← College Algebra

College Algebra

Average Rate of Change

Lesson

The average rate of changeof a function over an interval is the slope of the straight line connecting the endpoints. It generalizes “slope” from lines to any function.

The formula

ARC=f(b)−f(a)b−a\text{ARC} = \frac{f(b) - f(a)}{b - a}

Compute f(a)f(a) and f(b)f(b), subtract, then divide by b−ab - a.

Worked example 1

f(x)=x2f(x) = x^2, from x=1x = 1 to x=3x = 3.

f(3)−f(1)3−1=9−12=4\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4

Worked example 2 — interpret as velocity

s(t)=t2+1s(t) = t^2 + 1 in meters. Average velocity from t=2t = 2 to t=5t = 5:

s(5)−s(2)5−2=26−53=7 m/s\frac{s(5) - s(2)}{5 - 2} = \frac{26 - 5}{3} = 7 \text{ m/s}

For a linear function

A line has the SAME slope over every interval. So if f(x)=mx+bf(x) = mx + b, the average rate of change over any interval is just mm.

Real-world cue: ARC is the answer to “how fast on average did y change per unit of x?” — speed, growth rate, profit per unit, etc.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x)=x2; ARC from x=1 to x=3f(x) = x^2; \text{ ARC from } x = 1 \text{ to } x = 3

Problem 2

f(x)=x2; ARC from x=0 to x=2f(x) = x^2; \text{ ARC from } x = 0 \text{ to } x = 2

Problem 3

f(x)=2x+1; ARC from x=1 to x=4f(x) = 2x + 1; \text{ ARC from } x = 1 \text{ to } x = 4

Problem 4

f(x)=x3; ARC from x=1 to x=2f(x) = x^3; \text{ ARC from } x = 1 \text{ to } x = 2

Practice

Standard problems matching the lesson.

Problem 5

f(x)=x2; ARC from −1 to 2f(x) = x^2; \text{ ARC from } -1 \text{ to } 2

Problem 6

f(x)=x2; ARC from 0 to 4f(x) = x^2; \text{ ARC from } 0 \text{ to } 4

Problem 7

f(x)=3x−2; ARC from 0 to 5f(x) = 3x - 2; \text{ ARC from } 0 \text{ to } 5

Problem 8

f(x)=−x2+5; ARC from −1 to 2f(x) = -x^2 + 5; \text{ ARC from } -1 \text{ to } 2

Problem 9

f(x)=x2−3x; ARC from 1 to 4f(x) = x^2 - 3x; \text{ ARC from } 1 \text{ to } 4

Problem 10

f(x)=x3; ARC from −2 to 2f(x) = x^3; \text{ ARC from } -2 \text{ to } 2

Problem 11

f(x)=2x; ARC from 0 to 3f(x) = 2^x; \text{ ARC from } 0 \text{ to } 3

Problem 12

f(x)=x; ARC from 4 to 9f(x) = \sqrt{x}; \text{ ARC from } 4 \text{ to } 9

Problem 13

f(x)=x2; ARC from −3 to 1f(x) = x^2; \text{ ARC from } -3 \text{ to } 1

Problem 14

f(x)=x2+2x; ARC from 0 to 3f(x) = x^2 + 2x; \text{ ARC from } 0 \text{ to } 3

Problem 15

f(x)=1x; ARC from 1 to 2f(x) = \frac{1}{x}; \text{ ARC from } 1 \text{ to } 2

Problem 16

f(x)=x2−4x+1; ARC from 1 to 5f(x) = x^2 - 4x + 1; \text{ ARC from } 1 \text{ to } 5

Problem 17

s(t)=t^2+1; avg v from 2 to 5

Problem 18

P(x)=50x-x^2; ARC 10 to 20

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

f(x)=x3−x; ARC from 1 to 3f(x) = x^3 - x; \text{ ARC from } 1 \text{ to } 3

Problem 20

f(x)=2x2+x; ARC from −2 to 1f(x) = 2x^2 + x; \text{ ARC from } -2 \text{ to } 1

Problem 21

f(x)=x+1; ARC from 3 to 8f(x) = \sqrt{x + 1}; \text{ ARC from } 3 \text{ to } 8

Problem 22

f(x)=3x; ARC from 0 to 2f(x) = 3^x; \text{ ARC from } 0 \text{ to } 2

Problem 23

f(x)=x2−5x+6; ARC from 0 to 6f(x) = x^2 - 5x + 6; \text{ ARC from } 0 \text{ to } 6

Problem 24

f(x)=x3+3; ARC from −1 to 1f(x) = x^3 + 3; \text{ ARC from } -1 \text{ to } 1

Problem 25

T(t)=60+2t^2; ARC 0 to 5

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.