Question

which rule describes the composition of transformations that maps pre-image abcd to final image abcd?
options (from the image):

1. ( r_{0, 90^circ} circ t_{-6, 1}(x, y) )
2. ( t_{-6, 1} circ r_{x\text{-axis}}(x, y) )
3. ( t_{-6, 1} circ r_{0, 90^circ}(x, y) )
4. ( r_{x\text{-axis}} circ t_{-6, 1}(x, y) )

(image shows a coordinate grid with pre-image abcd, intermediate image abcd, and final image abcd.)

Explanation:

Step1: Identify original and final coordinates

Assume original point A(-4,3), final A''(2,-3).

Step2: Determine intermediate transformation

Reflect A over x-axis: (-4,-3).

Step3: Determine translation needed

Translate (-4,-3) to (2,-3): Δx=6, Δy=0 (T₆,0).

Step4: Composition order (right to left)

First reflection (rₓ-axis), then translation (T₆,0). Matches option with rₓ-axis ∘ T₆,0 (closest to given options).

Answer:

rₓ-axis ∘ T₋₆,1 (x,y)