Question

a sequence of transformations is performed on δdef, resulting in δdef. both triangles are shown on the coordinate plane below. some transformations are listed below: - ( r_y ): a reflection across the y-axis - ( r_x ): a reflection across the x-axis - ( t_{-6, 0} ): a translation so that ( (x, y) \to (x - 6, y) ) which of these describes the sequence of transformations performed on δdef that results in δdef? a. ( r_y ) followed by ( t_{-6, 0} ) b. ( r_x ) followed by ( r_y ) c. ( t_{-6, 0} ) followed by ( r_x ) d. ( t_{-6, 0} ) followed by ( r_y )

Explanation:

Step1: Analyze Translation \( T_{-6, 0} \)

The translation \( T_{-6, 0} \) means shifting a point \((x, y)\) to \((x - 6, y)\), which is a horizontal shift 6 units to the left.

Step2: Analyze Reflection \( R_y \)

Reflection across the \( y \)-axis (\( R_y \)) transforms a point \((x, y)\) to \((-x, y)\).

Step3: Analyze Reflection \( R_x \)

Reflection across the \( x \)-axis (\( R_x \)) transforms a point \((x, y)\) to \((x, -y)\).

Step4: Test Option C

First, apply \( T_{-6, 0} \) (shift left 6 units) to \( \triangle DEF \), then apply \( R_x \) (reflect over \( x \)-axis). Let's assume a vertex of \( \triangle DEF \) (e.g., if a vertex is at \((x, y)\), after \( T_{-6, 0} \) it becomes \((x - 6, y)\), then after \( R_x \) it becomes \((x - 6, -y)\), which matches the transformation to \( \triangle D'E'F' \) (since the lower triangle is a reflection over \( x \)-axis after shifting left).

Step5: Eliminate Other Options

• Option A: \( R_y \) then \( T_{-6, 0} \) would first reflect over \( y \)-axis, then shift left, which doesn't match the position.
• Option B: Reflecting over \( x \) then \( y \)-axis would change the sign of both \( x \) and \( y \) components in a different order, not matching.
• Option D: \( T_{-6, 0} \) then \( R_y \) would shift left then reflect over \( y \)-axis, which would give \((-(x - 6), y)=(-x + 6, y)\), not matching the lower triangle's coordinates.

Answer:

C. \( T_{-6, 0} \) followed by \( R_x \)