To determine which point maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex], we need to understand the geometric properties of reflection in this line. Specifically, when a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], its new coordinates will be [tex]\((-y, -x)\)[/tex].
We will analyze each of the given points to see which one remains the same after the reflection.
1. Point [tex]\((-4, -4)\)[/tex]:
- Original coordinates: [tex]\((-4, -4)\)[/tex]
- Reflected coordinates: [tex]\(( -(-4), -(-4) ) = (4, 4)\)[/tex]
- This point does not map onto itself because [tex]\((-4, -4) \neq (4, 4)\)[/tex].
2. Point [tex]\((-4, 0)\)[/tex]:
- Original coordinates: [tex]\((-4, 0)\)[/tex]
- Reflected coordinates: [tex]\((-(0), -(-4)) = (0, 4)\)[/tex]
- This point does not map onto itself because [tex]\((-4, 0) \neq (0, 4)\)[/tex].
3. Point [tex]\((0, -4)\)[/tex]:
- Original coordinates: [tex]\((0, -4)\)[/tex]
- Reflected coordinates: [tex]\(( -(-4), -(0) ) = (4, 0)\)[/tex]
- This point does not map onto itself because [tex]\((0, -4) \neq (4, 0)\)[/tex].
4. Point [tex]\((4, -4)\)[/tex]:
- Original coordinates: [tex]\((4, -4)\)[/tex]
- Reflected coordinates: [tex]\(( -(-4), -(4) ) = (4, -4)\)[/tex]
- This point maps onto itself because [tex]\((4, -4) = (4, -4)\)[/tex].
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex].
