Question

identifying the rule for a composition of transformations. which rule describes the composition of transformations that maps rectangle pqrs to pqrs? options: ( r_{o,2} circ t_{0,180^circ}(x,y) ), ( t_{0,2} circ r_{o,270^circ}(x,y) ), ( r_{o,270^circ} circ t_{0,2}(x,y) ) (marked with x), ( r_{o,180^circ} circ t_{2,0}(x,y) )

Explanation:

Step1: Recall Transformation Rules

• Rotation \( R_{O, 270^\circ}(x, y) \): Rotates a point \((x, y)\) 270° counterclockwise about the origin, rule is \((x, y) \to (y, -x)\).
• Translation \( T_{a, b}(x, y) \): Translates a point \((x, y)\) by \(a\) units horizontally and \(b\) units vertically, rule is \((x, y) \to (x + a, y + b)\).
• For composition \( A \circ B \), we apply \( B \) first, then \( A \).

Step2: Analyze the Rectangle's Transformation

• Let's take a vertex of \( PQRS \), say \( P \). From the graph, \( P \) has coordinates (let's assume) \( (-3, -5) \) (estimating from the grid).
• First, check the translation then rotation vs rotation then translation.
• Let's test \( T_{0, 2} \circ R_{O, 270^\circ}(x, y) \):
• Apply \( R_{O, 270^\circ} \) first: \( (x, y) \to (y, -x) \).
• Then apply \( T_{0, 2} \): \( (y, -x) \to (y, -x + 2) \).
• Test with \( P(-3, -5) \):
• \( R_{O, 270^\circ}(-3, -5) = (-5, 3) \) (since \( x=-3, y=-5 \), so \( (y, -x)=(-5, 3) \)).
• Then \( T_{0, 2}(-5, 3) = (-5, 3 + 2) = (-5, 5) \), which matches \( P'' \) (from the graph, \( P'' \) is at (-5, 5)).
• Let's check the other option we had wrong: \( R_{O, 270^\circ} \circ T_{0, 2}(x, y) \) would apply translation first: \( (x, y) \to (x, y + 2) \), then rotation: \( (x, y + 2) \to (y + 2, -x) \). For \( P(-3, -5) \), translation gives \( (-3, -3) \), rotation gives \( (-3, 3) \), which is not \( P'' \).
• Now check the composition order: \( T_{0, 2} \circ R_{O, 270^\circ}(x, y) \) means rotate first, then translate up 2 units. Which worked for \( P \).

Answer:

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