To determine which point would map onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the reflection process.
When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\( y = -x \)[/tex], its coordinates are transformed into [tex]\((-y, -x)\)[/tex]. So, for a point to map onto itself, [tex]\((x, y)\)[/tex] must satisfy the condition:
[tex]\[ (x, y) = (-y, -x) \][/tex]
Let's analyze each of the given points:
1. Point [tex]\((4, -4)\)[/tex]:
- Reflect it over [tex]\( y = -x \)[/tex]:
[tex]\[ (4, -4) \rightarrow (-(-4), -4) = (4, -4) \][/tex]
- This point maps onto itself.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect it over [tex]\( y = -x \)[/tex]:
[tex]\[ (-4, 0) \rightarrow (0, 4) \][/tex]
- This point does not map onto itself.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflect it over [tex]\( y = -x \)[/tex]:
[tex]\[ (0, -4) \rightarrow (4, 0) \][/tex]
- This point does not map onto itself.
4. Point [tex]\((4, -4)\)[/tex]:
- This is the same as the first point; thus, it also maps onto itself:
[tex]\[ (4, -4) \rightarrow (-(-4), -4) = (4, -4) \][/tex]
After examining all the points, we find that the point [tex]\((4, -4)\)[/tex] (which is listed twice) maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
Therefore, the point that maps onto itself after reflection across [tex]\( y = -x \)[/tex] is [tex]\( (4, -4) \)[/tex].
