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To determine the transformations that map Figure \( C \) onto Figure \( D \), let's break down the process step by step by analyzing the coordinates and orientation of both figures:

1. **Reflection**: 
   - Notice that the orientation of Figure \( D \) is flipped relative to Figure \( C \), which suggests that a reflection has occurred.
   - Both figures are positioned on opposite sides of the x-axis, meaning the reflection is likely over the x-axis.

2. **Translation**:
   - After reflecting, we need to move the figure so that the points on Figure \( C \) match those on Figure \( D \). This indicates a translation in the x- and y-directions.

### Steps:

1. **Reflect over the x-axis**: 
   - Reflect Figure \( C \) over the x-axis. This changes the sign of the y-coordinates of each point of Figure \( C \), which gives us a mirrored image of Figure \( C \) below the x-axis.

2. **Translate the reflected figure**:
   - We then translate the figure to the right and downward so that it matches the position of Figure \( D \). Specifically, we translate it 4 units right (positive x-direction) and 6 units downward (negative y-direction).

Thus, the **series of transformations** would be:
- **Step 1**: Reflect Figure \( C \) over the **x-axis**.
- **Step 2**: Translate the reflected figure by \( 4 \) units to the **right** and \( 6 \) units **downward**.

Would you like more details on this solution? Here are some questions you might consider next:

1. How do we reflect a figure over the x-axis or y-axis using coordinates?
2. How can you determine the direction and distance of translation mathematically?
3. What other rigid motions (like rotations) could also map Figure \( C \) to Figure \( D \)?
4. How can you check if a transformation preserves the size and shape of a figure?
5. What is the difference between rigid and non-rigid transformations?

**Tip**: Always check both the orientation and distance between corresponding points to identify the right sequence of transformations!

Determine a series of transformations that would map Figure C onto Figure D.

Discover how to map Figure C onto Figure D using transformations. This problem focuses on rigid motions, including reflection over the x-axis and translation in both x and y directions. Suitable for students in Grades 8-10.

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