Apply translations and stretches to graphs and equations

Alberta Grade 12 Mathematics

Relations and Functions

How changes to an equation move or resize its graph

A graph can be moved or resized without changing its basic shape. For example, the graph of

y=x^2

can move right, move up, or become narrower. These changes are called transformations. This lesson focuses on translations, which move a graph, and stretches, which change its size. A reliable method is to track a point

(x,y)

on the original graph and determine where it moves.

What you will learn

Identify translations and stretches from a function equation.
Describe how a point on a graph moves under a translation or stretch.
Write a transformed equation from a given parent graph.
Check transformed graphs using points and domain restrictions.

1. Review: functions and points

A function rule gives an output for each allowed input. We often write the rule as

y=f(x)

. The graph contains points

(x,y)

that satisfy the rule. For instance, if

f(x)=x^2

, then

f(2)=4

, so

(2,4)

is on its graph.

The domain is the set of input values for which the rule is defined. For

f(x)=\sqrt{x}

, the domain is

x\geq 0

, because the square root of a negative real number is not defined. When a graph moves or stretches horizontally, its domain may change too.

y=f(x)

A point $(x,y)$ on a graph satisfies its equation.
The domain lists the allowed input values.

2. Translations: moving a graph

A translation moves every point the same distance in the same direction. If each point moves right by

h

units and up by

k

units, then

(x,y)

moves to

(x+h,y+k)

. The graph keeps its shape and size.

In the equation, a horizontal move is made by changing the input inside the function. The rule

y=f(x-h)

moves the graph right by

h

units when

h>0

. The rule

y=f(x+h)

moves it left by

h

units. The sign inside the brackets appears opposite to the direction of the horizontal move.

A vertical move is shown outside the function. The rule

y=f(x)+k

moves the graph up by

k

units when

k>0

, and down when

k<0

. For example,

y=(x-3)^2+2

is the graph of

y=x^2

moved right 3 units and up 2 units.

(x,y)\mapsto(x+h,y+k)

Inside change: $f(x-h)$ moves right by $h$ .
Outside change: $f(x)+k$ moves up by $k$ .

3. Stretches: changing a graph's size

A vertical stretch multiplies every output by the same factor. In

y=af(x)

, a point

(x,y)

moves to

(x,ay)

. If

a>1

, distances from the horizontal axis are multiplied by

a

. For example,

y=3f(x)

triples every

y

-coordinate.

A horizontal stretch multiplies every input-coordinate by a factor. To stretch a graph horizontally by a factor of

c

, where

c>1

, use

y=f(x/c)

. A point

(x,y)

on the original graph moves to

(cx,y)

. The division by

c

inside the function causes the horizontal stretch.

These rules can be combined. For

y=af((x-h)/c)+k

, with positive

a

and

c

, an original point

(x,y)

moves to

(cx+h,ay+k)

. This describes a horizontal stretch by

c

, then a right translation by

h

, a vertical stretch by

a

, and an up translation by

k

. In practice, identify each change from the equation and check the resulting point coordinates.

y=af\left(\frac{x-h}{c}\right)+k

Outside multiplier $a$ : vertical stretch factor $a$ .
Input $x/c$ : horizontal stretch factor $c$ .
Check a point to confirm the direction and size of each change.

4. Reading graphs and equations together

A graph provides a visual check on the equation. A translated parabola has the same width as the original. A stretched square-root graph still begins at an endpoint, but that endpoint and the distances between plotted points may change.

For a horizontal stretch, the input values are multiplied by the stretch factor. This also affects the domain. The original rule

f(x)=\sqrt{x}

has domain

x\geq0

. Replacing its input with

x/3

still requires

x/3\geq0

, so the transformed domain remains

x\geq0

.

When a question gives a graph rather than an equation, choose a clear point such as a vertex or endpoint. Record its new location after each transformation. Then use the same point movement for other points, or use the transformed equation to check the graph.

x/c\geq0\Longleftrightarrow x\geq0

A translation preserves shape and size.
A stretch changes distances from an axis.
Check domain restrictions after changing the input.

Worked example

Translate a parabola

The graph of

f(x)=x^2

is moved 2 units right and 1 unit up. Write the new equation and locate the original vertex.

Parabola translation

The transformed graph is shifted right and up. The window shows only part of the full domain.

Apply the horizontal move
A move 2 units right uses $x-2$ inside the function. This opposite sign is the standard horizontal translation rule.
$f(x-2)=(x-2)^2$
Apply the vertical move
A move 1 unit up adds 1 outside the function. The original vertex $(0,0)$ therefore moves to $(2,1)$ .
$y=(x-2)^2+1$

Answer: The transformed equation is

y=(x-2)^2+1

. The vertex moves from

(0,0)

to

(2,1)

.

Check: At the new vertex input

x=2

, the equation gives

y=1

, confirming the vertex location.

Worked example

Stretch a square-root graph

Starting with

f(x)=\sqrt{x}

, apply a horizontal stretch by a factor of 3 and then move the graph down 2 units. Write the equation and state its domain.

Square-root horizontal stretch

Both curves have domain $x\geq0$. The graph window shows only part of each graph.

Make the horizontal stretch
For a horizontal stretch by 3, replace the input with $x/3$ . This sends each original input-coordinate to three times its value.
$f\left(\frac{x}{3}\right)=\sqrt{\frac{x}{3}}$
Translate down
Subtract 2 outside the square root to move every point down 2 units. The expression under the root must still be non-negative.
$y=\sqrt{\frac{x}{3}}-2$
Find the domain
The radicand must be at least zero. Since 3 is positive, this condition gives the same non-negative input restriction.
$\frac{x}{3}\geq0\Longrightarrow x\geq0$

Answer: The equation is

y=\sqrt{x/3}-2

, with domain

x\geq0

. The endpoint moves from

(0,0)

to

(0,-2)

.

Check: At

x=3

, the transformed output is

\sqrt{1}-2=-1

. This matches the original point

(1,1)

stretched to

(3,1)

and moved down to

(3,-1)

.

Worked example

Combine horizontal and vertical stretches

The graph of

f(x)=|x|

is horizontally stretched by a factor of 2, vertically stretched by a factor of 2, then moved right 1 unit and up 1 unit. Write the equation and find the image of the point

(2,2)

.

Absolute-value stretches and translations

The transformed graph has vertex $(1,1)$. The window shows only part of the full domain.

Set up the input change
A horizontal stretch by 2 uses $(x-1)/2$ : divide by 2 for the stretch, and subtract 1 for the right translation.
$\frac{x-1}{2}$
Apply the vertical changes
Multiply the function output by 2 for the vertical stretch, then add 1 for the upward translation.
$y=2\left|\frac{x-1}{2}\right|+1$
Move the stated point
The point rule is $(x,y)\mapsto(2x+1,2y+1)$ . Applying it to $(2,2)$ gives its new coordinates.
$(2,2)\mapsto(5,5)$

Answer: The equation is

y=2\left|(x-1)/2\right|+1

. The point

(2,2)

moves to

(5,5)

.

Check: Substituting

x=5

gives

y=2|4/2|+1=5

, so the image point is on the transformed graph.

Common mistakes and how to avoid them

Treating

f(x-4)

as a shift left 4 units.

Correction: The minus sign inside the function shifts the graph right 4 units.

Using

f(3x)

for a horizontal stretch by 3.

Correction:

f(3x)

compresses horizontally by a factor of 3. Use

f(x/3)

for a stretch by 3.

Changing the equation but not checking the domain.

Correction: For square roots, require the expression inside the root to be at least zero.

Lesson summary

A translation moves every point by the same amount.
In $f(x-h)+k$ , the graph moves right by $h$ and up by $k$ .
A vertical multiplier changes the $y$ -coordinates; an input divisor creates a horizontal stretch.
Use point coordinates and domain restrictions to check a transformed equation.

Check your understanding

Question 1

The graph of

y=f(x)

is moved left 5 units and down 2 units. Which equation represents the new graph?

$y=f(x+5)-2$
$y=f(x-5)-2$
$y=f(x+5)+2$
$y=f(5x)-2$

Show answer and explanation

y=f(x+5)-2

A left shift uses

x+5

inside the function, and a downward shift subtracts 2 outside it.

Question 2

Which equation gives a horizontal stretch of

y=\sqrt{x}

by a factor of 4, followed by an upward shift of 3?

$y=\sqrt{4x}+3$
$y=\sqrt{x/4}+3$
$y=4\sqrt{x}+3$
$y=\sqrt{x/4}-3$

Show answer and explanation

y=\sqrt{x/4}+3

A horizontal stretch by 4 replaces

x

with

x/4

. Adding 3 moves the graph up. Its domain is

x\geq0

.

Question 3

Under the transformation

y=2f(x-3)+1

, where does an original point

(4,5)

move?

$(7,11)$
$(1,11)$
$(7,6)$
$(8,10)$

Show answer and explanation

(7,11)

The point rule is

(x,y)\mapsto(x+3,2y+1)

. Thus

(4,5)

becomes

(7,11)

.

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 move that shifts every point of a graph the same distance and direction.
Stretch: A change that multiplies distances from an axis by a fixed factor.

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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 RF4. 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

Report a correction or ask a question

RF4 · Apply translations and stretches to graphs and equations

Alberta Grade 12 Mathematics

How changes to an equation move or resize its graph

What you will learn

1. Review: functions and points

2. Translations: moving a graph

3. Stretches: changing a graph's size

4. Reading graphs and equations together

Translate a parabola

Stretch a square-root graph

Combine horizontal and vertical stretches

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