AP® Precalculus · Unit 1 · Topic 1.3
Rates of Change in Linear Functions
A linear function changes by the same amount for every equal step in x, so its average rate of change is just its slope, no matter which interval you pick.
Average rate of change between two points is rise over run: the change in output divided by the change in input. For a line, that ratio is always the same number, the slope m, because a line is built from constant growth. Every time x increases by some amount h, y increases by exactly mh, no matter where you start. Pick the interval near x = 0 or way out at x = 1000, the rise-over-run calculation collapses to the same m every time, since the rise is always m times the run by definition of the line.
This is really the defining feature of linearity. Any function where equal steps in x give equal jumps in y has to be linear, and any linear function has this property. That is why slope works as a single number describing the whole line instead of something you'd need to recompute piece by piece.
Because the slope never changes, the rate of change of the rate of change is zero for a line. Pushed further, this is the bridge into curved functions next: a quadratic's average rate of change is not constant, but changes at a constant rate itself, which is exactly the idea you check with second differences when comparing linear, quadratic, and higher-degree behavior.
Key terms
- average rate of change
- the change in output divided by the change in input over an interval.
- slope
- the constant ratio of rise to run for a linear function, usually called m.
- constant rate of change
- the property that equal steps in input always produce equal steps in output.
- second difference
- the rate of change of the rate of change, used to detect curvature in a function.
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End of topic.