# How to Graph Transformation on the Coordinate Plane: Rotation

This article teaches you how to graph Rotations on the coordinate plane in a few simple steps.

## Step by step guide to graph Transformation: Rotation

A rotation is a transformation in which the object is rotated about a fixed point. The direction of rotation can be clockwise or counterclockwise. In this case, an image and its pre-image have the same shape and size, but the pre-image may be turned in different directions

We can rotate shapes on the coordinate 90, 180, or 270 degrees counterclockwise around the origin using three basic rules:

• For rotating a shape 90 degrees counterclockwise:$$(x, y)→(-y, x)$$
• For rotating a shape 180 degrees: $$(x, y)→(-x, -y)$$
• For rotating a shape 270 degrees counterclockwise: $$(x, y)→(y, -x)$$

Remember that:

• You should be able to assume the center of rotation to be the origin when working on the coordinate plane unless otherwise stated.
• You should be able to assume that, unless otherwise stated, a positive angle of rotation rotates the figure counterclockwise and a negative angle rotates it clockwise.
• You need to be able to recognize angles of certain sizes when working with rotation. $$(90^{\circ}, 180^{\circ}, 270^{\circ}, …)$$
• You must be able to understand the directionality of a unit circle. (the circle with a radius length of 1 unit)
• You must know that rotation on a coordinate grid is considered to be counterclockwise unless otherwise stated.

### Transformation: Rotation Example 1:

Triangle $$ABC$$ has vertices $$A=(3, 4), B=(4, 1), C=(1, 2)$$. Graph triangle $$ABC$$ and its image after a rotation of $$90^{\circ}$$ about the origin.

Solution:

The rule for rotating a shape 90 degrees is $$(x, y)→(-y, x)$$

$$A=(3, 4)→A^\prime=(-4, 3)$$

$$B=(4, 1)→B^\prime=(-1, 4)$$

$$C=(1, 2)→C^\prime=(-2, 1)$$

Graph Triangle $$ABC$$ and its image $$A^\prime B^\prime C^\prime$$.

### Transformation: Rotation Example 2:

Graph the image of the figure after a rotation of $$270^{\circ}$$ about the origin.

Solution:

The rule for rotating a shape 270 degrees is $$(x, y)→(y, -x)$$

$$A=(-4, 1)→A^\prime=(1, 4)$$

$$B=(-3, 3)→B^\prime=(3, 3)$$

$$C=(-1, 4)→C^\prime=(4, 1)$$

$$D=(0, 2)→D^\prime=(2, 0)$$

$$C=(-2, 1)→C^\prime=(1, 2)$$

Graph the figure $$ABCDE$$ and its image $$A^\prime B^\prime C^\prime D^\prime E^\prime$$.

## Exercises for Transformation: Rotation

### Graph the image of the figure using the transformation given.

1.$$\color{blue}{Rotation 180^{\circ}}$$

2.$$\color{blue}{Rotation 90^{\circ}}$$

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