Answer :
To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the properties of reflection across this line. A point [tex]\((a, b)\)[/tex] when reflected across the line [tex]\( y = -x \)[/tex] will map to the point [tex]\((-b, -a)\)[/tex].
For a point to map onto itself when reflected across the line [tex]\( y = -x \)[/tex], the point must satisfy the condition:
[tex]\[ (a, b) = (-b, -a) \][/tex]
This implies that:
[tex]\[ a = -b \][/tex]
[tex]\[ b = -a \][/tex]
Therefore, we see that the only points that satisfy this condition are those where both coordinates are equal in magnitude but opposite in sign. Specifically for our question, since we want the point to remain unchanged:
[tex]\[ a = -a \][/tex]
[tex]\[ b = -b \][/tex]
This further simplifies to:
[tex]\[ a = -a \Rightarrow a = 0 \][/tex]
[tex]\[ b = -b \Rightarrow b = 0 \][/tex]
So, a point that maps onto itself must satisfy [tex]\( x = -y \)[/tex]. Now, let's check each given point:
1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = 0 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[ x = 0 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x = -y \)[/tex], so this point does satisfy the condition.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
So, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
Thus, the correct choice number is:
[tex]\[ \boxed{4} \][/tex]
For a point to map onto itself when reflected across the line [tex]\( y = -x \)[/tex], the point must satisfy the condition:
[tex]\[ (a, b) = (-b, -a) \][/tex]
This implies that:
[tex]\[ a = -b \][/tex]
[tex]\[ b = -a \][/tex]
Therefore, we see that the only points that satisfy this condition are those where both coordinates are equal in magnitude but opposite in sign. Specifically for our question, since we want the point to remain unchanged:
[tex]\[ a = -a \][/tex]
[tex]\[ b = -b \][/tex]
This further simplifies to:
[tex]\[ a = -a \Rightarrow a = 0 \][/tex]
[tex]\[ b = -b \Rightarrow b = 0 \][/tex]
So, a point that maps onto itself must satisfy [tex]\( x = -y \)[/tex]. Now, let's check each given point:
1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = 0 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[ x = 0 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x = -y \)[/tex], so this point does satisfy the condition.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
So, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
Thus, the correct choice number is:
[tex]\[ \boxed{4} \][/tex]
