Question

which rule describes the composition of transformations that maps δabc to δa\b\c\?
t_{-6, -2} \circ r_{x - axis}(x, y)
r_{x - axis} \circ t_{-6, -2}(x, y)
t_{-6, -2} \circ r_{0, 90\circ}(x, y)
r_{0, 90\circ} \circ t_{-6, -2}(x, y)

Explanation:

Step1: Analyze the transformation from $\triangle ABC$ to $\triangle A'B'C'$

First, observe that $\triangle ABC$ is reflected over the $x -$axis to get $\triangle A'B'C'$. The rule for reflection over the $x -$axis is $r_{x - axis}(x,y)=(x, - y)$.

Step2: Analyze the transformation from $\triangle A'B'C'$ to $\triangle A''B''C''$

Then, $\triangle A'B'C'$ is translated. The translation rule $T_{-6,-2}(x,y)=(x - 6,y - 2)$. The composition of transformations is first the reflection and then the translation. In function - composition notation, we write the reflection first and then the translation, so the rule is $T_{-6,-2}\circ r_{x - axis}(x,y)$.

Answer:

$T_{-6,-2}\circ r_{x - axis}(x,y)$