Answer :
To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the reflection process.
When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the coordinates of the reflected point become [tex]\((-y, -x)\)[/tex].
We need to check whether each of the given points maps onto itself after performing this reflection.
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflect [tex]\((-4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (-4, -4) \rightarrow (4, 4) \][/tex]
- Since [tex]\((4, 4) \neq (-4, -4)\)[/tex], this point does not map onto itself.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect [tex]\((-4, 0)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (-4, 0) \rightarrow (0, 4) \][/tex]
- Since [tex]\((0, 4) \neq (-4, 0)\)[/tex], this point does not map onto itself.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflect [tex]\((0, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (0, -4) \rightarrow (4, 0) \][/tex]
- Since [tex]\((4, 0) \neq (0, -4)\)[/tex], this point does not map onto itself.
4. Point [tex]\((4, -4)\)[/tex]:
- Reflect [tex]\((4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (4, -4) \rightarrow (4, -4) \][/tex]
- Since [tex]\((4, -4) = (4, -4)\)[/tex], this point maps onto itself.
Therefore, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is [tex]\((4, -4)\)[/tex].
When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the coordinates of the reflected point become [tex]\((-y, -x)\)[/tex].
We need to check whether each of the given points maps onto itself after performing this reflection.
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflect [tex]\((-4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (-4, -4) \rightarrow (4, 4) \][/tex]
- Since [tex]\((4, 4) \neq (-4, -4)\)[/tex], this point does not map onto itself.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect [tex]\((-4, 0)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (-4, 0) \rightarrow (0, 4) \][/tex]
- Since [tex]\((0, 4) \neq (-4, 0)\)[/tex], this point does not map onto itself.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflect [tex]\((0, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (0, -4) \rightarrow (4, 0) \][/tex]
- Since [tex]\((4, 0) \neq (0, -4)\)[/tex], this point does not map onto itself.
4. Point [tex]\((4, -4)\)[/tex]:
- Reflect [tex]\((4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[ (4, -4) \rightarrow (4, -4) \][/tex]
- Since [tex]\((4, -4) = (4, -4)\)[/tex], this point maps onto itself.
Therefore, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is [tex]\((4, -4)\)[/tex].
