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 transformation from ABCD to A'B'C'D'

First, observe that from ABCD to A'B'C'D', the figure is reflected over the x - axis. The rule for reflection over the x - axis is $r_{x - axis}(x,y)=(x, - y)$.

Step2: Analyze the transformation from A'B'C'D' to A''B''C''D''

Then, from A'B'C'D' to A''B''C''D'', the figure is translated. The translation rule $T_{-6,1}(x,y)=(x - 6,y + 1)$. In a composition of transformations, the order matters. The reflection happens first and then the translation. The notation for composition $T_{-6,1}\circ r_{x - axis}(x,y)$ means we first apply the reflection $r_{x - axis}(x,y)$ and then apply the translation $T_{-6,1}$ to the result.

Answer:

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