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The problem involves translating each vertex of a shape on a coordinate plane. The transformation rule given is:

\[
(x, y) \rightarrow (x + 1, y - 3)
\]

This means we add 1 to the \( x \)-coordinate and subtract 3 from the \( y \)-coordinate of each vertex. Let's find the new coordinates for each point.

### Step-by-Step Solution

1. **Identify the initial coordinates** of each vertex (R, S, T) from the graph:
   - **R** is located at (3, 5).
   - **S** is located at (7, 4).
   - **T** is located at (6, 7).

2. **Apply the translation** \((x + 1, y - 3)\) to each vertex.

   - **For \( R \rightarrow R' \):**
     \[
     (3 + 1, 5 - 3) = (4, 2)
     \]
   - **For \( S \rightarrow S' \):**
     \[
     (7 + 1, 4 - 3) = (8, 1)
     \]
   - **For \( T \rightarrow T' \):**
     \[
     (6 + 1, 7 - 3) = (7, 4)
     \]

### Final Answers

- **R'**: (4, 2)
- **S'**: (8, 1)
- **T'**: (7, 4)

Would you like a detailed explanation of any specific part, or have any questions on transformations?

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**Follow-Up Questions:**

1. How does a translation affect the orientation of a shape?
2. What would be the effect of the transformation \((x - 2, y + 3)\) on each vertex?
3. How could you confirm that a translation preserves the size and shape of the original figure?
4. What are other types of transformations, besides translations, that can be applied to figures on a coordinate plane?
5. How would this translation change if applied to a shape positioned at negative coordinates?

**Tip:** When translating points, remember that translations simply shift the shape without altering its size, shape, or orientation.

Use the graph and the translation (x, y) → (x + 1, y - 3) to find the images of each vertex of a shape after translation.

Explore how to apply the translation rule (x + 1, y - 3) to determine the new positions of vertices on a coordinate plane. This solution includes step-by-step calculations, making it perfect for middle school students learning translations in coordinate geometry.

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