Question

which rule describes the composition of transformations that maps △jkl to △jkl? options: $t_{0,-2} \circ r_{0,90^{\circ}}(x, y)$, $r_{0,90^{\circ}} \circ t_{-2,0}(x, y)$, $t_{-2,0} \circ r_{0,90^{\circ}}(x, y)$, $r_{0,90^{\circ}} \circ t_{0,-2}(x, y)$ (accompanied by a grid with triangles)

Explanation:

To solve this, we analyze the composition of transformations (rotation \( R_{0,90^\circ} \) and translation \( T \)):

Step 1: Recall Transformation Order

In function composition \( f \circ g(x) \), we apply \( g \) first, then \( f \). For a rotation \( R_{0,90^\circ} \) (90° counterclockwise about the origin) and translation \( T_{h,k}(x,y)=(x+h,y+k) \), we need to determine the order (rotate first or translate first) that maps \( \triangle JKL \) to \( \triangle J'K'L' \).

Step 2: Analyze Rotation Then Translation

A 90° counterclockwise rotation about the origin has the rule \( R_{0,90^\circ}(x,y)=(-y,x) \). After rotating, we translate. The translation \( T_{0,-2} \) means shifting down 2 units ( \( (x,y) \to (x,y-2) \) ).

For the composition \( T_{0,-2} \circ R_{0,90^\circ}(x,y) \):

1. Apply \( R_{0,90^\circ} \): \( (x,y) \to (-y,x) \).
2. Apply \( T_{0,-2} \): \( (-y,x) \to (-y, x - 2) \).

Step 3: Analyze Other Options (Incorrect Order)

• \( R_{0,90^\circ} \circ T_{-2,0} \): Translate left 2, then rotate. This does not align with the graph.
• \( T_{-2,0} \circ R_{0,90^\circ} \): Rotate, then translate left 2. Incorrect direction.
• \( R_{0,90^\circ} \circ T_{0,-2} \): Translate down 2, then rotate. Incorrect order (rotation should come before translation here).

The correct composition is \( T_{0,-2} \circ R_{0,90^\circ}(x,y) \), as it matches the transformation order (rotate first, then translate down 2).

Answer:

\( T_{0,-2} \circ R_{0,90^\circ}(x, y) \) (the first option in the given choices)