How to Reflect a Shape in y = −x Using Cartesian Coordinates

Reflecting a Shape in y = −x Using Cartesian Coordinates

A shape can be reflected in the line y = −x.

If point on a shape is reflected in the line y = −x:

The image below shows a point on a shape being reflected in the line y = −x:

  • The point A has Cartesian coordinates (−3, 5).

  • The reflected point A' has Cartesian coordinates (−5, 3).

    The x-coordinate of A (−3) has its sign changed (3) and becomes the y-coordinate of A'. The y-coordinate of A (5) has its sign changed (−5) and becomes the x-coordinate of A'.

How to Reflect a Shape in y = −x Using Cartesian Coordinates

Reflecting a shape in the line y = −x using Cartesian coordinates is easy.

Question

Reflect the shape below in the line y = −x.

Step-by-Step:

1

Find the Cartesian coordinates of each point on the shape.

Write the x-coordinates and y-coordinates of each point.

Point x-coordinate y-coordinate
A −6 3
B −3 2
C −6 1

2

Change the sign of both coordinates.

Make them negative if they are positive and positive if they are negative.

Point x-coordinate y-coordinate −x-coordinate −y-coordinate
A −6 3 6 −3
B −3 2 3 −2
C −6 1 6 −1

3

Using the new coordinates from Step 2 (that have had their signs changed), make the x-coordinate the y-coordinate and the y-coordinate the x-coordinate.

−x-coordinate −y-coordinate Point x-coordinate y-coordinate
6 −3 A' −3 6
3 −2 B' −2 3
6 −1 C' −1 6

4

Plot the reflected points and draw in the shape.

Answer:

We have reflected the shape in the line y = −x.

A Formula to Reflect a Point in y = −x Using Cartesian Coordinates

In general, we write Cartesian coordinates as:

x is the x-coordinate. y is the y-coordinate. x and y can taken any number.

The reflected point has Cartesian coordinates:

The image below shows a general Cartesian coordinate being reflected in the line y = −x:

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The slider below shows another real example of how to reflect a shape in the line y = −x using Cartesian coordinates.

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See Also

What is geometry? What are transformations? What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate?