To graph the function $y = \sin(x - \pi)$, let's break it down:

Step 1: Understand the transformation

• The basic sine function is $y = \sin(x)$, which has a period of $2\pi$ and oscillates between -1 and 1.
• The expression $x - \pi$ means the function is shifted right by $\pi$ units.

Step 2: Key points for one period

• The period of $y = \sin(x - \pi)$ remains the same as $y = \sin(x)$, which is $2\pi$. This means the function will complete one cycle over an interval of length $2\pi$.
• For the sine wave, the key points are:
  1. At $x = 0$, the sine wave starts at 0 (but shifted to the right).
  2. The function reaches its maximum value (1) at $x = \frac{3\pi}{2}$.
  3. The function crosses 0 again at $x = 2\pi$.
  4. It reaches its minimum value (-1) at $x = \frac{5\pi}{2}$.
  5. Then, it crosses 0 again at $x = 2\pi$.

Step 3: Graphing the function

Let me now create a graph of one period of this sine function.

I’ll go ahead and generate the plot. numpy as np import matplotlib.pyplot as plt

Define the function

x = np.linspace(0, 2 * np.pi, 400) y = np.sin(x - np.pi)

Plot the graph

plt.figure(figsize=(8, 4)) plt.plot(x, y, label=r'$y = \sin(x - \pi)$', color='b')

Mark key points

plt.scatter([np.pi, 3np.pi/2, 2np.pi], [0, 1, 0], color='r') # points at x=pi, 3pi/2, 2pi

Labels and title

plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.title(r"Graph of $y = \sin(x - \pi)$") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.grid(True)

Show the plot

plt.show()Here is the graph of one period of the function $y = \sin(x - \pi)$.

Key Observations:

• The sine wave starts at $x = \pi$, crossing the x-axis (value = 0) and oscillating between -1 and 1.
• The peak (maximum value of 1) is at $x = \frac{3\pi}{2}$.
• The sine function completes one full cycle from $x = 0$ to $x = 2\pi$.

Would you like further details on any part of the graph or the function?

Here are a few follow-up questions:

1. How would the graph change if the sine function was $y = \sin(x + \pi)$?
2. What would happen if the coefficient of the sine function was $2$, making it $y = 2\sin(x - \pi)$?
3. Can you explain how the phase shift affects the graph of trigonometric functions?
4. What would the graph look like if the sine function had a vertical shift, e.g., $y = \sin(x - \pi) + 2$?
5. How does the period of the sine function change if the equation was $y = \sin(2x - \pi)$?

Tip: The amplitude of the sine wave is determined by the coefficient in front of the sine function. For example, $y = 2\sin(x - \pi)$ would have an amplitude of 2.