To find the original coordinates of point [tex]\( P \)[/tex], given that [tex]\( P' = (3, 0) \)[/tex] is the transformed point after applying a series of transformations, we need to work backward through each transformation step by step.
1. Reflection over the line [tex]\( y = x \)[/tex]:
When you reflect a point over the line [tex]\( y = x \)[/tex], the coordinates of the point are swapped. So if the image point after reflection (let's call it [tex]\( P'_{\text{reflect}} \)[/tex]) has coordinates [tex]\( (3, 0) \)[/tex], then the point before reflection (let's call it [tex]\( P_{\text{before reflect}} \)[/tex]) has coordinates [tex]\( (0, 3) \)[/tex].
2. Inverse the translation:
Next, we need to consider the translation transformation [tex]\( (x, y) \rightarrow (x-2, y-2) \)[/tex]. To find the original coordinates [tex]\( (x, y) \)[/tex], we need to reverse this translation. If [tex]\( P_{\text{translated}} = (0, 3) \)[/tex], then reversing the translation means adding 2 to each coordinate:
[tex]\[
x = 0 + 2 = 2
\][/tex]
[tex]\[
y = 3 + 2 = 5
\][/tex]
So, the original coordinates [tex]\( P \)[/tex] before any transformations are [tex]\( (2, 5) \)[/tex].
Therefore, the coordinates of [tex]\( P \)[/tex] are [tex]\( (2, 5) \)[/tex].
