https://file.aiquickdraw.com/mathbot/file/1728595601441u847bowz.jpg

In the image provided, the task is to write rules for the composition of translations based on the diagram that shows multiple shapes and their translations across the coordinate plane.

### Step-by-step translation process:

#### Translation 1: 
Looking at the first translation, let's focus on the shift from the shape labeled with double primes (e.g., \( G'' \)) to the shape with single primes (e.g., \( G' \)). This movement seems to be:
- **Horizontally**: 3 units to the right.
- **Vertically**: 1 unit down.

Thus, the rule for **Translation 1** can be written as:
\[
(x, y) \rightarrow (x + 3, y - 1)
\]

#### Translation 2:
Now, consider the second translation, from the shape labeled with single primes (e.g., \( G' \)) to the unprimed shape (e.g., \( G \)). This movement appears to be:
- **Horizontally**: 1 unit to the left.
- **Vertically**: 3 units up.

Thus, the rule for **Translation 2** can be written as:
\[
(x, y) \rightarrow (x - 1, y + 3)
\]

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Would you like more detailed explanations on any specific part of the process? Feel free to ask! Here are five additional related questions that could deepen your understanding:

1. How would the final coordinates of a point change if you applied both translations successively?
2. What is the combined rule if both translations are applied as a composition?
3. How can you verify the correctness of the translation rules using specific points from the shapes?
4. How does a translation affect the size and orientation of a shape?
5. Can you write a general rule for any translation on the coordinate plane?

**Tip:** When writing translation rules, always check both the horizontal and vertical shifts independently to ensure accuracy.

Write rules for the composition of translations based on the diagram that shows multiple shapes and their translations across the coordinate plane.

Learn how to write rules for the composition of translations on a coordinate plane. This geometry problem involves determining horizontal and vertical shifts for translating shapes and is suitable for students in Grades 7-9.

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