Question

For Exercises 1-3, reflect square ABCDEwith vertices $\mathbf{A}\left(1,2\right)$, $\mathbf{B}\left(4,-1\right)$, $\mathbf{C}\left(1,-4\right)$and$\mathbf{D}\left(-2,-1\right)$ over the y-axis.

2. Find the coordinates of $\mathbf{A}\mathbf{\text{'}}\mathbf{B}\mathbf{\text{'}}\mathbf{C}\mathbf{\text{'}}\mathbf{D}\mathbf{\text{'}}$.

Short answer

The coordinates of$A\text{'}B\text{'}C\text{'}D\text{'}$ are:$\mathbf{A}\mathbf{\text{'}}\left(-1,2\right)$ ,$\mathbf{B}\mathbf{\text{'}}\left(-4,-1\right)$ ,$\mathbf{C}\mathbf{\text{'}}\left(-1,-4\right)$ and .$\mathbf{D}\mathbf{\text{'}}\left(2,-1\right)$

Step-by-step solution

­- Description of step.

The vertex matrix for the square is given by:

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

­- Description of step.

As, the given square$ABCD$ is to be reflected over the Y-axis, therefore multiply the matrix $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$on the left with the vertex matrix for the squareABCD to get the vertex matrix for square $A\text{'}B\text{'}C\text{'}D\text{'}$.

Therefore, it can be obtained that:

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

­- Description of step.

Therefore, the coordinates of$A\text{'}B\text{'}C\text{'}D\text{'}$ are: $A\text{'}\left(-1,2\right)$,$B\text{'}\left(-4,-1\right)$ ,$C\text{'}\left(-1,-4\right)$ and .$D\text{'}\left(2,-1\right)$