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.

Try it

Average Rate of Change Calculator

Open the full calculator →

Type the problems from this lesson and check each step.

Tutor-style math help

Find Average Rate of Change of a Function: what to notice and how to work it

Functions skill

A function is a rule that gives each input exactly one output. Function notation, tables, graphs, and equations are different ways to show the same input-output relationship.

What to notice first

Ask what kind of input you are given. Sometimes you substitute a number, sometimes you read a graph, and sometimes you combine two rules.

Common student mistake

Do not read \(f(4)\) as multiplication. It means the output of f when the input is 4.

Key formulas and cues

\(f(a)\text{ means replace }x\text{ with }a\)

\((f\circ g)(x)=f(g(x))\)

\(f^{-1}(x)\text{ reverses }f(x)\)

A reliable path

Identify the inputFind the x-value, expression, or inner function being used.
Apply the ruleSubstitute with parentheses so signs and powers stay clear.
Interpret the outputState the value, point, interval, domain, range, or inverse relationship.

Worked examples

Evaluate a function

Example: \(f(x)=4x-3\), find \(f(2)\)

Replace x with 2.
Compute 4(2) – 3.
Simplify.

Answer: \(5\)

Compose functions

Example: \(f(x)=x+1\), \(g(x)=2x\), find \(f(g(3))\)

Find g(3) = 6.
Use that as the input for f.
f(6) = 7.

Answer: \(7\)

Open pop-up practice

Try one before moving on

Try: If \(h(x)=2x^2\), find \(h(-3)\).

Answer: \(18\). Use parentheses: \(2(-3)^2=18\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

What Is the Average Rate of Change?

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

Algebra I for Beginners 2027 The Ultimate Step by Step Guide to Acing Algebra I

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

Write down the function f(x) and the interval [a, b].
Compute f(a): substitute \(\color{blue}{x = a}\) into the formula.
Compute f(b): substitute \(\color{blue}{x = b}\) into the formula.
Plug into \(\color{blue}{\text{ ARC } = \frac{(f(b) – f(a))}{(b – a)}}\).
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. &checkmark;)

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.

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

Answers

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

The Ultimate Algebra Bundle 2027 From Pre-Algebra to Algebra II

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.

Download Slope and Rate of Change Worksheet

Frequently Asked Questions

How do you find the average rate of change of a function over an interval?

Use the formula [f(b) − f(a)] ÷ (b − a) for the interval from x = a to x = b. Find both outputs, subtract them, then divide by the change in x. For f(x) = x² + 1 on [1, 4], f(4) = 17 and f(1) = 2, so the average rate of change is (17 − 2) ÷ 3 = 5.

How do you find the average rate of change of a linear function?

For a linear function, the average rate of change is simply the slope, and it is the same on every interval. For f(x) = 3x − 7, the rate is 3 whether you use [0, 2] or [5, 10]. You can confirm it with the formula: f(2) − f(0) = −1 − (−7) = 6, and 6 ÷ 2 = 3.

How do you find the average rate of change for a quadratic function?

Use the same formula, but expect a different answer for each interval, because a parabola’s steepness keeps changing. For f(x) = x² on [−1, 3], f(3) = 9 and f(−1) = 1, so the rate is (9 − 1) ÷ 4 = 2. On [3, 5] it would be (25 − 9) ÷ 2 = 8.

How do you find the average rate of change for exponential functions?

Plug the interval’s endpoints into the function and use [f(b) − f(a)] ÷ (b − a). For f(x) = 2ˣ on [0, 3], f(3) = 8 and f(0) = 1, so the rate is 7 ÷ 3 ≈ 2.33. Exponential functions grow faster and faster, so later intervals of the same width give larger rates.

How do you find the average rate of change of a function from a graph?

Read the two points on the graph at the ends of the interval, then find rise over run between them. If a graph passes through (1, 4) and (5, 12), the average rate of change is (12 − 4) ÷ (5 − 1) = 2. That value is the slope of the straight secant line joining the two points.

How do you find average rate of change without a function?

Use any two data points from a table or a word problem. Subtract the starting value from the ending value, then divide by the change in time or input. If the temperature is 52°F at 8 a.m. and 70°F at 2 p.m., the average rate of change is 18 ÷ 6 = 3°F per hour. Always include units.

Related to This Article

Average Functions Rate

What people say about "How to Find Average Rate of Change of a Function"?

No one replied yet.

How to Find Average Rate of Change of a Function?