To determine which point maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex], we need to reflect each given point across this line and check if the reflected point is the same as the original point.
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 apply this reflection to each of the given points:
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflect [tex]\((-4, -4)\)[/tex] across [tex]\(y = -x\)[/tex]:
[tex]\[(-4, -4) \rightarrow (4, 4)\][/tex]
- The reflected point is [tex]\((4, 4)\)[/tex], which is not the same as the original point.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect [tex]\((-4, 0)\)[/tex] across [tex]\(y = -x\)[/tex]:
[tex]\(( -4, 0) \rightarrow (0, 4)\)[/tex]
- The reflected point is [tex]\((0, 4)\)[/tex], which is not the same as the original point.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflect [tex]\((0, -4)\)[/tex] across [tex]\(y = -x\)[/tex]:
[tex]\((0, -4) \rightarrow (4, 0)\)[/tex]
- The reflected point is [tex]\((4, 0)\)[/tex], which is not the same as the original point.
4. Point [tex]\((4, -4)\)[/tex]:
- Reflect [tex]\((4, -4)\)[/tex] across [tex]\(y = -x\)[/tex]:
[tex]\((4, -4) \rightarrow (4, -4)\)[/tex]
- The reflected point is [tex]\((4, -4)\)[/tex], which is the same as the original point.
Therefore, the point [tex]\((4, -4)\)[/tex] maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex].
So the answer is:
[tex]\((4, -4)\)[/tex]
