The given trigonometric function is:

$y = \frac{3}{2} \sin(x)$

Important Points to Plot:

This is a sine function with an amplitude of $\frac{3}{2}$, which means it oscillates between $\frac{3}{2}$ and $-\frac{3}{2}$.

1. Amplitude: The amplitude is $\frac{3}{2}$, so the maximum value of $y$ is $\frac{3}{2}$ and the minimum value of $y$ is $-\frac{3}{2}$.

2. Period: The sine function has a period of $2\pi$, meaning it completes one full cycle over an interval of $2\pi$.

3. X-Intercepts: The sine function is zero at $x = 0, \pi, 2\pi$, etc. So, the x-intercepts will be at these points within one cycle.

4. Maximum and Minimum:

   • Maximum occurs at $x = \frac{\pi}{2}$ where $y = \frac{3}{2}$.
   • Minimum occurs at $x = \frac{3\pi}{2}$ where $y = -\frac{3}{2}$.

Key Points for the Graph:

• At $x = 0$, $y = 0$ (x-intercept).
• At $x = \frac{\pi}{2}$, $y = \frac{3}{2}$ (maximum).
• At $x = \pi$, $y = 0$ (x-intercept).
• At $x = \frac{3\pi}{2}$, $y = -\frac{3}{2}$ (minimum).
• At $x = 2\pi$, $y = 0$ (x-intercept).

These points will complete one cycle of the sine curve.

Graphing Instructions:

1. Plot the x-intercepts: $(0, 0)$, $(\pi, 0)$, and $(2\pi, 0)$.
2. Plot the maximum point: $\left(\frac{\pi}{2}, \frac{3}{2}\right)$.
3. Plot the minimum point: $\left(\frac{3\pi}{2}, -\frac{3}{2}\right)$.
4. Sketch the curve of the sine function that passes through these points, making sure it oscillates smoothly between them.

Would you like further details or any clarifications?

Here are some related questions:

1. How do you determine the amplitude of a sine function?
2. How do you find the period of a trigonometric function?
3. What role does the coefficient of sine play in stretching or compressing the graph?
4. Can you explain how the sine function's x-intercepts are related to its period?
5. How do the maximum and minimum values change if the function is shifted vertically?

Tip: When plotting sine and cosine functions, always identify the amplitude, period, phase shifts, and vertical shifts to ensure accuracy.