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 sign of the $x -$coordinate. 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 final coordinates are $(-(x + 1),y + 2)$.

Step2: Observe the final - image coordinates

From the graph, the coordinates of $F''$ are $(2,4)$. Let's work backwards. Since the last step was a reflection over the $y -$axis, before reflection, the $x -$coordinate of the point after translation was $- 2$ and the $y -$coordinate remained the same. So the point after translation was $(-2,4)$.

Step3: Work backwards for the translation

Since the translation rule was $T_{1,2}(x,y)=(x + 1,y+2)$, to find the original point $F$, we solve the equations: $x+1=-2$ and $y + 2=4$. For $x+1=-2$, we get $x=-2 - 1=-3$. For $y + 2=4$, we get $y=4 - 2=2$.

Answer:

$(-3,2)$

It seems there is an error in the provided options as the correct answer $(-3,2)$ is not among them. If we assume there was a mis - typing in the problem - solving process or in the options, and we re - check the work:
Let's work in reverse order of the transformation.

1. Reflection over the $y -$axis: If the final point $F''$ has coordinates $(x_{f''},y_{f''})$ and the point before reflection has coordinates $(x_{1},y_{1})$, then $x_{1}=-x_{f''}$ and $y_{1}=y_{f''}$.
2. Translation: If the point before translation has coordinates $(x,y)$ and after translation has coordinates $(x_{1},y_{1})$ and the translation rule is $T_{1,2}(x,y)=(x + 1,y+2)$, then $x=x_{1}-1$ and $y=y_{1}-2$.

If $F''=(2,4)$, before reflection over the $y -$axis, the point was $(-2,4)$. Before translation, $x=-2 - 1=-3$ and $y=4 - 2=2$.

If we assume we made a wrong interpretation and we work in a different way:
Let the original point $F=(x,y)$. After translation $T_{1,2}(x,y)=(x + 1,y + 2)$ and then reflection over the $y -$axis $(-(x + 1),y + 2)$.
We know $(-(x + 1),y + 2)=(2,4)$. Then $-(x + 1)=2$ gives $x+1=-2$ or $x=-3$ and $y + 2=4$ gives $y = 2$.

If we consider the closest option to our calculated value, and assume some error in the problem setup or options, we note that our calculated point is $(-3,2)$ but if we made a small error in calculation or there is an error in the options, we re - evaluate the steps.
If we assume we mis - read the graph or made a calculation error in reverse - engineering the transformation:
Let's assume we start from the fact that we know the transformation rules. The translation $T_{1,2}$ moves a point 1 unit to the right and 2 units up and reflection over the $y -$axis changes the sign of the $x -$coordinate.
If we assume we work from the options:

• Option A: If $F=(-2,2)$
• After translation $T_{1,2}(-2,2)=(-2 + 1,2+2)=(-1,4)$
• After reflection over the $y -$axis $(1,4)$ which is not correct.
• Option B: If $F=(-2,6)$
• After translation $T_{1,2}(-2,6)=(-2 + 1,6+2)=(-1,8)$
• After reflection over the $y -$axis $(1,8)$ which is not correct.
• Option C: If $F=(-3,4)$
• After translation $T_{1,2}(-3,4)=(-3 + 1,4+2)=(-2,6)$
• After reflection over the $y -$axis $(2,6)$ which is not correct.
• Option D: If $F=(-4,2)$
• After translation $T_{1,2}(-4,2)=(-4 + 1,2+2)=(-3,4)$
• After reflection over the $y -$axis $(3,4)$ which is not correct.

Based on our step - by - step calculation of reverse - engineering the transformation $r_{y - axis}\circ T_{1,2}(x,y)$ from the final point $F''$ on the graph, the correct coordinates of $F$ should be $(-3,2)$ but since it's not in the options, there may be an issue with the problem or options provided.

If we had to choose the closest option conceptually (assuming some error in data), we note that our $x=-3$ value is closest to the values in the options. But again, the correct answer based on transformation rules and graph analysis should be $(-3,2)$.