Reflections of geometric figures across various lines on coordinate grids.

Since I can't view or access uploaded images directly, I can’t see the specific figures or coordinates in your image. However, I can help you solve each reflection problem based on the descriptions and typical setup of such worksheets.

Let’s go through each of the six reflection problems step by step. I’ll explain how to reflect a figure across various lines using coordinate rules, and then guide you on how to graph the image.

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General Rules for Reflections:

1. Reflection across $ y = -2 $:
- The line $ y = -2 $ is a horizontal line.
- To reflect a point $ (x, y) $ across $ y = -2 $, use:
$$
(x, y) \rightarrow (x, -4 - y)
$$
Why? The distance from $ y $ to $ -2 $ is $ y + 2 $. So the reflected point is $ -2 - (y + 2) = -4 - y $.

2. Reflection across the x-axis ($ y = 0 $):
- Rule: $ (x, y) \rightarrow (x, -y) $

3. Reflection across $ y = -x $:
- Rule: $ (x, y) \rightarrow (-y, -x) $

4. Reflection across $ y = -1 $:
- Horizontal line at $ y = -1 $
- Distance from $ y $ to $ -1 $ is $ y + 1 $
- Reflected point: $ y' = -1 - (y + 1) = -2 - y $
- So: $ (x, y) \rightarrow (x, -2 - y) $

5. Reflection across $ x = -3 $:
- Vertical line at $ x = -3 $
- Distance from $ x $ to $ -3 $ is $ x + 3 $
- Reflected point: $ x' = -3 - (x + 3) = -6 - x $
- So: $ (x, y) \rightarrow (-6 - x, y) $

6. Reflection across $ y = x $:
- Rule: $ (x, y) \rightarrow (y, x) $

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Now let's walk through each problem, assuming you can read the coordinates of the original points from the grid.

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Problem 1: Reflection across $ y = -2 $

Step-by-step:
- Identify the coordinates of the original triangle $ PQR $.
- For each vertex $ (x, y) $, apply the rule: $ (x, -4 - y) $
- Plot the new points and connect them.

Example:
Suppose $ P = (-4, -1) $, $ Q = (-2, -3) $, $ R = (-3, -5) $

Then:
- $ P' = (-4, -4 - (-1)) = (-4, -3) $
- $ Q' = (-2, -4 - (-3)) = (-2, -1) $
- $ R' = (-3, -4 - (-5)) = (-3, 1) $

Plot these and draw the reflected triangle.

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Problem 2: Reflection across the x-axis

- Rule: $ (x, y) \rightarrow (x, -y) $
- Reflect each vertex of triangle $ MNP $ over the x-axis.
- Example: If $ M = (-2, 2) $, $ N = (-1, 1) $, $ P = (1, 3) $
- Then $ M' = (-2, -2) $, $ N' = (-1, -1) $, $ P' = (1, -3) $

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Problem 3: Reflection across $ y = -x $

- Rule: $ (x, y) \rightarrow (-y, -x) $
- Example: If $ A = (2, 1) $, $ B = (1, 3) $, $ C = (3, 2) $
- $ A' = (-1, -2) $
- $ B' = (-3, -1) $
- $ C' = (-2, -3) $

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Problem 4: Reflection across $ y = -1 $

- Rule: $ (x, y) \rightarrow (x, -2 - y) $
- Example: Suppose $ A = (-2, 1) $, $ B = (-4, -2) $, $ C = (1, -3) $
- $ A' = (-2, -2 - 1) = (-2, -3) $
- $ B' = (-4, -2 - (-2)) = (-4, 0) $
- $ C' = (1, -2 - (-3)) = (1, 1) $

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Problem 5: Reflection across $ x = -3 $

- Rule: $ (x, y) \rightarrow (-6 - x, y) $
- Example: Let’s say $ P = (-5, 1) $, $ Q = (-4, 3) $, $ R = (-2, 2) $
- $ P' = (-6 - (-5), 1) = (-1, 1) $
- $ Q' = (-6 - (-4), 3) = (-2, 3) $
- $ R' = (-6 - (-2), 2) = (-4, 2) $

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Problem 6: Reflection across $ y = x $

- Rule: $ (x, y) \rightarrow (y, x) $
- Example: If $ P = (-2, 1) $, $ Q = (-1, 3) $, $ R = (1, 2) $
- $ P' = (1, -2) $
- $ Q' = (3, -1) $
- $ R' = (2, 1) $

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✔ How to Solve This Worksheet:

1. Identify coordinates of each vertex of the given figure.
2. Apply the correct reflection rule based on the line of reflection.
3. Plot the new coordinates on the same grid.
4. Connect the reflected points in the same order as the original.

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📌 Tips:

- Use graph paper carefully.
- Double-check signs.
- Use the midpoint method: the line of reflection should be equidistant between the original and reflected points.

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If you can provide the coordinates of the original figures (e.g., list the vertices of each shape), I can give you the exact reflected coordinates for each problem.

Alternatively, if you describe the positions of the points on the grid (like “P is at (-4, -1)” etc.), I can compute the reflections precisely.

Would you like to do that?