences only, "fast = exponential", add vs multiply)

Questions worked in the video

0:29Sequences 2,4,6,8,10 and 2,4,8,16,32 — which is linear, which is exponential?
1:095, 10, 20, 40 — linear or exponential, and what is the rule?

Worked examples

The questions the video works, written out: the setup, each step, the answer and the trap.

0:29Example 1

Two sequences both start 2, 4. Sequence A continues 2, 4, 6, 8, 10. Sequence B continues 2, 4, 8, 16, 32. Which is linear and which is exponential, and what is the rule for each?

Test A for a constant difference: 4 - 2 = 2, 6 - 4 = 2, 8 - 6 = 2, 10 - 8 = 2. The same amount is added each step, so A is linear.
Its rule adds 2 per step from a start of 2. With x counting steps from 0, $y = 2x + 2$ .
Test B: the differences are 2, 4, 8, 16, not constant. Now test the ratios: $\displaystyle \frac{4}{2} = 2$ , $\displaystyle \frac{8}{4} = 2$ , $\displaystyle \frac{16}{8} = 2$ , $\displaystyle \frac{32}{16} = 2$ . The same factor each step, so B is exponential.
Its rule multiplies by 2 each step from a start of 2: $y = 2 \cdot 2^x$ .
Check the 5th term, x = 4: A gives $2(4) + 2 = 10$ and B gives $2 \cdot 2^4 = 32$ . Graphed together they start on top of each other, then the curve peels away; by x = 10 the line is at 22 and the curve is at 2048.

Answer: A is linear, $y = 2x + 2$ ; B is exponential, $y = 2 \cdot 2^x$

Linear means a constant difference, add the same amount each step; exponential means a constant ratio, multiply by the same factor each step. The first two terms cannot tell you which, only the pattern can. The trap is checking differences only and, seeing they are not constant, calling B not linear without naming what it is; the second trap is assuming any fast-growing sequence is exponential, when a steep line grows fast too.

1:09Example 2

The sequence 5, 10, 20, 40 continues the same way. Is it linear or exponential, and what is the rule?

Differences: 10 - 5 = 5, 20 - 10 = 10, 40 - 20 = 20. Not constant, so it is not linear.
Ratios: $\displaystyle \frac{10}{5} = 2$ , $\displaystyle \frac{20}{10} = 2$ , $\displaystyle \frac{40}{20} = 2$ . A constant ratio of 2, so it is exponential.
Rule: start at 5, multiply by 2 each step. $y = 5 \cdot 2^x$ , with x = 0 giving the first term.
Check: $5 \cdot 2^3 = 5 \cdot 8 = 40$ , the 4th term. The next term would be 80.

Answer: exponential, $y = 5 \cdot 2^x$

In $y = a \cdot b^x$ , a is the starting value and b is the ratio, so both numbers come straight off the sequence. The trap is writing $y = 2 \cdot 5^x$ with the two numbers swapped, or $y = 5x + 5$ after noticing only the first difference of 5; test any rule against the third term before you commit.

One more, same method

A sample starts at 80 grams, and the readings at the end of each hour are 80, 40, 20, 10. Is this linear or exponential, and what is the rule?

Differences: 40 - 80 = -40, 20 - 40 = -20, 10 - 20 = -10. Not constant, so not linear.
Ratios: $\displaystyle \frac{40}{80} = \frac{1}{2}$ , $\displaystyle \frac{20}{40} = \frac{1}{2}$ , $\displaystyle \frac{10}{20} = \frac{1}{2}$ . A constant ratio of $\displaystyle \frac{1}{2}$ , so exponential; because the ratio is less than 1, it is exponential decay.
Rule: start at 80, multiply by $\displaystyle \frac{1}{2}$ each hour: $\displaystyle y = 80 \left(\frac{1}{2}\right)^x$ , with x the number of hours.
Check: $\displaystyle 80 \cdot \left(\frac{1}{2}\right)^3 = 80 \cdot \frac{1}{8} = 10$ , the reading after 3 hours.

Answer: exponential decay, $\displaystyle y = 80 \left(\frac{1}{2}\right)^x$

Decay is the same test as growth: a constant ratio, just one below 1, so the amount shrinks by the same fraction each step instead of by the same number of grams. The trap is calling it linear because it goes down by a steady amount; the drops are 40, 20, 10, which halve each time, not a fixed 40. A linear decrease would read 80, 40, 0, -40.

Chapters

Lesson transcript

The narration of the video, word for word, under its chapter headings.

0:00Linear or exponential?

Welcome to SAT Climb. Linear or exponential? The SAT hides this everywhere. One adds the same amount each step; the other multiplies. Tell them apart, and the question's half solved.

0:16The setup: two sequences

Two sequences start the same: 2, then 4. But watch what happens next. One is linear, one is exponential. Which is which?

0:29Difference vs ratio

Check the pattern. Linear adds a constant: 2, 4, 6, 8 — +2 every time. Exponential multiplies: 2, 4, 8, 16 — ×2 every time. Constant difference = linear; constant ratio = exponential.

0:49Watch them diverge (Desmos)

Graph both, and it's obvious. They start together, but the exponential curve peels away, leaving the line far below. Same start — wildly different finish.

1:09Your turn

Your turn. 5, 10, 20, 40. Linear or exponential — and what's the rule? Check the ratios.

1:35Three traps

Three traps. Checking only the differences — you'll miss exponential. Assuming fast growth means exponential — a steep line grows fast too. And confusing the rule: adding vs multiplying.

1:56Recap

Growth: solved. Constant difference, or constant ratio. Start free at satclimb.com.

Read the written version: the Nonlinear functions strategy guide, then try 25 hard Nonlinear functions questions with full explanations. Both are free, no account needed.

Watched it? Now drill it.

SAT Climb turns a 30-question diagnostic into a study plan across all 31 skills, with a 7-day free trial on every plan.

Start free trial →All video lessons