Graph one period of the given function.

y equals sine left parenthesis x minus pi right parenthesis

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, 3*np.pi/2, 2*np.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.

Graph one period of the given function. y = sin(x - π)

Learn how to graph one period of the sine function y = sin(x - π), including phase shifts and key points. Understand how the transformation affects the standard sine wave and visualize the function's behavior over a full cycle. Suitable for students in Grades 9-12.

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