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 analyze each option:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-4, 6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-6, -4)\)[/tex].
- The reflected coordinates [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex] do not match the desired endpoints [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 the point [tex]\((-x, y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((6, 4)\)[/tex].
- The reflected coordinates [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] do match the desired endpoints [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 the point [tex]\((y, x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((-6, -4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((4, -6)\)[/tex].
- The reflected coordinates [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/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 the point [tex]\((-y, -x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((6, 4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((-4, -6)\)[/tex].
- The reflected coordinates [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
From this analysis, it is clear that:
- Reflecting the points [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis produces the desired image points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Therefore, the correct reflection is:
- A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
The correct answer is:
```
a reflection of the line segment across the y-axis
```
