To determine what transformation is represented by the rule [tex]\((x, y) \rightarrow (-x, -y)\)[/tex], let's analyze the options given:
1. Rotation of [tex]\(90^\circ\)[/tex] counterclockwise about the origin:
For a point [tex]\((x, y)\)[/tex] rotated [tex]\(90^\circ\)[/tex] counterclockwise about the origin, the new coordinates would be [tex]\((-y, x)\)[/tex]. Clearly, this does not match our given transformation [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
2. Reflection across the [tex]\(x\)[/tex]-axis:
For a point [tex]\((x, y)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis, the new coordinates would be [tex]\((x, -y)\)[/tex]. This does not match our transformation either.
3. Rotation of [tex]\(90^\circ\)[/tex] clockwise about the origin:
For a point [tex]\((x, y)\)[/tex] rotated [tex]\(90^\circ\)[/tex] clockwise about the origin, the new coordinates would be [tex]\((y, -x)\)[/tex]. This is not the same as [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
4. Rotation of [tex]\(180^\circ\)[/tex] about the origin:
For a point [tex]\((x, y)\)[/tex] rotated [tex]\(180^\circ\)[/tex] about the origin, the new coordinates would be [tex]\((-x, -y)\)[/tex]. This transformation matches the given rule exactly.
Based on this analysis, the correct transformation represented by the rule [tex]\((x, y) \rightarrow (-x, -y)\)[/tex] is a rotation of [tex]\(180^\circ\)[/tex] about the origin.
