Transformation of Graphs Using Matrices - Rotations

A rotation is a transformation in a plane that turns every point of a preimage through a specified angle and direction about a fixed point. The fixed point is called the center of rotation . The amount of rotation is called the angle of rotation and it is measured in degrees.

A rotation maps every point of a preimage to an image rotated about a center point, usually the origin, using a rotation matrix.

Use the following rules to rotate the figure for a specified rotation. To rotate counterclockwise about the origin, multiply the vertex matrix by the given matrix.

$\begin{array}{|cccc|}\hline \hfill \text{Angle}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Rotation}& \hfill 90°& \hfill 180°& \hfill 270°\\ \hfill \begin{array}{l}\hfill \text{Rotation}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Matrix}\\ \hfill \left(\text{Multiply}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{left}\right)\end{array}& \hfill \left[\begin{array}{cc}\hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]& \hfill \left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]& \hfill \left[\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill -1& \hfill 0\end{array}\right]\\ \hline\end{array}$

Example:

Find the coordinates of the vertices of the image $\Delta XYZ$ with $X\left(1,2\right),Y\left(3,5\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Z\left(-3,4\right)$ after it is rotated $180°$ counterclockwise about the origin.

Write the ordered pairs as a vertex matrix.

$\left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill -3\\ \hfill 2& \hfill 5& \hfill 4\end{array}\right]$

To rotate the $\Delta XYZ$ 180° counterclockwise about the origin, multiply the vertex matrix by the rotation matrix, $\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]$ .

$\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]\cdot \left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill -3\\ \hfill 2& \hfill 5& \hfill 4\end{array}\right]=\left[\begin{array}{ccc}\hfill -1& \hfill -3& \hfill 3\\ \hfill -2& \hfill -5& \hfill -4\end{array}\right]$

Therefore, the coordinates of the vertices of $\Delta {X}^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ are ${X}^{\text{'}}\left(-1,-2\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Y}^{\text{'}}\left(-3,-5\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Z}^{\text{'}}\left(3,-4\right)$ .

Notice that the image $\Delta {X}^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ is congruent to the preimage $\left(\Delta XYZ\right)$ . Both figures have the same size and same shape.