Question

what are the coordinates of the vertices of the pre - image given? $r_{y = -x}circ t_{1,-2}(x,y)$ a b c (3,4) d

Explanation:

Step1: Analyze the transformation

The transformation is a composition of a translation $T_{1,-2}(x,y)=(x + 1,y-2)$ followed by a reflection $r_{y=-x}$ which has the rule $(x,y)\to(-y,-x)$. Let the coordinates of a vertex of the pre - image be $(x,y)$. After translation, the coordinates become $(x + 1,y - 2)$. After reflection, the coordinates become $-(y - 2),-(x + 1)=(2 - y,-x - 1)$.

Step2: For point C with image coordinates $(3,4)$

Let the pre - image coordinates be $(x,y)$. We know that $(2 - y,-x - 1)=(3,4)$. So we have the system of equations:
Set up equations:
$2 - y=3$ gives $y=-1$.
$-x - 1=4$ gives $x=-5$.
Let's work backward from the image points of the other vertices.
For a reflection $r_{y=-x}$ followed by a translation. If we want to find the pre - image from the image, we first reverse the reflection and then reverse the translation.
The reverse of reflection $r_{y=-x}$ is itself. The reverse of translation $T_{1,-2}$ is $T_{-1,2}(x,y)=(x - 1,y+2)$.
Let the image point be $(x_{i},y_{i})$ and pre - image be $(x_{p},y_{p})$. After reflection and translation, $x_{i}=2 - y_{p}$ and $y_{i}=-x_{p}-1$. Reversing, if we start from the image point $(x_{i},y_{i})$:
First, reverse the reflection: if $(x_{i},y_{i})$ is the image after reflection and translation, then after just the translation the point was $(-y_{i},-x_{i})$. Then reverse the translation: the pre - image point $(x,y)=(-y_{i}-1,-x_{i}+2)$.
For point C with image $(3,4)$:
Reverse reflection: we get $(-4,-3)$. Reverse translation: $(-4 - 1,-3+2)=(-5,-1)$.
Let's assume the image of point A is $A''(x_{A''},y_{A''})$, point B is $B''(x_{B''},y_{B''})$ and point D is $D''(x_{D''},y_{D''})$.
If we assume $A''(-4,-2)$, reverse reflection gives $(2,4)$ and reverse translation gives $(2 - 1,4+2)=(1,6)$.
If $B''(-2,-2)$, reverse reflection gives $(2,2)$ and reverse translation gives $(2 - 1,2+2)=(1,4)$.
If $D''(-4,-4)$, reverse reflection gives $(4,4)$ and reverse translation gives $(4 - 1,4+2)=(3,6)$.

Answer:

A. $(1,6)$
B. $(1,4)$
C. $(-5,-1)$
D. $(3,6)$