Question

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

Explanation:

Step1: Analyze the translation

First, observe that to get from the original figure ABCD to an intermediate - step figure (say A'B'C'D'), it seems that a translation is involved. The translation $T_{- 6,1}(x,y)=(x - 6,y + 1)$ moves each point of the pre - image 6 units to the left and 1 unit up.

Step2: Analyze the reflection

Then, to get from the intermediate - step figure to the final figure A''B''C''D'', a reflection over the x - axis is performed. The rule for a reflection over the x - axis is $r_{x - axis}(x,y)=(x,-y)$. When we perform the composition of transformations, the order matters. We first perform the translation and then the reflection. So the composition of transformations is $r_{x - axis}\circ T_{-6,1}(x,y)$.

Answer:

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