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To determine which reflection transforms the endpoints of the line segment from [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], we need to analyze each possible reflection scenario:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (-1, -4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (4, -1) \][/tex]
- The reflected points are [tex]\((-1, -4)\)[/tex] and [tex]\( (4, -1) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (1, 4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (-4, 1) \][/tex]
- The reflected points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
3. Reflection across the line [tex]\( y=x \)[/tex]:
- When reflecting across the line [tex]\(y=x\)[/tex], the coordinates swap their positions.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (4, -1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (1, 4) \][/tex]
- The reflected points are [tex]\((4, -1)\)[/tex] and [tex]\( (1, 4) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
4. Reflection across the line [tex]\( y=-x \)[/tex]:
- When reflecting across the line [tex]\(y=-x\)[/tex], the coordinates swap their positions and change signs.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-4, 1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-1, -4) \][/tex]
- The reflected points are [tex]\((-4, 1)\)[/tex] and [tex]\( (-1, -4) \)[/tex], which match the given reflected points exactly.
Thus, the correct reflection that produces the endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original points [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] is a reflection across the line [tex]\( y = -x \)[/tex].
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (-1, -4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (4, -1) \][/tex]
- The reflected points are [tex]\((-1, -4)\)[/tex] and [tex]\( (4, -1) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (1, 4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (-4, 1) \][/tex]
- The reflected points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
3. Reflection across the line [tex]\( y=x \)[/tex]:
- When reflecting across the line [tex]\(y=x\)[/tex], the coordinates swap their positions.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (4, -1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (1, 4) \][/tex]
- The reflected points are [tex]\((4, -1)\)[/tex] and [tex]\( (1, 4) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
4. Reflection across the line [tex]\( y=-x \)[/tex]:
- When reflecting across the line [tex]\(y=-x\)[/tex], the coordinates swap their positions and change signs.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-4, 1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-1, -4) \][/tex]
- The reflected points are [tex]\((-4, 1)\)[/tex] and [tex]\( (-1, -4) \)[/tex], which match the given reflected points exactly.
Thus, the correct reflection that produces the endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original points [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] is a reflection across the line [tex]\( y = -x \)[/tex].
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
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