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 the order of transformations

First, consider the rotation \(R_{0,270^{\circ}}\) which rotates a point \((x,y)\) to \((y, - x)\). Then, consider the translation \(T_{-2,0}(x,y)=(x - 2,y)\).
If we first rotate a point \((x,y)\) by \(270^{\circ}\) around the origin \(R_{0,270^{\circ}}(x,y)=(y,-x)\), and then translate it by \(T_{-2,0}(y,-x)=(y - 2,-x)\).
If we first translate \(T_{-2,0}(x,y)=(x - 2,y)\) and then rotate \(R_{0,270^{\circ}}(x - 2,y)=(y,-(x - 2))=(y,-x + 2)\) which is not the correct transformation.
The reflection \(r_{y - axis}(x,y)=(-x,y)\), then \(R_{0,270^{\circ}}(-x,y)=(y,x)\) which is not correct.
If we first rotate \(R_{0,270^{\circ}}(x,y)=(y,-x)\) and then reflect over \(y - axis\) \(r_{y - axis}(y,-x)=(-y,-x)\) which is not correct.

Step2: Check the composition \(T_{-2,0}\circ R_{0,270^{\circ}}(x,y)\)

Let's take a vertex of the pre - image. Suppose \(P=(1,1)\).
First, \(R_{0,270^{\circ}}(1,1)=(1,-1)\) (using the rotation formula \((x,y)\to(y,-x)\) for \(270^{\circ}\) rotation about the origin).
Then \(T_{-2,0}(1,-1)=(1 - 2,-1)=(-1,-1)\)

Answer:

\(T_{-2,0}\circ R_{0,270^{\circ}}(x,y)\)