How do we measure how much a function changes? Take the change in the function, and divide it by the interval the change happened over. That ratio is the average rate of change — and it is the idea differentiation is built on.

Concept The formula

Take the function's value at the right-hand point, . Subtract its value at the left-hand point, — that is how much the function changed. Divide by , the interval you measured across.

In words: the difference in the y-values, divided by the difference in the x-values.

In shorthand:

Concept Reading f(x) off the graph

If you already know the function's value at a point, use it directly. If you don't, take the value, go up to the curve, and read across — that is what substituting into looks like on a graph.

Concept The secant line

Join the two points with a straight line, and you get the secant line. Its slope is written — sec from secant — and that slope is the average rate of change.

The tall side of the triangle is the change in the function.

The flat side is the interval in

The sloped side joining the two points is the secant — its slope is the answer.

Note Two special cases

Flat function: no change over the interval, so the secant is flat too —

Constant-rate function: a line that already rises at a fixed rate — its secant lies exactly on the graph itself.

Note Where this leads

Remember this line when differentiation arrives. Move the two points closer and closer together, and watch how the secant line behaves — that limiting slope is the instantaneous rate of change, the idea the derivative is built from.

Summary

Average rate of change = the change in a function, divided by the interval it changed over.
Formula: , also written .
Joining the two points gives the secant line; its slope is the average rate of change.
A flat function gives a secant with slope 0; a constant-rate function's secant lies on the graph itself.
Bring the two points together and the secant becomes the tangent — the instantaneous rate of change.

The Secant Line and Average Rate of Change