Answer :
To determine which reflection produces the specified endpoints, let's review various transformations:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-6, -4)\)[/tex].
- The endpoints after reflection are: [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((6, 4)\)[/tex].
- The endpoints after reflection are: [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((-4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
We want the reflected line segment to have endpoints [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Let's compare this with our results:
- Reflection across the [tex]\(x\)[/tex]-axis gives endpoints [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis gives endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] gives endpoints [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] gives endpoints [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
None of the reflections match the desired endpoints of [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Therefore, it is concluded that the specified reflection does not produce the required endpoints.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-6, -4)\)[/tex].
- The endpoints after reflection are: [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((6, 4)\)[/tex].
- The endpoints after reflection are: [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((-4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
We want the reflected line segment to have endpoints [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Let's compare this with our results:
- Reflection across the [tex]\(x\)[/tex]-axis gives endpoints [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis gives endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] gives endpoints [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] gives endpoints [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
None of the reflections match the desired endpoints of [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Therefore, it is concluded that the specified reflection does not produce the required endpoints.
