To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the effect of such a reflection.
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], the new coordinates of the point become [tex]\((-y, -x)\)[/tex]. For a point to map onto itself during this reflection, the original coordinates must equal the new coordinates after the transformation. That is, the point [tex]\((x, y)\)[/tex] must satisfy:
[tex]\[
(x, y) = (-y, -x)
\][/tex]
This condition implies:
1. [tex]\( x = -y \)[/tex]
2. [tex]\( y = -x \)[/tex]
Both conditions are inherently the same equation. Let's evaluate each given point to see if it satisfies [tex]\( x = -y \)[/tex].
1. [tex]\( (-4, -4) \)[/tex]:
[tex]\[
x = -4 \quad \text{and} \quad y = -4
\][/tex]
[tex]\[
x = -y \implies -4 = -(-4) \implies -4 = 4 \quad (\text{False})
\][/tex]
2. [tex]\( (-4, 0) \)[/tex]:
[tex]\[
x = -4 \quad \text{and} \quad y = 0
\][/tex]
[tex]\[
x = -y \implies -4 = -(0) \implies -4 = 0 \quad (\text{False})
\][/tex]
3. [tex]\( (0, -4) \)[/tex]:
[tex]\[
x = 0 \quad \text{and} \quad y = -4
\][/tex]
[tex]\[
x = -y \implies 0 = -(-4) \implies 0 = 4 \quad (\text{False})
\][/tex]
4. [tex]\( (4, -4) \)[/tex]:
[tex]\[
x = 4 \quad \text{and} \quad y = -4
\][/tex]
[tex]\[
x = -y \implies 4 = -(-4) \implies 4 = 4 \quad (\text{True})
\][/tex]
Thus, the point [tex]\( (4, -4) \)[/tex] satisfies [tex]\( x = -y \)[/tex] and maps onto itself after being reflected across the line [tex]\( y = -x \)[/tex].
Therefore, the point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is:
[tex]\[
(4, -4)
\][/tex]
