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 step by step:
1. Domain
The cotangent function, , is undefined whenever , where is an integer. For the argument , we have: Thus, the domain of the function is:
2. Periodicity
The cotangent function has a period of . For , the period becomes: Hence, the function repeats every 2 units along the -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:
4. Vertical Asymptotes
Vertical asymptotes occur where the function is undefined, i.e., at .
5. Symmetry
The cotangent function is odd: . Thus, the function is also odd: This implies rotational symmetry about the origin.
6. Behavior
- At (midpoint between consecutive vertical asymptotes), (since ).
- As or , , indicating vertical asymptotes.
Sketching Two Periods of the Graph
I will plot the graph of for two periods (e.g., ). 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 for two periods ():
- Vertical Asymptotes: At and .
- Zeros: At , where .
- The function alternates between near vertical asymptotes.
Would you like a detailed explanation of the sketch or further analysis?
5 Questions to Expand Understanding:
- What happens to the function as or ?
- How does scaling by a constant (like 2 here) affect other trigonometric functions?
- How does the period change if the argument is modified further?
- Can this function be expressed in terms of sine and cosine? If yes, how?
- How would the graph change if were replaced by ?
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