Question

parallelogram fghj is the final image after the rule $r_{y - axis}circ t_{1,2}(x,y)$ was applied to parallelogram fghj. what are the coordinates of vertex f of parallelogram fghj? (-2,2) (-2,6) (-3,4) (-4,2)

Explanation:

Step1: Analyze the transformation rules

The transformation $r_{y - axis}\circ T_{1,2}(x,y)$ means first a translation $T_{1,2}(x,y)=(x + 1,y+2)$ and then a reflection over the $y - axis$ which changes the $x$ - coordinate sign, so the overall transformation is $(-(x + 1),y + 2)$.

Step2: Find the coordinates of $F''$

From the graph, the coordinates of $F''$ are $(2,4)$. Let the original coordinates of $F$ be $(x,y)$. Then applying the transformation $(-(x + 1),y + 2)=(2,4)$.
We have the following two - equations:
For the $x$ - coordinate: $-(x + 1)=2$, which implies $x+1=-2$ and $x=-3$.
For the $y$ - coordinate: $y + 2=4$, which implies $y = 2$.

Answer:

$(-3,2)$ (Note: There is an error in the provided options as the correct answer $(-3,2)$ is not among them. But following the steps of transformation analysis we found the correct pre - image coordinates.)