Question

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

Explanation:

Step1: Analyze translation

First, observe the horizontal and vertical shifts. The figure seems to be translated. If we consider the general form of a translation $T_{a,b}(x,y)=(x + a,y + b)$. By looking at the position of a point (e.g., a vertex) before and after the transformation, we can see that there is a translation.

Step2: Analyze reflection

Then, check for reflection. The line of reflection can be determined by seeing how the orientation of the figure changes. The notation $r_{y=-x}(x,y)=(-y,-x)$ represents a reflection over the line $y = -x$ and $r_{y - axis}(x,y)=(-x,y)$ represents a reflection over the y - axis. By comparing the original and final figures, we can identify the reflection.

Step3: Determine the order

The order of transformations matters. When we have a composition of transformations $A\circ B$, we first perform transformation $B$ and then transformation $A$. By analyzing the position of the vertices of $\triangle BCD$ and $\triangle B''C''D''$, we find that first, a translation $T_{5,-6}(x,y)=(x + 5,y-6)$ is applied and then a reflection over the line $y=-x$ is applied. The composition is written as $r_{y=-x}\circ T_{5,-6}(x,y)$.

Answer:

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