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All courses Algebra I · F-IF.6 24 of 76
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 change quotient

\[\text{average}~\text{rate}~=~(\text{change}~\text{in}~y)/(\text{change}~\text{in}~x)\]

Average rate of change between two points is the slope of their secant line, so output change must be divided by input change.

02

Substitute in one consistent order

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

Using the second point minus the first point in both numerator and denominator preserves the sourced coordinate pairing.

03

Compute both changes

\[\text{change}~\text{in}~y~=~12;~\text{change}~\text{in}~x~=~4;~\text{rate}~=~12/4~=~3\]

The output rises 12 units while the input rises 4 units, giving 3 output units per input unit.

04

State the average rate

\[3\]

Thus the average rate of change between (2,5) and (6,17) is exactly 3.

+

Another way

  1. From (2,5) to (6,17), count a run of 4 and rise of 12; rise divided by run is 3.

!

Common mistake

Do not divide by 6 alone or subtract coordinates in opposite orders. The denominator is the input change 6-2.

Solution walkthrough video