To determine which reflection produces the given image, we need to analyze the transformations of the endpoints of the given line segment. We have the original endpoints at [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1)\)[/tex] and the new endpoints after reflection are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
Let's consider each type of reflection and see which one matches the new endpoints.
1. Reflection across the [tex]\( x \)[/tex]-axis: This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
[tex]\[
(-1, 4) \rightarrow (-1, -4) \\
(4, 1) \rightarrow (4, -1)
\][/tex]
The endpoints after this reflection do not match [tex]\((-4,1)\)[/tex] and [tex]\((-1,-4)\)[/tex].
2. Reflection across the [tex]\( y \)[/tex]-axis: This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
[tex]\[
(-1, 4) \rightarrow (1, 4) \\
(4, 1) \rightarrow (-4, 1)
\][/tex]
After reflection, we get endpoints [tex]\( (1, 4) \)[/tex] and [tex]\((-4, 1)\)[/tex]. This does not match our target endpoints.
3. Reflection across the line [tex]\( y = x \)[/tex]: This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
[tex]\[
(-1, 4) \rightarrow (4, -1) \\
(4, 1) \rightarrow (1, 4)
\][/tex]
These points do not match our target endpoints.
4. Reflection across the line [tex]\( y = -x \)[/tex]: This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
[tex]\[
(-1, 4) \rightarrow (-4, -1) \\
(4, 1) \rightarrow (-1, -4)
\][/tex]
For [tex]\((-1, 4)\)[/tex], reflection gives us [tex]\( (-4, -1) \)[/tex]. But [tex]\( (4, 1) \)[/tex] reflects to [tex]\((-1, -4)\)[/tex] which matches one of our desired endpoints. Revising the process:
[tex]\[
(-1, 4) \rightarrow (-4, 1) \\
(4, 1) \rightarrow (-1, -4)
\][/tex]
The correct reflection transforms [tex]\((-1, 4)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-1, -4)\)[/tex]. Therefore, the reflection that produces the required image is the reflection across the line [tex]\( y = -x \)[/tex].
[tex]\[
\boxed{\text{a reflection of the line segment across the line } y = -x}
\][/tex]
