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.