https://file.aiquickdraw.com/mathbot/file/17254999569763vfybjwc.jpg

The problem in the image is asking to reflect the parallelogram ABCD over the line \( y = 1 \). Let's break down how to solve this:

### Step-by-Step Reflection Process:

1. **Identify the coordinates of the vertices of the parallelogram**:
   - A: (-4, 3)
   - B: (-2, 6)
   - C: (3, 6)
   - D: (1, 3)

2. **Reflection rule over the line \( y = 1 \)**:
   - When reflecting a point over a horizontal line \( y = c \), the y-coordinate of each point changes according to the formula:
   \[
   y_{\text{new}} = 2c - y_{\text{old}}
   \]
   where \( c = 1 \) in this case.

3. **Apply the reflection formula** for each point:
   - A(-4, 3): Reflect over \( y = 1 \):
     \[
     y_{\text{new}} = 2(1) - 3 = 2 - 3 = -1
     \]
     So, A' = (-4, -1).
   
   - B(-2, 6): Reflect over \( y = 1 \):
     \[
     y_{\text{new}} = 2(1) - 6 = 2 - 6 = -4
     \]
     So, B' = (-2, -4).
   
   - C(3, 6): Reflect over \( y = 1 \):
     \[
     y_{\text{new}} = 2(1) - 6 = 2 - 6 = -4
     \]
     So, C' = (3, -4).
   
   - D(1, 3): Reflect over \( y = 1 \):
     \[
     y_{\text{new}} = 2(1) - 3 = 2 - 3 = -1
     \]
     So, D' = (1, -1).

4. **Plot the reflected points**: 
   The new coordinates for the reflected image of ABCD are:
   - A'(-4, -1)
   - B'(-2, -4)
   - C'(3, -4)
   - D'(1, -1)

Now, the reflected parallelogram can be drawn with these new coordinates.

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Do you want further details on how the reflection works or how to graph this on a coordinate plane?

Here are 5 questions that could help expand your understanding:
1. What happens if you reflect a shape over a different line, like \( y = -1 \)?
2. How would you reflect a shape over a vertical line, such as \( x = 2 \)?
3. What is the geometric significance of reflecting over the line \( y = 1 \)?
4. Can this method be used for reflecting any polygon, or just parallelograms?
5. How do you verify if the reflected points are correct?

**Tip**: When reflecting a shape over any horizontal or vertical line, visualize the distance each point is from the line of reflection to help with accuracy.

Given the image of vertices ABCD, find the image of ABCD after a reflection over the line y = 1.

Learn how to reflect a parallelogram ABCD over the line y = 1 using the reflection formula on a coordinate plane. This geometry problem involves identifying vertices and applying the reflection theorem, suitable for students in Grades 8-10.

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