Question

The rotation function mapping for a 90-degree clockwise rotation about the origin is equivalent to which other transformation mapping?

a. a reflection over the line y = x
b. a dilation with a scale factor of 1/4
c. a translation
d. a 270 degrees counterclockwise rotation

Answer 1

To analyze the transformation mapping of a 90-degree clockwise rotation about the origin, we can express this mathematically. A point \((x, y)\) after a 90-degree clockwise rotation transforms as follows:

\[
(x, y) \rightarrow (y, -x)
\]

Now, let's consider the options provided:

a. **Reflection over the line \( y = x \)** transforms points as follows:

\[
(x, y) \rightarrow (y, x)
\]

This is not equivalent to a 90-degree clockwise rotation.

b. **Dilation with a scale factor of 1/4** transforms points as follows:

\[
(x, y) \rightarrow \left(\frac{x}{4}, \frac{y}{4}\right)
\]

This does not change the angle of the points and is also not equivalent to a 90-degree clockwise rotation.

c. **Translation** shifts all points by a certain amount, transforming points as follows:

\[
(x, y) \rightarrow (x + a, y + b)
\]

This also does not reorient points in a way that resembles rotation.

d. **270 degrees counterclockwise rotation** transforms points as follows:

\[
(x, y) \rightarrow (-y, x)
\]

This is equivalent to a 90-degree clockwise rotation due to the following reasoning: if you perform a 270-degree counterclockwise rotation, it is the same as a 90-degree clockwise rotation.

Thus, the equivalent transformation mapping for a 90-degree clockwise rotation about the origin is:

**d. a 270 degrees counterclockwise rotation.**