Average Rate of Change: Definition, Formula & Examples

Definitionnoun phrase

What is the average rate of change of a function?

The average rate of change of a function over an interval is the change in output divided by the change in input: f(b) − f(a)b − a. It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.

In Spanish: tasa de cambio promedio
Also called: average rate, slope of the secant line, difference quotient
Grade level: Algebra 1

Algebra 1Algebra 2PrecalculusCCSS HSF-IF.B.6SATACTPSATAP

Average Rate of Change in a picture

On [1, 3], y = x² rises 8 while x moves 2, so its average rate of change is 4: the slope of the secant line.

Formal definition and notation

For a function f defined on an interval containing a and b, with a ≠ b, the average rate of change of f from a to b is f(b) − f(a)b − a. It equals the slope of the secant line through (a, f(a)) and (b, f(b)), and its units are output units per input unit. For a linear function it is the same on every interval.

Symbol	Read it as	Meaning
Δ yΔ x	delta y over delta x	change in output divided by change in input
f(b) − f(a)	f of b minus f of a	how much the output changes over the interval
[a, b]	the interval from a to b	the inputs where you start and stop

Key facts

Only the two endpoints matter. What the function does in between does not change the average rate.
A positive average rate means the output went up over the interval; negative means it went down; 0 means it ended where it started.
For a line, the average rate of change on any interval is the slope. For a curve, it changes with the interval.
In a table, pick the two rows for the interval, subtract the outputs, subtract the inputs, and divide.
As the interval shrinks toward one point, the average rate of change approaches the instantaneous rate, which calculus calls the derivative.

Average rate of change formula

average rate of change = f(b) − f(a)b − a

a, b: the starting and ending inputs of the interval
f(a), f(b): the outputs at those inputs (read from a rule, table or graph)
units: output units per input unit, such as miles per hour or dollars per year

Subtract in the same order on top and bottom. Starting with b on top means starting with b on the bottom.

Average Rate of Change examples, step by step

Example 1: From a table

Warm-up

A plant is 6 cm tall on day 2 and 18 cm tall on day 8. What is its average rate of growth over that time?

Change in height: 18 − 6 = 12 cm.
Change in time: 8 − 2 = 6 days.
Divide: 126 = 2.

Answer: The plant grew at an average of 2 cm per day.

Example 2: From a function rule

Core

Find the average rate of change of f(x) = x² − 4x from x = −1 to x = 2.

Evaluate the endpoints: f(−1) = 1 + 4 = 5 and f(2) = 4 − 8 = −4.
Change in output: −4 − 5 = −9.
Change in input: 2 − (−1) = 3.
Divide: −93 = −3.

Answer: The average rate of change is -3: the output falls 3 units per unit of x, on average.

Example 3: Compare two intervals

Test-style

A ball’s height is h(t) = −5t² + 20t meters after t seconds. What is its average speed of climb from t = 0 to t = 1?

h(0) = 0 and h(1) = −5 + 20 = 15.
Change in height: 15 − 0 = 15 m. Change in time: 1 s.
Average rate: 151 = 15 m per second.
Compare [1, 2]: h(2) = 20, so 20 − 151 = 5. The ball slows as it rises.

Answer: 15 meters per second (and only 5 meters per second from t = 1 to t = 2).

Common mistakes

✗ MistakeFor (2, 7) and (5, 1): 5 − 21 − 7 = −12.

✓ Instead1 − 75 − 2 = −63 = −2.

Output change goes on top and input change on the bottom. Flipping them gives the reciprocal.

✗ MistakeFor f(x) = 3x² on [1, 4]: 4 − 13 = 1, using the inputs only.

✓ Insteadf(1) = 3, f(4) = 48, so 48 − 34 − 1 = 15.

You need the outputs f(a) and f(b), not just a and b. Evaluate the function at both endpoints first.

✗ Mistakef(5) − f(2)2 − 5, mixing the order.

✓ Insteadf(5) − f(2)5 − 2.

Subtracting in opposite orders flips the sign of the answer. Keep the same point first on top and bottom.

Not to be confused with

Average Rate of Changevs.Rate of change

Rate of change is the general idea; for a line it is constant. Average rate of change is the version for any function over a chosen interval, such as y = x² on [1, 3].

Average Rate of Changevs.Average value

The average of the outputs, f(a) + f(b)2, is a height. The average rate of change, f(b) − f(a)b − a, is a slope.

Average Rate of Changevs.Instantaneous rate of change

That is the rate at a single point (the derivative). The average rate uses two points: on [1, 3] y = x² averages 4, while at exactly x = 3 its rate is 6.

Quick check: 3 questions

Question 1 of 3

g(x) = 2x². What is the average rate of change of g from x = 1 to x = 3?

Show answer and explanation

Answer: A. g(1) = 2 and g(3) = 18, so 18 − 23 − 1 = 162 = 8. 16 forgot to divide by the change in x, 18 is upside down, and 4 used x² without the 2.

Question 2 of 3

A car’s odometer reads 120 miles at 1 p.m. and 270 miles at 4 p.m. What is its average rate of change in distance?

Show answer and explanation

Answer: C. 270 − 1204 − 1 = 1503 = 50 miles per hour. 90 and 67.5 divided a single reading by the hours, and 150 is the distance change without dividing by time.

Question 3 of 3

A function has outputs f(0) = 10, f(2) = 4 and f(6) = 10. What is its average rate of change on [0, 6]?

Show answer and explanation

Answer: D. Only the endpoints count: 10 − 106 − 0 = 0. The dip to 4 in between does not matter. -3 is the rate on [0, 2], and the others mixed up outputs and inputs.

Average Rate of Change: frequently asked questions

Is average rate of change the same as slope?

It is the slope of the secant line through the two endpoints. For a linear function that slope is the same everywhere, so the two words mean the same thing. For a curve, the slope keeps changing, and the average rate of change only describes the interval you picked, as if the function had moved in a straight line between the endpoints.

What are the units of average rate of change?

Output units per input unit. If height is in feet and time in seconds, the rate is in feet per second. If cost is in dollars and items are counted, it is dollars per item. Writing the units helps you check that the subtraction on top uses outputs and the subtraction on the bottom uses inputs.

Can the average rate of change be negative or zero?

Yes. Negative means the output was lower at the end of the interval than at the start, so the function decreased overall. Zero means the output ended at the same value it started with, even if it rose and fell in between. A ball thrown up and caught at the same height has an average rate of 0.

Practice and tools

Average Rate of Change CalculatorEnter a function and an interval to see each step Difference Quotient CalculatorThe algebraic form of average rate of change

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.B.6

Cite this entry

Effortless Math Team. “Average Rate of Change.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/average-rate-of-change/

Flashcard

TermAverage Rate of Changef(b) − f(a)b − a

DefinitionChange in output over change in input on an interval. y = x² from 1 to 3: 9 − 12 = 4.

Study the full deck

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Average Rate of Change

What is the average rate of change of a function?

Average Rate of Change in a picture

Formal definition and notation

Key facts

Average rate of change formula

Average Rate of Change examples, step by step

Example 1: From a table

Example 2: From a function rule

Example 3: Compare two intervals

Common mistakes

Not to be confused with

Quick check: 3 questions

Average Rate of Change: frequently asked questions

Is average rate of change the same as slope?

What are the units of average rate of change?

Can the average rate of change be negative or zero?

Related terms

Practice and tools

Sources

Flashcard

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