The given equation $y = \sqrt{x - 1} - 3$ is a transformation of the parent function $y = \sqrt{x}$. Here are the steps to graph this transformation and interpret the shifts:

1. Horizontal Shift: The term $x - 1$ inside the square root shifts the graph 1 unit to the right. This is because we set $x - 1 = 0$, and the graph starts at $x = 1$.

2. Vertical Shift: The term $-3$ outside the square root shifts the graph 3 units downward. This lowers every point of the graph by 3.

3. Key Points:

   • The starting point (vertex) of the transformed graph is shifted from $(0, 0)$ (for $y = \sqrt{x}$) to $(1, -3)$. To summarize the transformations for the function $y = \sqrt{x - 1} - 3$:

4. Parent Function: The parent function is $y = \sqrt{x}$.

5. Horizontal Shift: The graph is shifted 1 unit to the right due to the $x - 1$ term.

6. Vertical Shift: The graph is shifted 3 units downward due to the $-3$ outside the square root.

Steps to Graph:

• Start at the new vertex point $(1, -3)$.
• The general shape of the square root function remains the same: gradually increasing, with a steep slope near the vertex and flattening as $x$ increases.
• Use key points from the parent function and apply the transformations. For example:
  • At $x = 1$, $y = \sqrt{1 - 1} - 3 = -3$.
  • At $x = 2$, $y = \sqrt{2 - 1} - 3 = -2$.
  • At $x = 5$, $y = \sqrt{5 - 1} - 3 = -1$.

Would you like a detailed graph or further assistance with graphing this function? Here are some related questions for further exploration:

1. How does the graph of $y = \sqrt{x - h} + k$ change with different $h$ and $k$ values?
2. What is the domain and range of $y = \sqrt{x - 1} - 3$?
3. How can transformations be used to graph functions more efficiently?
4. Can you explain why the graph starts at $x = 1$ for this function?
5. How would additional transformations, like stretching or reflecting, affect the graph?

Tip: Always identify shifts and starting points first when dealing with transformed functions!