Chapter 8.5, Problem 38E

To determine

To find: the vertices of the image of the square with the vertices $\left(1,2\right)$ , $\left(3,2\right)$ , $\left(1,4\right)$ and $\left(3,4\right)$ after the transformation; reflection in the x -axis

Answer to Problem 38E

  $\left(1,-2\right)$ , $\left(3,-2\right)$ , $\left(1,-4\right)$ and $\left(3,-4\right)$

Explanation of Solution

First write the vertices $\left(1,2\right)$ , $\left(3,2\right)$ , $\left(1,4\right)$ and $\left(3,4\right)$ as column matrices as shown below:

  $\left[\begin{array}{c}1\\ 2\end{array}\right]$ , $\left[\begin{array}{c}3\\ 2\end{array}\right]$ , $\left[\begin{array}{c}1\\ 4\end{array}\right]$ and $\left[\begin{array}{c}3\\ 4\end{array}\right]$

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

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}1+0\\ 0+\left(-2\right)\end{array}\right]\\ =\left[\begin{array}{c}1\\ -2\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}3\\ 2\end{array}\right]=\left[\begin{array}{c}3+0\\ 0+\left(-2\right)\end{array}\right]\\ =\left[\begin{array}{c}3\\ -2\end{array}\right]\end{array}$

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

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

So, the vertices of the image of the given square are $\left(1,-2\right)$ , $\left(3,-2\right)$ , $\left(1,-4\right)$ and $\left(3,-4\right)$ .

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