https://r2.erweima.ai/mathbot/image/1723522704083f3eqoc8n.jpg

The image you uploaded contains a series of mathematical functions that need to be graphed by hand, using transformations based on the graphs of basic functions.

The specific function you've highlighted is:

\[
y = \sqrt{x - 2} - 1
\]

### To graph this function, follow these steps:

1. **Start with the basic function \( y = \sqrt{x} \):**
   - This is the standard square root function.
   - The graph of \( y = \sqrt{x} \) starts at the origin (0, 0) and increases slowly to the right, forming a curve.

2. **Apply a horizontal shift:**
   - The expression \( \sqrt{x - 2} \) indicates a shift to the right by 2 units.
   - So, move the graph of \( y = \sqrt{x} \) two units to the right. The starting point (vertex) of the graph is now at (2, 0).

3. **Apply a vertical shift:**
   - The expression \( \sqrt{x - 2} - 1 \) indicates a shift downward by 1 unit.
   - This means you move the entire graph 1 unit down. The new vertex of the graph is at (2, -1).

### Final Graph Characteristics:
- **Vertex:** The graph starts at the point (2, -1).
- **Shape:** The graph is a curved line that moves upwards to the right, starting from the vertex.

If you plot the graph, you'll see that it maintains the shape of the square root function, but with the vertex at (2, -1).

Would you like further details on how to graph this function, or do you have any other questions?

### Related Questions:
1. How do you apply transformations to other basic functions like \( y = |x| \) or \( y = x^2 \)?
2. What are the steps to identify horizontal and vertical shifts in a function?
3. Can you explain how to graph a function involving reflection, such as \( y = -\sqrt{x} \)?
4. How does the transformation \( y = \sqrt{x + 3} + 2 \) differ from \( y = \sqrt{x - 2} - 1 \)?
5. What are common mistakes when graphing transformed functions?
6. How can transformations be combined to produce more complex graphs?
7. How would you graph \( y = \sqrt{4x - 8} \) using transformations?
8. What is the significance of the vertex in the graph of a square root function?

**Tip:** Always start by identifying the base function before applying any transformations. This will make it easier to understand how the graph will shift and stretch.

Learn how to graph the function sqrt(x - 2) - 1 using transformations. This problem involves understanding the square root function and applying horizontal and vertical shifts. Suitable for high school students.

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