Question

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

Explanation:

Step1: Understand transformation notations

The notation $T_{a,b}(x,y)=(x + a,y + b)$ represents a translation, $r_{x - axis}(x,y)=(x,-y)$ is reflection over the x - axis and $R_{0,90^{\circ}}(x,y)=(-y,x)$ is a 90 - degree counter - clockwise rotation about the origin.

Step2: Analyze the pre - image to image transformation visually

First, observe that the pre - image ABCD is translated and then reflected over the x - axis. A translation of 6 units to the left and 1 unit up is represented by $T_{-6,1}(x,y)=(x-6,y + 1)$. Then a reflection over the x - axis changes the sign of the y - coordinate. The correct order is translation first and then reflection. The composition $T_{-6,1}\circ r_{x - axis}(x,y)$ means we first apply the reflection $r_{x - axis}(x,y)$ and then the translation $T_{-6,1}$, which is incorrect. The correct composition is $r_{x - axis}\circ T_{-6,1}(x,y)$.

Step3: Check each option

For option 1: $r_{x - axis}\circ T_{-6,1}(x,y)$ first translates the point $(x,y)$ to $(x - 6,y + 1)$ and then reflects it over the x - axis to $(x - 6,-(y + 1))$. This is the correct composition.
For option 2: $T_{-6,1}\circ r_{x - axis}(x,y)$ first reflects $(x,y)$ over the x - axis to $(x,-y)$ and then translates it to $(x - 6,-y + 1)$, which is incorrect.
For option 3: $R_{0,90^{\circ}}\circ T_{-6,1}(x,y)$ involves a rotation instead of a reflection, so it is incorrect.
For option 4: $T_{-6,1}\circ R_{0,90^{\circ}}(x,y)$ also involves a rotation and incorrect order of operations, so it is incorrect.

Answer:

$r_{x - axis}\circ T_{-6,1}(x,y)$