# Linear vs. exponential growth on the SAT: differences or factors

> Distinguish linear from exponential growth on the SAT using tables, wording, equations, equal differences, equal factors, and percent change.

Published 2026-07-17 | Math | StudyHall

Canonical: https://trystudyhall.com/blog/sat-linear-vs-exponential

Linear and exponential models can both increase, but they preserve different patterns. Linear growth adds the same amount over equal input intervals, while exponential growth multiplies by the same factor.

To distinguish linear from exponential growth on the SAT, check equal input steps: constant first differences mean linear, and constant nonzero ratios mean exponential.

**Key takeaways:**
- Linear growth has equal differences and commonly uses the form $y=mx+b$.
- Exponential growth has equal factors and commonly uses the form $y=a(b)^x$.
- “Increases by 3” is additive; “increases by 3 percent” multiplies by $1.03$; “triples” multiplies by $3$.
- Percent change compounds because each new change is calculated from the latest amount.

## How can you tell linear from exponential growth?

| Feature | Linear | Exponential |
| --- | --- | --- |
| Repeated change | Add the same difference | Multiply by the same factor |
| Common form | $y=mx+b$ | $y=a(b)^x$ |
| Meaning of parameter | $m$ is change per input unit | $b$ is the factor per input unit |
| Table test | First differences are constant | Ratios are constant when outputs are nonzero |
| Typical wording | Increases by 4 each year | Increases by 4 percent each year |
*Use equal input intervals. Unequal spacing can hide both patterns with surprising efficiency.*

The [SAT linear functions guide](https://trystudyhall.com/blog/sat-linear-functions) develops slope, intercepts, and additive models. This child focuses on choosing between additive and multiplicative growth. The [SAT exponent rules guide](https://trystudyhall.com/blog/sat-exponent-rules) is a sibling about simplifying powers; those laws support exponential algebra, but model recognition begins with differences, factors, and wording.

## How do tables reveal equal differences or factors?

1. **Check the input spacing.** Confirm that $x$ increases by equal amounts, usually $1$.
2. **Subtract consecutive outputs.** A constant result identifies a linear pattern.
3. **Divide consecutive outputs.** If differences change, test $\frac{y_{n+1}}{y_n}$. A constant nonzero ratio identifies an exponential pattern.
4. **Match the model.** Use the starting output for $a$ or $b$, then use the difference or factor for the rate parameter.

> **Watch out: Growing faster does not define exponential** A table with increasing differences could be quadratic or another nonlinear pattern. Exponential requires a constant multiplicative factor across equal input steps.

A linear table might show outputs $8, 11, 14, 17$, with a difference of $3$. An exponential table might show $8, 12, 18, 27$, with a factor of $1.5$. The exponential differences are $4, 6, 9$, so subtraction will not stay constant even though the table begins modestly.

## Which words signal linear or exponential change?

- **Increases by 3:** add $3$ each interval, so the model is linear.
- **Decreases by 3:** subtract $3$ each interval, also linear.
- **Increases by 3 percent:** multiply by $1+0.03=1.03$ each interval, so the model is exponential.
- **Decreases by 3 percent:** multiply by $1-0.03=0.97$ each interval, so the model is exponential decay.
- **Triples:** multiply by $3$ each interval, so the model is exponential.

The unit phrase controls the interval. A population that increases by $6\%$ each year has model $P(t)=P_0(1.06)^t$ when $t$ is measured in years. If $t$ counts months, the stated annual factor cannot be copied directly without converting the interval.

## Why does percent change compound?

A repeated percent change uses a new base each time. Starting from $200$, a $10\%$ increase gives $200(1.10)=220$. The next increase is $10\%$ of $220$, so the result is $220(1.10)=242$. Equal percentages therefore produce unequal absolute increases, $20$ and then $22$.

This is why multiplying the starting value by $1+rt$ generally fails for repeated percent growth. That expression adds the same original-base amount each period. The correct repeated model applies the factor once per interval: $a(1+r)^t$. The [SAT percentages guide](https://trystudyhall.com/blog/sat-percentages) covers percent multipliers, percent change, and reverse-percent questions in depth.

## Can you classify one growth table?

**Example: Linear and exponential functions.**

> A quantity follows the table: when $x$ is $0,1,2,3$, the corresponding values of $y$ are $5,10,20,40$.

Which statement best describes the relationship between $x$ and $y$?

- A) It is linear because the outputs increase as x increases.
- B) It is exponential because each output is twice the preceding output.
- C) It is linear because each output is 5 more than the input.
- D) It is quadratic because the first differences increase.

**Answer:** B. For equal steps of 1 in x, the outputs have a constant ratio: $10/5=2$, $20/10=2$, and $40/20=2$. That fixed factor identifies an exponential relationship, so choice B is correct. The first differences are not constant, and increasing differences alone do not prove a quadratic model.

**[Practice growth models](https://trystudyhall.com/learn/linear-functions)**: Check the interval, test differences and factors, and translate each rate with its units attached.

## FAQ

### How do you tell linear from exponential growth on the SAT?

For equal input steps, constant differences between outputs indicate linear growth, while a constant multiplicative ratio indicates exponential growth.

### Does increases by a percent mean exponential?

Yes, when the same percentage is applied repeatedly over equal intervals. Each interval multiplies the current amount by the same growth factor.

### What does triples mean in a growth model?

Triples means the quantity is multiplied by three during each stated interval, which is an exponential factor rather than an additive change.

### Why does percent growth compound?

Each percentage change is calculated from the newest amount, so the absolute amount added changes while the multiplicative factor stays constant.

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