Question

Помогите исследовать функцию и построить график,пожалуйста)y= -x^4+8x^2+9

Answer 1

Это обычная парабола (как график квадратной функции) с вершиной в точке (0;9), ветвями вверх.

Answer 2

Function Analysis and Graph Plotting

To analyze the function y = -x^4 + 8x^2 + 9 and plot its graph, we can start by examining the key features of the function.

1. Degree of the Function: The highest power of the variable x in the function is 4, which makes it a fourth-degree polynomial function.

2. Leading Coefficient: The leading coefficient of the function is -1, which means that the graph of the function will be concave down.

3. Symmetry: The function is not symmetric with respect to the y-axis (x = 0), nor is it symmetric with respect to the origin (x = 0, y = 0).

4. x-Intercepts: To find the x-intercepts, we set y = 0 and solve for x. However, since this is a fourth-degree polynomial, finding the exact solutions algebraically can be challenging. We can use numerical methods or graphing software to approximate the x-intercepts.

5. y-Intercept: To find the y-intercept, we set x = 0 and evaluate the function. In this case, the y-intercept is (0, 9).

6. Turning Points: To find the turning points (local extrema) of the function, we can take the derivative of the function and solve for x when the derivative equals zero. Then, we can substitute these x-values back into the original function to find the corresponding y-values.

7. Behavior at Infinity: As x approaches positive or negative infinity, the function approaches negative infinity. This is because the leading term, -x^4, dominates the function.

Now, let's plot the graph of the function using Python and the matplotlib library:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 100)  # Generate 100 x-values from -10 to 10
y = -x4 + 8*x2 + 9  # Evaluate the function for each x-value

plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of y = -x^4 + 8x^2 + 9')
plt.grid(True)
plt.show()

The resulting graph will show the shape of the function and its key features, such as the x-intercepts, y-intercept, and turning points.

Please note that the graph is plotted based on the provided function, and the accuracy of the graph depends on the accuracy of the function itself.

Let me know if there's anything else I can help you with!