Chapter 12.5, Problem 32PPSE

Solution

To find: The coordinates of image after the reflection.

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

Given information:

Given, to use the $△ABC$

below:

  

The figure is reflected across the x -axis.

Calculation:

The coordinates of vertices are given by $A\left(-3,1\right),B\left(1,2\right),C\left(-1,4\right)$

The coordinates in the matrix form is given by:

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

The reflection of a figure across x -axis is given by multiplying the matrix with:

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

The coordinates after the reflection are given by:

  $\begin{array}{l}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{ccc}-3& 1& -1\\ 1& 2& 4\end{array}\right]=\left[\begin{array}{ccc}\left(1\right)\left(-3\right)+\left(0\right)\left(1\right)& \left(1\right)\left(1\right)+\left(0\right)\left(2\right)& \left(1\right)\left(-1\right)+\left(0\right)\left(4\right)\\ \left(0\right)\left(-3\right)+\left(-1\right)\left(1\right)& \left(0\right)\left(1\right)+\left(-1\right)\left(2\right)& \left(0\right)\left(-1\right)+\left(-1\right)\left(4\right)\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{ccc}-3& 1& -1\\ 1& 2& 4\end{array}\right]=\left[\begin{array}{ccc}-3+0& 1+0& -1+0\\ 0-1& 0-2& 0-4\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{ccc}-3& 1& -1\\ 1& 2& 4\end{array}\right]=\left[\begin{array}{ccc}-3& 1& -1\\ -1& -2& -4\end{array}\right]\end{array}$

The coordinates after reflection aregiven by the matrix:

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

Conclusion:

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