Describe horizontal and vertical stretches of functions

Alberta Grade 12 Mathematics

Relations and Functions

How multiplying outside or inside a function changes its graph

A stretch changes the size of a graph without changing its basic shape. The key is where the factor appears: a factor outside a function changes vertical distances, while a factor inside changes horizontal distances. In this lesson, a stretch factor is greater than 1. We will use point coordinates and function notation to describe these changes.

What you will learn

Describe how a vertical stretch changes a function's graph.
Describe how a horizontal stretch changes a function's graph.
Use coordinates and function notation to identify stretch factors.
Distinguish a horizontal stretch from multiplication outside the function.

1. Review: functions and points

A function assigns an output to each allowed input. We often write the output as

y=f(x)

. A point

(x,y)

on its graph means that the function's output at input

x

is

y

.

For example, if

f(2)=3

, then

(2,3)

is on the graph of

y=f(x)

. The first coordinate is horizontal, and the second is vertical. A stretch changes one or both of these coordinates in a consistent way.

A domain is the set of inputs for which a function is defined. When a transformation changes the input used in the function, check that the new input is still allowed.

The first coordinate of a point is horizontal; the second is vertical.
The graph of $y=f(x)$ contains $(x,f(x))$ for every allowed input $x$ .

2. Vertical stretches

A vertical stretch makes every point farther from the horizontal axis by the same factor. If the original point is

(x,y)

and the vertical stretch factor is

a

, the new point is

(x,ay)

. The input stays the same, while the output is multiplied by

a

.

The corresponding function is

y=af(x)

. For a vertical stretch,

a>1

. For example, a point

(2,3)

becomes

(2,6)

when the vertical stretch factor is

2

. Its horizontal position has not changed.

Points below the horizontal axis stay below it under a positive vertical stretch, but their vertical distances from that axis increase. A point on the horizontal axis remains there because multiplying zero still gives zero.

y=af(x),\quad a>1

Multiply the output by the vertical stretch factor.
Coordinate rule: $(x,y)\mapsto(x,ay)$ , where $a>1$ .

3. Horizontal stretches

A horizontal stretch makes every point farther from the vertical axis by the same factor. If the original point is

(x,y)

and the horizontal stretch factor is

k

, the new point is

(kx,y)

. The output stays the same, while the horizontal coordinate is multiplied by

k

.

In function notation, a horizontal stretch by factor

k

is written

y=f\left(\frac{x}{k}\right)

, where

k>1

. To see why the input is divided by

k

, let the new horizontal coordinate be

X=kx

. Then the input to the original function is

X/k=x

, so the output remains

f(x)

.

For instance, if

(2,3)

is on the original graph, a horizontal stretch by factor

3

sends it to

(6,3)

. The output is unchanged. This is why a factor written inside the function has the reciprocal effect on the input:

f(kx)

compresses horizontally by factor

k

, rather than stretching. Do not confuse it with

f(x/k)

.

The domain changes with the horizontal stretch. If the original input

x

is allowed, then the stretched graph has the corresponding input

kx

. For a square-root function with original domain

x\geq 0

, a positive horizontal stretch still has domain

x\geq 0

.

y=f\left(\frac{x}{k}\right),\quad k>1

Multiply the original point's horizontal coordinate by the stretch factor.
A horizontal stretch by $k>1$ is written $y=f(x/k)$ .
The factor inside the function works in the opposite way from the resulting horizontal coordinate.

4. Reading and describing a stretch

To describe a transformation from an equation, first identify whether the factor is outside the function or inside its input. An outside factor multiplies outputs. An input divided by a number produces a horizontal stretch by that number.

A table makes the coordinate changes clear. The original point

(1,2)

becomes

(1,6)

under a vertical stretch by

3

, but becomes

(4,2)

under a horizontal stretch by

4

. These are different transformations even though both use a factor greater than one.

A graph can show only part of a function's domain. The axis window in a diagram is a viewing choice, not a restriction on the function itself. When describing a stretch, use the stated equation and coordinate rule, rather than estimating a factor from a small portion of a graph.

Outside factor: outputs change; horizontal coordinates stay fixed.
Input divided by a factor: horizontal coordinates change; outputs stay fixed.
Use corresponding points to check whether a description matches an equation.

Coordinate changes for the point (1, 2)

Transformation	New point	What changes?
Vertical stretch by factor 3	(1, 6)	Output is multiplied by 3
Horizontal stretch by factor 4	(4, 2)	Input coordinate is multiplied by 4

Worked example

Identify a vertical stretch

The function

f

contains the point

(2,-3)

. Describe the transformation and find the corresponding point on

y=4f(x)

.

Identify the factor
The factor $4$ is outside the function. Therefore, it multiplies each output and creates a vertical stretch by factor $4$ .
$y=4f(x)$
Transform the point
Keep the horizontal coordinate at $2$ . Multiply the vertical coordinate, $-3$ , by $4$ . The negative output remains negative and becomes four times as far from the horizontal axis.
$(2,-3)\mapsto(2,-12)$

Answer: The graph is vertically stretched by factor

4

. The corresponding point is

(2,-12)

.

Check: The input coordinate stays fixed, as expected for a vertical stretch.

Worked example

Describe a horizontal stretch of a parabola

Let

f(x)=x^2

. Describe the transformation represented by

g(x)=f(x/2)

, and give the image of the point

(3,9)

on the graph of

f

.

Original and stretched parabolas

The wider curve is the horizontal stretch. The graph shows only a window of the full domain.

Read the input
The input is divided by $2$ . This gives a horizontal stretch by factor $2$ . The point rule multiplies the original horizontal coordinate by $2$ and leaves the output unchanged.
$(x,y)\mapsto(2x,y)$
Transform the point
Apply the point rule to $(3,9)$ . The new horizontal coordinate is $6$ , and the vertical coordinate remains $9$ .
$(3,9)\mapsto(6,9)$
Check using the equation
Since $f(x)=x^2$ , the transformed function can also be written by substituting $x/2$ for the input. At the new input $6$ , it gives the original output at input $3$ .
$g(6)=f(6/2)=f(3)=9$

Answer: The graph is horizontally stretched by factor

2

. The image of

(3,9)

is

(6,9)

.

Check: The point rule and the transformed equation both give output

9

at the new input

6

.

Worked example

Compare vertical and horizontal stretches

The original function is

f(x)=\sqrt{x}

, with domain

x\geq 0

. Describe

p(x)=3f(x)

and

q(x)=f(x/4)

. Find each transformed graph's point corresponding to

(4,2)

on the original graph.

Square-root stretches

The curves show only a window of the full domain, which continues to the right.

Describe p
The factor $3$ is outside $f$ , so $p$ is a vertical stretch by factor $3$ . It multiplies the output coordinate by $3$ and keeps the input coordinate fixed.
$(4,2)\mapsto(4,6)$
Describe q
The input is divided by $4$ , so $q$ is a horizontal stretch by factor $4$ . It multiplies the horizontal coordinate by $4$ and keeps the output coordinate fixed.
$(4,2)\mapsto(16,2)$
Check the domains
For $p$ , the square-root input remains $x$ , so it must satisfy $x\geq 0$ . For $q$ , the square-root input is $x/4$ , which is also nonnegative exactly when $x\geq 0$ . Both transformed functions have domain $x\geq 0$ .
$x\geq 0$

Answer:

p

is a vertical stretch by factor

3

and has corresponding point

(4,6)

.

q

is a horizontal stretch by factor

4

and has corresponding point

(16,2)

. Both domains are

x\geq 0

.

Check: The output changes for

p

; the input changes for

q

. The domain condition is valid for both square-root expressions.

Common mistakes and how to avoid them

Calling

f(3x)

a horizontal stretch by factor

3

.

Correction:

f(3x)

uses a larger input at each chosen

x

, so it compresses horizontally. A horizontal stretch by factor

3

is written

f(x/3)

.

Changing both coordinates during a vertical stretch.

Correction: A vertical stretch changes only the output coordinate. Use

(x,y)\mapsto(x,ay)

.

Changing the output during a horizontal stretch.

Correction: A horizontal stretch changes only the input coordinate. Use

(x,y)\mapsto(kx,y)

.

Assuming a graph window gives the complete domain.

Correction: A graph window shows only the displayed portion. Determine the domain from the function's allowed inputs.

Lesson summary

A vertical stretch by factor $a>1$ is $y=af(x)$ and sends $(x,y)$ to $(x,ay)$ .
A horizontal stretch by factor $k>1$ is $y=f(x/k)$ and sends $(x,y)$ to $(kx,y)$ .
An outside factor changes outputs; division inside the function produces a horizontal stretch.
Check the domain when the input to a function changes.

Check your understanding

Question 1

A point

(2,5)

is on

y=f(x)

. What is its corresponding point after the graph is stretched horizontally by factor

3

?

$(6,5)$
$(2,15)$
$(2/3,5)$
$(6,15)$

Show answer and explanation

(6,5)

A horizontal stretch multiplies the horizontal coordinate by

3

and leaves the output unchanged.

Question 2

Which equation represents a vertical stretch of

y=f(x)

by factor

2

?

$y=f(x/2)$
$y=2f(x)$
$y=f(2x)$
$y=f(x)+2$

Show answer and explanation

y=2f(x)

A vertical stretch multiplies each output by the factor, so the factor appears outside the function.

Question 3

The graph of

y=f(x)

includes

(4,7)

. Which point corresponds to it on

y=f(x/5)

?

$(4,35)$
$(20,7)$
$(4/5,7)$
$(20,35)$

Show answer and explanation

(20,7)

The input is divided by

5

, which makes a horizontal stretch by factor

5

. The point's horizontal coordinate becomes

20

; its output stays

7

.

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.
Vertical stretch: A change that multiplies each output by a factor greater than one.
Horizontal stretch: A change that multiplies each input coordinate by a factor greater than one.
Corresponding point: The point on a transformed graph that comes from a given point on the original graph.

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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 RF3. 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.

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RF3 · Describe horizontal and vertical stretches of functions

Alberta Grade 12 Mathematics

How multiplying outside or inside a function changes its graph

What you will learn

1. Review: functions and points

2. Vertical stretches

3. Horizontal stretches

4. Reading and describing a stretch

Coordinate changes for the point (1, 2)

Identify a vertical stretch

Describe a horizontal stretch of a parabola

Compare vertical and horizontal 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