To determine which reflection will produce the desired image of the line segment, we need to analyze the given endpoints and how their coordinates change under various reflections.
Given endpoints:
- Initial point 1: [tex]\((-1, 4)\)[/tex]
- Initial point 2: [tex]\((4, 1)\)[/tex]
Desired endpoints:
- Reflected point 1: [tex]\((-4, 1)\)[/tex]
- Reflected point 2: [tex]\((-1, -4)\)[/tex]
Let's consider each reflection option and see how the endpoints would change.
### Reflection across the [tex]\(x\)[/tex]-axis
When reflecting a point across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign, but the [tex]\(x\)[/tex]-coordinate remains the same.
[tex]\[
(x, y) \rightarrow (x, -y)
\][/tex]
Applying this to our initial points:
- Point 1: [tex]\((-1, 4) \rightarrow (-1, -4)\)[/tex]
- Point 2: [tex]\((4, 1) \rightarrow (4, -1)\)[/tex]
The resulting points [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex] do not match the desired reflected points.
### Reflection across the [tex]\(y\)[/tex]-axis
When reflecting a point across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign, but the [tex]\(y\)[/tex]-coordinate remains the same.
[tex]\[
(x, y) \rightarrow (-x, y)
\][/tex]
Applying this to our initial points:
- Point 1: [tex]\((-1, 4) \rightarrow (1, 4)\)[/tex]
- Point 2: [tex]\((4, 1) \rightarrow (-4, 1)\)[/tex]
The resulting points [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex] do not match the desired reflected points.
### Reflection across the line [tex]\(y = x\)[/tex]
When reflecting a point across the line [tex]\(y = x\)[/tex], the coordinates invert positions.
[tex]\[
(x, y) \rightarrow (y, x)
\][/tex]
Applying this to our initial points:
- Point 1: [tex]\((-1, 4) \rightarrow (4, -1)\)[/tex]
- Point 2: [tex]\((4, 1) \rightarrow (1, 4)\)[/tex]
The resulting points [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex] do not match the desired reflected points.
### Reflection across the line [tex]\(y = -x\)[/tex]
When reflecting a point across the line [tex]\(y = -x\)[/tex], the coordinates swap and both change their signs.
[tex]\[
(x, y) \rightarrow (-y, -x)
\][/tex]
Applying this to our initial points:
- Point 1: [tex]\((-1, 4) \rightarrow (-4, 1)\)[/tex]
- Point 2: [tex]\((4, 1) \rightarrow (-1, -4)\)[/tex]
The resulting points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] match perfectly with the desired reflected points.
Therefore, the correct reflection that will produce the given image is:
a reflection of the line segment across the line [tex]\(y = -x\)[/tex].
So the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
