Answer :
To determine which point maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex], we need to apply the reflection transformation to each point.
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], its image is [tex]\((-y, -x)\)[/tex].
Let’s examine each given point:
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflecting [tex]\((-4, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -(-4)) = (4, 4) \][/tex]
- The reflected point is [tex]\((4, 4)\)[/tex], which is not equal to the original point [tex]\((-4, -4)\)[/tex].
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflecting [tex]\((-4, 0)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-0, -(-4)) = (0, 4) \][/tex]
- The reflected point is [tex]\((0, 4)\)[/tex], which is not equal to the original point [tex]\((-4, 0)\)[/tex].
3. Point [tex]\((0, -4)\)[/tex]:
- Reflecting [tex]\((0, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -0) = (4, 0) \][/tex]
- The reflected point is [tex]\((4, 0)\)[/tex], which is not equal to the original point [tex]\((0, -4)\)[/tex].
4. Point [tex]\((4, -4)\)[/tex]:
- Reflecting [tex]\((4, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -4) = (4, -4) \][/tex]
- The reflected point is [tex]\((4, -4)\)[/tex], which is equal to the original point [tex]\((4, -4)\)[/tex].
Thus, the point that maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex] is [tex]\((4, -4)\)[/tex].
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], its image is [tex]\((-y, -x)\)[/tex].
Let’s examine each given point:
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflecting [tex]\((-4, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -(-4)) = (4, 4) \][/tex]
- The reflected point is [tex]\((4, 4)\)[/tex], which is not equal to the original point [tex]\((-4, -4)\)[/tex].
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflecting [tex]\((-4, 0)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-0, -(-4)) = (0, 4) \][/tex]
- The reflected point is [tex]\((0, 4)\)[/tex], which is not equal to the original point [tex]\((-4, 0)\)[/tex].
3. Point [tex]\((0, -4)\)[/tex]:
- Reflecting [tex]\((0, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -0) = (4, 0) \][/tex]
- The reflected point is [tex]\((4, 0)\)[/tex], which is not equal to the original point [tex]\((0, -4)\)[/tex].
4. Point [tex]\((4, -4)\)[/tex]:
- Reflecting [tex]\((4, -4)\)[/tex] across the line [tex]\(y = -x\)[/tex], we get:
[tex]\[ (-(-4), -4) = (4, -4) \][/tex]
- The reflected point is [tex]\((4, -4)\)[/tex], which is equal to the original point [tex]\((4, -4)\)[/tex].
Thus, the point that maps onto itself after a reflection across the line [tex]\(y = -x\)[/tex] is [tex]\((4, -4)\)[/tex].
