Key takeaway: The average rate of change over an interval measures how much a function’s output moves per unit of input, calculated as rise over run between the interval’s two endpoints.
Grade 9 – Algebra 1F-IF.B.6Functions and Graphs
Definition of Average Rate of Change
The average rate of change tells how fast a quantity changes, on average, between two inputs. A car trip shows the idea: the average speed from noon to 3 p.m. is the distance covered divided by 3 hours, even if the car sped up and slowed down along the way.
For a function, pick an interval from \(x = a\) to \(x = b\), find the two outputs, subtract, and divide by the width of the interval. On a graph, connect the two points with a straight segment; the steepness of that segment is the answer. For a linear function the average rate of change is the same on every interval, but for a curve it depends on which interval you choose.
Formula and notation
Over the interval \([a, b]\):
\[\frac{f(b) – f(a)}{b – a}\]
It has units of output per unit of input, such as feet per second. A positive result means the function increases on average, and a negative one means it decreases.
Key facts
- Geometrically, it is the steepness of the straight segment joining the interval’s endpoints (a secant).
- For a linear function, the rate is constant and equals the slope.
- Different intervals on a curve usually give different values for the average rate of change.
- A result of zero means the output ended where it started.
Average Rate of Change example
Mia pops a ball straight up from a roof ledge, and its height in feet at time \(t\) seconds follows \(h(t) = -16t^{2} + 80t\). How quickly does the height change, on average, between the 1-second mark and the 3-second mark?
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Heights: \(h(1) = -16 + 80 = 64\) and \(h(3) = -144 + 240 = 96\).
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Change in height: \(96 – 64 = 32\) feet. Change in time: \(3 – 1 = 2\) seconds.
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Divide: \(\frac{32}{2} = 16\).
The ball rises at an average rate of 16 feet per second over that interval.
Common mistake
Watch out
Cleo divides in the wrong order: \(\frac{3 – 1}{96 – 64} = \frac{2}{32} = 0.0625\) and reports 0.0625 feet per second.
Why it is wrong
The output change belongs on top and the input change on the bottom, because the rate is height per second. The correct quotient is \(\frac{32}{2} = 16\). A ball that climbs 32 feet in 2 seconds cannot be moving at a fraction of a foot per second.
Related terms and lessons
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