Function Transformation Examples in Math

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

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 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.

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 transformation shifts, reflects, stretches, or compresses a known parent graph by editing its formula.

Common stuck point: The procedure for function transformation is the easy part; the trap is shifting the wrong direction inside the function. Asking "Is a known parent graph being moved, scaled, or flipped by added or multiplied constants?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is a known parent graph being moved, scaled, or flipped by added or multiplied constants?

Worked Examples

Example 1

easy

Describe all transformations applied to

f(x) = x^2

to obtain

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

Answer

Shift right

3

, stretch vertically by

2

, shift up

1

; vertex at

(3,1)

First step

Write in standard form

y = a\cdot f(b(x-h))+k

: here

a=2

b=1

h=3

k=1

Full solution

2
Horizontal shift: $h=3$ shifts the parabola $3$ units to the right (vertex moves from $(0,0)$ to $(3,0)$ ).
3
Vertical stretch: $a=2$ stretches vertically by factor $2$ (makes parabola narrower). Vertical shift: $k=1$ shifts the entire graph $1$ unit up. Final vertex: $(3, 1)$ .

The transformation

y = a\cdot f(b(x-h))+k

encodes four independent transformations. Reading off

a

b

h

k

allows systematic description without re-deriving the graph from scratch.

Example 2

medium

Starting from

f(x) = \sqrt{x}

, apply the transformation

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

step by step and identify the key point transformations.

Example 3

medium

Starting from

f(x) = x^2

, write the equation that reflects across the

x

-axis, stretches vertically by

4

, and shifts right

1

Example 4

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 5

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 6

medium

The graph of

f(x) = 2^x

is shifted right

1

, reflected over the

y

-axis, then shifted up

3

. Write the resulting equation.

Example 7

hard

Rewrite

g(x) = 9x^2 - 18x + 7

in the form

a(x - h)^2 + k

and describe the transformations from

f(x) = x^2

Example 8

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 9

hard

A function

f

has

f(0) = 1

f(2) = 5

, and

f(4) = 9

. Build a function

g

that shifts

f

left

2

, reflects over the

x

-axis, and stretches vertically by

3

. Find

g(0)

g(-2)

, and

g(2)

Example 10

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

Practice Problems

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

Example 1

easy

The graph of

y = f(x)

passes through

(1, 4)

. Where does the transformed graph

y = f(x+2) - 3

pass through?

Example 2

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 3

easy

Describe the transformation from

f(x)

f(x)+3

Example 4

easy

Describe the transformation from

f(x)

f(x-2)

Example 5

easy

Describe the transformation from

f(x)

-f(x)

Example 6

easy

Describe the transformation from

f(x)

f(-x)

Example 7

easy

Describe the transformation from

f(x)

2f(x)

Example 8

easy

Describe the transformation from

f(x)

f(2x)

Example 9

easy

Starting from

f(x)=x^2

, what is the vertex of

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

Example 10

easy

From

f(x)=x^2

, where is the vertex of

g(x)=x^2+5

Example 11

medium

Describe all transformations from

f(x)

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

Example 12

medium

Write the equation after shifting

f(x)=x^2

right 2 and up 3.

Example 13

medium

From

f(x)=\sqrt{x}

, describe

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

Example 14

medium

From

f(x)=|x|

, describe

g(x)=\tfrac{1}{2}|x|

Example 15

medium

Apply transformations in order to

f(x)=x^2

: reflect over

x

-axis, then shift up 4. Write

g(x)

Example 16

medium

The point

(2,5)

is on

y=f(x)

. Where does it move on

y=f(x)+3

Example 17

medium

The point

(2,5)

is on

y=f(x)

. Where does it move on

y=f(x-1)

Example 18

medium

From

f(x)=x^2

, describe

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

Example 19

medium

The point

(4,2)

lies on

y=f(x)

. Where does it move on

y=f(2x)

Example 20

challenge

Given

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

, list every transformation in correct order.

Example 21

challenge

Find

g(x)

if the graph of

f(x)=x^2

is reflected over the

y

-axis, compressed horizontally by factor 2, then shifted up 1.

Example 22

challenge

f(x)

has a maximum at

(1,4)

, where is the maximum of

g(x)=3f(x-2)

Example 23

easy

Describe the transformation from

f(x)

f(x) - 4

Example 24

easy

Describe the transformation from

f(x)

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

Example 25

easy

Describe the transformation from

f(x)

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

Example 26

easy

The point

(2, 5)

is on

y = f(x)

. Where is the corresponding point on

y = f(x) + 3

Example 27

easy

Describe the transformation from

f(x)

3f(x) - 2

Example 28

medium

Given

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

and

f(4) = 7

, find

g(5)

Example 29

medium

The point

(3, -4)

is on

y = f(x)

. Where is the corresponding point on

y = -f(x - 1) + 5

Example 30

medium

Find the equation of the function obtained by stretching

f(x) = \cos(x)

vertically by

3

, compressing horizontally by

2

, and shifting up

1

Example 31

medium

From

f(x) = \ln(x)

, write

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

and state the vertical asymptote.

Example 32

hard

The graph of

y = f(x)

has domain

[-2, 6]

. What is the domain of

y = f(2x - 4)

Example 33

hard

The graph of

y = f(x)

passes through

(2, 5)

and

(4, 1)

. Find two points on

y = -\tfrac{1}{2}f(2(x + 1)) + 3

Example 34

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

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Related Concepts

Function Coordinate Plane Parent Functions

Background Knowledge

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

function definitioncoordinate plane