Q32. - Algebra 2

Question

For exercise 29-32, use rectangle $ABCD$ with vertices $A\left(-4,4\right),B\left(4,4\right),C\left(4,-4\right),\text{ and }D\left(-4,-4\right)$

32. What do you observe about these three matrices? Explain.

Step-by-Step Solution

Answer

It is observed that vertex matrix of image after reflection over the x-axis followed by reflection over the y-axis is same as that of matrix after $180°$ rotation about origin

Step 1 - Vertex matrix in question 29

Vertex matrix of image after reflection over the x-axis followed by reflection over the y-axis is 

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

Step 2 - Vertex matrix in question 30

Vertex matrix of image after $180°$ rotation about origin is 

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

Step 3 - Vertex matrix in question 31

Vertex matrix of image after reflection over the line $y=x$ 

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

By comparing these three matrices, it is observed that vertex matrix of image after reflection over the x-axis followed by reflection over the y-axis is same as that of matrix after $180°$ rotation about origin