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. (Picture below)

Answer 1

The coordinates of the vertices of the reflected figure are :

R' is (3 , 7), S' is (-7 , 2), T' is (-5 , -3) ⇒the right answer is figure (d)

Explanation:

* Lets study the matrices of the reflection

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

`\left[\begin{array}{ccc}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}{ccc}-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 y-axis

by each point to find the image of each point

- The dimension of the matrix of the reflection across the y-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}{ccc}-1&0\\0&1\end{array}\right] \left[\begin{array}{ccc}-3\\7\end{array}\right]=`

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

∴ R' is (3 , 7)

∵ Point S is (7 , 2)

∴ S' =
`\left[\begin{array}{ccc}-1&0\\0&1\end{array}\right]\left[\begin{array}{ccc}2\\7\end{array}\right]=`

`\left[\begin{array}{ccc}(-1)(7)+(0)(2)\\(0)(7)+(1)(2)\end{array}\right]=\left[\begin{array}{ccc}-7\\2\end{array}\right]`

∴ S' is (-7 , 2)

∵ Point T is (5 , -3)

∴ T' =
`\left[\begin{array}{ccc}-1&0\\0&1\end{array}\right]\left[\begin{array}{ccc}5\\-3\end{array}\right]=`

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

∴ T' is (-5 , -3)

* Look to the answer and find the correct figure

- In figure (d) R' is (3 , 7), S' is (-7 , 2), T' is (-5 , -3)

∴ The right answer is figure (d)