Question

Постройте график функции y=x^4-3x^2-4/x^2-4

Answer 1

Решение задания приложено

Answer 2

Graph of the Function y=x^4-3x^2-4/x^2-4

To graph the function y=x^4-3x^2-4/x^2-4, we can start by analyzing its behavior and identifying key points.

Behavior Analysis: The behavior of the function can be understood by analyzing its components and identifying any vertical or horizontal asymptotes, intercepts, and critical points.

Vertical Asymptotes: The vertical asymptotes occur where the denominator of the function becomes zero. In this case, the vertical asymptotes can be found by solving the equation x^2-4=0.

Horizontal Asymptotes: The horizontal asymptote can be determined by analyzing the behavior of the function as x approaches positive or negative infinity.

Intercepts: The x-intercepts can be found by setting y=0 and solving for x. The y-intercept can be found by setting x=0 and evaluating y.

Critical Points: The critical points can be found by taking the derivative of the function and solving for points where the derivative is zero or undefined.

Graphing the Function

Let's start by analyzing the components of the function and identifying the key points.

Vertical Asymptotes: The vertical asymptotes can be found by solving the equation x^2-4=0. - Solving x^2-4=0 gives x=2 and x=-2.

Horizontal Asymptotes: To determine the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive or negative infinity.

Intercepts: To find the x-intercepts, we can set y=0 and solve for x. To find the y-intercept, we can set x=0 and evaluate y.

Critical Points: The critical points can be found by taking the derivative of the function and solving for points where the derivative is zero or undefined.

Now, let's plot the identified key points and asymptotes on the graph to visualize the function.

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return (x4 - 3*x2 - 4) / (x**2 - 4)

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

# Generate y values
y = f(x)

# Plot the function
plt.figure(figsize=(8, 6))
plt.plot(x, y, label='y=x^4-3x^2-4/x^2-4')
plt.title('Graph of the Function y=x^4-3x^2-4/x^2-4')
plt.xlabel('x')
plt.ylabel('y')
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.grid(True)
plt.legend()
plt.show()

The graph of the function y=x^4-3x^2-4/x^2-4 will display the behavior of the function, including the identified key points and asymptotes.

Remember to consider the behavior of the function near the asymptotes and critical points for a comprehensive understanding of its graph.