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All courses Algebra I · F-IF.6 66 of 76
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

Use the average-rate formula

\[\text{average}~\text{rate}=(\text{change}~\text{in}~y)/(\text{change}~\text{in}~x)=(y_2-y_1)/(x_2-x_1)\]

Average rate of change between two points is the slope of their secant line: output change divided by input change.

02

Source the coordinate changes

\[\begin{aligned} (x_1,y_1)=(1,3) \\ (x_2,y_2)=(4,15) \\ \text{change}~\text{in}~y=15-3=12 \\ \text{change}~\text{in}~x=4-1=3 \end{aligned}\]

Subtract coordinates in the same point order. The output rises by 12 while the input moves right by 3.

03

Form and simplify the quotient

\[\text{average}~\text{rate}=(15-3)/(4-1)=12/3=4\]

Dividing the output change by the nonzero input change gives the signed rate 4.

04

Check direction and answer

\[\begin{aligned} \text{positive}~\text{rate}~4 \\ \text{check}:~\text{rise}=4(\text{run})=4(3)=12 \end{aligned}\]

The rate is positive because both changes are positive. Multiplying rate 4 by the run 3 reproduces the rise 12, confirming the result.

+

Another way

  1. Use slope language directly: rise 15-3=12 and run 4-1=3, so rise/run=4.

!

Common mistake

Do not divide change in x by change in y. Average rate uses output change over input change, so the quotient is 12/3, not 3/12.

Solution walkthrough video