To find the mapping rule for a 180-degree rotation about the origin, let's consider what happens to a point [tex]\((x, y)\)[/tex] when it undergoes this transformation.
Imagine rotating the point [tex]\((x, y)\)[/tex] by 180 degrees. The point would end up on the opposite side of the origin, maintaining the same distance but in the opposite direction. This results in the new coordinates being the negatives of the original coordinates.
Thus, under a 180-degree rotation, the point [tex]\((x, y)\)[/tex] would map to [tex]\((-x, -y)\)[/tex].
Therefore, the correct mapping rule for a 180-degree rotation about the origin is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
