# How to find A Reflection Image

To find a reflection image of a point or a shape,  there are some rules you can follow although it is not always possible to follow a rule.

First, let us start with all situations where you can just follow a rule to quickly find the location of the reflected image on the coordinate system.

Keep in mind that once you know how to find a reflection image for a point, you can easily find it for a shape. For example, if the shape has 3 points, just find the reflected image for all 3 points and then connect these points with a line.

## How to find a reflection image using the x-axis or the y-axis

Reflection of a point across the x-axis

The coordinate of point P is (3, 3) and the coordinate of the reflected image P’ is (3, -3)

The coordinate of point A is (-2, 4) and the coordinate of the reflected image A’ is (-2, -4)

What do you notice about the x-coordinate and y-coordinate of the preimage and image?

We see that the x-coordinate of the preimage stays the same, but the y-coordinate of the image is the opposite of the y-coordinate of the preimage.

In general, when reflecting a point across the x-axis, if the coordinate of the preimage is (x , y), then the coordinate of the reflected image is (x , -y)

Reflection of a point across the y-axis

The coordinate of point P is (-4, 3) and the coordinate of the reflected image P’ is (4, 3)

The coordinate of point A is (2, -4) and the coordinate of the reflected image A’ is (-2, -4)

What do you notice about the x-coordinate and y-coordinate of the preimage and image?

We see that the y-coordinate of the preimage stays the same, but the x-coordinate of the image is the opposite of the x-coordinate of the preimage.

In general, when reflecting a point across the y-axis,  if the coordinate of the preimage is (x , y), then the coordinate of the reflected image is (-x , y)

## How to find a reflection image using the lines y = x and y = -x

Reflection of a point across the line y = x

The coordinate of point P is (1, 4) and the coordinate of the reflected image P’ is (4, 1)

The coordinate of point A is (-5, -2) and the coordinate of the reflected image A’ is (-2, -5)

Just swap the x-coordinate with the y-coordinate

In general, when reflecting a point across the line y = x,  if the coordinate of the preimage is (x , y), then the coordinate of the reflected image is (y, x)

Reflection of a point across the line y = -x

The coordinate of point P is (2, 4) and the coordinate of the reflected image P’ is (-4, -2)

The coordinate of point A is (4, -1) and the coordinate of the reflected image A’ is (1, -4)

Therefore, swap the x-coordinate with the y-coordinate and then take the opposite of both coordinates.

In general, when reflecting a point across the line y = -x,  if the coordinate of the preimage is (x , y), then the coordinate of the reflected image is (-y, -x)

## Reflection of a point in the origin

The coordinate of point P is (5, 2) and the coordinate of the reflected image P’ is (-5, -2)

The coordinate of point A is (-2, 3) and the coordinate of the reflected image A’ is (2, -3)

Just take the opposite of both coordinates.

In general, when reflecting a point in the origin,  if the coordinate of the preimage is (x , y), then the coordinate of the reflected image is (-x, -y)

When reflecting a point across any line, just keep this mind. The distance between the point and the line and the reflected point and the line is exactly the same.

When reflecting a point A in any point P, just keep this mind these two things.

The distance between point A and point P and the reflected point A’ and point P is exactly the same.

Point P, the reflected point A’ and point A are colinear.