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To reflect the given polygon across the line \( y = x \), we apply the following transformation rule for each vertex:

- Any point \( (x, y) \) reflected over the line \( y = x \) will map to \( (y, x) \).

Given the vertices of the polygon:
- \( A(2, -1) \)
- \( B(-1, 2) \)
- \( C(2, 3) \)
- \( D(4, 2) \)

Let's reflect each point:

- Reflect \( A(2, -1) \) → \( A'(-1, 2) \)
- Reflect \( B(-1, 2) \) → \( B'(2, -1) \)
- Reflect \( C(2, 3) \) → \( C'(3, 2) \)
- Reflect \( D(4, 2) \) → \( D'(2, 4) \)

So, the reflected vertices are:
- \( A'(-1, 2) \)
- \( B'(2, -1) \)
- \( C'(3, 2) \)
- \( D'(2, 4) \)

Now, you can plot these new vertices on the graph to obtain the polygon reflected across the line \( y = x \).

Would you like a step-by-step guide on how to plot this reflection, or do you have other questions?

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Here are 5 related questions to deepen understanding:

1. How do you find the reflection of a point across the x-axis or y-axis?
2. What is the formula for reflecting a point across the origin?
3. How would the transformation change if reflecting over a different line (e.g., \( y = -x \))?
4. What is the general geometric meaning of a reflection transformation?
5. How can you verify whether two polygons are congruent after a reflection?

**Tip:** When reflecting points across the line \( y = x \), remember to swap the coordinates!

Graph the polygon with the given vertices and its image after a reflection in the line y = x. A(2, -1), B(-1, 2), C(2, 3), D(4, 2).

This lesson covers the reflection of a polygon with vertices A(2, -1), B(-1, 2), C(2, 3), and D(4, 2) across the line y = x. Learn how to apply the reflection transformation rule to swap the coordinates of each vertex.

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