Answer :
To determine which reflection will produce an image with endpoints at [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex], let's examine the effects of reflecting the line segment across different axes and lines.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-4, 6)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-6, -4)\)[/tex].
- The endpoints would be [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
Hence, reflecting across the [tex]\(x\)[/tex]-axis does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((4, -6)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((6, 4)\)[/tex].
- The endpoints would be [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Therefore, reflecting across the [tex]\(y\)[/tex]-axis yields the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((y, x)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((-6, -4)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((4, -6)\)[/tex].
- The endpoints would be [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
Hence, reflecting across the line [tex]\(y = x\)[/tex] does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((6, 4)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-4, -6)\)[/tex].
- The endpoints would be [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
Hence, reflecting across the line [tex]\(y = -x\)[/tex] does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Based on this evaluation, the correct reflection that produces the desired endpoints is:
a reflection of the line segment across the [tex]\(y\)[/tex]-axis.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-4, 6)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-6, -4)\)[/tex].
- The endpoints would be [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
Hence, reflecting across the [tex]\(x\)[/tex]-axis does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((4, -6)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((6, 4)\)[/tex].
- The endpoints would be [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Therefore, reflecting across the [tex]\(y\)[/tex]-axis yields the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((y, x)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((-6, -4)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((4, -6)\)[/tex].
- The endpoints would be [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
Hence, reflecting across the line [tex]\(y = x\)[/tex] does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
- Reflecting [tex]\((-4, -6)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((6, 4)\)[/tex].
- Reflecting [tex]\((-6, 4)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-4, -6)\)[/tex].
- The endpoints would be [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
Hence, reflecting across the line [tex]\(y = -x\)[/tex] does not yield the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Based on this evaluation, the correct reflection that produces the desired endpoints is:
a reflection of the line segment across the [tex]\(y\)[/tex]-axis.
