Practice Function Transformation in Math

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

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Quick Recap

A function transformation shifts, stretches, compresses, or reflects the graph of a parent function by modifying its formula in a systematic way.

Moving or reshaping a graph without changing its basic shape.

Showing a random 20 of 50 problems.

Example 1

medium

Write

g(x) = \sqrt{4x - 8} + 2

in the form

a\sqrt{b(x - h)} + k

and identify each transformation from

f(x) = \sqrt{x}

Example 2

easy

Starting from

f(x)=x^2

, what is the vertex of

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

Example 3

medium

The point

(2,5)

is on

y=f(x)

. Where does it move on

y=f(x)+3

Example 4

hard

The graph of

y = f(x)

has domain

[-2, 6]

. What is the domain of

y = f(2x - 4)

Example 5

hard

Given

f(x) = \sqrt{x}

, find a single function

g(x)

obtained by reflecting

f

over the line

y = x

. State the result.

Example 6

easy

From

f(x) = x^2

, where is the vertex of

g(x) = (x+5)^2 - 1

Example 7

easy

Describe the transformation from

f(x)

f(x-2)

Example 8

easy

The graph of

y = f(x)

passes through

(1, 4)

. Where does the transformed graph

y = f(x+2) - 3

pass through?

Example 9

easy

Describe the transformation from

f(x)

2f(x)

Example 10

medium

From

f(x)=\sqrt{x}

, describe

g(x)=\sqrt{x+4}

Example 11

hard

Write the equation of the function obtained by reflecting

f(x)=\log_2(x)

over the

y

-axis, compressing horizontally by a factor of

3

, and shifting up

5

Example 12

easy

Describe the transformation from

f(x)

f(-x)

Example 13

medium

Starting from

f(x) = |x|

, sketch the key features of

g(x) = -2|x + 3| + 4

. Give the vertex and direction of opening.

Example 14

challenge

The function

g(x) = a f(b(x - h)) + k

takes the point

(0, 0)

f

to the point

(3, -2)

, and the point

(2, 4)

f

(5, -10)

. If

b = 1

, find

a

h

, and

k

Example 15

easy

Describe all transformations applied to

f(x) = x^2

to obtain

g(x) = 2(x-3)^2 + 1

Example 16

easy

From

f(x)=x^2

, where is the vertex of

g(x)=x^2+5

Example 17

hard

For what value of

c

is the graph of

y = f(x + c)

symmetric about the

y

-axis, given that

y = f(x)

is symmetric about the line

x = 4

Example 18

medium

Write the equation after shifting

f(x)=x^2

right 2 and up 3.

Example 19

medium

From

f(x) = \ln(x)

, write

g(x) = \ln(x - 4) - 2

and state the vertical asymptote.

Example 20

medium

Compared with

y = \sin(x)

, what is the period of

y = \sin(3x)

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Related Concepts

Function Coordinate Plane Parent Functions