Practice Shifting 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

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.

Showing a random 20 of 50 problems.

Example 1

hard

The function

f(x)

has horizontal asymptote

y = 3

. What is the horizontal asymptote of

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

Example 2

medium

Express

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

as a shift of

f(x) = x^2

and find its

x

-intercepts.

Example 3

easy

Describe the transformation from

f(x)

f(x - 3)

Example 4

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 5

easy

For

f(x) = x^2

, write

f(x - 1) + 4

explicitly.

Example 6

easy

Which shift gives

f(x) + 6

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

Example 7

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 8

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 9

easy

f(x)

has the point

(2, 9)

, where does it go under

f(x - 3)

Example 10

hard

For

f(x) = \sqrt{x}

shifted to start at the point

(7, -2)

, write the equation.

Example 11

medium

f(x) = 2^x

is shifted right

3

and down

1

. Write

g(x)

and find the horizontal asymptote.

Example 12

hard

Given

f(2) = 7

and

f(5) = -1

, find

g(7)

and

g(10)

for

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

Example 13

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 14

hard

What single equation describes shifting

y = \cos(x)

left

\tfrac{\pi}{2}

? Compare to

\sin(x)

Example 15

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 16

medium

What shifts take

f(x) = (x - 1)^2 + 2

g(x) = (x + 3)^2 - 5

Example 17

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 18

medium

The vertex of

y = (x - 4)^2 + 1

comes from shifting

y = x^2

. State the shift and the vertex.

Example 19

easy

f(x)

has the point

(1, 7)

, where does that point go under

f(x) - 5

Example 20

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?

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Related Concepts

Function Transformation