Answer :
To determine which point maps onto itself after a reflection across the line \( y = -x \), we need to understand how points transform when reflected over this line.
The rule for reflecting a point \((x, y)\) across the line \( y = -x \) is to swap and negate both coordinates, turning \((x, y)\) into \((-y, -x)\).
Let's apply this transformation to each point and check if any point remains unchanged:
1. Point \((-4, -4)\):
- Reflecting \((-4, -4)\) results in:
[tex]\[ (-(-4), -(-4)) = (4, 4) \][/tex]
So, \((-4, -4)\) does not map onto itself.
2. Point \((-4, 0)\):
- Reflecting \((-4, 0)\) results in:
[tex]\[ (-(0), -(-4)) = (0, 4) \][/tex]
So, \((-4, 0)\) does not map onto itself.
3. Point \((0, -4)\):
- Reflecting \((0, -4)\) results in:
[tex]\[ (-(-4), -(0)) = (4, 0) \][/tex]
So, \((0, -4)\) does not map onto itself.
4. Point \((4, -4)\):
- Reflecting \((4, -4)\) results in:
[tex]\[ (-(-4), -(4)) = (4, -4) \][/tex]
Here, \((4, -4)\) remains \((4, -4)\) after reflection.
Thus, the point \((4, -4)\) maps onto itself after a reflection across the line \( y = -x \).
Therefore, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
The rule for reflecting a point \((x, y)\) across the line \( y = -x \) is to swap and negate both coordinates, turning \((x, y)\) into \((-y, -x)\).
Let's apply this transformation to each point and check if any point remains unchanged:
1. Point \((-4, -4)\):
- Reflecting \((-4, -4)\) results in:
[tex]\[ (-(-4), -(-4)) = (4, 4) \][/tex]
So, \((-4, -4)\) does not map onto itself.
2. Point \((-4, 0)\):
- Reflecting \((-4, 0)\) results in:
[tex]\[ (-(0), -(-4)) = (0, 4) \][/tex]
So, \((-4, 0)\) does not map onto itself.
3. Point \((0, -4)\):
- Reflecting \((0, -4)\) results in:
[tex]\[ (-(-4), -(0)) = (4, 0) \][/tex]
So, \((0, -4)\) does not map onto itself.
4. Point \((4, -4)\):
- Reflecting \((4, -4)\) results in:
[tex]\[ (-(-4), -(4)) = (4, -4) \][/tex]
Here, \((4, -4)\) remains \((4, -4)\) after reflection.
Thus, the point \((4, -4)\) maps onto itself after a reflection across the line \( y = -x \).
Therefore, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
