Let's analyze the given function $y = 2 \cot\left(\frac{\pi}{2}x\right)$ step by step:

1. Domain

The cotangent function, $\cot(z)$, is undefined whenever $z = n\pi$, where $n$ is an integer. For the argument $\frac{\pi}{2}x$, we have: $\frac{\pi}{2}x = n\pi \implies x = 2n \quad (n \in \mathbb{Z})$ Thus, the domain of the function is: $x \in \mathbb{R} \setminus \{2n \mid n \in \mathbb{Z}\}$

2. Periodicity

The cotangent function $\cot(z)$ has a period of $\pi$. For $\frac{\pi}{2}x$, the period becomes: $\text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2$ Hence, the function $y = 2 \cot\left(\frac{\pi}{2}x\right)$ repeats every 2 units along the $x$-axis.

3. Range

The cotangent function can take all real values except where it is undefined (vertical asymptotes). Multiplying by 2 scales the output, but the range remains: $y \in \mathbb{R}$

4. Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, i.e., at $x = 2n$.

5. Symmetry

The cotangent function is odd: $\cot(-z) = -\cot(z)$. Thus, the function $y = 2 \cot\left(\frac{\pi}{2}x\right)$ is also odd: $y(-x) = -y(x)$ This implies rotational symmetry about the origin.

6. Behavior

• At $x = 2n+1$ (midpoint between consecutive vertical asymptotes), $y = 0$ (since $\cot\left(\frac{\pi}{2}(2n+1)\right) = 0$).
• As $x \to 2n^+$ or $x \to 2n^-$, $y \to \pm\infty$, indicating vertical asymptotes.

Sketching Two Periods of the Graph

I will plot the graph of $y = 2 \cot\left(\frac{\pi}{2}x\right)$ for two periods (e.g., $x \in [-2, 2]$). np import matplotlib.pyplot as plt

Define the function

def func(x): return 2 / np.tan((np.pi / 2) * x)

Define the range of x, avoiding vertical asymptotes

x = np.linspace(-2, 2, 1000) x = x[(x % 2 != 0)] # Remove points at multiples of 2

Plot the function

plt.figure(figsize=(10, 6)) plt.plot(x, func(x), label=r"$y = 2 \cot\left(\frac{\pi}{2}x\right)$", color="blue")

Add vertical asymptotes at x = -2, 0, 2

for asymptote in [-2, 0, 2]: plt.axvline(asymptote, color="red", linestyle="--", alpha=0.7, label="Asymptote" if asymptote == -2 else "")

Add labels and grid

plt.axhline(0, color="black", linewidth=0.8, linestyle="--") plt.title("Graph of $y = 2 \cot(\frac{\pi}{2}x)$ for Two Periods") plt.xlabel("$x$") plt.ylabel("$y$") plt.ylim(-10, 10) plt.grid(alpha=0.5) Here is the graph of $y = 2 \cot\left(\frac{\pi}{2}x\right)$ for two periods ($x \in [-2, 2]$):

• Vertical Asymptotes: At $x = -2, 0,$ and $2$.
• Zeros: At $x = -1, 1$, where $y = 0$.
• The function alternates between $\pm\infty$ near vertical asymptotes.

Would you like a detailed explanation of the sketch or further analysis?

5 Questions to Expand Understanding:

1. What happens to the function as $x \to \infty$ or $x \to -\infty$?
2. How does scaling by a constant (like 2 here) affect other trigonometric functions?
3. How does the period change if the argument is modified further?
4. Can this function be expressed in terms of sine and cosine? If yes, how?
5. How would the graph change if $\cot$ were replaced by $\tan$?

Tip:

When sketching periodic functions, always identify the period, amplitude, and key points like zeros and asymptotes for a clear visualization.