Answer :
To determine which reflection will produce the image with endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original points [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex], let's analyze the transformation.
We are given the original points:
[tex]\[ (-4, -6) \quad \text{and} \quad (-6, 4) \][/tex]
We are given the reflected points:
[tex]\[ (4, -6) \quad \text{and} \quad (6, 4) \][/tex]
We need to determine which reflection method results in these transformed points.
### 1. Reflection across the [tex]\( x \)[/tex]-axis:
A reflection across the [tex]\( x \)[/tex]-axis changes [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (-4, 6) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (-6, -4) \][/tex]
After this reflection, the points are:
[tex]\[ (-4, 6) \quad \text{and} \quad (-6, -4) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 2. Reflection across the [tex]\( y \)[/tex]-axis:
A reflection across the [tex]\( y \)[/tex]-axis changes [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (4, -6) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (6, 4) \][/tex]
After this reflection, the points are:
[tex]\[ (4, -6) \quad \text{and} \quad (6, 4) \][/tex]
These match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 3. Reflection across the line [tex]\( y = x \)[/tex]:
A reflection across the line [tex]\( y = x \)[/tex] changes [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (-6, -4) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (4, -6) \][/tex]
After this reflection, the points are:
[tex]\[ (-6, -4) \quad \text{and} \quad (4, -6) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 4. Reflection across the line [tex]\( y = -x \)[/tex]:
A reflection across the line [tex]\( y = -x \)[/tex] changes [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (6, 4) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (-4, 6) \][/tex]
After this reflection, the points are:
[tex]\[ (6, 4) \quad \text{and} \quad (-4, 6) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Given the detailed analysis, the only reflection that correctly transforms the original points into the reflected points is:
[tex]\[ \boxed{\text{a reflection of the line segment across the } y \text{-axis}} \][/tex]
We are given the original points:
[tex]\[ (-4, -6) \quad \text{and} \quad (-6, 4) \][/tex]
We are given the reflected points:
[tex]\[ (4, -6) \quad \text{and} \quad (6, 4) \][/tex]
We need to determine which reflection method results in these transformed points.
### 1. Reflection across the [tex]\( x \)[/tex]-axis:
A reflection across the [tex]\( x \)[/tex]-axis changes [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (-4, 6) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (-6, -4) \][/tex]
After this reflection, the points are:
[tex]\[ (-4, 6) \quad \text{and} \quad (-6, -4) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 2. Reflection across the [tex]\( y \)[/tex]-axis:
A reflection across the [tex]\( y \)[/tex]-axis changes [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (4, -6) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (6, 4) \][/tex]
After this reflection, the points are:
[tex]\[ (4, -6) \quad \text{and} \quad (6, 4) \][/tex]
These match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 3. Reflection across the line [tex]\( y = x \)[/tex]:
A reflection across the line [tex]\( y = x \)[/tex] changes [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (-6, -4) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (4, -6) \][/tex]
After this reflection, the points are:
[tex]\[ (-6, -4) \quad \text{and} \quad (4, -6) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
### 4. Reflection across the line [tex]\( y = -x \)[/tex]:
A reflection across the line [tex]\( y = -x \)[/tex] changes [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting gives:
[tex]\[ (6, 4) \][/tex]
- For [tex]\((-6, 4)\)[/tex], reflecting gives:
[tex]\[ (-4, 6) \][/tex]
After this reflection, the points are:
[tex]\[ (6, 4) \quad \text{and} \quad (-4, 6) \][/tex]
These do not match the reflected points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Given the detailed analysis, the only reflection that correctly transforms the original points into the reflected points is:
[tex]\[ \boxed{\text{a reflection of the line segment across the } y \text{-axis}} \][/tex]
