Average Rate of Change
In short: Average rate of change tells how much a function’s output changes per unit of input over an interval, computed as the change in y divided by the change in x.
Algebra 1Functions and GraphsF-IF.B.6
What is Average Rate of Change?
A road trip rarely keeps one speed, yet you can still say you averaged 55 miles per hour, and the average rate of change does the same for any function on an interval from \(x = a\) to \(x = b\).
Its formula is \(\frac{f(b) – f(a)}{b – a}\). On a graph, it is the slope of the straight segment, called a secant line, joining the two points. For a linear function it equals the constant slope; for a curve it changes with the chosen interval.
How to spot it
- The question names an interval, such as from 2020 to 2024 or between x = 1 and x = 4.
- Words like average speed, average growth per year or per hour appear.
- A table gives function values at two inputs and asks how fast the output changed.
- A curve is not a straight line and you are asked to compare the steepness over two intervals.
The method
- Identify the interval endpoints a and b and find the two outputs \(f(a)\) and \(f(b)\).
- Subtract the outputs in order: \(f(b) – f(a)\).
- Subtract the inputs in the same order: \(b – a\).
- Divide, and attach units such as dollars per month or feet per second.
Average Rate of Change: solved example
Problem
A town’s population was 12,400 in 2010, 15,100 in 2020 and 15,820 in 2024. Find the average rate of change per year from 2010 to 2020, then from 2020 to 2024.
Solution
From 2010 to 2020: \(\frac{15{,}100 – 12{,}400}{2020 – 2010} = \frac{2{,}700}{10} = 270\) people per year.
From 2020 to 2024: \(\frac{15{,}820 – 15{,}100}{4} = \frac{720}{4} = 180\) people per year. Growth slowed from 270 to 180 people per year.
Try it
Question 1
Find the average rate of change of \(f(x) = x^2 + 1\) from \(x = 2\) to \(x = 5\).
Show the answer
\(f(2) = 5\) and \(f(5) = 26\), so \(\frac{26 – 5}{5 – 2} = \frac{21}{3} = 7\).
Question 2
A car’s odometer reads 3,120 miles at 1:00 p.m. and 3,184 miles at 2:30 p.m. What was the average speed?
Show the answer
\(\frac{64}{1.5} \approx 42.7\) miles per hour.
Average Rate of Change vs. Rate of Change
A rate of change can describe any change per unit, in general. The average version is a specific calculation over a stated interval, blending everything that happened between the two endpoints into one number. It says nothing about the speed at one exact instant inside that interval.
Keep going
Lessons
Worksheets and quizzes
- Slope and Rate of Change – Worksheet
- Slope and Rate of Change – Quiz
- Functions and Function Notation – Worksheet
- Functions and Function Notation – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Functions and Graphs