Question

which rule describes a composition of transformations that maps pre - image pqrs to image pqrs?
$r_{0,270^{circ}}circ t_{- 2,0}(x,y)$
$t_{-2,0}circ r_{0,270^{circ}}(x,y)$
$r_{0,270^{circ}}circ r_{y - axis}(x,y)$
$r_{y - axis}circ r_{0,270^{circ}}(x,y)$

Explanation:

Step1: Analyze rotation

First, consider the rotation $R_{0,270^{\circ}}(x,y)$. A $270 - degree$ counter - clockwise rotation about the origin has the rule $(x,y)\to(y, - x)$.

Step2: Analyze reflection

Then, consider the reflection $r_{y - axis}(x,y)$ which has the rule $(x,y)\to(-x,y)$. If we first rotate a point $(x,y)$ by $270^{\circ}$ counter - clockwise about the origin to get $(y, - x)$ and then reflect it over the $y$ - axis, we get $(-y,-x)$. By observing the pre - image $PQRS$ and image $P''Q''R''S''$, this sequence of transformations matches. If we were to first translate or perform other combinations, they would not map the pre - image to the image as shown.

Answer:

D. $r_{y - axis}\circ R_{0,270^{\circ}}(x,y)$