How can you tell from data whether something grows linearly or exponentially? Compare it with arithmetic sequences (add the same amount) and exponential functions (multiply by the same factor).
Why it works
Over equal steps in $x$, a linear function changes by equal differences: $y = 3x + 2$ adds $3$ each step. An exponential function changes by equal ratios: $y = 2 \cdot 3^x$ multiplies by $3$ each step. That is the test. It also explains the long run: a linear function always adds the same amount, but an exponential function adds an amount proportional to what it already has, so its yearly gain keeps growing. Eventually any exponential growth passes any linear growth, however far behind it starts.
| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ |
|---|---|---|---|---|---|
| linear $y = 10x + 1$ | $1$ | $11$ | $21$ | $31$ | $41$ |
| exponential $y = 3^x$ | $1$ | $3$ | $9$ | $27$ | $81$ |
Key idea
Equal differences over equal intervals mean linear; equal ratios mean exponential. Exponential growth eventually exceeds linear growth.
Worked examples
Worked example: Test a table
Is the data $(0, 5), (1, 10), (2, 20), (3, 40)$ linear or exponential?
- Differences: $5, 10, 20$. Not equal, so not linear.
- Ratios: $\frac{10}{5} = 2$, $\frac{20}{10} = 2$, $\frac{40}{20} = 2$. Equal.
Answer: Exponential: $y = 5 \cdot 2^x$.
Worked example: When does the exponential win?
$f(x) = 2^x$ and $g(x) = 25x$. For which whole $x$ does $f$ first pass $g$?
- $x = 6$: $64$ vs $150$. $x = 7$: $128$ vs $175$.
- $x = 8$: $256$ vs $200$.
Answer: At $x = 8$, and $f$ stays ahead from then on.
Common mistake
Judging from the first few values. At first $25x$ is far ahead of $2^x$, so it can look as if the linear function wins. Keep the table going, or compare the step-by-step gains: the line gains $25$ each time, while $2^x$ gains $2^x$ itself, which soon exceeds $25$.