Let's analyze the reflection of the given line segment endpoints step-by-step.
We have a line segment with endpoints at [tex]\((3, 2)\)[/tex] and [tex]\((2, -3)\)[/tex].
The question is asking for a reflection that will transform the endpoint [tex]\((2, -3)\)[/tex] to [tex]\((2, 3)\)[/tex].
To determine which reflection produces this image, let's review the effect of each type of reflection on a point [tex]\((x, y)\)[/tex]:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- The [tex]\(x\)[/tex]-coordinate remains the same.
- The [tex]\(y\)[/tex]-coordinate changes sign.
- Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex].
- For [tex]\((2, -3)\)[/tex], the transformation will be:
[tex]\[
(2, -3) \rightarrow (2, -(-3)) = (2, 3)
\][/tex]
2. Reflection across the [tex]\(y\)[/tex]-axis:
- The [tex]\(x\)[/tex]-coordinate changes sign.
- The [tex]\(y\)[/tex]-coordinate remains the same.
- Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
- For [tex]\((2, -3)\)[/tex], the transformation will be:
[tex]\[
(2, -3) \rightarrow (-2, -3)
\][/tex]
3. Reflection across the line [tex]\(y = x\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate swap places.
- Transformation: [tex]\((x, y) \rightarrow (y, x)\)[/tex].
- For [tex]\((2, -3)\)[/tex], the transformation will be:
[tex]\[
(2, -3) \rightarrow (-3, 2)
\][/tex]
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate swap places and change signs.
- Transformation: [tex]\((x, y) \rightarrow (-y, -x)\)[/tex].
- For [tex]\((2, -3)\)[/tex], the transformation will be:
[tex]\[
(2, -3) \rightarrow (3, -2)
\][/tex]
After analyzing the transformations, we see that reflecting a point [tex]\((x, y) = (2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the desired endpoint [tex]\((2, 3)\)[/tex].
Therefore, the reflection that will transform the endpoint [tex]\((2, -3)\)[/tex] to [tex]\((2, 3)\)[/tex] is:
a reflection of the line segment across the [tex]\(x\)[/tex]-axis.
Thus, the correct answer is:
[tex]\[
\boxed{\text{a reflection of the line segment across the $x$-axis}}
\][/tex]
