numpy as np import matplotlib.pyplot as plt

Define the range for x

x = np.linspace(-2np.pi, 2np.pi, 400)

Define the functions

y1 = 2 * np.cos(x) y2 = np.sin(2 * x) y3 = 2 * np.sin(x) + 3

Create the figure and axes

fig, axes = plt.subplots(3, 1, figsize=(8, 10))

Plot each function

axes[0].plot(x, y1, label=r'$y = 2\cos(x)$', color='b') axes[0].set_title(r'$y = 2\cos(x)$') axes[0].grid(True) axes[0].legend()

axes[1].plot(x, y2, label=r'$y = \sin(2x)$', color='g') axes[1].set_title(r'$y = \sin(2x)$') axes[1].grid(True) axes[1].legend()

axes[2].plot(x, y3, label=r'$y = 2\sin(x) + 3$', color='r') axes[2].set_title(r'$y = 2\sin(x) + 3$') axes[2].grid(True) axes[2].legend()

Set common labels

for ax in axes: ax.set_xlabel('x') ax.set_ylabel('y')

Adjust layout

plt.tight_layout() plt.show()Here are the graphs of the functions you requested:

1. $y = 2 \cos(x)$
2. $y = \sin(2x)$
3. $y = 2 \sin(x) + 3$

Let me know if you'd like further details on these graphs or any specific aspects explained. Here are a few related questions that could expand your understanding:

1. How does the amplitude of a trigonometric function affect its graph?
2. What is the impact of the frequency of $\cos(x)$ and $\sin(x)$ functions on their graphs?
3. How would the graph of $y = 2\sin(x) + 3$ change if we replaced the coefficient of 2 with 1?
4. How does the phase shift influence the sine and cosine functions?
5. What is the significance of the vertical shift in the function $y = 2\sin(x) + 3$?

Tip: When analyzing trigonometric functions, always pay attention to the amplitude, frequency, and phase shifts, as they significantly impact the appearance of the graph!