write down the transformations implied by sqrt(x+2)+3. find the x transformation, y transformation, and the parent function

The function \( f(x) = \sqrt{x + 2} + 3 \) can be analyzed in terms of transformations applied to its parent function.

### 1. **Parent Function:**
   The parent function here is \( \sqrt{x} \).

### 2. **Transformations:**

- **X-Transformation (Horizontal Shift):**
  - The term \( x + 2 \) indicates a horizontal shift.
  - Since the expression is \( x + 2 \), it implies a **left shift** by 2 units. 
  - Transformation: \( x' = x - (-2) = x + 2 \).

- **Y-Transformation (Vertical Shift):**
  - The term \( +3 \) outside the square root indicates a vertical shift.
  - It implies a **shift upwards** by 3 units.
  - Transformation: \( y' = y + 3 \).

### Summary:

- **Parent Function:** \( \sqrt{x} \)
- **Horizontal Shift:** Left by 2 units.
- **Vertical Shift:** Up by 3 units.

This means the function \( f(x) = \sqrt{x + 2} + 3 \) is obtained by shifting the graph of the parent function \( \sqrt{x} \) 2 units to the left and 3 units up.

Would you like a detailed graph of this transformation or have any further questions?

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### 5 Related Questions:
1. How does the graph of \( \sqrt{x} \) compare to \( \sqrt{x+2} \)?
2. What happens to the domain and range of the function after the transformation?
3. How would the function change if the transformation was \( f(x) = \sqrt{x - 2} + 3 \)?
4. Can you derive the inverse function of \( f(x) = \sqrt{x + 2} + 3 \)?
5. What would be the effect of a vertical stretch by a factor of 2 on the function?

### Tip:
When dealing with horizontal shifts, remember that adding a value inside the function moves the graph to the left, while subtracting moves it to the right.

Learn how to analyze the transformations of the function sqrt(x + 2) + 3, including horizontal and vertical shifts. This problem covers function transformations and square root functions, suitable for students in Grades 9-11.

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