describe the transformations from the parent graph g(x)=-square root of x-3+2

The given function is \( g(x) = -\sqrt{x - 3} + 2 \). To describe the transformations from the parent graph \( f(x) = \sqrt{x} \), we break down the changes step by step:

1. **Horizontal Translation (Right Shift)**: 
   - The term inside the square root, \( (x - 3) \), indicates a horizontal shift.
   - Since it's \( x - 3 \), the graph shifts **3 units to the right**.

2. **Reflection (Vertical Flip)**:
   - The negative sign in front of the square root, \( -\sqrt{x - 3} \), reflects the graph **across the x-axis**. This means the graph will now point downward instead of upward.

3. **Vertical Translation (Up Shift)**:
   - The \( +2 \) outside the square root shifts the graph **2 units up**.

### Summary of Transformations:
- The parent graph \( f(x) = \sqrt{x} \) is first shifted **3 units to the right**.
- It is then **reflected across the x-axis** (flipped upside down).
- Finally, it is shifted **2 units up**.

Would you like more details on any of these transformations?

### 5 Related Questions:
1. How would the graph change if the function was \( g(x) = \sqrt{x + 3} \)?
2. What does a vertical stretch or compression look like in a square root function?
3. How can you graph \( g(x) = -\sqrt{x - 3} + 2 \) by hand?
4. What effect does a horizontal reflection (over the y-axis) have on the square root function?
5. How do horizontal translations differ from vertical translations in general?

### Tip:
When analyzing transformations, always start with the operations inside the function (like shifts and reflections), then move to outside operations (like stretching and translations).

Understand how the square root function g(x) = -√(x - 3) + 2 is transformed. This problem covers graph transformations, including horizontal shifts, vertical reflections, and vertical translations, and is suitable for students in Grades 9-12.

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