To identify which reflection will produce an image with the given endpoints, let's first analyze the original and reflected points.
The original endpoints of the line segment are [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]. The reflected endpoints are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
To determine the type of reflection, we will examine the changes to the coordinates:
1. Reflection across the [tex]\(x\)[/tex]-axis: This changes [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
- [tex]\((-4, -6)\)[/tex] would become [tex]\((-4, 6)\)[/tex], not [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] would become [tex]\((-6, -4)\)[/tex], not [tex]\((6, 4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis: This changes [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
- [tex]\((-4, -6)\)[/tex] would become [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] would become [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y=x\)[/tex]: This changes [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- [tex]\((-4, -6)\)[/tex] would become [tex]\((-6, -4)\)[/tex], not [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] would become [tex]\((4, -6)\)[/tex], not [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y=-x\)[/tex]: This changes [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
- [tex]\((-4, -6)\)[/tex] would become [tex]\((6, 4)\)[/tex], not [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] would become [tex]\((-4, -6)\)[/tex], not [tex]\((6, 4)\)[/tex].
From the above analysis, we see that the second type of reflection (across the [tex]\(y\)[/tex]-axis) changes the endpoints from [tex]\((-4, -6)\)[/tex] to [tex]\((4, -6)\)[/tex] and from [tex]\((-6, 4)\)[/tex] to [tex]\((6, 4)\)[/tex].
Therefore, the correct reflection that produces the image with endpoints at [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] is:
[tex]\[
\boxed{\text{a reflection of the line segment across the } y\text{-axis}}
\][/tex]
