To determine which point would map onto itself after a reflection across the line [tex]\( y = x \)[/tex], we need to understand how reflection works. A point [tex]\((a, b)\)[/tex] reflected across the line [tex]\( y = x \)[/tex] moves to the point [tex]\((b, a)\)[/tex].
So, for a point to map onto itself after reflection across the line [tex]\( y = x \)[/tex], the original point [tex]\((a, b)\)[/tex] must satisfy the condition:
[tex]\[ (a, b) = (b, a) \][/tex]
This implies that [tex]\( a \)[/tex] must be equal to [tex]\( b \)[/tex].
Now, let's analyze each given point:
1. [tex]\((-4, -4)\)[/tex]:
- Reflected point: [tex]\((-4, -4) \rightarrow (-4, -4)\)[/tex]
- Since [tex]\( -4 = -4 \)[/tex], this point maps onto itself.
2. [tex]\((-4, 0)\)[/tex]:
- Reflected point: [tex]\((-4, 0) \rightarrow (0, -4)\)[/tex]
- Since [tex]\( -4 \neq 0 \)[/tex], this point does not map onto itself.
3. [tex]\( (0, -4)\)[/tex]:
- Reflected point: [tex]\((0, -4) \rightarrow (-4, 0)\)[/tex]
- Since [tex]\( 0 \neq -4 \)[/tex], this point does not map onto itself.
4. [tex]\( (4, -4)\)[/tex]:
- Reflected point: [tex]\( (4, -4) \rightarrow (-4, 4)\)[/tex]
- Since [tex]\( 4 \neq -4 \)[/tex], this point does not map onto itself.
From the analysis, we can see that the only point that maps onto itself after the reflection across the line [tex]\( y = x \)[/tex] is [tex]\((-4, -4)\)[/tex].
Therefore, the point that would map onto itself after a reflection across the line [tex]\( y = x \)[/tex] is:
[tex]\[ \boxed{(-4, -4)} \][/tex]
