Question

identifying the rule for a composition of transformations. which rule describes the composition of transformations that maps rectangle pqrs to pqrs? r₀, 180 ∘ t₂,0 (x, y); t₀,₂ ∘ r₀,270 (x, y); r₀,₂ ∘ t₀,180 (x, y); r₀,270 ∘ t₀,₂ (x, y)

Explanation:

Step1: Analyze translation

Translation $T_{a,b}(x,y)=(x + a,y + b)$. Rotation $R_{o,\theta}$ is rotation about the origin by an angle $\theta$. First, observe the horizontal - shift. Rectangle $PQRS$ seems to be shifted horizontally first. If we consider a translation $T_{2,0}(x,y)=(x + 2,y)$, it moves the rectangle 2 units to the right.

Step2: Analyze rotation

After the translation, a rotation of $180^{\circ}$ about the origin $R_{o,180}(x,y)=(-x,-y)$ will map the translated rectangle to $P''Q''R''S''$. The composition of transformations is first translation and then rotation. The notation for composition is read from right - to - left. So the rule is $R_{o,180}\circ T_{2,0}(x,y)$.

Answer:

$R_{o,180}\circ T_{2,0}(x,y)$