numpy as np import matplotlib.pyplot as plt

Parameters for the function

amplitude = 2 period = 4 * np.pi max_val = -1 min_val = -5

Vertical shift and midline

midline = (max_val + min_val) / 2

Define the function

x = np.linspace(0, 4 * period, 1000) # x values over two full periods y = amplitude * np.sin((2 * np.pi / period) * x) + midline

Plot the function

plt.figure(figsize=(8, 4)) plt.plot(x, y, label=f"y = 2*sin((π/{period})*x) - 3") plt.axhline(y=max_val, color='red', linestyle='--', label="Max = -1") plt.axhline(y=min_val, color='blue', linestyle='--', label="Min = -5") plt.axhline(y=midline, color='green', linestyle='--', label="Midline = -3") plt.title("Graph of a Sinusoidal Function") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.grid(True) plt.show()Here is the graph of the sinusoidal function. It has:

1. A period of $4\pi$,
2. An amplitude of $2$,
3. A maximum value of $-1$,
4. A minimum value of $-5$.

The function is vertically shifted downward by $-3$, resulting in the midline at $y = -3$.

Would you like a detailed explanation of how these properties were set up? Here are some related questions:

1. How is the period of a sinusoidal function calculated?
2. How does amplitude affect the graph of a sine or cosine function?
3. What does the midline of a sinusoidal function represent?
4. How do vertical shifts affect maximum and minimum values?
5. Can this graph be written as a cosine function instead of a sine function?

Tip: To find the equation of a sinusoidal function, start by identifying the amplitude, period, midline, and phase shift (if any).