Final answer:
The mapping rule for a 180-degree rotation about the origin is (x, y) → (-x, -y), which is described by option (c). This transforms each point to the quadrant diagonally opposite its initial position.
Step-by-step explanation:
The student is asking about the mapping rule for a 180-degree rotation about the origin of the coordinate system. When a point (x, y) is rotated 180 degrees about the origin, both the x and y coordinates are inverted. This means that the point is mapped to (-x, -y). Therefore, the correct mapping rule for a 180-degree rotation about the origin is (x, y) → (-x, -y), which is an option (c).
To visualize this transformation, imagine that each point on the graph is pinned to the page, and the page itself is flipped over both the x-axis and the y-axis. The point would end up in the quadrant diagonally opposite from where it started. For example, a point in the first quadrant would end up in the third quadrant after a 180-degree rotation.