Chapter 4.4, Problem 37PPS

To determine

To find: The coordinates of the image after transformation.

Answer to Problem 37PPS

The new coordinate of the image after transformation are $X\left(5,-8\right),Y\left(3,-6\right),Z\left(7,-5\right)$ .

Explanation of Solution

Given:

The given image is shown in Figure 1

  

Figure 1

The image is reflected 4 units to the left and 2 units in the upward direction.

Calculation:

Consider the vertices of the image are,

  $X\left(-4,3\right),Y\left(-2,1\right),Z\left(-1,5\right)$

The vertices in the matrix form are,

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

Consider the transformation matrix is,

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

Then, the vertices after transformation is,

  $\left[\begin{array}{ccc}-4& -2& -1\\ 3& 1& 5\end{array}\right]+\left[\begin{array}{ccc}-4& -4& -4\\ 2& 2& 2\end{array}\right]=\left[\begin{array}{ccc}-8& -6& -5\\ 5& 3& 7\end{array}\right]$

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 is,

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}-8& -6& -5\\ 5& 3& 7\end{array}\right]=\left[\begin{array}{ccc}5& 3& 7\\ -8& -6& -5\end{array}\right]$

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

  

Figure 1

Then, the coordinates of the transformed image is,

  $X\left(5,-8\right),Y\left(3,-6\right),Z\left(7,-5\right)$