To find which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the transformation that happens during such a reflection.
A point [tex]\((x, y)\)[/tex] will reflect to a point [tex]\((-y, -x)\)[/tex] when reflected across the line [tex]\( y = -x \)[/tex]. Essentially, we swap the coordinates and change their signs.
Now, we need to determine which point remains unchanged under this transformation. This means we need to solve for the point [tex]\((x, y)\)[/tex] such that:
[tex]\[ (x, y) = (-y, -x) \][/tex]
Let's test each given point:
1. For [tex]\((-4, -4)\)[/tex]:
[tex]\[ (x, y) = (-4, -4) \][/tex]
[tex]\[ (-y, -x) = (--4, --4) = (4, 4) \][/tex]
After reflection: [tex]\((4, 4)\)[/tex]. This does not map onto itself.
2. For [tex]\((-4, 0)\)[/tex]:
[tex]\[ (x, y) = (-4, 0) \][/tex]
[tex]\[ (-y, -x) = (0, 4) \][/tex]
After reflection: [tex]\((0, 4)\)[/tex]. This does not map onto itself.
3. For [tex]\((0, -4)\)[/tex]:
[tex]\[ (x, y) = (0, -4) \][/tex]
[tex]\[ (-y, -x) = (4, 0) \][/tex]
After reflection: [tex]\((4, 0)\)[/tex]. This does not map onto itself.
4. For [tex]\((4, -4)\)[/tex]:
[tex]\[ (x, y) = (4, -4) \][/tex]
[tex]\[ (-y, -x) = (4, -4) \][/tex]
After reflection: [tex]\((4, -4)\)[/tex]. This point maps onto itself.
Therefore, the correct point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is:
[tex]\[
(4, -4)
\][/tex]
So, the point that would map onto itself is the fourth point:
[tex]\[
(4, -4)
\][/tex]
