Question

Use matrices to determine the coordinates of the vertices of the reflected figure. Then graph the pre-image and the image on the same coordinate grid.

Answer 1

Answer:

The coordinates of the vertices of the reflected figure are :

R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5) ⇒the right answer is figure (a)

Step-by-step explanation:

* Lets study the matrices of the reflection

- The matrix of the reflection across the x-axis is

 $\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]$

- Because when we reflect any point across the x-axis we

 change the sign of the y-coordinate

- The matrix of the reflection across the y-axis is

 $\left[\begin{array}{cc}-1&0\\0&1\end{array}\right]$

- Because when we reflect any point across the y-axis we

 change the sign of the x-coordinate

* Now lets solve the problem

- We will multiply the matrix of the reflection across the x-axis

 by each point to find the image of each point

- The dimension of the matrix of the reflection across the x-axis

 is 2×2 and the dimension of the matrix of each point is 2×1,

then the dimension of the matrix of each image is 2×1

∵ Point R is (-3 , 7)

∴ $R'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}-3\\7\end{array}\right]=$

  $\left[\begin{array}{c}(1)(-3)+(0)(7)\\(0)(-3)+(-1)(7)\end{array}\right]=\left[\begin{array}{c}-3\\-7\end{array}\right]$

∴ R' is (-3 , -7)

∵ Point S is (5 , 3)

∴ $S'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}5\\3\end{array}\right]=$

   $\left[\begin{array}{c}(1)(5)+(0)(3)\\(0)(5)+(-1)(3)\end{array}\right]=\left[\begin{array}{c}5\\-3\end{array}\right]$

∴ S' is (5 , -3)

∵ Point T is (6 , -5)

∴ $T'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}6\\-5\end{array}\right]=$

   $\left[\begin{array}{c}(1)(6)+(0)(-5)\\(0)(6)+(-1)(-5)\end{array}\right]=\left[\begin{array}{c\pi }6\\5\end{array}\right]$

∴ T' is (6 , 5)

* Look to the answer and find the correct figure

- In figure (d) R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5)

∴ The right answer is figure (a)