Question

6 describe, in coordinate notation, the following sequence of transformations a rotation clockwise 90 degrees and a reflection across the y - axis. use the diagram below as a reference.

Explanation:

Step1: Rotation formula

The formula for a \(90^{\circ}\) clockwise rotation is \((x,y)\to(y, -x)\).

Step2: Reflection formula

The formula for a reflection across the \(y -\)axis is \((x,y)\to(-x,y)\).
Let's first apply the \(90^{\circ}\) clockwise rotation: \((x,y)\to(y, -x)\). Then apply the reflection across the \(y -\)axis: \((y,-x)\to(-y,-x)\). But if we consider the composition of these two transformations in terms of coordinate rules.
Another way:
Let's take a general point \((x,y)\).
For a \(90^{\circ}\) clockwise rotation:
The rotation matrix for a \(90^{\circ}\) clockwise rotation is \(

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

\). If we have a point \(

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

\), after rotation it becomes \(

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

\).
For a reflection across the \(y -\)axis, the transformation matrix is \(

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

\). Applying this to \(

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

\) gives \(

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

\). But if we consider the net - effect on the original point \((x,y)\) in terms of the given options.
Let's use a sample point. Suppose we have a point \((1,0)\).
After \(90^{\circ}\) clockwise rotation: \((1,0)\to(0, - 1)\). After reflection across the \(y -\)axis: \((0,-1)\to(0, - 1)\).
Let's check each option:

• Option 1: \((x,y)\): If we take \((1,0)\), it remains \((1,0)\) (not correct).
• Option 2: \((-x,y)\): For \((1,0)\) it would be \((-1,0)\) (not correct).
• Option 3: \((y,x)\): For \((1,0)\) it would be \((0,1)\) (not correct).
• Option 4: Let's use the rotation and reflection rules.

The rule for a \(90^{\circ}\) clockwise rotation: \((x,y)\to(y,-x)\).

Answer:

\((y, -x)\)