Chapter 8.5, Problem 39E

To determine

To find: the vertices of the image of the square with the vertices $\left(4,3\right)$ , $\left(5,3\right)$ , $\left(4,4\right)$ and $\left(5,4\right)$ after the transformation ; reflection in the y -axis

Answer to Problem 39E

  $\left(-4,3\right)$ , $\left(-5,3\right)$ , $\left(-4,4\right)$ and $\left(-5,4\right)$

Explanation of Solution

First write the vertices $\left(4,3\right)$ , $\left(5,3\right)$ , $\left(4,4\right)$ and $\left(5,4\right)$ as column matrices as shown below:

  $\left[\begin{array}{c}4\\ 3\end{array}\right]$ , $\left[\begin{array}{c}5\\ 3\end{array}\right]$ , $\left[\begin{array}{c}4\\ 4\end{array}\right]$ and $\left[\begin{array}{c}5\\ 4\end{array}\right]$

Now, the transformation matrix for reflection in the y -axis is $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ .

Now, one by one multiply all the column matrices obtained above by the matrix $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ .

  $\begin{array}{c}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}4\\ 3\end{array}\right]=\left[\begin{array}{c}-4+0\\ 0+3\end{array}\right]\\ =\left[\begin{array}{c}-4\\ 3\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}5\\ 3\end{array}\right]=\left[\begin{array}{c}-5+0\\ 0+3\end{array}\right]\\ =\left[\begin{array}{c}-5\\ 3\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}4\\ 4\end{array}\right]=\left[\begin{array}{c}-4+0\\ 0+4\end{array}\right]\\ =\left[\begin{array}{c}-4\\ 4\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}5\\ 4\end{array}\right]=\left[\begin{array}{c}-5+0\\ 0+4\end{array}\right]\\ =\left[\begin{array}{c}-5\\ 4\end{array}\right]\end{array}$

So, the vertices of the image of the given square are $\left(-4,3\right)$ , $\left(-5,3\right)$ , $\left(-4,4\right)$ and $\left(-5,4\right)$ .

The sketch of the give square and its image is given below: