What transformation is represented by the rule (x, y)→(y, − x) ? reflection across the x-axis rotation of 180° about the origin reflection across the y-axis rotation of 90° clockwise about the origin

The given transformation is (x, y) -> (y, -x).

The choices are
A) Reflection across the x-axis
B) Rotation of  180° about the origin
C) Reflection across the y-axis
D) Rotation of 90° clockwise about the origin

D) Rotation of 90 clockwise about the origin.

Explanation:
Refer to the diagram shown below.
Consider the point A (3, 5) subject to the rule (x,y) -> (y, -x).
Then the point A transforms into B (5, -3).

If a line is drawn from the orgin to A (3,5), then the slope of the line is
m₁ = 5/3
If a line is drawn from the origin to B (5, -3),  then the slope of the line is
m₂ = -3/5

Note:
When two lines are perpendicular to each other, the product of their slopes equals -1.

Because (5/3)*(-3/5) = -1, the two lines that are shown in the figure are perpendicular. Therefore the point A (3,5) was rotated 90° clockwise into the point B (5,-3).