Average Rate of Change

Average rate of change measures output change per input change across an interval and equals the slope of a secant line.

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Slope compares vertical change with horizontal change. A line contains two marked points joined by a right-angled rise-and-run triangle. The vertical and horizontal legs make the signed ratio behind slope visible.

Average rate of change answers: “Across this interval, how much did the output change per one unit of input?”

Formula

For a function $f$ on the interval from $x=a$ to $x=b$,

\text{AROC}=\frac{f(b)-f(a)}{b-a},\qquad a\ne b.

This is change in output divided by change in input. The order must match in numerator and denominator.

Secant-line meaning

The points $(a,f(a))$ and $(b,f(b))$ lie on the graph. The line through them is a secant line, and its slope is the average rate of change.

The function may curve between the endpoints; AROC summarizes the net change with one constant rate.

Worked example from a formula

The rate does not claim the function's slope equals $3$ at every point.

Units

Units are output units per input unit. If distance is in kilometres and time in hours, average rate of change is kilometres per hour. If cost is dollars and quantity is items, it is dollars per item.

Units help expose a reversed fraction and give the number a contextual meaning.

Application example

A car's position changes from $s(2)=35$ km to $s(5)=245$ km. Its average velocity is

\frac{245-35}{5-2}=70\text{ km/h}.

The car need not have travelled at exactly $70$ km/h at every instant; this is a net interval rate.

Common mistakes

Dividing outputs instead of subtracting. Rate of change uses differences.

Reversing only one subtraction. Keep endpoint order consistent.

Using $f(b-a)$. Evaluate $f(a)$ and $f(b)$ separately.

Calling AROC the instantaneous slope. It belongs to an interval and secant line.

Omitting units or interval. Both are part of a meaningful rate.