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To determine which reflection produces the given image of a line segment with the endpoints changing from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], let's analyze the possible reflections one by one.
### Original Endpoints:
1. [tex]\((-4, -6)\)[/tex]
2. [tex]\((-6, 4)\)[/tex]
### New Endpoints:
1. [tex]\((4, -6)\)[/tex]
2. [tex]\((6, 4)\)[/tex]
### Possible Reflections:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Transformation rule: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (-4, 6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-6, -4)\)[/tex]
The resulting points [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex] do not match the given new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Hence, this is not the correct reflection.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Transformation rule: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (4, -6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (6, 4)\)[/tex]
The resulting points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] match the given new endpoints perfectly. Therefore, this is the correct reflection.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Transformation rule: [tex]\((x, y) \rightarrow (y, x)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (-6, -4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (4, -6)\)[/tex]
The resulting points [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex] do not match the given new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Hence, this is not the correct reflection.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Transformation rule: [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (6, 4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-4, -6)\)[/tex]
The resulting points [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex] might appear to match, but the order and signs of the points are incorrect compared to our desired new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Therefore, this is not the correct reflection.
### Conclusion:
The reflection that transforms the endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to the new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] is a reflection across the [tex]\(y\)[/tex]-axis.
Thus, the correct answer is:
A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
### Original Endpoints:
1. [tex]\((-4, -6)\)[/tex]
2. [tex]\((-6, 4)\)[/tex]
### New Endpoints:
1. [tex]\((4, -6)\)[/tex]
2. [tex]\((6, 4)\)[/tex]
### Possible Reflections:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Transformation rule: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (-4, 6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-6, -4)\)[/tex]
The resulting points [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex] do not match the given new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Hence, this is not the correct reflection.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Transformation rule: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (4, -6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (6, 4)\)[/tex]
The resulting points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] match the given new endpoints perfectly. Therefore, this is the correct reflection.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Transformation rule: [tex]\((x, y) \rightarrow (y, x)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (-6, -4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (4, -6)\)[/tex]
The resulting points [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex] do not match the given new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Hence, this is not the correct reflection.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Transformation rule: [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
Applying this to the original points:
- [tex]\((-4, -6) \rightarrow (6, 4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-4, -6)\)[/tex]
The resulting points [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex] might appear to match, but the order and signs of the points are incorrect compared to our desired new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Therefore, this is not the correct reflection.
### Conclusion:
The reflection that transforms the endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to the new endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] is a reflection across the [tex]\(y\)[/tex]-axis.
Thus, the correct answer is:
A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
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