To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand how reflection works in this context. When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], its coordinates become [tex]\((-y, -x)\)[/tex].
Let's reflect each given point and see which one maps onto itself.
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflect [tex]\((-4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((-(-4), -(-4)) = (4, 4)\)[/tex].
- [tex]\((-4, -4) \neq (4, 4)\)[/tex].
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect [tex]\((-4, 0)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((0, 4)\)[/tex].
- [tex]\((-4, 0) \neq (0, 4)\)[/tex].
3. Point [tex]\((0, -4)\)[/tex]:
- Reflect [tex]\((0, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((4, 0)\)[/tex].
- [tex]\((0, -4) \neq (4, 0)\)[/tex].
4. Point [tex]\((4, -4)\)[/tex]:
- Reflect [tex]\((4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((4, -4)\)[/tex].
- [tex]\((4, -4) = (4, -4)\)[/tex].
Thus, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is [tex]\((4, -4)\)[/tex].
Therefore, the answer is the fourth point: [tex]\((4, -4)\)[/tex].
