Linear Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Linear Functions.

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

Concept Recap

A function whose graph is a straight line, characterized by a constant rate of change between any two points.

Every step right changes $y$ by the same amount—like climbing stairs at a constant pace.

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: A linear function adds the same amount to output for each equal input step.

Common stuck point: The procedure for linear functions is the easy part; the trap is calling every equation with $x$ linear. Asking "Does the output change by the same amount each time the input step is the same?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the output change by the same amount each time the input step is the same?

Worked Examples

Example 1

easy

Write the equation of a line with slope

2

and

y

-intercept

-3

Answer

y = 2x - 3

First step

The slope-intercept form is

y = mx + b

Full solution

2
Substitute $m = 2$ and $b = -3$ .
3
The equation is $y = 2x - 3$ .

In slope-intercept form

y = mx + b

, the parameter

m

is the slope (rate of change) and

b

is the

y

-intercept (the value of

y

when

x = 0

Example 2

medium

Find the equation of the line through

(1, 3)

and

(4, 12)

Example 3

medium

Write the equation of a line with slope

3

that passes through

(1, 7)

Example 4

medium

Find the slope of the line through

(-2, 5)

and

(3, -5)

Example 5

medium

Convert

4x - 5y = 20

to slope-intercept form.

Example 6

medium

A taxi charges $3 to start and $2 per mile. Write a linear function

C(m)

for the total cost after

m

miles.

Example 7

medium

The points

(2, 7)

and

(5, 16)

lie on a line. Write its slope-intercept equation.

Example 8

medium

A linear function satisfies

f(0) = -3

and

f(4) = 5

. Find

f(x)

Example 9

hard

Find the equation of the line through

(-3, 4)

parallel to

y = -2x + 7

Example 10

hard

A linear function

g

satisfies

g(2) = 11

and

g(7) = -4

. Find

g(0)

Example 11

hard

Find the intersection of

y = 2x - 1

and

y = -x + 8

Example 12

challenge

A linear function

f

satisfies

f(f(x)) = 4x + 3

for all

x

, with positive slope. Find

f(x)

Practice Problems

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

Example 1

easy

What are the slope and

y

-intercept of

y = -5x + 7

Example 2

hard

Convert

3x + 2y = 12

to slope-intercept form.

Example 3

easy

What is the slope of

y = 3x + 2

Example 4

easy

What is the

y

-intercept of

y = -2x + 5

Example 5

easy

y = x^2 + 1

a linear function?

Example 6

easy

For

f(x) = 2x - 1

, find

f(3)

Example 7

easy

x

increases by

1

y = 4x

, how does

y

change?

Example 8

easy

Does

y = 7

describe a line?

Example 9

easy

Write a line with slope

2

through the

y

-intercept

-3

Example 10

easy

What is the slope of a vertical line?

Example 11

medium

Find the slope between

(1, 2)

and

(4, 11)

Example 12

medium

Write the equation of the line through

(0, 1)

with slope

-2

Example 13

medium

Find the equation through

(2, 5)

with slope

3

Example 14

medium

Where does

y = 2x - 6

cross the

x

-axis?

Example 15

medium

Are

y = 2x + 1

and

y = 2x - 4

parallel?

Example 16

medium

A line has slope

\frac{1}{2}

. What is the slope of a line perpendicular to it?

Example 17

medium

A plumber charges $50 plus $30/hour. Write the cost function

C(h)

Example 18

challenge

Lines

y = 3x + 1

and

y = -x + 9

intersect where?

Example 19

challenge

A line passes through

(1, 4)

and

(3, 4)

. Describe it fully.

Example 20

challenge

Find

k

so that

(2, k)

lies on

y = 3x - 4

Example 21

medium

Does the point

(3, 5)

lie on

y = 2x - 1

Example 22

medium

Convert

2x + y = 6

to slope-intercept form.

Example 23

easy

What is the slope of the line

y = -4x + 9

Example 24

easy

For

f(x) = -3x + 5

, find

f(4)

Example 25

easy

A line has slope

\tfrac{1}{2}

and

y

-intercept

-4

. Write its equation.

Example 26

easy

Does the point

(2, 1)

lie on the line

y = 3x - 5

Example 27

medium

Write the equation of the line through

(0, -2)

with slope

\tfrac{3}{4}

Example 28

medium

A line passes through

(2, 3)

with slope

-1

. Write its equation in slope-intercept form.

Example 29

medium

For

f(x) = -2x + 7

, find the value of

x

that makes

f(x) = 1

Example 30

medium

A line has slope

-2

and passes through

(3, 4)

. Find its

y

-intercept.

Example 31

medium

For the linear function

y = \tfrac{2}{3}x + 1

, how much does

y

change when

x

increases by

6

Example 32

hard

Find the equation of the line through

(2, -1)

perpendicular to

y = \tfrac{1}{3}x + 4

Example 33

hard

For what value of

k

does the line

y = (k - 1)x + 6

pass through the point

(2, 4)

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Related Concepts

Slope Equations Coordinate Plane Systems of Equations Quadratic Functions

Background Knowledge

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

slopeequationscoordinate plane