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 rule

The rule $r_{y - axis}\circ T_{1,2}(x,y)$ means first a translation $T_{1,2}(x,y)=(x + 1,y+2)$ and then a reflection over the $y - axis$ which changes the sign of the $x -$coordinate, i.e., $(x,y)\to(-x,y)$. Let the original coordinates of $F$ be $(x,y)$. After translation, the coordinates become $(x + 1,y + 2)$. After reflection over the $y -$axis, the coordinates of $F'$ are $(-(x + 1),y + 2)$.

Step2: Identify the coordinates of $F'$ from the graph

From the graph, the coordinates of $F'$ are $(2,4)$. So we have the equations: $-(x + 1)=2$ and $y + 2=4$.

Step3: Solve for $x$

From $-(x + 1)=2$, we get $x+1=-2$, then $x=-3$.

Step4: Solve for $y$

From $y + 2=4$, we get $y=2$.

Answer:

$(-3,4)$