To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], let's analyze how each given point behaves under this reflection. When a point [tex]\((a, b)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], the coordinates of the reflected point become [tex]\((-b, -a)\)[/tex]. We will test each point to check if it maps onto itself.
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflecting [tex]\((-4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex]:
- The coordinates become [tex]\((-( -4), -( -4))\)[/tex].
- This simplifies to [tex]\((4, 4)\)[/tex].
[tex]\((-4, -4)\)[/tex] does not map onto itself after the reflection.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflecting [tex]\((-4, 0)\)[/tex] across the line [tex]\( y = -x \)[/tex]:
- The coordinates become [tex]\((-(0), -(-4))\)[/tex].
- This simplifies to [tex]\((0, 4)\)[/tex].
[tex]\((-4, 0)\)[/tex] does not map onto itself after the reflection.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflecting [tex]\((0, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex]:
- The coordinates become [tex]\((4, 0)\)[/tex].
[tex]\((0, -4)\)[/tex] does not map onto itself after the reflection.
4. Point [tex]\((4, -4)\)[/tex]:
- Reflecting [tex]\((4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex]:
- The coordinates become [tex]\((4, -4)\)[/tex].
[tex]\((4, -4)\)[/tex] maps onto itself after the reflection.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
