To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the transformation involved.
When a point [tex]\((a, b)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], it is transformed to the point [tex]\((-b, -a)\)[/tex].
Let's apply this transformation to each of the given points to see which one maps onto itself:
1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[
(-4, -4) \rightarrow (-(-4), -(-4)) = (4, 4)
\][/tex]
Therefore, [tex]\((-4, -4)\)[/tex] does not map onto itself.
2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[
(-4, 0) \rightarrow (0, 4)
\][/tex]
Therefore, [tex]\((-4, 0)\)[/tex] does not map onto itself.
3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[
(0, -4) \rightarrow (4, 0)
\][/tex]
Therefore, [tex]\((0, -4)\)[/tex] does not map onto itself.
4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[
(4, -4) \rightarrow (4, -4)
\][/tex]
Therefore, [tex]\((4, -4)\)[/tex] maps onto itself.
Hence, the point that maps onto itself after reflecting across the line [tex]\( y = -x \)[/tex] is [tex]\((4, -4)\)[/tex].
Thus, the answer is:
[tex]\[
\boxed{4}
\][/tex]
