Question

what is the mapping rule for a 180 degree rotation about the origin?
$(x, y) \
ightarrow (x, -y)$
$(x, y) \
ightarrow (-x, -y)$
$(x, y) \
ightarrow (-y, x)$
$(x, y) \
ightarrow (-y, -x)$

Explanation:

Step1: Recall rotation rule

A 180° rotation about the origin changes the sign of both the x - coordinate and the y - coordinate of a point \((x,y)\). The rule is \((x,y)\to(-x,-y)\).

Step2: Match with options

We check each option:

• First option: \((x,y)\to(x,-y)\) (only y - coordinate sign changes, not 180° rotation).
• Second option: \((x,y)\to(-x,-y)\) (matches the 180° rotation rule).
• Third option: \((x,y)\to(-y,x)\) (this is a 90° counter - clockwise rotation rule).
• Fourth option: \((x,y)\to(-y,-x)\) (not the 180° rotation rule).

Answer:

The mapping rule for a 180 degree rotation about the origin is \((x,y)\to(-x,-y)\) (the second option among the given choices).