To graph $g(x) = f(x - 3) - 1$ using the graph of $y = f(x)$, we’ll apply two transformations to the graph of $f(x)$:

1. Horizontal Shift: The expression $f(x - 3)$ indicates a horizontal shift. Specifically, it shifts the graph of $f(x)$ 3 units to the right.

2. Vertical Shift: The expression $f(x - 3) - 1$ indicates a vertical shift. The $-1$ at the end means we shift the entire graph 1 unit down.

Steps to Graph $g(x) = f(x - 3) - 1$

1. Start with the graph of $y = f(x)$.
2. Shift every point on $y = f(x)$ 3 units to the right. This will give the graph of $f(x - 3)$.
3. Take the graph of $f(x - 3)$ and shift every point down by 1 unit to obtain the graph of $g(x) = f(x - 3) - 1$.

By following these transformations, you will get the graph of $g(x)$ as described.

Would you like more details on transformations, or do you have any specific questions?

Here are some related questions to expand on this topic:

1. How would you graph $g(x) = f(x + 2) + 4$?
2. What transformations occur in $h(x) = -f(x) + 5$?
3. How would a vertical stretch affect the function $g(x) = 2f(x - 1)$?
4. How can we determine if a transformation is horizontal or vertical based on the function notation?
5. How does a reflection over the y-axis change the graph of $f(x)$?

Tip: For horizontal shifts, focus on the value inside the function’s argument $f(x - h)$; a negative shifts right, while a positive shifts left.