Dilations Revision Notes for VCE SSCE Mathematical Methods

Dilations

Introduction

A dilation is a transformation that stretches or shrinks a graph away from or towards a coordinate axis. Think of it like pulling or pushing the graph in one direction while keeping it anchored to one of the axes.

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To understand dilations, we'll start with a simple example using a circle centered at the origin. This makes it easy to see how points move during the transformation.

A dilation can stretch a graph (make it larger) or compress it (make it smaller), depending on the dilation factor used.

Dilation from the x-axis

A dilation from the x-axis is a vertical transformation. It changes the y-coordinates of points while leaving the x-coordinates unchanged.

Understanding vertical dilations

When we dilate a graph by a factor from the x-axis:

Factor greater than 1: The graph stretches vertically (becomes taller)
Factor between 0 and 1: The graph compresses vertically (becomes shorter)

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For example:

A dilation of factor 2 from the x-axis makes the graph twice as tall
A dilation of factor $\frac{1}{2}$ from the x-axis makes the graph half as tall

The general rule

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For $b$ a positive constant, a dilation of factor $b$ from the x-axis follows the rule:

(x, y) \rightarrow (x, by)

This means:

$x' = x$ (x-coordinate stays the same)
$y' = by$ (y-coordinate is multiplied by $b$ )

Finding the equation after dilation

For the graph of $y = f(x)$ , there are two equivalent ways to find the equation after dilation from the x-axis:

Apply the transformation rule $(x, y) \rightarrow (x, by)$ to the graph
Replace $y$ with $\frac{y}{b}$ in the original equation to get $y = bf(x)$

Both methods produce the same result.

Worked example: Dilation from the x-axis

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Worked Example: Dilating y = √x from the x-axis

Consider the curve $y = \sqrt{x}$ with a dilation of factor 2 from the x-axis.

Let $(x', y')$ be the image of a point $(x, y)$ on the original curve.

Step 1: Using the transformation rule:

$x' = x$
$y' = 2y$

Step 2: Solving for $x$ and $y$ :

$x = x'$
$y = \frac{y'}{2}$

Step 3: Since the original point satisfies $y = \sqrt{x}$ , we substitute:

\frac{y'}{2} = \sqrt{x'}

Step 4: Multiplying both sides by 2:

y' = 2\sqrt{x'}

Result: The curve $y = \sqrt{x}$ transforms to the curve y = 2√x.

Dilation from the y-axis

A dilation from the y-axis is a horizontal transformation. It changes the x-coordinates of points while leaving the y-coordinates unchanged.

Understanding horizontal dilations

When we dilate a graph by a factor from the y-axis:

Factor greater than 1: The graph stretches horizontally (becomes wider)
Factor between 0 and 1: The graph compresses horizontally (becomes narrower)

infoNote

For example:

A dilation of factor 2 from the y-axis makes the graph twice as wide
A dilation of factor $\frac{1}{2}$ from the y-axis makes the graph half as wide

The general rule

chatImportant

For $a$ a positive constant, a dilation of factor $a$ from the y-axis follows the rule:

(x, y) \rightarrow (ax, y)

This means:

$x' = ax$ (x-coordinate is multiplied by $a$ )
$y' = y$ (y-coordinate stays the same)

Finding the equation after dilation

For the graph of $y = f(x)$ , there are two equivalent ways to find the equation after dilation from the y-axis:

Apply the transformation rule $(x, y) \rightarrow (ax, y)$ to the graph
Replace $x$ with $\frac{x}{a}$ in the original equation to get $y = f\left(\frac{x}{a}\right)$

Both methods produce the same result.

Worked example: Dilation from the y-axis

lightbulbExample

Worked Example: Dilating y = √x from the y-axis

Consider the curve $y = \sqrt{x}$ with a dilation of factor 2 from the y-axis.

Let $(x', y')$ be the image of a point $(x, y)$ on the original curve.

Step 1: Using the transformation rule:

$x' = 2x$
$y' = y$

Step 2: Solving for $x$ and $y$ :

$x = \frac{x'}{2}$
$y = y'$

Step 3: Since the original point satisfies $y = \sqrt{x}$ , we substitute:

y' = \sqrt{\frac{x'}{2}}

Result: This can be written as y = √(x/2).

Worked example: Finding equations after dilation

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Worked Example: Dilating y = 1/x²

Question: Determine the rule of the image when the graph of $y = \frac{1}{x^2}$ is dilated by a factor of 4:

a) from the x-axis

b) from the y-axis

Solution to part a) - Dilation from the x-axis

The transformation rule is $(x, y) \rightarrow (x, 4y)$ .

Let $(x', y')$ be the coordinates of the image of $(x, y)$ :

$x' = x$
$y' = 4y$

Solving for $x$ and $y$ :

$x = x'$
$y = \frac{y'}{4}$

Substituting into the original equation $y = \frac{1}{x^2}$ :

\frac{y'}{4} = \frac{1}{(x')^2}

Multiplying both sides by 4:

y' = \frac{4}{(x')^2}

Answer: The equation of the transformed function is y = 4/x²

lightbulbExample

Solution to part b) - Dilation from the y-axis

The transformation rule is $(x, y) \rightarrow (4x, y)$ .

Let $(x', y')$ be the coordinates of the image of $(x, y)$ :

$x' = 4x$
$y' = y$

Solving for $x$ and $y$ :

$x = \frac{x'}{4}$
$y = y'$

Substituting into the original equation $y = \frac{1}{x^2}$ :

y' = \frac{1}{\left(\frac{x'}{4}\right)^2}

y' = \frac{1}{\frac{(x')^2}{16}}

y' = \frac{16}{(x')^2}

Answer: The equation of the transformed function is y = 16/x²

Worked example: Finding the dilation factor

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Worked Example: Finding Dilation Factors

Question: Determine the factor of dilation when the graph of $y = \sqrt{3x}$ is obtained by dilating the graph of $y = \sqrt{x}$ :

a) from the y-axis

b) from the x-axis

Solution to part a) - From the y-axis

A dilation from the y-axis affects x-values. We need to write the transformed function showing how x has changed.

Write: $y' = \sqrt{3x'}$ where $(x', y')$ are the coordinates of the image of $(x, y)$ .

This means $x = 3x'$ and $y = y'$ (note that x is changed).

Solving for $x'$ and $y'$ :

$x' = \frac{x}{3}$
$y' = y$

The transformation rule is $(x, y) \rightarrow \left(\frac{x}{3}, y\right)$ .

Answer: The graph of $y = \sqrt{x}$ is dilated by a factor of 1/3 from the y-axis to produce the graph of $y = \sqrt{3x}$ .

lightbulbExample

Solution to part b) - From the x-axis

A dilation from the x-axis affects y-values. We need to write the transformed function showing how y has changed.

Write: $\frac{y'}{\sqrt{3}} = \sqrt{x'}$ where $(x', y')$ are the coordinates of the image of $(x, y)$ .

This means $x = x'$ and $y = \frac{y'}{\sqrt{3}}$ (note that y is changed).

Solving for $x'$ and $y'$ :

$x' = x$
$y' = \sqrt{3} \cdot y$

The transformation rule is $(x, y) \rightarrow (x, \sqrt{3} \cdot y)$ .

Answer: The graph of $y = \sqrt{x}$ is dilated by a factor of √3 from the x-axis to produce the graph of $y = \sqrt{3x}$ .

Remember!

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Key Points to Remember:

Dilation from the x-axis $(x, y) \rightarrow (x, by)$ creates vertical stretching (if $b > 1$ ) or compression (if $0 \< b \< 1$ ). The equation becomes $y = bf(x)$ .
Dilation from the y-axis $(x, y) \rightarrow (ax, y)$ creates horizontal stretching (if $a > 1$ ) or compression (if $0 \< a \< 1$ ). The equation becomes $y = f\left(\frac{x}{a}\right)$ .
To find the equation after dilation: Use the substitution method - express the primed coordinates in terms of unprimed, then solve for unprimed coordinates and substitute into the original equation.
x-axis dilations change y-coordinates only; y-axis dilations change x-coordinates only. A helpful reminder: "x-axis dilations touch y, y-axis dilations touch x."
Factor greater than 1 means stretching; factor between 0 and 1 means compression. The larger the factor, the more stretched the graph becomes.

Dilations (VCE SSCE Mathematical Methods): Revision Notes

Dilations

Introduction

Dilation from the x-axis

Understanding vertical dilations

The general rule

Finding the equation after dilation

Worked example: Dilation from the x-axis

Dilation from the y-axis

Understanding horizontal dilations

The general rule

Finding the equation after dilation

Worked example: Dilation from the y-axis

Worked example: Finding equations after dilation

Worked example: Finding the dilation factor

Remember!

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