Question

Identify the rotation function mapping used to turn A (-7, 4) to A' (-4, -7).

a. (x, y) -> (-y, x)
b. (x, y) -> (2x, 2y)
c. (x, y) -> (y, x)
d. (x, y) -> (x, -y)

Answer 1

To identify the rotation function mapping used to turn the point \( A (-7, 4) \) into \( A' (-4, -7) \), we can analyze the changes in coordinates.

The original point \( A \) is \( (-7, 4) \), and the transformed point \( A' \) is \( (-4, -7) \).

Let's break down the transition from \( (x, y) \) to \( (x', y') \):

1. The x-coordinate changes from \( -7 \) to \( -4 \).
2. The y-coordinate changes from \( 4 \) to \( -7 \).

Next, we can inspect each of the provided options to see if they match the transformation.

- **Option a**: \( (x, y) \to (-y, x) \)
For \( A \):
\[ (-y, x) = (-4, -7) \] (when \( y = 4 \) and \( x = -7 \)).
This option works, since we get \( (-4, -7) \).

- **Option b**: \( (x, y) \to (2x, 2y) \)
For \( A \):
\[ (2 \cdot -7, 2 \cdot 4) = (-14, 8) \]
This option does not work.

- **Option c**: \( (x, y) \to (y, x) \)
For \( A \):
\[ (4, -7) \]
This option does not work.

- **Option d**: \( (x, y) \to (x, -y) \)
For \( A \):
\[ (-7, -4) \]
This option does not work.

From our analysis, only Option a matches the transformation that changes \( (-7, 4) \) to \( (-4, -7) \).

Thus, the correct answer is:
**a. \( (x, y) \to (-y, x) \)**.