Q42. - Algebra 2

Question

CRITICAL THINKING 

Do you think a matrix exist that would represent a reflection over the line $x=3$?  If so, make a conjecture and verify it.

Step-by-Step Solution

Answer

Yes there is a matrix exist that would represent a reflection over the line $x=3$

Conjecture is:

• Let $\left(x,y\right)$ be an ordered pair then its vertex matrix will be $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$
• Before reflection over the line $x=3$, translate this line to $x=0$ or 3 units to the left. With the line all points are also shifted 3 units to the left. If the matrix is translated to 3 units left then 3 is subtracted from first row.
  Vertex matrix after translation becomes $\left[\begin{array}{c}x-3\\ y\\ \end{array}\right]$
• Now reflect the point over the line $x=0$ or y-axis. Vertex matrix of new point after reflection over the y -axis is product matrix $\left[\begin{array}{cc}-1& 0\\ 0& 1\\ & \end{array}\right]$ with vertex matrix of point as shown below
  $\left[\begin{array}{cc}-1& 0\\ 0& 1\\ & \end{array}\right]\times \left[\begin{array}{c}x-3\\ y\\ \end{array}\right]=\left[\begin{array}{c}-x+3\\ y\\ \end{array}\right]$

• Finally translate line $x=0$ to the line $x=3$ or 3 units to the right. With the line all points are also shifted 3 units to the right. If the matrix is translated to 3 units right then 3 is added in first row. Vertex matrix after translation becomes
  $\left[\begin{array}{c}-x+3+3\\ y\\ \end{array}\right]=\left[\begin{array}{c}-x+6\\ y\\ \end{array}\right]$
  Thus coordinate of a point $\left(x,y\right)$ after reflection over the line $x=3$ becomes $\left(-x+6,y\right)$
  Now to verify, let the point to be reflected over the line $x=3$ is $\left(1,2\right)$
  Then after reflection its new coordinate $\left(x,y\right)$ will be calculated as
  $\begin{array}{c}\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-1+6\\ 2\end{array}\right]\\ =\left[\begin{array}{c}5\\ 2\end{array}\right]\end{array}$
  Thus new coordinate is $\left(5,2\right)$
  Now plot these point on coordinate plane to check it is correct or not

From graph it is clear that point $\left(5,2\right)$ is reflection of point $\left(1,2\right)$ over the line $x=3$ 

Hence verified.

Step 1 - Vertex matrix

Let $\left(x,y\right)$ be an ordered pair then its vertex matrix will be $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$

Step 2 - Translate original point

Before reflection over the line $x=3$, translate this line to $x=0$ or 3 units to the left. With the line all points are also shifted 3 units to the left

If the matrix is translated to 3 units left then 3 is subtracted from first row.

Vertex matrix after translation becomes 

$\left[\begin{array}{c}x-3\\ y\\ \end{array}\right]$

Step 3 - Vertex matrix after reflection over the y - a x i s

Now reflect the point over the line $x=0$ or y-axis.

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

$\left[\begin{array}{cc}-1& 0\\ 0& 1\\ & \end{array}\right]\times \left[\begin{array}{c}x-3\\ y\\ \end{array}\right]=\left[\begin{array}{c}-x+3\\ y\\ \end{array}\right]$

Step 4 - Translate resulting point

Finally translate line $x=0$ to the line $x=3$ or 3 units to the right. With the line all points are also shifted 3 units to the right

If the matrix is translated to 3 units right then 3 is added in first row.

Vertex matrix after translation becomes 

$\left[\begin{array}{c}-x+3+3\\ y\\ \end{array}\right]=\left[\begin{array}{c}-x+6\\ y\\ \end{array}\right]$

Thus coordinate of a point $\left(x,y\right)$ after reflection over the line $x=3$ becomes $\left(-x+6,y\right)$

Step 5 - Verification

Let the point to be reflected over the line $x=3$ is $\left(1,2\right)$

Then after reflection its new coordinate $\left(x,y\right)$ will be calculated as

 $\begin{array}{c}\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-1+6\\ 2\end{array}\right]\\ =\left[\begin{array}{c}5\\ 2\end{array}\right]\end{array}$

Thus new coordinate is $\left(5,2\right)$

Now plot these point on coordinate plane to check it is correct or not

From graph it is clear that point $\left(5,2\right)$ is reflection of point $\left(1,2\right)$ over the line $x=3$