Average Rate of Change
How fast is something changing? If a car's odometer climbs from
100 to 220 miles over
2 hours, its average speed is
120 \div 2 = 60 miles per hour. That is an
average rate of change: total change in the output divided by the
change in the input.
For a function f over an interval from
x = a to x = b, the average rate
of change is the change in y divided by the change in
x:
\text{average rate of change} = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}
That fraction should look familiar — it is exactly the
slope of a line,
just written with function values.
It is the slope of the secant line
Mark the two points \big(a, f(a)\big) and
\big(b, f(b)\big) on the graph and draw the straight line
through them. That line is called a secant line, and the average rate
of change is its slope — rise over run between the two points.
Drag the sliders to move the two endpoints along the parabola
f(x) = x^2. The bold line is the secant; watch its slope (the
average rate of change) update as you change the interval.
A worked example
Take f(x) = x^2 on the interval from
x = 1 to x = 3. The outputs are
f(1) = 1 and f(3) = 9, so:
\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4
Over this stretch the function rises, on average, 4 units of
output for every 1 unit of input. Notice the average can be
positive, negative, or zero: if the function ends lower than it started, the rise is
negative and so is the rate.
Watch it explained
Sal Khan introduces the average rate of change of a function over an interval, and shows
how it connects to slope.