To sketch the graph of the function $y = -2 - 4\sin\left(\frac{2x}{9}\right)$, let's break down the key components of the sine function and how they transform the graph:

1. Amplitude: The coefficient of the sine function is $-4$, which means the amplitude of the sine wave is 4. The negative sign reflects the graph across the x-axis.

2. Period: The period of a sine function is calculated as $\frac{2\pi}{\text{frequency}}$. Here, the frequency is $\frac{2}{9}$, so the period of the function is $\frac{2\pi}{\frac{2}{9}} = 9\pi$.

3. Vertical Shift: The entire graph is shifted downward by 2 units because of the $-2$ at the beginning of the equation.

4. Phase Shift: There is no phase shift (horizontal shift), as there is no term added or subtracted inside the sine function.

Key Points of the Graph:

• The graph oscillates between $-2 - 4 = -6$ (minimum) and $-2 + 4 = 2$ (maximum).
• The period of the sine wave is $9\pi$, so one full cycle occurs over an interval of $9\pi$.
• The sine wave starts from its midline at $y = -2$, goes down to the minimum, back to the midline, up to the maximum, and returns to the midline within one period.

Would you like an actual sketch of this graph? Let me know if you want the image or need further details.

Five Questions to Explore:

1. What effect does changing the amplitude have on the sine function graph?
2. How does the vertical shift affect the position of the graph?
3. What is the impact of the frequency on the period of a sine wave?
4. How does reflecting the sine function across the x-axis change its shape?
5. What would happen if there was a phase shift in this equation?

Tip: The period of a sine function is inversely proportional to the frequency. The higher the frequency, the shorter the period.