import numpy as np import matplotlib.pyplot as plt

Define the range and the function

x = np.linspace(-2 * np.pi, np.pi / 2, 1000) y = np.tan(x)

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$y = \tan(x)$", color='blue') plt.axhline(0, color='black', linewidth=0.8, linestyle="--") plt.axvline(0, color='black', linewidth=0.8, linestyle="--")

Mark the vertical asymptotes (where tan(x) is undefined)

asymptotes = np.arange(-2 * np.pi, np.pi, np.pi / 2) for asymptote in asymptotes: if -2 * np.pi < asymptote < np.pi / 2: plt.axvline(asymptote, color='red', linestyle='--', linewidth=0.8)

Add labels, legend, and title

plt.title(r"Graph of $y = \tan(x)$ for $-2\pi < x < \pi/2$") plt.xlabel("x") plt.ylabel("y") plt.ylim(-10, 10) # Limit y-axis for better visualization plt.legend() plt.grid(alpha=0.4)

Show the plot

plt.show() <<ImageDisplayed>>successHere is the graph of $y = \tan(x)$ for the domain $-2\pi < x < \pi/2$. The vertical red dashed lines represent the asymptotes where the tangent function is undefined (e.g., at $x = -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0$).

Would you like a detailed explanation about any specific part of this graph or function?

Here are 5 relative questions for further exploration:

1. What causes the vertical asymptotes in the tangent function?
2. How can the periodicity of the tangent function be identified from its graph?
3. How is the behavior of $y = \tan(x)$ different from $y = \sin(x)$ or $y = \cos(x)$?
4. How can transformations (shifts, stretches, or compressions) affect the tangent graph?
5. What are the key applications of the tangent function in real-world problems?

Tip: The tangent function has a period of $\pi$, so you can predict its behavior in both positive and negative directions by observing one period.