To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to analyze what happens to a point [tex]\((a, b)\)[/tex] when it is reflected over this line.
The formula for reflecting a point [tex]\((a, b)\)[/tex] across the line [tex]\( y = -x \)[/tex] results in the new coordinates [tex]\((-b, -a)\)[/tex].
For a point to map onto itself, its coordinates must satisfy the condition:
[tex]\[
(a, b) = (-b, -a)
\][/tex]
This leads to the following system of equations:
[tex]\[
a = -b
\][/tex]
[tex]\[
b = -a
\][/tex]
These are two expressions of the same condition, essentially meaning [tex]\( a \)[/tex] must be the negative of [tex]\( b \)[/tex].
Let's now analyze each point given in the problem:
1. Point (-4, -4):
- Reflect [tex]\((-4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[
(-(-4), -(-4)) = (4, 4)
\][/tex]
[tex]\((-4, -4)\)[/tex] does not map onto itself.
2. Point (-4, 0):
- Reflect [tex]\((-4, 0)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[
(0, 4)
\][/tex]
[tex]\((-4, 0)\)[/tex] does not map onto itself.
3. Point (0, -4):
- Reflect [tex]\((0, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[
(4, 0)
\][/tex]
[tex]\((0, -4)\)[/tex] does not map onto itself.
4. Point (4, -4):
- Reflect [tex]\( (4, -4)\)[/tex] across [tex]\( y = -x \)[/tex]:
[tex]\[
(4, -4) \rightarrow (4, -4)
\][/tex]
We see that [tex]\( (4, -4) \)[/tex] maps back to itself.
By analyzing each point, we conclude that the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is:
[tex]\( \boxed{(4, -4)} \)[/tex]
