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Math101learn.math101.caAverage Rate of Change
Average rate of change measures output change per input change across an interval and equals the slope of a secant line.
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 $x=b$,
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.
Reading from a table
Choose the rows containing the interval endpoints and compute
If exact endpoint values are absent, interpolation would add an assumption. State when a value is estimated rather than given.
For unevenly spaced inputs, still divide by the actual input change; do not merely subtract consecutive outputs.
Reading from a graph
Read the two endpoint coordinates, draw or imagine the secant, and calculate its rise over run. Graph values may be approximate, so report suitable precision.
A positive AROC means the endpoint output increased overall; a negative AROC means it decreased; zero means equal endpoint outputs, even if the graph moved in between.
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.
Linear versus nonlinear functions
A linear function has the same AROC on every interval; that constant is its slope. A nonlinear function usually has different average rates over different intervals.
Comparing interval rates reveals where growth is speeding up, slowing down, or changing direction.
Difference quotient
From $x=a$ to $x=a+h$, the average rate is
This difference quotient is the bridge to instantaneous rate of change. As $h$ approaches zero, the secant line approaches a tangent line when the limit exists.
Application example
A car's position changes from $s(2)=35$ km to $s(5)=245$ km. Its average velocity is
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.
Quick self-check
- What are the two endpoint inputs and outputs?
- Are numerator and denominator orders consistent?
- Does the sign match the graph's net endpoint change?
- What are the output-per-input units?
- Is this an interval average rather than a tangent rate?
Related topics
Explore the idea
Tangent and accumulation explorer
Change one quantity at a time and connect what moves to Average Rate of Change.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the average rate of change of f(x) = x² − 3x + 2 on [1, 5].
- f(1) = 0 and f(5) = 12.
- AROC = (12 − 0)/(5 − 1)
- AROC = 3.
End of lesson
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