To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand how reflections across this line work. The reflection of a point [tex]\((a, b)\)[/tex] across the line [tex]\( y = -x \)[/tex] transforms the point into [tex]\((-b, -a)\)[/tex].
A point [tex]\((a, b)\)[/tex] will map onto itself if and only if [tex]\((a, b) = (-b, -a)\)[/tex]. This relationship holds true if [tex]\( a = -b \)[/tex] and [tex]\( b = -a \)[/tex].
Let's verify each given point to see if it maps onto itself:
1. Point (-4, -4)
- Reflecting [tex]\((-4, -4)\)[/tex]: It becomes [tex]\((4, 4)\)[/tex]
- Clearly, [tex]\((-4, -4) \ne (4, 4)\)[/tex], so this point does not map onto itself.
2. Point (-4, 0)
- Reflecting [tex]\((-4, 0)\)[/tex]: It becomes [tex]\((0, 4)\)[/tex]
- [tex]\((-4, 0) \ne (0, 4)\)[/tex], so this point does not map onto itself.
3. Point (0, -4)
- Reflecting [tex]\((0, -4)\)[/tex]: It becomes [tex]\((4, 0)\)[/tex]
- [tex]\((0, -4) \ne (4, 0)\)[/tex], so this point does not map onto itself.
4. Point (4, -4)
- Reflecting [tex]\((4, -4)\)[/tex]: It becomes [tex]\((4, -4)\)[/tex]
- [tex]\((4, -4) = (4, -4)\)[/tex], meaning this point maps onto itself.
Therefore, the point [tex]\((4, -4)\)[/tex] maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
So, the correct answer is:
(4, -4).
