To determine which reflection will produce an image with endpoints at [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the initial endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex], let's analyze each type of reflection in detail:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis changes its coordinates to [tex]\((x, -y)\)[/tex].
- Applying this to the initial points:
[tex]\[
(-4, -6) \rightarrow (-4, 6)
\][/tex]
[tex]\[
(-6, 4) \rightarrow (-6, -4)
\][/tex]
- The reflected points are [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], which do not match [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 changes its coordinates to [tex]\((-x, y)\)[/tex].
- Applying this to the initial points:
[tex]\[
(-4, -6) \rightarrow (4, -6)
\][/tex]
[tex]\[
(-6, 4) \rightarrow (6, 4)
\][/tex]
- The reflected points are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], which do not match [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] changes its coordinates to [tex]\((y, x)\)[/tex].
- Applying this to the initial points:
[tex]\[
(-4, -6) \rightarrow (-6, -4)
\][/tex]
[tex]\[
(-6, 4) \rightarrow (4, -6)
\][/tex]
- The reflected points are [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], which do not match [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] changes its coordinates to [tex]\((-y, -x)\)[/tex].
- Applying this to the initial points:
[tex]\[
(-4, -6) \rightarrow (6, 4)
\][/tex]
[tex]\[
(-6, 4) \rightarrow (-4, -6)
\][/tex]
- The reflected points are [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], which match neither [tex]\((4, 6)\)[/tex] nor [tex]\((6, 4)\)[/tex].
Based on the analysis above, no reflection produces the image points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the initial points [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]. Therefore, the answer is that there is no matching reflection from the given options.
