Average Rate of Change and the Secant Line

How the slope of a line extends to curves, how to calculate an average rate of change, what it means in real situations like average speed, and how shrinking the interval leads to calculus.

3 min read Supports 3 lessons

A straight line has one slope, the same everywhere. A curve keeps changing direction, so it has no single slope. But you can still ask how fast it changes on average between two points.

Average rate of change

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

Change in output over change in input: rise over run between two points on the curve.

Geometrically, it is the slope of the secant line, the straight line through the two points on the curve.

The secant through $(1, 1)$ and $(3, 9)$ has slope $\frac{9 - 1}{3 - 1} = 4$ .

A real example

If $f(t)$ is a position at time $t$ , the average rate of change is the average velocity: total change in position over time taken.

$h(t) = 20t - 5t^2$
A ball's height in metres after $t$ seconds.

1/4

Key idea

An average rate of change hides what happens in between. The ball was moving fast the whole time; it just ended where it started.

Shrinking the interval

Average speed over an hour says little about your speed at one moment. Average over a minute says more; over a second, more still. Bring the second point closer and closer to the first and the secant swings round towards the tangent line, which just touches the curve.

Press the button and watch the secant close in on the tangent.

Key idea

The slope of that tangent is the instantaneous rate of change: the derivative. Calculus is the study of what happens to $\frac{f(b) - f(a)}{b - a}$ as $b$ slides towards $a$ .

Watch out

Keep the order consistent: if $f(b)$ comes first on top, $b$ comes first on the bottom.

Tip

Units help. If $f$ is in metres and $x$ in seconds, the rate of change is in metres per second.

Common questions

Is average rate of change the same as slope?: It is the slope of the straight line joining two points on a graph. For a straight line it equals the line's slope; for a curve it depends on which two points you choose.
What is a secant line?: A straight line that passes through two points of a curve. Its slope is the average rate of change between those points.

Practise average rate of change

The Average Rate of Change lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Is average rate of change the same as slope?

It is the slope of the straight line joining two points on a graph. For a straight line it equals the line's slope; for a curve it depends on which two points you choose.

What is a secant line?

A straight line that passes through two points of a curve. Its slope is the average rate of change between those points.