Rate of Change (Algebraic) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rate of Change (Algebraic).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The ratio of how much one quantity changes to how much another quantity changes — measured over an interval.

Miles per hour, dollars per item, degrees per minute — change per unit.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Average rate of change is $\frac{\Delta y}{\Delta x}$ , the change in output divided by the change in input over an interval.

Common stuck point: The procedure for rate of change (algebraic) is the easy part; the trap is subtracting the points in inconsistent order. Asking "Am I dividing a change in output by a change in input over an interval?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I dividing a change in output by a change in input over an interval?

Worked Examples

Example 1

easy

A car travels 150 miles in 3 hours. What is its average rate of change (speed)?

Answer

50 \text{ mph}

First step

Rate of change =

\frac{\Delta y}{\Delta x} = \frac{\text{distance}}{\text{time}}

Full solution

2
Rate = $\frac{150}{3} = 50$ miles per hour.
3
The car's average speed is 50 mph.

The average rate of change measures how quickly one quantity changes relative to another. For distance vs. time, rate of change is speed.

Example 2

medium

Find the average rate of change of

f(x) = x^2

from

x = 1

x = 4

Example 3

medium

A streaming service charges \$10 plus \$0.05 per minute streamed. What is the rate of change of total cost with respect to minutes?

Example 4

hard

Why can't

f(x)=x^2

have a constant rate of change?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A plant grows from 5 cm to 17 cm in 4 weeks. What is the average growth rate?

Example 2

hard

Find the average rate of change of

g(x) = 2x^2 - 3

from

x = -1

x = 3

Example 3

easy

A line passes through

(1,2)

and

(3,8)

. Find its rate of change.

Example 4

easy

y

increases by 10 when

x

increases by 2, what is the rate of change?

Example 5

easy

A car goes 120 miles in 2 hours. What is its rate (speed)?

Example 6

easy

y=4x+1

, what is the rate of change?

Example 7

easy

Find the rate of change between

(0,3)

and

(2,3)

Example 8

easy

A plant grows from 5 cm to 11 cm in 3 days. What is its growth rate per day?

Example 9

easy

The cost rises by \$6 for every 2 extra items. Find the cost rate per item.

Example 10

easy

Between

(2,5)

and

(2,9)

, what is the rate of change?

Example 11

medium

Find the rate of change of the line through

(-2,4)

and

(3,-6)

Example 12

medium

A tank drains from 50 L to 20 L in 6 minutes. Find the rate of change of volume.

Example 13

medium

For

f(x)=x^2

, find the average rate of change from

x=1

x=4

Example 14

medium

A line has rate of change 3 and passes through

(2,5)

. Find

y

when

x=6

Example 15

medium

Two points give

\frac{\Delta y}{\Delta x}=\frac{8}{-4}

. Simplify the rate and interpret its sign.

Example 16

medium

A phone plan costs \$20 plus \$0.10 per minute. What is the rate of change of cost with respect to minutes?

Example 17

challenge

For

f(x)=x^2

, find the average rate of change from

x=a

x=a+h

and simplify.

Example 18

challenge

A car goes 30 mph for 1 hour then 60 mph for 2 hours. Find the average rate of change of distance over the whole trip.

Example 19

challenge

A line through

(1,k)

and

(4,10)

has rate of change 2. Find

k

Example 20

medium

Find the rate of change of the line through

(0,1)

and

(4,9)

Example 21

medium

A savings account grows from \$200 to \$260 in 3 months. Find the monthly rate of change.

Example 22

medium

A line has rate of change

-3

through

(1,7)

. Find

y

x=4

Example 23

easy

Find the rate of change of the line through

(0,4)

and

(5,14)

Example 24

easy

y=-2x+9

, what is the rate of change?

Example 25

easy

A water tank gains 24 gallons in 8 minutes. Find the rate of change.

Example 26

easy

Find the rate of change of the line through

(-1,5)

and

(3,5)

Example 27

easy

A bike rider covers 15 km in 1.5 hours. Average rate of change of distance?

Example 28

easy

y=\dfrac{1}{3}x-2

, what is the rate of change?

Example 29

medium

Find the average rate of change of

f(x)=x^2+x

from

x=2

x=5

Example 30

medium

A tank goes from 80 L to 25 L in 11 minutes. Find the rate of change.

Example 31

medium

Find the average rate of change of

f(x)=2^x

from

x=1

x=4

Example 32

medium

A line passes through

(3,10)

and

(7,k)

with rate of change 2. Find

k

Example 33

medium

A college's tuition rose from \$8{,}000 to \$11{,}600 over 4 years. Find the average yearly rate of change.

Example 34

medium

A line has rate of change

\dfrac{2}{3}

and passes through

(3,1)

. Find

y

when

x=9

Example 35

medium

Find the average rate of change of

f(x)=\dfrac{1}{x}

from

x=1

x=4

Example 36

medium

Find the average rate of change of

f(x)=x^3

from

x=0

x=2

Example 37

medium

A line passes through

(a,3)

and

(a+5,18)

. Find the rate of change.

Example 38

hard

Find the average rate of change of

f(x)=x^2+3x

from

x=a

x=a+h

, in simplest form.

Example 39

hard

A car travels at 30 mph for 2 hours, then 60 mph for 1 hour. Find the average rate of change of distance over the full trip.

Example 40

hard

Find the average rate of change of

f(x)=\sqrt{x}

from

x=4

x=9

Example 41

hard

A population grows from 5000 to 8000 in 6 years. Express the average growth rate as a percent per year.

Example 42

hard

A line through

(2,5)

and

(6,k)

has rate of change

-3

. Find

k

Example 43

hard

Find the average rate of change of

f(x)=x^2-4x+1

[-1,3]

Example 44

challenge

On what interval

[1,b]

does

f(x)=x^2

have average rate of change equal to

7

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Related Concepts

Slope Derivative

Background Knowledge

These ideas may be useful before you work through the harder examples.

slope