A figure is rotated 90 degrees counterclockwise about the origin. Which of the following function mappings was applied?

Question

option #1: (x,y) --> (y,-x)
option #2: (x,y) --> (-y,x)
option #3: (x,y) --> (-x,-y)
option #4: (x,y) --> (y,x)

(1 point)
Responses

Option #1
Option #1

Option #2
Option #2

Option #3
Option #3

Option #4

Answer 1

To determine which function mapping corresponds to a 90-degree counterclockwise rotation about the origin, let's analyze the transformation.

When a point \((x, y)\) is rotated 90 degrees counterclockwise, it moves to a new position. The new coordinates can be derived from the following reasoning:

1. The point \((x, y)\) will have:
   • The x-coordinate move to the negative y-coordinate,
   • The y-coordinate move to the x-coordinate.

This gives us the transformation: \[ (x, y) \rightarrow (-y, x) \]

Looking at your options:

• Option #1: \((x,y) \rightarrow (y,-x)\) (this corresponds to a 90-degree clockwise rotation)
• Option #2: \((x,y) \rightarrow (-y,x)\) (this corresponds to a 90-degree counterclockwise rotation)
• Option #3: \((x,y) \rightarrow (-x,-y)\) (this corresponds to a rotation of 180 degrees)
• Option #4: \((x,y) \rightarrow (y,x)\) (this does not correspond to any standard rotation)

Thus, the correct answer is Option #2: \((x,y) \rightarrow (-y,x)\).