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 $r_{y - axis}(x,y)=(-x,y)$. Let the coordinates of $F$ be $(x,y)$. After translation, the coordinates become $(x + 1,y + 2)$, and after reflection over the y - axis, they become $(-(x + 1),y + 2)$.

Step2: Identify the coordinates of $F''$

From the graph, the coordinates of $F''$ are $(2,4)$.

Step3: Set up equations to find original coordinates

We know that $-(x + 1)=2$ and $y + 2=4$.
For $-(x + 1)=2$, we solve for $x$:
\[

$$\begin{align*} -(x + 1)&=2\\ x+1&=- 2\\ x&=-3 \end{align*}$$

\]
For $y + 2=4$, we solve for $y$:
\[

$$\begin{align*} y+2&=4\\ y&=2 \end{align*}$$

\]

Answer:

$(-3,4)$