Practice Scaling Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

easy

For

f(x) = x^2

, write

5 f(x)

explicitly.

Example 3

medium

A sine wave

y = \sin x

has period

2\pi

. Write the function with period

\pi

(and unchanged amplitude).

Example 4

medium

f(x)

has the point

(6, 9)

, where does it map under

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

(horizontal scaling)?

Example 5

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 6

medium

Does the transformation

f(x) \to f(-x)

change the y-values of points on the graph?

Example 7

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 8

medium

For

f(x) = |x|

, write the function obtained by vertical compression by factor

\tfrac{1}{4}

and reflection across the

x

-axis.

Example 9

medium

Write the function that vertically compresses

f(x) = 4x^2

by a factor of

\tfrac14

, and identify its leading coefficient.

Example 10

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 11

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

Example 12

easy

Write the function that vertically reflects

f(x) = x^2 + 1

Example 13

medium

f(x)

has a zero at

x = -2

. What is the zero of

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

Example 14

hard

g(x) = c f(x)

shares all of

f

's x-intercepts, what can

c

be?

Example 15

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 16

easy

f(x)

has a y-intercept at

(0, 7)

, where is the y-intercept of

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

Example 17

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

Example 18

easy

Describe the transformation from

f(x)

3f(x)

Example 19

easy

For

f(x) = x^2

, write

2f(x)

explicitly.

Example 20

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?

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Related Concepts

Function Transformation Amplitude