For a line, the slope tells how fast $y$ changes, and it is the same everywhere. A curve's steepness keeps changing, but you can still ask how fast it changed on average between two inputs. That builds on reading where a graph rises and falls.
Why it works
Average speed works this way: if you drive $120$ miles in $2$ hours, your average speed is $\frac{120}{2} = 60$ miles per hour, even if you sped up and slowed down along the way. For a function, the “distance” is the change in output and the “time” is the change in input. Geometrically, it is the slope of the straight line (called a secant line) joining the two points on the graph.
Key idea
The average rate of change of $f$ from $x = a$ to $x = b$ is $$\frac{f(b) - f(a)}{b - a}$$ the change in output divided by the change in input, in units of output per unit of input.
Worked examples
Worked example: From a rule
Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.
- Outputs: $f(1) = 1$ and $f(3) = 9$.
- $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.
Answer: $4$: on average, $f$ rises $4$ for each $1$ that $x$ increases on this interval.
Worked example: In context
A plant is $12$ cm tall on day $3$ and $33$ cm tall on day $10$. Find its average growth rate.
- Change in height: $33 - 12 = 21$ cm. Change in time: $10 - 3 = 7$ days.
- $\frac{21 \text{ cm}}{7 \text{ days}} = 3$.
Answer: $3$ cm per day.
Common mistake
Dividing the change in $x$ by the change in $y$, or using the $x$-values where the outputs belong. The outputs go on top: $\frac{f(b) - f(a)}{b - a}$. Also keep the same order on top and bottom. Check the units: output units per input unit, such as cm per day.