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To reflect the given figure over the line \( y = x + 2 \), we need to follow these steps:

### 1. Equation of the reflection line:
The line \( y = x + 2 \) has a slope of 1, which means it is at a 45-degree angle. The process for reflecting a point across a line involves finding the perpendicular distance from each point to the line and placing the reflected point the same distance on the opposite side of the line.

### 2. Steps for reflecting points:
Let’s label the vertices of the shape. From the diagram, we can estimate the coordinates of the vertices of the quadrilateral as approximately:
- \( A (-7, 4) \)
- \( B (-5, 6) \)
- \( C (-3, 8) \)
- \( D (-5, 2) \)

We will reflect each of these points across the line \( y = x + 2 \).

To reflect a point \( (x_1, y_1) \) over the line \( y = mx + b \) (in this case, \( m = 1 \), and \( b = 2 \)), we use the formula for reflecting a point over a line.

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Let me calculate and provide the coordinates of the reflected points next.The reflected coordinates of the points across the line \( y = x + 2 \) are as follows:

- \( A (-7, 4) \rightarrow A' (2, -5) \)
- \( B (-5, 6) \rightarrow B' (4, -3) \)
- \( C (-3, 8) \rightarrow C' (6, -1) \)
- \( D (-5, 2) \rightarrow D' (0, -3) \)

You can now plot these points on the grid to visualize the reflected figure.

Would you like a detailed explanation of how the reflection process works, or any other assistance with this problem?

Here are 5 related questions:
1. How do you reflect a point over the line \( y = -x + b \)?
2. What is the general formula for reflecting a point over a line?
3. How do translations affect reflections of shapes?
4. What is the relationship between a figure's original and reflected coordinates?
5. How can you use matrix transformations to reflect a figure?

**Tip:** When reflecting a point across a line, it helps to simplify the reflection by using symmetry, such as reflecting over \( y = x \) and adjusting for vertical shifts.

Reflect the figure over the line y = x + 2.

This problem focuses on reflecting a given figure over the line y = x + 2 in a coordinate plane. Key mathematical concepts such as reflection formulas and geometric transformations are explored. Suitable for high school students studying coordinate geometry.

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