import numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return 2 * np.cos(2 * x - np.pi / 2) - 1

Generate x values for two periods

x = np.linspace(0, 2 * np.pi, 1000) y = f(x)

Determine the period of the function

period = 2 * np.pi / 2 # Period = 2π / coefficient of x in cos

Extend x to cover two periods

x_two_periods = np.linspace(0, 2 * period, 1000) y_two_periods = f(x_two_periods)

Plot the function

plt.figure(figsize=(10, 6)) plt.plot(x_two_periods, y_two_periods, label=r"$f(x) = 2\cos(2x-\frac{\pi}{2})-1$")

Add labels for clarity

ticks = np.arange(0, 2 * period + period / 4, period / 4) tick_labels = [f"{round(tick, 2)}" for tick in ticks] plt.xticks(ticks, tick_labels) plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--')

plt.title("Graph of $f(x) = 2\cos(2x - \frac{\pi}{2}) - 1$") plt.xlabel("$x$") plt.ylabel("$f(x)$") plt.grid(True, linestyle='--', alpha=0.7) plt.legend() plt.show() <<ImageDisplayed>>successHere is the graph of $f(x) = 2\cos(2x - \frac{\pi}{2}) - 1$ over two periods. The x-axis ticks are labeled at quarter-period intervals for clarity. Let me know if you need further adjustments or explanations!

Do you have questions or need additional details? Here are some related topics:

1. How is the period of a trigonometric function determined?
2. What is the phase shift in this function, and how does it affect the graph?
3. How does the amplitude of a trigonometric function influence its graph?
4. How to interpret transformations like vertical shifts in trigonometric functions?
5. Can this function be rewritten in terms of sine or another trigonometric identity?

Tip: When graphing trigonometric functions, always note key parameters like amplitude, period, and phase shift—they guide the shape and position of the graph.