Shifting Functions Examples in Math

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

Shifting a function translates its graph horizontally or vertically without changing its shape: $f(x - h) + k$ shifts right by $h$ and up by $k$ .

Shifting is like sliding the entire graph on the coordinate plane — the function's shape is completely unchanged, only its position moves.

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: Shifting adds a constant to move the entire graph left/right or up/down without changing its shape at all.

Common stuck point: The procedure for shifting functions is the easy part; the trap is reading $f(x-3)$ as a left shift. Asking "Is the graph the same shape just moved by an added constant (not stretched or flipped)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the graph the same shape just moved by an added constant (not stretched or flipped)?

Worked Examples

Example 1

easy

Starting from

f(x)=x^2

, describe and sketch the transformations for

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

Answer

Right

3

, up

2

; vertex at

(3, 2)

First step

Horizontal shift:

x-3

inside the function shifts the parabola

3

units to the right. Vertex moves from

(0,0)

(3,0)

Full solution

2
Vertical shift: $+2$ outside shifts the graph $2$ units up. Vertex moves from $(3,0)$ to $(3,2)$ .
3
Result: upward-opening parabola with vertex at $(3,2)$ ; same shape as $f(x)=x^2$ .

Horizontal shifts come from changes inside the function argument:

f(x-h)

shifts right by

h

(left if

h<0

). Vertical shifts come from adding a constant outside:

f(x)+k

shifts up by

k

. The vertex formula is

(h,k)

Example 2

medium

The graph of

f(x)=e^x

is shifted left

2

units and down

5

units to give

g(x)

. Write

g(x)

, find

g(0)

, and determine if the horizontal asymptote changes.

Example 3

medium

Write the equation of

y = \sqrt{x}

shifted right

5

and up

3

. State the domain and the

y

-value at

x = 9

Example 4

medium

Starting from

f(x) = \ln(x)

, write

g(x)

that shifts left

1

and up

4

. Find the vertical asymptote.

Example 5

medium

The domain of

y = f(x)

[0, 10]

. Find the domain of

y = f(x - 3) + 1

Example 6

medium

f(x) = 2^x

is shifted right

3

and down

1

. Write

g(x)

and find the horizontal asymptote.

Example 7

hard

Rewrite

y = x^2 + 6x + 11

as a shift of

y = x^2

Example 8

hard

A function

f

has its only zero at

x = 4

. Find the zeros of

g(x) = f(x + 1) - 0

and of

h(x) = f(x - 3)

Example 9

hard

For

f(x) = \sqrt{x}

shifted to start at the point

(7, -2)

, write the equation.

Example 10

challenge

A graph

y = f(x)

has area

A

between it and the

x

-axis on

[0, 10]

. What is the corresponding area for

y = f(x - 5) + 2

[5, 15]

, between the graph and

y = 2

Practice Problems

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

Example 1

easy

The point

(4, 7)

is on the graph of

y=f(x)

. Find the corresponding point on each shifted graph: (a)

y=f(x-1)+3

, (b)

y=f(x+5)-2

Example 2

medium

Write the equation of the function whose graph is

y=\sqrt{x}

shifted right

9

and reflected over the

x

-axis. State the domain.

Example 3

easy

Describe the transformation from

f(x)

f(x) + 2

Example 4

easy

Describe the transformation from

f(x)

f(x - 3)

Example 5

easy

Describe the transformation from

f(x)

f(x + 4)

Example 6

easy

f(x)

has the point

(1, 7)

, where does that point go under

f(x) - 5

Example 7

easy

f(x)

has the point

(2, 9)

, where does it go under

f(x - 3)

Example 8

easy

For

f(x) = x^2

, write

f(x - 1) + 4

explicitly.

Example 9

easy

The graph

f(x) = |x|

has its corner at the origin. Where is the corner of

f(x - 2)

Example 10

easy

Which shift gives

f(x) + 6

: up, down, left, or right, and by how much?

Example 11

medium

The vertex of

y = (x - 4)^2 + 1

comes from shifting

y = x^2

. State the shift and the vertex.

Example 12

medium

Express the transformation that moves

f(x)

left 2 and down 5 as a single formula in terms of

f

Example 13

medium

Starting from

g(x) = f(x) + 3

, suppose

f(x) = 2x

. What is

g(x)

, and is the result the same as shifting

2x

up 3 or right 3?

Example 14

medium

The function

h(x) = (x + 1)^2 - 4

x^2

shifted. Find its x-intercepts.

Example 15

medium

Given

f(x)

with a maximum at

(0, 5)

, where is the maximum of

f(x - 3) + 2

Example 16

medium

A cost model

C(x)

is shifted to

C(x - 100)

to represent a fixed 100-unit startup before billing begins. Which way and how far does the graph move, and what does it mean?

Example 17

medium

Rewrite

y = x^2 - 6x + 11

in shifted (vertex) form to reveal the horizontal and vertical shift from

x^2

Example 18

challenge

A function is transformed by FIRST shifting right 2, THEN up 3. Write the result in terms of

f

, and explain whether the order of these two shifts matters.

Example 19

challenge

Transform

f(x)

by stretching vertically by 2 and THEN shifting up 1, versus shifting up 1 and THEN stretching by 2. Write both results and show they differ.

Example 20

challenge

Find

h

and

k

so that

f(x - h) + k

moves the point

(2, 3)

f

exactly onto

(7, -1)

Example 21

medium

A graph

f(x) = \sqrt{x}

(starting at the origin) is shifted to

f(x - 9) = \sqrt{x - 9}

. Where does the graph now start, and what is the new domain?

Example 22

medium

Express the transformation that takes

f(x)

up 4 and right 1 as

f(x - h) + k

, then apply it to

f(x) = x^2

to get an explicit formula.

Example 23

easy

Describe the shift from

f(x)

f(x) - 7

Example 24

easy

Describe the shift from

f(x)

f(x + 6)

Example 25

easy

The corner of

f(x) = |x|

is at the origin. Where is the corner of

f(x) + 5

Example 26

easy

The vertex of

y = x^2

(0, 0)

. What is the vertex of

y = (x + 2)^2 - 7

Example 27

easy

The point

(3, 8)

is on

y = f(x)

. Where is the corresponding point on

y = f(x - 2) + 5

Example 28

medium

The graph of

y = \sin(x)

is shifted right

\pi

and down

1

. Write the equation.

Example 29

medium

The graph of

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

has asymptotes at

x = 0

and

y = 0

. Find the asymptotes of

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

Example 30

medium

The range of

y = f(x)

[-2, 5]

. Find the range of

y = f(x + 1) - 4

Example 31

medium

Express

g(x) = (x + 4)^2 - 9

as a shift of

f(x) = x^2

and find its

x

-intercepts.

Example 32

hard

Given

f(2) = 7

and

f(5) = -1

, find

g(7)

and

g(10)

for

g(x) = f(x - 5) + 4

Example 33

hard

What single equation describes shifting

y = \cos(x)

left

\tfrac{\pi}{2}

? Compare to

\sin(x)

Example 34

hard

The function

f(x)

has horizontal asymptote

y = 3

. What is the horizontal asymptote of

g(x) = f(x - 4) + 6

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Related Concepts

Function Transformation

Background Knowledge

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

transformation