Question

which rule describes the composition of transformations that maps △bcd to △bcd?
t_{5, - 6} \circ r_{y=-x}(x,y)
r_{y=-x} \circ t_{5, - 6}(x,y)
t_{6, - 5} \circ r_{y - axis}(x,y)
r_{y - axis} \circ t_{6, - 5}(x,y)

Explanation:

Step1: Analyze reflection

First, observe that the shape is reflected over the line $y = -x$. The rule for reflecting a point $(x,y)$ over the line $y=-x$ is $r_{y = -x}(x,y)=(-y,-x)$.

Step2: Analyze translation

After reflection, the shape is translated. Counting the units of translation, we see that it is translated 5 units to the right and 6 units down. The rule for translation $T_{a,b}(x,y)=(x + a,y + b)$, here $a = 5$ and $b=-6$, so $T_{5,-6}(x,y)=(x + 5,y-6)$.

Step3: Determine composition order

The reflection over $y=-x$ occurs first and then the translation. The composition of transformations is written with the transformation that occurs first on the right - hand side of the composition symbol $\circ$. So the correct composition is $T_{5,-6}\circ r_{y=-x}(x,y)$.

Answer:

$T_{5,-6}\circ r_{y = -x}(x,y)$