Q38. - Algebra 2

Question

FOOTPRINTS

For exercises 38-41, use the following information.

The combination of a reflection and a translation is called a glide reflection.

An example is a set of footprints.

38. Describe the reflection and transformation combination shown at the right.

Step-by-Step Solution

Answer

From image, point is reflected over x-axis and then translated 6 units to the right.

Step 1 - Vertex matrix of a point

An ordered pair $\left(x,y\right)$ is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ which is known as vertex matrix of an ordered pair.

For the coordinates $A\left(5,-2\right)$ and $B\left(11,2\right)$, vertex matrices are $\left[\begin{array}{c}5\\ -2\\ \end{array}\right]\text{ and }\left[\begin{array}{c}11\\ 2\\ \end{array}\right]$ respectively.

Step 2 - Vertex matrix after reflection over the x - a x i s

From image point is reflected over x-axis.

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

$\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]\times \left[\begin{array}{c}x\\ y\\ \end{array}\right]=\left[\begin{array}{c}x\\ -y\\ \end{array}\right]$

So for point $A\left(5,-2\right)$, vertex matrix after reflection over the x -axis is $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]\times \left[\begin{array}{c}5\\ -2\\ \end{array}\right]=\left[\begin{array}{c}5\\ 2\\ \end{array}\right]$

Step 3 - Vertex matrix after translation

If the matrix is translated to $x\text{'}$ units right or left then $x\text{'}$ is added or subtracted from first row respectively.

If the matrix is translated to $y\text{'}$ units up or down then $y\text{'}$ is added or subtracted from second row respectively.

Thus translation matrix is of the form $\left[\begin{array}{c}x\text{'}\\ y\text{'}\\ \end{array}\right]$

As, vertex matrix after translation is sum of vertex matrix with translated matrix as shown below

$\left[\begin{array}{c}x\\ -y\\ \end{array}\right]+\left[\begin{array}{c}x\text{'}\\ y\text{'}\\ \end{array}\right]=\left[\begin{array}{c}x+x\text{'}\\ -y+y\text{'}\\ \end{array}\right]$

As for point $A\left(5,-2\right)$ vertex matrix after translation is sum of vertex matrix $\left[\begin{array}{c}5\\ 2\\ \end{array}\right]$ with translated matrix $\left[\begin{array}{c}x\text{'}\\ y\text{'}\\ \end{array}\right]$, but it is equal to vertex matrix for point $B\left(11,2\right)$, so

 $\begin{array}{c}\left[\begin{array}{c}5\\ 2\end{array}\right]+\left[\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right]=\left[\begin{array}{c}11\\ 2\end{array}\right]\\ \left[\begin{array}{c}5+x\text{'}\\ 2+y\text{'}\end{array}\right]=\left[\begin{array}{c}11\\ 2\end{array}\right]\end{array}$

Equating elements of theses to matrices to get value of $x\text{'}\text{ and }y\text{'}$

$\begin{array}{c}5+x\text{'}=11\\ x\text{'}=6\end{array}$ and $\begin{array}{c}2+y\text{'}=2\\ y\text{'}=0\end{array}$

Thus, given point is translated 6 units to the right.