Let's determine which reflection will transform the line segment with endpoints at [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to the endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
### Step 1: Understand the original and reflected points
- Original points: [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]
- Reflected points: [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]
### Step 2: Test each potential reflection
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Transformation rule: [tex]\( (x, y) \rightarrow (x, -y) \)[/tex]
- Apply to [tex]\((-1, 4)\)[/tex]: [tex]\( (-1, -4) \)[/tex]
- Apply to [tex]\((4, 1)\)[/tex]: [tex]\( (4, -1) \)[/tex]
- Resulting points: [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex], which are not [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Transformation rule: [tex]\( (x, y) \rightarrow (-x, y) \)[/tex]
- Apply to [tex]\((-1, 4)\)[/tex]: [tex]\( (1, 4) \)[/tex]
- Apply to [tex]\((4, 1)\)[/tex]: [tex]\( (-4, 1) \)[/tex]
- Resulting points: [tex]\( (1, 4) \)[/tex] and [tex]\((-4, 1)\)[/tex], which are not [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Transformation rule: [tex]\( (x, y) \rightarrow (y, x) \)[/tex]
- Apply to [tex]\((-1, 4)\)[/tex]: [tex]\( (4, -1) \)[/tex]
- Apply to [tex]\((4, 1)\)[/tex]: [tex]\( (1, 4) \)[/tex]
- Resulting points: [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex], which are not [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Transformation rule: [tex]\( (x, y) \rightarrow (-y, -x) \)[/tex]
- Apply to [tex]\((-1, 4)\)[/tex]: [tex]\( (-4, 1) \)[/tex]
- Apply to [tex]\((4, 1)\)[/tex]: [tex]\( (-1, -4) \)[/tex]
- Resulting points: [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], which match exactly with [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
### Conclusion
The reflection across the line [tex]\(y = -x\)[/tex] will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
Thus, the correct reflection is across the line [tex]\(y = -x\)[/tex].
