Dilation

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

Formula

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 1

Example

Solution

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.

Graph

This is the graph of (3, 2)
and its image
under the dilation of 2:
(2⋅3, 2⋅2).

Example 2

Example

Solution

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.

Graph

This is the graph of (3, 2)
and its image
under the dilation of 1/2:
([1/2]⋅3, [1/2]⋅2).

Example 3

Example

Solution

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.

Graph

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.