Scaling Functions Examples in Math

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

Scaling a function multiplies its output by a constant (vertical scaling) or compresses/stretches its input (horizontal scaling), changing amplitude or period without changing the shape.

Vertical scaling stretches or squishes the graph up/down; horizontal scaling stretches or squishes it left/right. Both change the function's measurements without altering its fundamental character.

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: Scaling multiplies a function's output (vertical) or input (horizontal) by a constant, changing its size or period but not its essential shape.

Common stuck point: The procedure for scaling functions is the easy part; the trap is applying the horizontal rule like the vertical one. Asking "Is the function the same shape multiplied by a constant on the output or input (not shifted)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the function the same shape multiplied by a constant on the output or input (not shifted)?

Worked Examples

Example 1

easy

Describe how

g(x)=3f(x)

and

h(x)=\frac{1}{2}f(x)

transform the graph of

f(x)=\sqrt{x}

. Evaluate both at

x=4

Answer

g(4)=6

(stretched);

h(4)=1

(compressed)

First step

g(x)=3\sqrt{x}

: vertical stretch by factor

3

. All

y

-values triple.

g(4)=3\cdot2=6

Full solution

2
$h(x)=\frac{1}{2}\sqrt{x}$ : vertical compression by factor $\frac{1}{2}$ . All $y$ -values halve. $h(4)=\frac{1}{2}\cdot2=1$ .
3
The shape of the graph (concave down, starting at origin) is preserved; only the vertical scale changes.

Multiplying a function by a constant

c

scales it vertically: if

|c|>1

, the graph stretches away from the

x

-axis; if

0<|c|<1

, it compresses toward the

x

-axis. The

x

-intercepts remain unchanged.

Example 2

medium

Explain the difference between

g(x)=f(2x)

(horizontal scaling) and

h(x)=2f(x)

(vertical scaling) for

f(x)=x^2

. Compare at

x=3

Example 3

medium

Compare

g(x) = f(3x)

and

h(x) = 3 f(x)

when

f(x) = \sqrt{x}

. Evaluate both at

x = 9

Example 4

hard

Starting from

f(x) = \cos x

, find

a

and

b

g(x) = a \cos(b x)

so that

g

has amplitude

2.5

and frequency

3

(i.e.,

3

cycles per

2\pi

Practice Problems

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

Example 1

easy

The graph of

f

has a maximum at

(2, 5)

. Where is the maximum of

y=4f(x)

? Of

y=f(3x)

Example 2

hard

Starting from

f(x)=\sin(x)

, write the equation and describe each transformation for

g(x)=3\sin(2x)

. State the amplitude and period of

g

Example 3

easy

f(x)

has the point

(2, 5)

, what point does

2f(x)

have at

x = 2

Example 4

easy

Describe the transformation from

f(x)

3f(x)

Example 5

easy

Describe the transformation from

f(x)

\frac{1}{2} f(x)

Example 6

easy

f(x)

has an x-intercept at

x = 4

(so

f(4) = 0

), where is the x-intercept of

5f(x)

Example 7

easy

Describe the transformation from

f(x)

f(2x)

Example 8

easy

For

f(x) = x^2

, write

2f(x)

explicitly.

Example 9

easy

For

f(x) = x^2

, write

f(3x)

explicitly.

Example 10

easy

A function

g(x) = 4f(x)

. By what factor are all the output values of

f

scaled?

Example 11

medium

A sine wave

y = \sin x

has amplitude 1 and period

2\pi

. Write the function with amplitude 3 and the same period.

Example 12

medium

A sine wave

y = \sin x

has period

2\pi

. Write the function with period

\pi

(and unchanged amplitude).

Example 13

medium

f(x)

has points

(1, 2)

and

(3, 0)

, list the corresponding points of

g(x) = \frac{1}{2}f(x)

Example 14

medium

Write the single function obtained by vertically stretching

f(x) = x^2 + 1

by a factor of 3.

Example 15

medium

Compare

f(2x)

and

2f(x)

for

f(x) = x + 3

. Write each and explain why they differ.

Example 16

medium

A graph of

f(x)

passes through

(4, 6)

. After horizontal compression

f(2x)

, which input now gives output 6?

Example 17

medium

A model

C(x)

gives cost for

x

items. To express cost when each input unit represents a dozen items, you use

C(12x)

. Is this horizontal stretch or compression, and by what factor?

Example 18

challenge

Find a single constant

a

so that

a \cdot f(x)

sends the point

(3, 8)

f

(3, -2)

. Then state what happens to the x-intercepts.

Example 19

challenge

The graph of

f(x) = x^2

is to be transformed so it has the SAME graph whether you apply

f(2x)

cf(x)

. Find

c

Example 20

challenge

A wave is modeled by

y = A\sin(Bx)

. Starting from

\sin x

, find

A

and

B

so the amplitude is 5 and the period is

\frac{2\pi}{3}

Example 21

medium

f(x)

has the point

(6, 9)

, where does it map under

f\left(\tfrac{1}{3}x\right)

(horizontal scaling)?

Example 22

medium

Write the function that vertically compresses

f(x) = 4x^2

by a factor of

\tfrac14

, and identify its leading coefficient.

Example 23

easy

For

f(x) = x^2

, write

5 f(x)

explicitly.

Example 24

easy

For

f(x) = \sqrt{x}

, write

f(4x)

explicitly.

Example 25

easy

f(x)

has a y-intercept at

(0, 7)

, where is the y-intercept of

\tfrac{1}{2} f(x)

Example 26

easy

f(x) = x^3

. Write

f(2x)

and identify the resulting leading coefficient.

Example 27

easy

f

has an x-intercept at

x = 5

, where is the x-intercept of

7 f(x)

Example 28

medium

f(x) = \sin x

has period

2\pi

. What is the period of

f(5x)

Example 29

medium

f(x)

has a maximum at

(4, 6)

. Find the maximum location of

-2 f(x)

Example 30

medium

f(x)

has a zero at

x = -2

. What is the zero of

f(\tfrac{1}{4} x)

Example 31

medium

Write a single

g(x) = A \sin(B x)

with amplitude

4

and period

\pi/2

Example 32

medium

f(x) = 2^x

. Write

f(x - 3)

as a vertical scaling of

f(x)

Example 33

medium

f(x)

passes through

(2, 8)

and

(5, 0)

. Find where the same y-values occur for

f(\tfrac{x}{2})

Example 34

medium

For

f(x) = |x|

, write the function obtained by vertical compression by factor

\tfrac{1}{4}

and reflection across the

x

-axis.

Example 35

hard

f(x) = \sqrt{x}

has graph

G

. Find constants

a, b

so that

a f(b x)

produces a vertical stretch by

6

AND a horizontal stretch by

9

Example 36

hard

Show that for

f(x) = x^n

, the transformation

f(c x)

equals

c^n f(x)

. Use this to convert

f(5x)

into a vertical scaling for

f(x) = x^4

Example 37

hard

f(x) = \sin x

. Write

3 \sin(2x + \pi)

as a sequence of transformations applied to

\sin x

Example 38

hard

f

has domain

[-2, 6]

and range

[0, 9]

. What are the domain and range of

g(x) = 4 f(2x)

Example 39

hard

For

f(x) = e^x

, show that

f(x + \ln 5) = 5 f(x)

, illustrating that for exponentials horizontal shifts equal vertical scalings.

Example 40

challenge

Find a single

g(x) = a f(b x)

that simultaneously sends

f

's point

(6, 4)

(2, 12)

for some function

f

. State

a

and

b

Example 41

challenge

Suppose

f

is a polynomial of degree

n

. Prove that

f(c x) = c^n f(x)

if and only if

f

has the form

f(x) = a x^n

for some constant

a

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Related Concepts

Function Transformation Amplitude

Background Knowledge

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

transformation