Question

For exercises 7-10, use the rectangle at the right

9. Find the coordinates of the image after a reflection over the $x$-axis

Short answer

Coordinates of the image after a reflection over x axis are $A\text{'}\left(0,-4\right),B\text{'}\left(5,-4\right),C\text{'}\left(5,0\right),\text{and}D\text{'}\left(0,0\right)$

Step-by-step solution

- Write Coordinates of rectangle 

From given image, coordinates of the vertices of rectangle are $A\left(0,4\right),B\left(5,4\right),C\left(5,0\right),\text{and}D\left(0,0\right)$

- Vertex matrix after reflection

Vertex matrix of given rectangle is $\left[\begin{array}{cccc}0& 5& 5& 0\\ 4& 4& 0& 0\\ & & & \end{array}\right]$

So vertex matrix of new rectangle after reflection over the x -axis is product matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$with vertex matrix of given rectangle as shown below

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

- Coordinates of image after reflection

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of image after reflection are

$A\text{'}\left(0,-4\right),B\text{'}\left(5,-4\right),C\text{'}\left(5,0\right),\text{and}D\text{'}\left(0,0\right)$