To determine which function mapping corresponds to a 90-degree counterclockwise rotation about the origin, we can analyze how the coordinates of a point \((x, y)\) change during this rotation.
When a point \((x, y)\) is rotated 90 degrees counterclockwise about the origin, the new coordinates can be derived as follows:
- The x-coordinate of the original point becomes the negative of the y-coordinate.
- The y-coordinate of the original point becomes the x-coordinate.
Thus, the new coordinates after the 90-degree counterclockwise rotation will be:
\[ (x, y) \to (-y, x) \]
Comparing this with the provided options:
a. \((x, y) \to (-y, -x)\)
b. \((x, y) \to (y, -x)\)
c. \((x, y) \to (y, x)\)
d. \((x, y) \to (-x, -y)\)
None of the options directly match the correct transformation. However, notice that if we look closely, the function that corresponds to the 90-degree counterclockwise rotation around the origin is actually \((x, y) \to (-y, x)\), but this option isn't among the choices given.
If you are looking for the closest correct answer, then keep in mind:
- The answer that corresponds to a 90-degree clockwise rotation would actually be \((x, y) \to (y, -x)\), which is option b; however, it is not what the question asked for.
To summarize, while none of the provided express the 90-degree counterclockwise rotation correctly, the mapping for \(90^\circ\) counterclockwise is indeed \((x, y) \to (-y, x)\). Hence, based on the given options, none are correct for a 90 degrees counterclockwise rotation.
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