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All courses Math II · F-IF.6 20 of 73
Calculate, estimate, and interpret average rate of change for quadratic functions.

Find the average rate of change from two given points

Problem
Calculate the average rate of change between points (1,3) and (4,15).
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Hint

Use the average-rate-of-change formula \(\frac{y_2-y_1}{x_2-x_1}\) with the two given points.

Average rate of change between two points is the slope of the line through those points.

Solution walkthrough

01

Write the two points

\[(1,3)~\text{and}~(4,15)\]

These give the starting and ending x- and y-values for the rate calculation.

02

Use the slope formula

\[\text{average}~\text{rate}~=~\frac{15-3}{4-1}\]

Subtract the y-values and divide by the change in x-values.

03

Simplify the fraction

\[\frac{12}{3}~=~4\]

The function increases \(12\) units in y over \(3\) units in x.

04

State the average rate of change

\[\text{average}~\text{rate}:~4\]

So the average rate of change between the two points is \(4\).

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Another way

  1. You can count the change directly: from \(x=1\) to \(x=4\) is \(+3\), and from \(y=3\) to \(y=15\) is \(+12\), so \(12/3=4\).

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Common mistake

A common mistake is to reverse only one subtraction order, such as using (3-15)/(4-1). That changes the sign because the y-change and x-change no longer use the same point order.