Question

select the sequence of transformations that are the inverse of rotating a figure 90° counterclockwise and then reflecting it over the x - axis.
a reflect it over the line y = - x and then reflect it over the x - axis.
b rotate the image 270° counterclockwise and then reflect it in the x - axis.
c rotate the image 180° counterclockwise and then reflect it in the x - axis.
d reflecting it over the x - axis, then over the y - axis and finally over the line y = x.

Explanation:

Step1: Find the inverse of rotation

The inverse of a \(90^{\circ}\) counter - clockwise rotation is a \(90^{\circ}\) clockwise rotation. A \(90^{\circ}\) clockwise rotation is equivalent to a \(270^{\circ}\) counter - clockwise rotation.

Step2: Find the inverse of reflection

The inverse of a reflection over the \(x\) - axis is a reflection over the \(x\) - axis itself (since reflecting a figure over the \(x\) - axis twice brings it back to its original position). But when we consider the combined transformation, we need to reverse the order of operations.

Let's consider the general transformation. If we have a transformation \(T = R_{x}\circ R_{90}\) (where \(R_{90}\) is a \(90^{\circ}\) counter - clockwise rotation and \(R_{x}\) is a reflection over the \(x\) - axis). The inverse transformation \(T^{-1}=(R_{x}\circ R_{90})^{-1}=R_{90}^{-1}\circ R_{x}^{-1}\).

The rotation matrix for a \(90^{\circ}\) counter - clockwise rotation of a point \((x,y)\) is \(

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

\to

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

\). The rotation matrix for a \(270^{\circ}\) counter - clockwise rotation (which is the inverse of \(90^{\circ}\) counter - clockwise rotation) of a point \((x,y)\) is \(

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

\to

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

\). The reflection of a point \((x,y)\) over the \(x\) - axis is \((x,y)\to(x, - y)\) and its inverse (since \(R_{x}\circ R_{x}(x,y)=(x,y)\)) is also \(R_{x}\).

If we first rotate a figure \(270^{\circ}\) counter - clockwise (inverse of \(90^{\circ}\) counter - clockwise rotation) and then reflect it over the \(x\) - axis (inverse of the second operation in the original transformation), we get the inverse of the original transformation \(R_{x}\circ R_{90}\).

Answer:

B. Rotate the image \(270^{\circ}\) counterclockwise and then reflect it in the \(x\) - axis.