Question

Determine whether each statement about the rotation (x,y) → (y,-x) is true or false.

A. Every point in Quadrant I is mapped to a point in Quadrant II
B. Points on the x-axis are mapped to points on the y-axis
C. The origin is a fixed point under the rotation
D. The rotation has the same effect as a 90-degree clockwise rotation
E. The angle of rotation is 180 degrees
F. A point on the line y = x is mapped to another point on the line y = x

Answer 1

The rotation (x,y) → (y,-x) does not map every point in Quadrant I to Quadrant II, points on the x-axis are mapped to the y-axis, the origin is a fixed point, the effect is the same as a 90-degree clockwise rotation, the angle of rotation is 90 degrees, and points on y = x are preserved.

Step-by-step explanation:

To determine whether each statement about the rotation (x,y) → (y,-x) is true or false, we can analyze the effects of the rotation on various points and lines.

1. A. Every point in Quadrant I is mapped to a point in Quadrant II: False. Points in Quadrant I, where both x and y coordinates are positive, are mapped to Quadrant II where x is negative and y is positive.
2. B. Points on the x-axis are mapped to points on the y-axis: True. Points on the x-axis have y-coordinate 0, so after rotation, they will have x-coordinate 0.
3. C. The origin is a fixed point under the rotation: True. The origin (0,0) remains the same after the rotation.
4. D. The rotation has the same effect as a 90-degree clockwise rotation: True. The given rotation is a 90-degree counterclockwise rotation, but if we look at it from the perspective of a clockwise rotation, the effect is the same.
5. E. The angle of rotation is 180 degrees: False. The angle of rotation is 90 degrees.
6. F. A point on the line y = x is mapped to another point on the line y = x: True. Points on the line y = x are preserved under the given rotation.