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All courses Math I · F-IF.6 24 of 59
Calculate, estimate, and interpret average rate of change over an interval.

Calculate average rate of change from two points

Problem
Calculate the average rate of change between (2,5) and (6,17).
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Hint

Use average rate of change as change in y over change in x: \((17-5)/(6-2)\).

Average rate of change between two points is the slope of the secant line through them. Subtract the y-values, then divide by the difference of the x-values.

Solution walkthrough

01

Write the average rate of change formula

\[\text{average}~\text{rate}~\text{of}~\text{change}~=~(y2-y1)/(x2-x1)\]

Between two points, the average rate of change is the slope of the line connecting them. That means change in output divided by change in input.

02

Substitute the coordinates

\[(17-5)/(6-2)\]

Use \((2,5)\) as \((x1,y1)\) and \((6,17)\) as \((x2,y2)\). The change in y is \(17-5\), and the change in x is \(6-2\).

03

Simplify the fraction

\[12/4~=~3\]

The output increases by 12 while the input increases by 4. Dividing gives an average rate of change of 3.

04

State the result

\[\text{average}~\text{rate}~\text{of}~\text{change}~=~3\]

For every 1 unit increase in x over this interval, y increases by 3 units on average.

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Another way

  1. You can think of this as slope between the two points: from \((2,5)\) to \((6,17)\), the rise is \(12\) and the run is \(4\), so the slope is \(12/4=3\).

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Common mistake

A common mistake is to divide by the wrong change in x, such as using \(6\) instead of \(6-2\). Average rate of change uses differences between the coordinates, not the coordinates by themselves.