The given function is:

$f(x) = -2 \sin\left(\frac{1}{3}x - \frac{\pi}{6}\right) - 1$

Let’s analyze the function step by step:

1. Amplitude

The amplitude of a sine function $A \sin(Bx + C) + D$ is given by the absolute value of the coefficient of sine, $|A|$.
Here, $A = -2$, so the amplitude is:

$\text{Amplitude} = |A| = 2$

2. Period

The period of a sine function is given by:

$\text{Period} = \frac{2\pi}{|B|}$

Here, $B = \frac{1}{3}$, so:

$\text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi$

3. Phase Shift

The phase shift is determined by solving $Bx + C = 0$ to find the shift of the sine wave along the $x$-axis. Here:

$Bx + C = \frac{1}{3}x - \frac{\pi}{6}$

Setting it to zero:

$\frac{1}{3}x - \frac{\pi}{6} = 0 \implies x = \frac{\pi}{2}$

So, the phase shift is:

$\text{Phase Shift} = \frac{\pi}{2} \text{ (to the right)}$

4. Vertical Shift

The vertical shift is given by $D$. Here, $D = -1$, so the graph is shifted down by 1 unit.

5. Reflection

The negative sign in front of $A$ ($-2$) reflects the sine wave across the horizontal axis.

Summary of the Function's Properties:

• Amplitude: $2$
• Period: $6\pi$
• Phase Shift: $\frac{\pi}{2}$ (to the right)
• Vertical Shift: $-1$ (downward)
• Reflection: Reflected across the horizontal axis.

Would you like further details on plotting or analyzing this function?

Here are 5 related questions for further exploration:

1. How would the graph change if the amplitude was $3$ instead of $-2$?
2. What is the value of $f(x)$ at $x = 0$?
3. How does the phase shift affect the starting point of one period of the sine wave?
4. What are the maximum and minimum values of $f(x)$?
5. How would the period change if $B = \frac{1}{6}$ instead of $\frac{1}{3}$?

Tip: To graph this function, start by plotting the parent function $\sin(x)$, then apply the transformations step by step (stretching, shifting, and reflecting).