https://file.aiquickdraw.com/mathbot/file/1726748000267he1wbeum.jpg

The question is asking to **sketch the graph** of the function:

\[
y = 3 \sin(x) - 1 \quad \text{for} \quad 0 \leq x \leq 2\pi
\]

### Breakdown of the function:
1. **Base function**: \( y = \sin(x) \) is a sine wave with a period of \( 2\pi \), amplitude 1, and no vertical shift.
2. **Amplitude change**: \( y = 3 \sin(x) \) changes the amplitude from 1 to 3, meaning the wave oscillates between -3 and 3.
3. **Vertical shift**: The term \( -1 \) shifts the entire graph downward by 1 unit. So the range of the graph changes to \([-4, 2]\) (since the sine wave was originally between -3 and 3).

### Key features to sketch:
- The period of \( \sin(x) \) remains \( 2\pi \).
- The graph will oscillate between \( y = -4 \) (minimum) and \( y = 2 \) (maximum).
- The vertical shift moves the whole graph downward by 1 unit.
- The zero crossings (where the graph intersects the x-axis) will be shifted downward as well.

### Critical Points:
- **At \(x = 0\)**: \( y = 3 \sin(0) - 1 = -1 \)
- **At \(x = \frac{\pi}{2}\)**: \( y = 3 \sin\left(\frac{\pi}{2}\right) - 1 = 3 - 1 = 2 \)
- **At \(x = \pi\)**: \( y = 3 \sin(\pi) - 1 = -1 \)
- **At \(x = \frac{3\pi}{2}\)**: \( y = 3 \sin\left(\frac{3\pi}{2}\right) - 1 = -3 - 1 = -4 \)
- **At \(x = 2\pi\)**: \( y = 3 \sin(2\pi) - 1 = -1 \)

This will give you the shape of the sine wave with a maximum of 2, minimum of -4, and centered at -1.

### Would you like me to sketch this graph for you?

### Related Questions:
1. What is the amplitude of the graph \( y = 3 \sin(x) - 1 \)?
2. How does shifting a sine graph vertically affect its zero crossings?
3. What is the period of the sine function, and how does it relate to its graph?
4. How would a phase shift affect the graph of \( y = 3 \sin(x) - 1 \)?
5. What are the maximum and minimum values of \( y = 3 \sin(x) - 1 \) and how are they calculated?

#### Tip:
Always check if the sine or cosine graph has been modified by amplitude changes or shifts before sketching, as these will affect the graph’s appearance and key points.

Sketch the graph of y = 3 sin(x) - 1 for 0 ≤ x ≤ 2π.

Learn how to sketch the graph of y = 3sin(x) - 1 for values of x between 0 and 2π. This problem explores key concepts in trigonometry, including amplitude changes, vertical shifts, and graphing the sine function. Suitable for students in Grades 10-12.

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