Answer :
To determine the type of reflection that changes the endpoints of a line segment, we need to compare the original and reflected coordinates.
Given endpoints of the original line segment:
[tex]\[ (-4, -6) \text{ and } (-6, 4) \][/tex]
Expected endpoints after reflection:
[tex]\[ (4, -6) \text{ and } (6, 4) \][/tex]
We need to find which type of reflection transforms the coordinates from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], respectively.
### Reflection across the x-axis
A reflection across the x-axis would invert the y-coordinates:
[tex]\[ (x, y) \rightarrow (x, -y) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (-4, 6) \\ (-6, 4) \rightarrow (-6, -4) \][/tex]
Clearly, this does not match the expected endpoints.
### Reflection across the y-axis
A reflection across the y-axis would invert the x-coordinates:
[tex]\[ (x, y) \rightarrow (-x, y) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (4, -6) \\ (-6, 4) \rightarrow (6, 4) \][/tex]
These transformed points exactly match the expected endpoints.
### Reflection across the line [tex]\(y = x\)[/tex]
A reflection across the line [tex]\(y = x\)[/tex] would swap the x- and y-coordinates:
[tex]\[ (x, y) \rightarrow (y, x) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (-6, -4) \\ (-6, 4) \rightarrow (4, -6) \][/tex]
Clearly, this does not match the expected endpoints.
### Reflection across the line [tex]\(y = -x\)[/tex]
A reflection across the line [tex]\(y = -x\)[/tex] would swap and invert the coordinates:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (6, 4) \\ (-6, 4) \rightarrow (-4, -6) \][/tex]
Clearly, this does not match the expected endpoints.
Based on this analysis, the reflection that produces the desired endpoints is:
[tex]\[ \boxed{\text{a reflection of the line segment across the y-axis}} \][/tex]
Given endpoints of the original line segment:
[tex]\[ (-4, -6) \text{ and } (-6, 4) \][/tex]
Expected endpoints after reflection:
[tex]\[ (4, -6) \text{ and } (6, 4) \][/tex]
We need to find which type of reflection transforms the coordinates from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], respectively.
### Reflection across the x-axis
A reflection across the x-axis would invert the y-coordinates:
[tex]\[ (x, y) \rightarrow (x, -y) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (-4, 6) \\ (-6, 4) \rightarrow (-6, -4) \][/tex]
Clearly, this does not match the expected endpoints.
### Reflection across the y-axis
A reflection across the y-axis would invert the x-coordinates:
[tex]\[ (x, y) \rightarrow (-x, y) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (4, -6) \\ (-6, 4) \rightarrow (6, 4) \][/tex]
These transformed points exactly match the expected endpoints.
### Reflection across the line [tex]\(y = x\)[/tex]
A reflection across the line [tex]\(y = x\)[/tex] would swap the x- and y-coordinates:
[tex]\[ (x, y) \rightarrow (y, x) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (-6, -4) \\ (-6, 4) \rightarrow (4, -6) \][/tex]
Clearly, this does not match the expected endpoints.
### Reflection across the line [tex]\(y = -x\)[/tex]
A reflection across the line [tex]\(y = -x\)[/tex] would swap and invert the coordinates:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
Applying this to the given points:
[tex]\[ (-4, -6) \rightarrow (6, 4) \\ (-6, 4) \rightarrow (-4, -6) \][/tex]
Clearly, this does not match the expected endpoints.
Based on this analysis, the reflection that produces the desired endpoints is:
[tex]\[ \boxed{\text{a reflection of the line segment across the y-axis}} \][/tex]
