# Dilation

How to find the image under a dilation: formula, 3 examples, and their solutions.

## Formula

The image of a point (x, y)

under the dilation of k is

(kx, ky).

k is like a scale factor.

If |k| > 1,

then the image shows an enlargement:

the image gets bigger

and goes away from the origin.

If |k| < 1,

then the image shows a reduction:

the image gets smaller

and goes toward the origin.

## Example(3, 2), Dilation: 2

The image of (3, 2) is

under the dilation of 2.

Then the image point is,

multiply 2 to both x and y,

(2⋅3, 2⋅2).

2⋅3 = 6

2⋅2 = 4

So (6, 4) is the answer.

This is the graph of (3, 2)

and its image

under the dilation of 2:

(2⋅3, 2⋅2).

## Example(3, 2), Dilation: 1/2

The image of (3, 2) is

under the dilation of 1/2.

Then the image point is,

multiply 1/2 to both x and y,

([1/2]⋅3, [1/2]⋅2).

[1/2]⋅3 = 3/2

[1/2]⋅2 = 1

So (3/2, 1) is the answer.

This is the graph of (3, 2)

and its image

under the dilation of 1/2:

([1/2]⋅3, [1/2]⋅2).

## Example(3, 2), Dilation: -2

The image of (3, 2) is

under the dilation of -2.

Then the image point is,

multiply -2 to both x and y,

(-2⋅3, -2⋅2).

-2⋅3 = -6

-2⋅2 = -4

So (-6, -4) is the answer.

This is the graph of (3, 2)

and its image

under the dilation of -2:

(-2⋅3, -2⋅2).

As you can see,

if k is minus,

the image goes to the opposite side of the origin.