Question

The coordinates for a point and its image are Q (-3,-5) and Q' (5, -3). Based on the mapping rules for rotations, which transformation occurred? (1 point)
Responses

a reflection across the x-axis
a reflection across the x-axis

a 90-degree (counterclockwise) rotation
a 90-degree (counterclockwise) rotation

a 270-degree (counterclockwise) rotation
a 270-degree (counterclockwise) rotation

a 180-degree rotation

Answer 1

To determine the transformation that occurred between the point \( Q(-3, -5) \) and its image \( Q'(5, -3) \), we can analyze the coordinate changes.

1. **Reflections:**
- A reflection across the **x-axis** would change the y-coordinate from negative to positive and leave the x-coordinate unchanged. Hence, \( Q(-3, -5) \) would become \( Q'(-3, 5) \), which is not the case here.
- A reflection across the **y-axis** would change the x-coordinate to its opposite sign and leave the y-coordinate unchanged. Hence, \( Q(-3, -5) \) would become \( Q'(3, -5) \), which is also not the case here.

2. **Rotations:**
- A **90-degree counterclockwise rotation** transforms a point \((x, y)\) into \((-y, x)\). So, for \( Q(-3, -5) \):
\[
Q' = (-(-5), -3) = (5, -3)
\]
This matches \( Q'(5, -3) \).

- A **270-degree counterclockwise rotation** (or equivalently a 90-degree clockwise rotation) transforms a point by \((x, y)\) into \((y, -x)\). Applying this on \( Q(-3, -5) \):
\[
Q' = (-5, 3)
\]
This does not match \( Q'(5, -3) \).

- A **180-degree rotation** transforms a point \((x, y)\) into \((-x, -y)\). Thus:
\[
Q' = (3, 5)
\]
This does not match \( Q'(5, -3) \).

Based on the analysis, the correct transformation that occurred is:

**A 90-degree (counterclockwise) rotation.**