To determine the transformation represented by the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to analyze the changes to the coordinates [tex]\((x, y)\)[/tex].
1. Initial Point Analysis: Start with a point [tex]\((x, y)\)[/tex].
2. Transformation Application: Apply the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex]. This changes the original coordinates [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
3. Effect of Transformation:
- For a point on the x-axis, [tex]\((x, 0)\)[/tex], the transformation changes this to [tex]\((0, -x)\)[/tex].
- For a point on the y-axis, [tex]\((0, y)\)[/tex], the transformation changes this to [tex]\((y, 0)\)[/tex].
4. Interpretation:
- This specific change, where each point [tex]\((x, y)\)[/tex] is mapped to [tex]\((y, -x)\)[/tex], indicates that there is a rotation involved.
- The point [tex]\((x, y)\)[/tex] after transformation lies 90 degrees clockwise from its initial position on the coordinate plane.
5. Conclusion:
- The rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex] corresponds to a rotation of 90 degrees clockwise about the origin.
Therefore, the transformation represented by the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is a rotation of 90 degrees clockwise about the origin.
