Comparing Linear and Exponential Growth

By the end of this lesson, you will be able to tell the two apart from a table, a graph or a description, know why exponential always wins eventually, and pick the right model for a situation.

Adding Versus Multiplying

That is the entire distinction, and everything else follows from it.

$$\textbf{linear: } y = mx + b \qquad \text{add } m \text{ each step}$$

$$\textbf{exponential: } y = a \cdot b^x \qquad \text{multiply by } b \text{ each step}$$

$x$	linear $+3$	exponential $\times 3$
$0$	$2$	$2$
$1$	$5$	$6$
$2$	$8$	$18$
$3$	$11$	$54$
$4$	$14$	$162$
$10$	$32$	$118{,}098$

Both started at $2$ and both used a $3$. By $x = 10$ one has reached $32$ and the other has passed a hundred thousand.

The linear column gains a fixed AMOUNT; the exponential column gains a fixed PERCENTAGE. Adding $3$ to $2$ is a large increase and adding $3$ to $1000$ is nothing, which is why linear growth feels slower and slower relative to the total. Tripling is the same relative jump forever.

Reading It From a Table

Two tests, and you should do both.

Subtract consecutive outputs. Constant differences mean linear.

$$4, 7, 10, 13 \quad \Rightarrow \quad +3, +3, +3 \quad \Rightarrow \quad \text{linear}$$

Divide consecutive outputs. Constant ratios mean exponential.

$$4, 8, 16, 32 \quad \Rightarrow \quad \times 2, \times 2, \times 2 \quad \Rightarrow \quad \text{exponential}$$

The $x$ values must be evenly spaced for either test to mean anything. A table jumping $x = 1, 2, 5$ will not show a constant anything even if the function is perfectly linear.

If neither test gives a constant, it is neither: a quadratic has constant SECOND differences, which is a third pattern you already met.

Sort It: Linear or Exponential

Test the outputs.

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