Transformation of Graphs Using Matrices - Translation

Example:

Find the coordinates of the vertices of triangle $TRI$ with $T(2,-1),R(4,3)\text{and}I(-3,-2)$ if it is to be moved $5$ units to the left and $2$ units up.

Write the coordinates of the triangle as a coordinate matrix.

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

Translating the triangle $5$ units to the left means that each $x$ -coordinate decreases by $5$ .

Translating the triangle $2$ units up means that each $y$ -coordinate increases by $2$ .

The translation matrix that will do this is

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

To find the coordinates of the vertices of the translated triangle $T^{\text{'}}{R}^{\text{'}}{I}^{\text{'}}$ add the translation matrix to the coordinate matrix:

    $\left[\begin{array}{ccc}\hfill 2& \hfill 4& \hfill -3\\ \hfill -1& \hfill 3& \hfill -2\end{array}\right]+\left[\begin{array}{ccc}\hfill -5& \hfill -5& \hfill -5\\ \hfill 2& \hfill 2& \hfill 2\end{array}\right]=\left[\begin{array}{ccc}\hfill -3& \hfill -1& \hfill -8\\ \hfill 1& \hfill 5& \hfill 0\end{array}\right]$

The coordinates of the vertices of $T^{\text{'}}{R}^{\text{'}}{I}^{\text{'}}$ are $T^{\text{'}}(-3,1),R^{\text{'}}(-1,5)\text{and}{I}^{\text{'}}(-8,0)$ .