To determine which point would map onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the effect of this type of reflection.
When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the coordinates of the point transform according to the rule:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
For a point to map onto itself, the coordinates after the transformation must be the same as the original coordinates. This means:
[tex]\[ (x, y) = (-y, -x) \][/tex]
Let's analyze each point given in the options:
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflecting [tex]\((-4, -4)\)[/tex] across [tex]\( y = -x \)[/tex] results in:
[tex]\[ (-4, -4) \rightarrow (-(-4), -(-4)) = (4, 4) \][/tex]
- Here, [tex]\((-4, -4)\)[/tex] does not map onto itself.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflecting [tex]\((-4, 0)\)[/tex] across [tex]\( y = -x \)[/tex] results in:
[tex]\[ (-4, 0) \rightarrow (0, 4) \][/tex]
- Here, [tex]\((-4, 0)\)[/tex] does not map onto itself.
3. Point [tex]\((0, -4)\)[/tex]:
- Reflecting [tex]\((0, -4)\)[/tex] across [tex]\( y = -x \)[/tex] results in:
[tex]\[ (0, -4) \rightarrow (4, 0) \][/tex]
- Here, [tex]\((0, -4)\)[/tex] does not map onto itself.
4. Point [tex]\((4, -4)\)[/tex]:
- Reflecting [tex]\((4, -4)\)[/tex] across [tex]\( y = -x \)[/tex] results in:
[tex]\[ (4, -4) \rightarrow (4, -4) \][/tex]
- Here, [tex]\((4, -4)\)[/tex] does map onto itself.
Thus, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is [tex]\((-4, -4)\)[/tex].
