Chapter 4.4, Problem 38PPS

To determine

To find: The coordinates of the image after transformation.

Answer to Problem 38PPS

The new coordinate of the image after transformation are $D\left(2,-3\right),E\left(5,-2\right),F\left(4,2\right),G\left(1,1\right)$ .

Explanation of Solution

Given:

The given image is shown in Figure 1

  

Figure 1

The image is rotated $90°$ about the origin and then reflected in the $y$ axis.

Calculation:

Consider the vertices of the image are,

  $D\left(-2,-3\right),E\left(-5,-2\right),F\left(-4,2\right),G\left(-1,1\right)$

The vertices in the matrix form are,

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

Consider the transformation matrix for the reflection across $x$ axis is,

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Consider the matrix for the reflection $y=x$ is,

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

Consider the matrix for the rotation of $90°$ about the origin is,

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

The graph for the coordinates after transformation is shown in Figure 1

  

Figure 1

Then, the coordinates of the transformed image is,

  $D\left(2,-3\right),E\left(5,-2\right),F\left(4,2\right),G\left(1,1\right)$