Let's analyze the given problem step-by-step.
First, we are given the original endpoints of a line segment:
[tex]\[
(3, 2) \quad \text{and} \quad (2, -3)
\][/tex]
We are also told the resultant endpoints of the line segment after reflection:
[tex]\[
(3, -2) \quad \text{and} \quad (2, 3)
\][/tex]
We need to determine which type of reflection will produce this image. We have four options to consider:
1. Reflection across the x-axis
2. Reflection across the y-axis
3. Reflection across the line \( y = x \)
4. Reflection across the line \( y = -x \)
Reflection across the x-axis:
The transformation rule for reflecting a point \((x, y)\) across the x-axis is:
[tex]\[
(x, y) \rightarrow (x, -y)
\][/tex]
Applying this rule to our original points:
[tex]\[
(3, 2) \rightarrow (3, -2) \quad \text{and} \quad (2, -3) \rightarrow (2, 3)
\][/tex]
This gives us the points:
[tex]\[
(3, -2) \quad \text{and} \quad (2, 3)
\][/tex]
These points match the resultant endpoints perfectly. Therefore, the reflection across the x-axis is the correct transformation.
Reflection across the y-axis:
The transformation rule for reflecting a point \((x, y)\) across the y-axis is:
[tex]\[
(x, y) \rightarrow (-x, y)
\][/tex]
Applying this rule to our original points:
[tex]\[
(3, 2) \rightarrow (-3, 2) \quad \text{and} \quad (2, -3) \rightarrow (-2, -3)
\][/tex]
This gives us the points:
[tex]\[
(-3, 2) \quad \text{and} \quad (-2, -3)
\][/tex]
These points do not match the resultant endpoints.
Reflection across the line \( y = x \):
The transformation rule for reflecting a point \((x, y)\) across the line \( y = x \) is:
[tex]\[
(x, y) \rightarrow (y, x)
\][/tex]
Applying this rule to our original points:
[tex]\[
(3, 2) \rightarrow (2, 3) \quad \text{and} \quad (2, -3) \rightarrow (-3, 2)
\][/tex]
This gives us the points:
[tex]\[
(2, 3) \quad \text{and} \quad (-3, 2)
\][/tex]
These points do not match the resultant endpoints.
Reflection across the line \( y = -x \):
The transformation rule for reflecting a point \((x, y)\) across the line \( y = -x \) is:
[tex]\[
(x, y) \rightarrow (-y, -x)
\][/tex]
Applying this rule to our original points:
[tex]\[
(3, 2) \rightarrow (-2, -3) \quad \text{and} \quad (2, -3) \rightarrow (3, -2)
\][/tex]
This gives us the points:
[tex]\[
(-2, -3) \quad \text{and} \quad (3, -2)
\][/tex]
These points do not match the resultant endpoints.
Therefore, the correct reflection that produces the image with endpoints \((3, -2)\) and \((2, 3)\) is the reflection across the x-axis. The answer is:
[tex]\[
\boxed{\text{a reflection of the line segment across the } x\text{-axis}}
\][/tex]
