Transformation of Graphs Using Matrices -Reflection

Example:

Find the coordinates of the vertices of the image of pentagon$ABCDE$with$A(2,4),B(4,3),C(4,0),D(2,-1),\text{and}E(0,2)$after a reflection across the$y$-axis.

Write the ordered pairs as a vertex matrix.

$\left[\begin{array}{ccccc}\hfill 2& \hfill 4& \hfill 4& \hfill 2& \hfill 0\\ \hfill 4& \hfill 3& \hfill 0& \hfill -1& \hfill 2\end{array}\right]$

To reflect the pentagon$ABCDE$across the$y$-axis, multiply the vertex matrix by the reflection matrix$\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right]$.

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

Therefore, the coordinates of the vertices of the image of pentagon$ABCDE$are$A^{\text{'}}(-2,4),B^{\text{'}}(-4,3),C^{\text{'}}(-4,0),D^{\text{'}}(-2,-1),\text{and}{E}^{\text{'}}(0,2)$.

Notice that, both figures have the same size and shape.