Average rate of change.

Step 1 of 4

From a line to a curve

A line has one slope, the same number everywhere: rise divided by run. A curve does not. The board shows $y = x^2$⁠, the parabola from the quadratic functions Unit: near $x = 0$⁠ it is almost flat, and further out it climbs faster and faster.

You can still ask how much it rises on average between two points. Stretch a straight line between them, a secant, and measure its slope exactly as for a line:

$\frac{\Delta y}{\Delta x} = \frac{f(x + h) - f(x)}{h}$$\frac{\Delta y}{\Delta x} =$$\frac{f(x + h) - f(x)}{h}$⁠

That is the average rate of change. On the board the point is at $x = 1$⁠ and the run is $h = 2$⁠.

Step 2 of 4

Bring the points together

The secant measures an average over a whole stretch. To know how steep the curve is at one point, bring the second point closer to the first: make Run h smaller. Predict what will happen to the secant’s slope, then watch.

Step 3 of 4

Either side of the vertex

An average hides what happened along the way. From $x = -1$⁠ to $x = 1$⁠ the parabola goes down to its vertex and then back up.

Step 4 of 4

Look for a rate

Run h is locked at $1$⁠, and Point x is free. The further right you move it, the steeper the secant.