Question

which rule describes the composition of transformations that maps pre-image abcd to final image abcd?
○ ( r_{x-\text{axis}} circ t_{-6, 1}(x, y) )
○ ( t_{-6, 1} circ r_{x-\text{axis}}(x, y) )
○ ( r_{0, 90^circ} circ t_{-6, 1}(x, y) )
○ ( t_{-6, 1} circ r_{0, 90^circ}(x, y) )

Explanation:

Step 1: Analyze the reflection over the x - axis

First, we consider the reflection of the pre - image \( ABCD \) over the \( x \) - axis. The rule for a reflection over the \( x \) - axis, \( r_{x - \text{axis}} \), is \( r_{x - \text{axis}}(x,y)=(x, - y) \). Let's take a point from \( ABCD \), say \( A \). If we assume the coordinates of \( A \) in \( ABCD \) are, for example, \( (4,4) \), after reflection over the \( x \) - axis, the coordinates become \( (4, - 4) \), which is the coordinates of \( A' \) (before translation).

Step 2: Analyze the translation

Then we need to translate the image after reflection ( \( A'B'C'D' \) before the final translation) to get the final image \( A''B''C''D'' \). The translation vector \( T_{-6,1} \) means we subtract 6 from the \( x \) - coordinate and add 1 to the \( y \) - coordinate. Using the point \( A'=(4, - 4) \) from the reflection, applying \( T_{-6,1} \): \( (4-6,-4 + 1)=(-2,-3) \), which should match the coordinates of \( A'' \) (we can verify with other points as well).

The composition of transformations is the translation \( T_{-6,1} \) applied after the reflection \( r_{x - \text{axis}} \), so the rule is \( T_{-6,1} \circ r_{x - \text{axis}}(x,y) \), which is the second option.

Answer:

\( T_{-6,1} \circ r_{x - \text{axis}}(x,y) \) (the second option: \( T_{-6,1} \circ r_{x - \text{axis}}(x,y) \))