Geometry

Question Analysis:
This question requires reversing two geometric transformations: a reflection and a translation.
The main focus is applying inverse operations in the correct order to determine the original coordinates.

Key Concept Explanation:
When a figure undergoes two transformations in sequence (reflection and translation), to recover the original point we must:
Reverse the translation by applying the opposite vector
Reverse the reflection over y = x by swapping the coordinates again

Step-by-Step Solution:
Given:
P′ = (–1, 8) is the final image after two transformations: reflection over y = x, then translation by (–3, 4)
Step 1 — Undo the translation:
Subtract the translation vector in reverse:
(–1 + 3, 8 – 4) = (2, 4)
This gives us the point that resulted from just the reflection over y = x
Step 2 — Undo the reflection over y = x:
Reflection over y = x swaps the x- and y-values.
So we swap (2, 4) → (4, 2)
Therefore, the original point P is (4, 2)