To find the original coordinates [tex]\( P \)[/tex] given the transformation and reflection information, follow these steps:
1. Starting Point:
We start with the point [tex]\( P' \)[/tex] given as [tex]\( P' = (1, -6) \)[/tex].
2. Reflection Across the Line [tex]\( y = x \)[/tex]:
When a point [tex]\( (a, b) \)[/tex] is reflected across the line [tex]\( y = x \)[/tex], the coordinates are swapped. Hence, reflecting [tex]\( P' \)[/tex]:
[tex]\[
P'' = (-6, 1)
\][/tex]
3. Reverse the Translation:
The transformation mentioned is [tex]\( (x, y) \rightarrow (x-3, y-3) \)[/tex], which means moving each x-coordinate and y-coordinate 3 units to the left and down, respectively. To find the original point before this transformation, we need to reverse it:
[tex]\[
(x', y') \rightarrow (x' + 3, y' + 3)
\][/tex]
Applying this to the reflected point:
[tex]\[
P = (-6 + 3, 1 + 3) = (-3, 4)
\][/tex]
Thus, the coordinates of [tex]\( P \)[/tex] are [tex]\( \boxed{-3, 4} \)[/tex].
