To graph the given trigonometric function $y = 10 \sin\left( \frac{2}{3}(x + \frac{3\pi}{2}) \right) - 8$, we need to identify the following elements:

Step 1: Identify the amplitude, period, phase shift, and vertical shift

• Amplitude: The amplitude is the coefficient in front of the sine function, which is $10$. This means the graph will oscillate between $y = 10$ and $y = -10$, before the vertical shift is applied.

• Period: The period of a sine function is given by $\frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$ inside the sine function. Here, $B = \frac{2}{3}$, so the period is:

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

• Phase shift: The phase shift is found by solving for $x$ in $x + \frac{3\pi}{2} = 0$. This gives:

  $x = -\frac{3\pi}{2}$

  So the graph is shifted left by $\frac{3\pi}{2}$.

• Vertical shift: The vertical shift is given by $-8$, meaning the graph is shifted down by 8 units.

Step 2: Choose appropriate scale for the graph

• Since the amplitude is $10$, the graph will oscillate between $y = 2$ (which is $-8 + 10$) and $y = -18$ (which is $-8 - 10$).
• The period is $3\pi$, so it would be good to label the x-axis in increments of $\frac{\pi}{2}$ or $\pi$ to capture the key points within one cycle.

Step 3: Key Points for One Cycle

Let's mark the key points of the sine function within one cycle:

1. Start at the phase shift: $x = -\frac{3\pi}{2}$
2. First quarter (maximum): The sine function reaches its maximum $10$ at $x = -\frac{3\pi}{2} + \frac{1}{4} \times \text{period} = -\frac{3\pi}{2} + \frac{3\pi}{4} = -\frac{3\pi}{4}$
3. Second quarter (crosses midline): The sine function crosses the midline at $x = -\frac{3\pi}{2} + \frac{1}{2} \times \text{period} = -\frac{3\pi}{2} + \frac{3\pi}{2} = 0$
4. Third quarter (minimum): The sine function reaches its minimum at $x = -\frac{3\pi}{2} + \frac{3}{4} \times \text{period} = -\frac{3\pi}{2} + \frac{9\pi}{4} = \frac{3\pi}{4}$
5. End of cycle: The function returns to the midline at $x = -\frac{3\pi}{2} + 1 \times \text{period} = -\frac{3\pi}{2} + 3\pi = \frac{3\pi}{2}$

Step 4: Plot the Points

For each of these x-values, calculate the corresponding y-values:

1. At $x = -\frac{3\pi}{2}$: $y = 10 \sin\left( \frac{2}{3} \times 0 \right) - 8 = -8$

2. At $x = -\frac{3\pi}{4}$: $y = 10 \sin\left( \frac{2}{3} \times \frac{\pi}{2} \right) - 8 = 10 \times 1 - 8 = 2$

3. At $x = 0$: $y = 10 \sin\left( \frac{2}{3} \times \frac{3\pi}{2} \right) - 8 = 10 \times 0 - 8 = -8$

4. At $x = \frac{3\pi}{4}$: $y = 10 \sin\left( \frac{2}{3} \times \frac{5\pi}{2} \right) - 8 = 10 \times -1 - 8 = -18$

5. At $x = \frac{3\pi}{2}$: $y = 10 \sin\left( \frac{2}{3} \times 3\pi \right) - 8 = 10 \times 0 - 8 = -8$

Step 5: Draw the Graph

Plot these points on a graph with appropriate scaling for both axes:

• The x-axis will range from $x = -\frac{3\pi}{2}$ to $x = \frac{3\pi}{2}$.
• The y-axis will range from $y = -18$ to $y = 2$.

Would you like me to generate a graph for you or provide further details on plotting?