To determine which point maps onto itself after a reflection across the line \( y = -x \), we need to understand the reflection process. For a point \((x, y)\), the reflection across the line \( y = -x \) is \((-y, -x)\).
Let's check each given point:
1. \((-4, -4)\)
- Reflect \((-4, -4)\) across \( y = -x \).
- The reflection is \((-(-4), -(-4)) = (4, 4)\).
- Thus, \((-4, -4)\) does not map onto itself.
2. \((-4, 0)\)
- Reflect \((-4, 0)\) across \( y = -x \).
- The reflection is \((0, -(-4)) = (0, 4)\).
- Thus, \((-4, 0)\) does not map onto itself.
3. \((0, -4)\)
- Reflect \((0, -4)\) across \( y = -x \).
- The reflection is \((-(-4), -(0)) = (4, 0)\).
- Thus, \((0, -4)\) does not map onto itself.
4. \((4, -4)\)
- Reflect \((4, -4)\) across \( y = -x \).
- The reflection is \((-(-4), -4) = (4, -4)\).
- Thus, \((4, -4)\) does map onto itself.
Therefore, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is [tex]\( (4, -4) \)[/tex].
