Question

the transformation $t = \

$$\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}$$

$ is applied the figure below.
graph of a triangle with vertices a, b, c on a coordinate plane
which transformation is applied to the figure?
reflect across the y-axis
rotate 90° counterclockwise about the origin
reflect across $y = -x$
rotate 180° counterclockwise about the origin

Explanation:

Step1: Recall transformation matrices

The general transformation matrix for a point \((x,y)\) is \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=T

$$\begin{bmatrix}x\\y\end{bmatrix}$$

\). For a rotation of \(90^{\circ}\) counter - clockwise about the origin, the transformation matrix is \(T =

$$\begin{bmatrix}0&- 1\\1&0\end{bmatrix}$$

\). When we apply this matrix to a point \((x,y)\), we get \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-y\\x\end{bmatrix}$$

\).

Step2: Analyze other transformation matrices

• Reflection across the \(y\) - axis: The matrix is \(

$$\begin{bmatrix}-1&0\\0&1\end{bmatrix}$$

\), and \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}-1&0\\0&1\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-x\\y\end{bmatrix}$$

\).

• Reflection across \(y=-x\): The matrix is \(

$$\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$$

\), and \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-y\\-x\end{bmatrix}$$

\).

• Rotation of \(180^{\circ}\) counter - clockwise about the origin: The matrix is \(

$$\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$$

\), and \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-x\\-y\end{bmatrix}$$

\).

Answer:

rotate \(90^{\circ}\) counterclockwise about the origin