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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?

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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.

Use the graph of y = f(x) to graph the function g(x) = f(x - 3) - 1.

Learn how to graph the function g(x) = f(x - 3) - 1 by using the graph of y = f(x). This problem involves function transformations, including horizontal and vertical shifts, and is suitable for high school students in Grades 9-11.

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