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import numpy as np
import matplotlib.pyplot as plt

# Define the x range for one period of y = 5 * cos(2x)
x = np.linspace(0, np.pi, 100)
y = 5 * np.cos(2 * x)

# Plot the function
plt.figure(figsize=(8, 4))
plt.plot(x, y, label=r'$y = 5 \cos(2x)$', color='blue')
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title('Graph of $y = 5 \cos(2x)$ for one period')
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(True, linestyle='--', alpha=0.7)
plt.legend(loc="upper right")

# Show key points
x_key_points = [0, np.pi / 4, np.pi / 2, 3 * np.pi / 4, np.pi]
y_key_points = 5 * np.cos(2 * np.array(x_key_points))
plt.scatter(x_key_points, y_key_points, color='red')
for (xk, yk) in zip(x_key_points, y_key_points):
    plt.text(xk, yk, f'({xk:.2f}, {yk:.1f})', ha='right')

plt.show()