https://file.aiquickdraw.com/mathbot/file/1727114809770ahm0egxw.jpg

The given problem shows the graph of a function \( y = f(x) \) with:

- A maximum turning point at \( (-2, 5) \)
- A minimum turning point at \( (3, -1) \)

You are asked to sketch the graph of \( y = f(x + 3) + 2 \).

### Explanation:
1. **Horizontal Shift:**
   - The expression \( f(x + 3) \) indicates a **horizontal shift** of the graph by 3 units to the **left**. This means every point on the original graph moves 3 units to the left along the x-axis.

2. **Vertical Shift:**
   - The \( +2 \) outside the function indicates a **vertical shift** of 2 units **upwards**. This moves every point on the graph up by 2 units.

### Transforming the key points:
- The maximum turning point \( (-2, 5) \) on the original graph:
  - Horizontal shift: \( (-2 - 3) = -5 \)
  - Vertical shift: \( 5 + 2 = 7 \)
  - New point: \( (-5, 7) \)

- The minimum turning point \( (3, -1) \) on the original graph:
  - Horizontal shift: \( (3 - 3) = 0 \)
  - Vertical shift: \( -1 + 2 = 1 \)
  - New point: \( (0, 1) \)

### Sketching the new graph:
The new graph will have:
- A maximum turning point at \( (-5, 7) \)
- A minimum turning point at \( (0, 1) \)
  
The overall shape remains the same as the original graph, but it's shifted to the left by 3 units and upwards by 2 units.

Would you like more details on sketching or have any questions?

Here are 5 relative questions to expand your understanding:
1. How do horizontal and vertical shifts affect the derivative of a function?
2. What happens to the symmetry of a graph after a transformation like \( f(x + 3) + 2 \)?
3. Can you explain how the turning points change after applying horizontal and vertical stretches or compressions?
4. What is the effect of a negative coefficient on the function \( f(x) \), like \( f(-x) \)?
5. How would the graph change if the transformation was \( y = f(x-3) - 2 \) instead?

**Tip:** When sketching transformations, always start by shifting key points like maximums, minimums, or intercepts to guide your drawing.

The diagram shows the graph of y = f(x) with a maximum turning point at (-2, 5) and a minimum turning point at (3, -1). Sketch the graph of y = f(x + 3) + 2.

Learn how to transform the graph of a function through horizontal and vertical shifts. This problem involves shifting the graph of y = f(x) with turning points at (-2, 5) and (3, -1) to sketch y = f(x + 3) + 2. Suitable for Grades 10-12, focusing on function transformations.

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