https://r2.erweima.ai/mathbot/image/1725456344343sa95d90x.jpg

The image contains a set of trigonometric functions that need to be graphed. Let's analyze and write down each function for clarity:

1. \( F(x) = 2 \sin(x - \pi) \)  over the interval \( [0, 3\pi] \)
2. \( F(x) = \sin\left(2x - \frac{\pi}{2}\right) + 2 \) over the interval \( [0, 2\pi] \)
3. \( F(x) = -1.5 \sin\left(x - \frac{\pi}{2}\right) + 2 \) over the interval \( [0, 2\pi] \)
4. \( F(x) = 2 \sin\left(x - \frac{\pi}{3}\right) \) over the interval \( [0, 6\pi] \)

### Summary of the Functions:
1. **Function 1:** A sine function with an amplitude of 2, a horizontal shift of \(\pi\), and over the interval \( [0, 3\pi] \).
2. **Function 2:** A sine function with a frequency multiplier of 2, a phase shift of \(\frac{\pi}{2}\), a vertical shift of +2, and over the interval \( [0, 2\pi] \).
3. **Function 3:** A sine function with an amplitude of -1.5 (indicating a reflection across the x-axis), a phase shift of \(\frac{\pi}{2}\), a vertical shift of +2, and over the interval \( [0, 2\pi] \).
4. **Function 4:** A sine function with an amplitude of 2, a phase shift of \(\frac{\pi}{3}\), and over the interval \( [0, 6\pi] \).

Would you like me to provide the detailed steps for graphing each function or to clarify any specific aspect of these functions?

### Follow-up Questions:
1. What effect does the amplitude have on the shape of the sine wave?
2. How does the phase shift affect the position of the sine function on the graph?
3. What is the significance of the frequency multiplier (e.g., 2 in function 2) on the graph?
4. How does a vertical shift modify the sine function graphically?
5. What role does the interval play in determining the portion of the sine function that is graphed?

### Tip:
When graphing trigonometric functions, it can be helpful to first sketch the basic function (e.g., \( \sin(x) \)) and then apply transformations step by step to better understand the effect of each parameter.

Graph the following trigonometric functions.

Learn how to graph trigonometric functions such as F(x) = 2 sin(x - π) over the interval [0, 3π]. This example covers key concepts including amplitude, phase shift, vertical shift, and the effect of these transformations on the sine function. Suitable for high school students in Grades 10-12.

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