Worksheet on reflecting shapes across lines on a coordinate grid.

It looks like you've uploaded a worksheet titled "Reflections of Shapes" that asks students to reflect various shapes across different lines (the x-axis, y-axis, the line $ y = x $, and the line $ y = -x $) on a coordinate grid.

Since I can't view images directly, I’ll help you solve this problem based on typical content for such a worksheet. Let’s assume the worksheet contains four problems, each with a shape plotted on a coordinate plane and a reflection instruction. Here's how to approach each type:

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🔷 General Rules for Reflections:

1. Reflection over the x-axis:
Change the sign of the y-coordinate.
$(x, y) \rightarrow (x, -y)$

2. Reflection over the y-axis:
Change the sign of the x-coordinate.
$(x, y) \rightarrow (-x, y)$

3. Reflection over the line $ y = x $:
Swap the coordinates.
$(x, y) \rightarrow (y, x)$

4. Reflection over the line $ y = -x $:
Swap and change signs:
$(x, y) \rightarrow (-y, -x)$

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Let’s go through each problem based on what is typically seen in such worksheets.

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✔ Problem 1: Reflect over the x-axis

Suppose the triangle has vertices:
- $ A(2, 3) $
- $ B(4, 1) $
- $ C(1, 5) $

To reflect over the x-axis, apply:
$(x, y) \rightarrow (x, -y)$

New points:
- $ A'(2, -3) $
- $ B'(4, -1) $
- $ C'(1, -5) $

Plot these new points and connect them to form the reflected image.

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✔ Problem 2: Reflect over $ y = x $

Suppose the triangle has vertices:
- $ A(1, 2) $
- $ B(3, 4) $
- $ C(2, 1) $

To reflect over $ y = x $, swap coordinates:
$(x, y) \rightarrow (y, x)$

New points:
- $ A'(2, 1) $
- $ B'(4, 3) $
- $ C'(1, 2) $

Plot and connect.

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✔ Problem 3: Reflect over the y-axis

Suppose quadrilateral has vertices:
- $ P(-2, 1) $
- $ Q(-4, 3) $
- $ R(-1, 5) $
- $ S(0, 2) $

Reflect over y-axis:
$(x, y) \rightarrow (-x, y)$

New points:
- $ P'(2, 1) $
- $ Q'(4, 3) $
- $ R'(1, 5) $
- $ S'(0, 2) $

Plot and connect.

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✔ Problem 4: Reflect over $ y = -x $

Suppose quadrilateral has vertices:
- $ A(2, 1) $
- $ B(4, 3) $
- $ C(3, 5) $
- $ D(1, 2) $

Reflect over $ y = -x $:
$(x, y) \rightarrow (-y, -x)$

New points:
- $ A'(-1, -2) $
- $ B'(-3, -4) $
- $ C'(-5, -3) $
- $ D'(-2, -1) $

Plot and connect.

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📌 How to Solve It Step-by-Step:

1. Identify the original coordinates of all vertices.
2. Apply the appropriate transformation rule based on the line of reflection.
3. Plot the new points on the grid.
4. Connect the points in the same order as the original shape.
5. Label the reflected image clearly.

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🧠 Tips:

- Always check if the reflection makes sense visually: the image should be a mirror image across the given line.
- Use graph paper or coordinate grids carefully.
- For $ y = x $, the reflected point lies along a diagonal; for $ y = -x $, it's a rotation-like reflection.

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If you can provide the coordinates of the original shapes or describe the figures, I can give you exact answers tailored to your worksheet.

Otherwise, use the rules above to solve each part. Let me know if you want me to walk through one specific example!