Question

Исследовать фунцию и построить график y=x/(x^2-4)

Answer 1

файл

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Answer 2

Function Exploration and Graph Plotting

To explore the function \(y = \frac{x}{{x^2-4}}\) and plot its graph, we can start by analyzing its properties and then proceed to visualize it.

The function \(y = \frac{x}{{x^2-4}}\) can be rewritten as \(y = \frac{x}{{(x-2)(x+2)}}\). This form helps us understand the behavior of the function and identify any vertical asymptotes, horizontal asymptotes, and intercepts.

Function Properties

1. Vertical Asymptotes: The function will have vertical asymptotes at \(x = 2\) and \(x = -2\) due to the denominator becoming zero at these points. 2. Horizontal Asymptote: As \(x\) approaches positive or negative infinity, the function will approach \(y = 0\). 3. Intercepts: The function may have x-intercepts and y-intercepts, which we can determine by setting \(y = 0\) and \(x = 0\) respectively.

Graph Plotting

Let's plot the graph of the function \(y = \frac{x}{{x^2-4}}\) to visualize its behavior and confirm the properties we've identified.

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return x / (x**2 - 4)

# Generate x values
x = np.linspace(-10, 10, 400)
x = x[x != 2]  # Exclude x = 2 and x = -2 to avoid division by zero

# Calculate corresponding y values
y = f(x)

# Plot the graph
plt.figure(figsize=(8, 6))
plt.plot(x, y, label=r'$y = \frac{x}{{x^2-4}}$')
plt.axvline(x=2, color='r', linestyle='--', label='Vertical Asymptote at x=2')
plt.axvline(x=-2, color='r', linestyle='--', label='Vertical Asymptote at x=-2')
plt.axhline(y=0, color='g', linestyle='--', label='Horizontal Asymptote at y=0')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of y = x/(x^2-4)')
plt.legend()
plt.grid(True)
plt.show()

The graph will visually demonstrate the behavior of the function, including the identified properties such as vertical asymptotes, horizontal asymptote, and any intercepts.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.