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The equation you've provided is:

\[
y = -\frac{3}{2} \tan\left(\frac{1}{2}x\right)
\]

### Step 1: Understand the Function
This is a transformation of the basic tangent function. Let's break it down:

1. **Vertical Stretch/Compression**: The factor \(-\frac{3}{2}\) indicates a vertical stretch by \(\frac{3}{2}\) and a reflection across the x-axis.
2. **Horizontal Stretch/Compression**: The factor \(\frac{1}{2}\) inside the tangent function compresses the graph horizontally by a factor of 2.

### Step 2: Identify Asymptotes
For the standard tangent function \(y = \tan(x)\), vertical asymptotes occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.

For your function \(y = -\frac{3}{2} \tan\left(\frac{1}{2}x\right)\), the asymptotes occur where the argument of the tangent function is undefined:

\[
\frac{1}{2}x = \frac{\pi}{2} + k\pi
\]

Solving for \(x\):

\[
x = \pi + 2k\pi
\]

So, the vertical asymptotes are at \(x = \pi(1 + 2k)\).

### Step 3: Create a Table of Values
Let's pick some points around one period to build the table. The period \(T\) of the function is:

\[
T = \frac{\pi}{\frac{1}{2}} = 2\pi
\]

Now let's calculate values for \(x\) and the corresponding \(y\) values. To choose useful points, select \(x\) values around the asymptote at \(x = \pi\), such as \(x = 0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).

\[
\begin{array}{|c|c|}
\hline
x & y = -\frac{3}{2} \tan\left(\frac{1}{2}x\right) \\
\hline
0 & 0 \\
\frac{\pi}{2} & -\frac{3}{2} \times 1 = -\frac{3}{2} \\
\pi & \text{Undefined (Asymptote)} \\
\frac{3\pi}{2} & \frac{3}{2} \\
2\pi & 0 \\
\hline
\end{array}
\]

### Summary
- **Vertical Asymptotes**: \(x = \pi(1 + 2k)\)
- **Period**: \(2\pi\)
- **Table of values**: See above for key points.

Would you like to explore this function further, such as plotting it or examining its behavior in more detail? 

---

### Follow-up Questions:
1. What is the significance of the vertical asymptotes for the tangent function?
2. How does the reflection across the x-axis affect the graph of the function?
3. Can you explain how the period of the function changes when a coefficient is added inside the tangent argument?
4. How would the graph change if the coefficient \(\frac{1}{2}\) inside the tangent was replaced by 2?
5. How does the vertical stretch by \(\frac{3}{2}\) alter the graph compared to the parent function \(y = \tan(x)\)?
6. What would the graph look like if the function had no vertical stretch/compression or reflection?
7. How do the asymptotes help in sketching the graph of a tangent function?
8. What happens to the graph when the function is modified to \(y = \frac{3}{2} \tan\left(\frac{1}{2}x\right)\) without the negative sign?

### Tip:
Remember, when dealing with trigonometric functions, especially those with transformations, it's crucial to identify key features like asymptotes, period, and amplitude to understand the graph's behavior.

Explore how to graph the function y = -3/2 tan(1/2x), including identifying vertical asymptotes and table of values. Learn about transformations of tangent functions and their implications. Suitable for Grades 11-12.

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