How to Find Average Rate of Change of a Function?

The average rate of change of a function measures how much the output changes compared to the input over a given interval. It is the slope of the secant line connecting two points on the graph and is one of the most important concepts bridging Algebra 1 and calculus. Once you know the formula, any average rate of change problem reduces to a two-step substitution and subtraction.

What Is the Average Rate of Change?

The average rate of change of a function f over the interval [a, b] is defined as:

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Average Rate of \(\color{blue}{\text{ Change } = \frac{(f(b) – f(a))}{(b – a)}}\)

This formula gives you the change in output (the “rise”) divided by the change in input (the “run”) — exactly the slope formula you already know from linear functions. For a linear function the average rate of change equals the slope everywhere; for a non-linear function it depends on the interval chosen.

How to Find the Average Rate of Change

Step 1 — Identify the interval [a, b]

The interval is given in the problem, for example [1, 3] or from \(\color{blue}{x = 2}\) to \(\color{blue}{x = 5}\).

Step 2 — Evaluate f(a) and f(b)

Substitute each endpoint into the function.

Step 3 — Apply the formula

Compute \(\color{blue}{\frac{(f(b) – f(a))}{(b – a)}}\) and simplify.

Example

Find the average rate of change of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 2x}\) on [1, 3].

\(\color{blue}{f(1) = 1 + 2 = 3}\);  \(\color{blue}{f(3) = 9 + 6 = 15}\)
\(\color{blue}{\text{ ARC } = \frac{(15 – 3)}{(3 – 1)} = \frac{12}{2}}\) = 6

Step-by-Step Summary

1. Write down the function f(x) and the interval [a, b].
2. Compute f(a): substitute \(\color{blue}{x = a}\) into the formula.
3. Compute f(b): substitute \(\color{blue}{x = b}\) into the formula.
4. Plug into \(\color{blue}{\text{ ARC } = \frac{(f(b) – f(a))}{(b – a)}}\).
5. Simplify the fraction completely.

Watch: Introduction to Average Rate of Change (Video Lesson)

Khan Academy explains the definition and formula for average rate of change of a function with clear visual examples:

Average Rate of Change – Worked Examples

Example 1: Find the average rate of change of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 2x}\) on [1, 3].

\(\color{blue}{f(1) = 1 + 2 = 3}\);  \(\color{blue}{f(3) = 9 + 6 = 15}\)
\(\color{blue}{\text{ ARC } = \frac{(15 – 3)}{(3 – 1)} = \frac{12}{2}}\) = 6

Example 2: Find the average rate of change of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 2x}\) on [1, 4].

\(\color{blue}{f(1) = 3}\);  \(\color{blue}{f(4) = 16 + 8 = 24}\)
\(\color{blue}{\text{ ARC } = \frac{(24 – 3)}{(4 – 1)} = \frac{21}{3}}\) = 7

Example 3: Find the average rate of change of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{3} – x}\) on [1, 4].

\(\color{blue}{f(1) = 1 – 1 = 0}\);  \(\color{blue}{f(4) = 64 – 4 = 60}\)
\(\color{blue}{\text{ ARC } = \frac{(60 – 0)}{(4 – 1)} = \frac{60}{3}}\) = 20

Example 4: Find the average rate of change of \(\color{blue}{f(x) = 2x + 3}\) on [1, 4].

\(\color{blue}{f(1) = 5}\);  \(\color{blue}{f(4) = 11}\)
\(\color{blue}{\text{ ARC } = \frac{(11 – 5)}{(4 – 1)} = \frac{6}{3}}\) = 2
(For a linear function, the ARC equals the slope, which is 2 here. ✓)

More Practice: Average Rate of Change Step-by-Step (Video Lesson)

eMATHinstruction provides a thorough Algebra 1 lesson on average rate of change with additional practice problems:

Exercises for Average Rate of Change

Find the average rate of change of each function on the given interval.

1. \(\color{blue}{f(x) = x}\)² on [2, 5]
2. \(\color{blue}{f(x) = 2x + 3}\) on [1, 4]
3. \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} – 3x}\) on [0, 4]
4. f(x) = −\(\color{blue}{x^{2} + 5}\) on [−1, 2]
5. \(\color{blue}{f(x) = x}\)³ on [1, 3]

Answers

1. \(\color{blue}{f(2)=4}\), \(\color{blue}{f(5)=25}\); \(\color{blue}{\text{ ARC } = \frac{(25-4)}{3}}\) = 7
2. \(\color{blue}{f(1)=5}\), \(\color{blue}{f(4)=11}\); \(\color{blue}{\text{ ARC } = \frac{(11-5)}{3}}\) = 2
3. \(\color{blue}{f(0)=0}\), \(\color{blue}{f(4)=16-12=4}\); \(\color{blue}{\text{ ARC } = \frac{(4-0)}{4}}\) = 1
4. \(\color{blue}{f(-1)=4}\), \(\color{blue}{f(2)=1}\); \(\color{blue}{\text{ ARC } = \frac{(1-4)}{3}}\) = −\(\color{blue}{\frac{3}{3}}\) = −1
5. \(\color{blue}{f(1)=1}\), \(\color{blue}{f(3)=27}\); \(\color{blue}{\text{ ARC } = \frac{(27-1)}{2}}\) = 13

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Free Average Rate of Change of a Function Worksheet

Ready to practice on your own? Download our free Average Rate of Change of a Function worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Average Rate of Change of a Function before a quiz or test.

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Frequently Asked Questions

What is the difference between average rate of change and slope?

For a linear function they are the same thing. For a non-linear function, the average rate of change over an interval is the slope of the secant line connecting the two endpoints, not the slope of the curve itself (which changes at every point).

Can the average rate of change be negative?

Yes. A negative average rate of change means the function is, on average, decreasing over that interval (the output goes down as the input goes up).

How is average rate of change related to calculus?

When you shrink the interval [a, b] until b approaches a, the average rate of change approaches the instantaneous rate of change — the derivative studied in calculus. Algebra 1 uses the average (secant-line) version as an introduction to that idea.

Related Topics

• Function Notation
• Finding Domain and Range of a Function
• Increasing and Decreasing Functions
• Maxima and Minima of a Function
• Slope