To determine which reflection produces the desired image for the endpoints of the line segment, we start by examining the endpoints of the original line segment and the desired image:
- Original endpoints: \((-1, 4)\) and \((4, 1)\)
- Desired endpoints: \((-4, 1)\) and \((-1, -4)\)
Now, let's explore each reflection option:
-
Reflection across the x-axis:
- \( (x, y) \) becomes \( (x, -y) \)
- Reflecting \((-1, 4)\) gives \((-1, -4)\)
- Reflecting \((4, 1)\) gives \((4, -1)\)
- Resulting points: \((-1, -4)\) and \( (4, -1)\) (not what we need)
-
Reflection across the y-axis:
- \( (x, y) \) becomes \( (-x, y) \)
- Reflecting \((-1, 4)\) gives \((1, 4)\)
- Reflecting \((4, 1)\) gives \((-4, 1)\)
- Resulting points: \((1, 4)\) and \((-4, 1)\) (not what we need)
-
Reflection across the line \(y = x\):
- \( (x, y) \) becomes \( (y, x) \)
- Reflecting \((-1, 4)\) gives \((4, -1)\)
- Reflecting \((4, 1)\) gives \((1, 4)\)
- Resulting points: \((4, -1)\) and \((1, 4)\) (not what we need)
-
Reflection across the line \(y = -x\):
- \( (x, y) \) becomes \((-y, -x)\)
- Reflecting \((-1, 4)\) gives \((-4, 1)\)
- Reflecting \((4, 1)\) gives \((-1, -4)\)
- Resulting points: \((-4, 1)\) and \((-1, -4)\) (which matches our desired endpoints)
Based on this analysis, the correct reflection is a reflection of the line segment across the line \(y = -x\).