Proportional Function Examples: 44 Problems with Answers

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

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 proportional function has the form $f(x) = kx$ for a constant $k \neq 0$ — it passes through the origin and the ratio $f(x)/x = k$ is constant.

Double the input, double the output. No offset—starts at zero.

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 proportional function multiplies the input by a fixed constant and nothing else, so output and input keep a constant ratio.

Common stuck point: The procedure for proportional function is the easy part; the trap is calling any straight line proportional. Asking "Does input $0$ give output $0$ , and is the ratio $y/x$ the same for every point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does input $0$ give output $0$ , and is the ratio $y/x$ the same for every point?

Worked Examples

Example 1

easy

The weight of water is proportional to its volume.

5

liters weighs

5

kg. Write the function, find the constant of proportionality

k

, and compute the weight of

8.5

liters.

Answer

W(V) = V

;

k = 1

kg/L;

W(8.5) = 8.5

First step

Proportional function:

W(V) = kV

Full solution

2
Find $k$ : $W(5) = k \cdot 5 = 5 \Rightarrow k = 1$ kg/L.
3
Compute: $W(8.5) = 1 \times 8.5 = 8.5$ kg.

Direct proportionality

y=kx

means the ratio

y/x

is constant. Here the density of water is

1

kg/L, making it a clean example where

k=1

Example 2

medium

Determine whether

y

is proportional to

x

given the table:

x: 2, 4, 6

and

y: 7, 14, 21

. If yes, find

k

and the equation.

Example 3

medium

A printer prints

24

pages in

3

minutes at a constant rate. Write the proportional function relating pages

p

to minutes

t

, and find pages printed in

11

minutes.

Example 4

medium

Hooke's Law: spring force is proportional to extension. A spring stretches

4

cm under

20

N. Find

k

in N/cm and the extension under

35

Example 5

hard

f

is proportional and

f(2) + f(5) = 49

, find

f(10)

Example 6

hard

A worker is paid $22.50 per hour proportionally to hours worked. Write the pay function

P(h)

and find the hours needed to earn $315.

Practice Problems

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

Example 1

easy

Hooke's Law states spring force is proportional to stretch:

F = kx

. If

F = 30

N when

x = 6

cm, find

k

and the force when

x = 10

cm.

Example 2

medium

Explain why

f(x) = 3x + 2

is NOT a proportional function, and find the value of

b

that would make

g(x) = 3x + b

proportional.

Example 3

easy

y=3x

a proportional function?

Example 4

easy

y=2x+5

proportional?

Example 5

easy

For

y=4x

, what is the constant of proportionality?

Example 6

easy

y=kx

and

y=10

when

x=2

, find

k

Example 7

easy

Must a proportional function pass through

(0,0)

Example 8

easy

Table

(2,6),(3,9),(4,12)

. Is

y/x

constant?

Example 9

easy

y=x

proportional?

Example 10

easy

Double the input of

y=5x

. What happens to the output?

Example 11

medium

A recipe uses

3

cups of flour for

12

cookies. Write the proportional function and find flour for

20

cookies.

Example 12

medium

(1,2),(2,5),(3,8)

proportional?

Example 13

medium

A car travels

150

miles on

5

gallons. Assuming proportionality, how far on

8

gallons?

Example 14

medium

y

is proportional to

x

and

y=21

when

x=7

, find

y

when

x=10

Example 15

medium

Which graph is proportional: line through

(0,0)

and

(2,4)

, or through

(0,3)

and

(2,7)

Example 16

medium

A spring stretches

2

cm per

5

N, proportionally. How much force for a

6

cm stretch?

Example 17

medium

Why is

y=2x+5

called linear but not proportional?

Example 18

medium

For a proportional function, what is the ratio

y/x

at every point?

Example 19

challenge

Suppose

y

is proportional to

x

with

y = kx

, and the input

x = 4

is tripled to

x = 12

. If this increases

y

24

, find

k

Example 20

challenge

Two quantities satisfy

y=kx

. Doubling

x

and the formula's

k

both. What is the net effect on

y

Example 21

challenge

Gas pressure is proportional to temperature:

P=kT

. At

T=300

P=2

atm. Find

P

T=450

Example 22

medium

A worker earns

\$90

for

6

hours. Assuming proportionality, pay for

10

hours?

Example 23

easy

For the proportional function

y = 6x

, find

y

when

x = 4

Example 24

easy

f(x) = kx

passes through

(3, 18)

, find

k

Example 25

easy

A table has

x: 1, 2, 3

and

y: 4, 8, 12

. Is

y

proportional to

x

Example 26

easy

For

y = \tfrac{1}{2}x

, find

y

when

x = 10

Example 27

medium

Determine whether

(2, 5), (4, 11), (6, 17)

lie on a proportional function.

Example 28

medium

y = kx

, doubling the input causes the output to change by what factor?

Example 29

medium

A graph of

y = kx

passes through

(-3, 12)

. Find

k

and write the equation.

Example 30

medium

A car uses

5

liters of fuel to travel

80

km at constant efficiency. Write the proportional model for fuel

f

in terms of distance

d

, and find fuel for

200

km.

Example 31

medium

Which of the following are proportional: (a)

y = 7x

, (b)

y = x + 3

, (c)

y = -x

, (d)

y = x^2

Example 32

hard

Suppose

y

is proportional to

x

. When

x

increases by

6

y

increases by

15

. Find

k

and

y

when

x = 8

Example 33

hard

f(x) = kx

and

g(x) = mx

are both proportional, is

h(x) = f(x) + g(x)

proportional? If so, find its constant.

Example 34

hard

Gas pressure

P

is proportional to temperature

T

(in kelvin) at constant volume. If

P = 1.2

atm at

T = 300

K, find

T

when

P = 1.8

atm.

Example 35

hard

For

y = kx

, the point

(t, 12)

lies on the graph and the point

(t+1, 15)

also lies on the graph. Find

k

and

t

Example 36

medium

Write the proportional function whose graph passes through

(5, -20)

and find

y

when

x = -2

Example 37

challenge

Suppose

y

is proportional to

x

and

z

is proportional to

y

, with constants

k_1

and

k_2

. Express

z

as a proportional function of

x

, and state the new constant.

Example 38

challenge

Let

f(x) = kx

be proportional. Show that

f(a) + f(b) = f(a + b)

for all real

a, b

, and explain why this fails for

f(x) = kx + c

with

c \neq 0

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Related Concepts

Linear Functions Proportional Relationships Direct Variation Constant of Proportionality

Background Knowledge

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

linear functionsproportionality