To determine which reflection will produce the specified image for the given line segment, we need to examine the transformations of the endpoints. The initial endpoints of the line segment are \((-1, 4)\) and \((4, 1)\). The target endpoints after reflection are \((-4, 1)\) and \((-1, -4)\).
Let's examine each of the reflection options to see which one matches the target endpoints:
1. Reflection across the \(x\)-axis:
- Reflection across the \(x\)-axis changes \((x, y)\) to \((x, -y)\).
- Initial endpoints: \((-1, 4)\) and \((4, 1)\)
- Reflected endpoints: \((-1, -4)\) and \((4, -1)\)
- These do not match the target endpoints \((-4, 1)\) and \((-1, -4)\).
2. Reflection across the \(y\)-axis:
- Reflection across the \(y\)-axis changes \((x, y)\) to \((-x, y)\).
- Initial endpoints: \((-1, 4)\) and \((4, 1)\)
- Reflected endpoints: \((1, 4)\) and \((-4, 1)\)
- These only partially match the target endpoints; \((-1, -4)\) is missing.
3. Reflection across the line \(y=x\):
- Reflection across the line \(y=x\) changes \((x, y)\) to \((y, x)\).
- Initial endpoints: \((-1, 4)\) and \((4, 1)\)
- Reflected endpoints: \((4, -1)\) and \((1, 4)\)
- These do not match the target endpoints \((-4, 1)\) and \((-1, -4)\).
4. Reflection across the line \(y=-x\):
- Reflection across the line \(y=-x\) changes \((x, y)\) to \((-y, -x)\).
- Initial endpoints: \((-1, 4)\) and \((4, 1)\)
- Reflected endpoints: \((-4, 1)\) and \((-1, -4)\)
- These match exactly with the target endpoints \((-4, 1)\) and \((-1, -4)\).
Given the options, the reflection across the line \(y=-x\) is the transformation that results in the correct target endpoints for the line segment. Therefore, the correct reflection is:
- a reflection of the line segment across the line [tex]\(y=-x\)[/tex].
