Math Problem Statement

Determine the properties of this function: y = 2 cot (π/2 x) then sketch two circles of the graph of each.

Solution

Let's analyze the given function y=2cot(π2x)y = 2 \cot\left(\frac{\pi}{2}x\right) step by step:

1. Domain

The cotangent function, cot(z)\cot(z), is undefined whenever z=nπz = n\pi, where nn is an integer. For the argument π2x\frac{\pi}{2}x, we have: π2x=nπ    x=2n(nZ)\frac{\pi}{2}x = n\pi \implies x = 2n \quad (n \in \mathbb{Z}) Thus, the domain of the function is: xR{2nnZ}x \in \mathbb{R} \setminus \{2n \mid n \in \mathbb{Z}\}

2. Periodicity

The cotangent function cot(z)\cot(z) has a period of π\pi. For π2x\frac{\pi}{2}x, the period becomes: Period=ππ2=2\text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2 Hence, the function y=2cot(π2x)y = 2 \cot\left(\frac{\pi}{2}x\right) repeats every 2 units along the xx-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: yRy \in \mathbb{R}

4. Vertical Asymptotes

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

5. Symmetry

The cotangent function is odd: cot(z)=cot(z)\cot(-z) = -\cot(z). Thus, the function y=2cot(π2x)y = 2 \cot\left(\frac{\pi}{2}x\right) is also odd: y(x)=y(x)y(-x) = -y(x) This implies rotational symmetry about the origin.

6. Behavior

  • At x=2n+1x = 2n+1 (midpoint between consecutive vertical asymptotes), y=0y = 0 (since cot(π2(2n+1))=0\cot\left(\frac{\pi}{2}(2n+1)\right) = 0).
  • As x2n+x \to 2n^+ or x2nx \to 2n^-, y±y \to \pm\infty, indicating vertical asymptotes.

Sketching Two Periods of the Graph

I will plot the graph of y=2cot(π2x)y = 2 \cot\left(\frac{\pi}{2}x\right) for two periods (e.g., x[2,2]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=2cot(π2x)y = 2 \cot\left(\frac{\pi}{2}x\right) for two periods (x[2,2]x \in [-2, 2]):

  • Vertical Asymptotes: At x=2,0,x = -2, 0, and 22.
  • Zeros: At x=1,1x = -1, 1, where y=0y = 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 xx \to \infty or xx \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\cot were replaced by tan\tan?

Tip:

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

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Periodic Functions
Properties of Cotangent Function
Graphing Trigonometric Functions

Formulas

y = a cot(bx + c) (General form of cotangent function)
Period of cotangent function: Period = π / |b|
Domain of cotangent function: x ≠ nπ/b where n ∈ ℤ

Theorems

Properties of Cotangent Function
Odd Function Symmetry

Suitable Grade Level

Grades 10-12