Describe horizontal and vertical translations of functions

Alberta Grade 12 Mathematics

Relations and Functions

How changes inside and outside a function move its graph

A translation moves a graph without changing its shape or orientation. It can move every point left or right, up or down, or both. The key is to notice whether a change is made outside the function or inside its input. This lesson reviews function notation, builds the translation rules, and applies them to graphs and points.

What you will learn

Describe how adding a value outside a function changes its graph.
Describe how adding or subtracting a value inside a function changes its graph.
Identify the direction and size of a translation from a function rule.
Write a translated function rule from a described movement.
Use corresponding points to check a translation.

1. Review: functions and points

A function rule assigns an output to each allowed input. In

f(x)

, the letter

x

is the input and

f(x)

is the output. A point on the graph has coordinates

(x,y)

, where

y=f(x)

.

For example, if

f(2)=5

, then

(2,5)

is on the graph of

y=f(x)

. A domain is the set of inputs for which the function is defined. When a graph moves horizontally, its allowed input values move with it.

y=f(x)

The first coordinate of a point is the input.
The second coordinate is the function output.
A translation moves all graph points by the same amount.

2. Vertical translations

Adding a constant outside the function changes every output by that constant. The graph moves vertically, while each point keeps the same first coordinate.

If

k

is positive,

y=f(x)+k

moves the graph up

k

units. If

k

is positive,

y=f(x)-k

moves it down

k

units. In either case, a point

(x,y)

becomes

(x,y+k)

for an upward move, or

(x,y-k)

for a downward move.

For example, adding

3

to each output changes a point

(2,5)

to

(2,8)

. Its input stays at

2

. The shape and orientation of the graph do not change.

y=f(x)+k\;\Rightarrow\;(x,y)\mapsto(x,y+k)

A change outside the function affects outputs.
The sign outside the function gives the vertical direction directly.
Vertical translations do not change the domain.

3. Horizontal translations

A horizontal translation changes the input used by the function. The direction may seem opposite to the sign inside the brackets, so check which old input gives the same output.

The graph of

y=f(x-h)

is the graph of

y=f(x)

moved right

h

units when

h

is positive. The old input

x

now occurs at the new input

x+h

. Thus a point

(x,y)

moves to

(x+h,y)

.

The graph of

y=f(x+h)

moves left

h

units when

h

is positive. Its points move from

(x,y)

to

(x-h,y)

. The output stays the same in each horizontal translation.

A useful way to read the rule is to find where the original input occurs. In

f(x-4)

, the input to the original function is

4

less than the displayed input. The same old function value therefore appears

4

units farther right.

y=f(x-h)\;\Rightarrow\;(x,y)\mapsto(x+h,y)

A change inside the input affects horizontal position.
$f(x-h)$ moves right by $h$ units.
$f(x+h)$ moves left by $h$ units.
A horizontal translation moves the domain by the same amount as the graph.

4. Combining translations and checking

A function can move both horizontally and vertically. In

y=f(x-h)+k

, the horizontal movement is right

h

units when

h

is positive, and the vertical movement is up

k

units when

k

is positive. A corresponding point changes from

(x,y)

to

(x+h,y+k)

.

The same idea works when either value is negative. For instance, a negative horizontal shift in this form moves left, and a negative vertical shift moves down. Read each change from its position: inside controls horizontal movement, outside controls vertical movement.

To check a description, choose a point on the original graph. Track its input and output separately. If the rule is horizontal, the output should stay the same. If it is vertical, the input should stay the same. For a combined translation, both coordinates change by the stated amounts.

y=f(x-h)+k\;\Rightarrow\;(x,y)\mapsto(x+h,y+k)

Inside changes move points horizontally; outside changes move them vertically.
Track coordinates to confirm direction and distance.
Translations preserve the graph's shape and orientation.

Worked example

Read a horizontal shift

The graph of

y=f(x)

contains the point

(2,5)

. Describe the translation represented by

g(x)=f(x-4)

, and give the image of the point.

A graph and its horizontal translation

The curves show a sample function and its shift right by four units. The window is only part of the full domain.

Identify the input change
The function input is $x-4$ . The same original input value is reached when the new displayed input is $4$ greater.
$x-4=2\;\Rightarrow\;x=6$
State the translation
The graph moves right $4$ units. A horizontal translation keeps the output unchanged, so the point's second coordinate remains $5$ .
$(2,5)\mapsto(6,5)$

Answer: The graph moves right

4

units. The point

(2,5)

becomes

(6,5)

.

Check: At the new input

6

, the rule gives

g(6)=f(6-4)=f(2)=5

, confirming the point.

Worked example

Translate vertically

The graph of

y=f(x)

contains

(1,-2)

. Describe

p(x)=f(x)+5

and find the image of that point.

A graph and its vertical translation

The curves show a sample function and its shift up by five units. The window is only part of the full domain.

Read the outside change
The $5$ is added after the function value is found. It increases each output by $5$ and leaves each input unchanged.
$-2+5=3$
Give the movement and image
The graph moves up $5$ units. The point's first coordinate remains $1$ , and its output becomes $3$ .
$(1,-2)\mapsto(1,3)$

Answer: The graph moves up

5

units, and

(1,-2)

becomes

(1,3)

.

Check: At input

1

,

p(1)=f(1)+5=-2+5=3

.

Worked example

Describe a combined translation

The graph of

y=f(x)

contains

(3,1)

. Describe

q(x)=f(x+2)-4

and find the image of the point.

A graph and its combined translation

The curves show a sample function shifted left by two and down by four. The window is only part of the full domain.

Find the horizontal movement
The input is $x+2$ . The same original input occurs when the new input is $2$ smaller, so the graph moves left $2$ units.
$x+2=3\;\Rightarrow\;x=1$
Find the vertical movement
The $-4$ is outside the function, so every output decreases by $4$ . The original output $1$ becomes $-3$ .
$1-4=-3$
State the image
Apply both coordinate changes: the input decreases by $2$ , and the output decreases by $4$ .
$(3,1)\mapsto(1,-3)$

Answer: The graph moves left

2

units and down

4

units. The point

(3,1)

becomes

(1,-3)

.

Check: At the new input,

q(1)=f(1+2)-4=f(3)-4=1-4=-3

.

Common mistakes and how to avoid them

Reading

f(x-3)

as a move left because the expression contains a minus sign.

Correction: The input

x-3

means the same old input occurs when the new input is three greater. The graph moves right.

Changing both coordinates for a vertical translation.

Correction: A vertical translation changes outputs only. Keep the first coordinate fixed.

Changing only the output for a horizontal translation.

Correction: A horizontal translation changes input positions. Keep the output fixed.

Lesson summary

Adding a value outside a function moves the graph vertically.
Adding or subtracting a value inside the input moves the graph horizontally in the opposite direction from the sign shown.
For $y=f(x-h)+k$ , the graph moves right by $h$ and up by $k$ when those values are positive.
Track a known point's input and output to verify the direction and distance.

Check your understanding

Question 1

The point

(4,2)

lies on

y=f(x)

. Where does it move under

y=f(x-3)

?

$(1,2)$
$(7,2)$
$(4,5)$
$(4,-1)$

Show answer and explanation

(7,2)

The graph moves right

3

units. The input becomes

7

and the output stays

2

.

Question 2

What translation is represented by

r(x)=f(x)-6

?

Left $6$ units
Right $6$ units
Up $6$ units
Down $6$ units

Show answer and explanation

Down

6

units

The subtraction is outside the function, so every output decreases by

6

. This is a downward translation.

Question 3

A point on

y=f(x)

is

(2,7)

. What is its image under

s(x)=f(x+5)+1

?

$(7,8)$
$(-3,8)$
$(-3,6)$
$(7,6)$

Show answer and explanation

(-3,8)

f(x+5)

moves the graph left

5

, and

+1

moves it up

1

. Therefore

(2,7)

becomes

(-3,8)

.

Key terms

Function: A rule that assigns an output to each allowed input.
Domain: The set of input values for which a function is defined.
Translation: A movement of a graph that shifts every point the same distance in a direction without changing its shape or orientation.
Horizontal translation: A left or right movement that changes point inputs while keeping outputs the same.
Vertical translation: An up or down movement that changes point outputs while keeping inputs the same.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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RF2 · Describe horizontal and vertical translations of functions

Alberta Grade 12 Mathematics

How changes inside and outside a function move its graph

What you will learn

1. Review: functions and points

2. Vertical translations

3. Horizontal translations

4. Combining translations and checking

Read a horizontal shift

Translate vertically

Describe a combined translation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review