https://r2.erweima.ai/mathbot/image/1726792330635q7py7r65.jpg

The problem asks for two transformations of the polygon: 

1. **Reflection across the horizontal axis (x-axis)**.
2. **Translation (sliding) by 8 units to the left**.

### Step 1: Reflection across the x-axis
To reflect a point across the x-axis, the rule is to multiply the y-coordinate by \(-1\), keeping the x-coordinate the same.

The vertices of the polygon are:
- \( A(0, -2) \)
- \( B(1, -4) \)
- \( C(5, -4) \)
- \( D(2, -2) \)

After reflection across the x-axis, the new coordinates will be:
- \( A'(0, 2) \)
- \( B'(1, 4) \)
- \( C'(5, 4) \)
- \( D'(2, 2) \)

### Step 2: Translation 8 units to the left
To translate a point 8 units to the left, subtract 8 from the x-coordinate of each point.

The reflected points will be translated as follows:
- \( A'(0, 2) \rightarrow A''(-8, 2) \)
- \( B'(1, 4) \rightarrow B''(-7, 4) \)
- \( C'(5, 4) \rightarrow C''(-3, 4) \)
- \( D'(2, 2) \rightarrow D''(-6, 2) \)

### Final Coordinates
The final coordinates of the polygon after both transformations are:
- \( A''(-8, 2) \)
- \( B''(-7, 4) \)
- \( C''(-3, 4) \)
- \( D''(-6, 2) \)

Would you like a more detailed explanation of each step or how to visualize this transformation?

Here are 5 related questions you might find interesting:
1. How do you reflect a point across the y-axis instead of the x-axis?
2. What would happen if we translated the polygon upwards by 8 units instead of to the left?
3. How does reflecting across a line other than the x-axis or y-axis affect the coordinates?
4. Can you reflect the polygon across both the x and y axes at the same time?
5. How do the properties of the shape change after reflection and translation?

**Tip**: When performing transformations like reflections and translations, always adjust one coordinate at a time to avoid confusion!

Faça a reflexão deslizante do polígono a seguir com relação ao eixo x na horizontal, e em 8 unidades para a esquerda.

Learn how to perform a reflection of a polygon across the x-axis and a translation of 8 units to the left. Understand the geometric transformations and the new coordinates of each vertex. This problem is suitable for students in Grades 6-9.

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