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Let's analyze the sine function graphs given in the problem and determine the functions in the form \( y = a \sin(bx - c) + d \) for each:

### 1. Analysis of the First Graph:
- Amplitude (\( a \)): The height from the centerline (0) to the peak is about 2, so \( a = 2 \).
- Period (\( \frac{2\pi}{b} \)): The period is the distance between two peaks, which appears to be \( \pi \), so \( b = 2 \).
- Phase shift (\( c \)): It looks like there is no horizontal shift, so \( c = 0 \).
- Vertical shift (\( d \)): The graph is centered around \( y = 0 \), so \( d = 0 \).

The function is:
\[
y = 2 \sin(2x)
\]

### 2. Analysis of the Second Graph:
- Amplitude (\( a \)): The amplitude is 3.
- Period (\( \frac{2\pi}{b} \)): The distance between two peaks is about 6, which gives \( b = \frac{\pi}{3} \).
- Phase shift (\( c \)): The graph starts at 0, indicating \( c = 0 \).
- Vertical shift (\( d \)): The graph is centered around \( y = 0 \).

The function is:
\[
y = 3 \sin\left(\frac{\pi}{3}x\right)
\]

### 3. Analysis of the Third Graph:
- This graph is described to have a maximum at \( \left(\frac{\pi}{2}, 7\right) \) and a minimum at \( \left(\frac{3\pi}{2}, -1\right) \).
- Amplitude (\( a \)): The difference between the maximum and minimum is \( 7 - (-1) = 8 \), so \( a = 4 \).
- Period (\( \frac{2\pi}{b} \)): The distance between \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) is \( \pi \), so \( b = 2 \).
- Phase shift (\( c \)): The maximum occurs at \( \frac{\pi}{2} \), suggesting a phase shift of \( \frac{\pi}{2} \).
- Vertical shift (\( d \)): The midline is at \( \frac{7 + (-1)}{2} = 3 \).

The function is:
\[
y = 4 \sin(2x - \frac{\pi}{2}) + 3
\]

Would you like more details or have any questions on these solutions?

### Relative Questions:
1. How do you determine the period of a sine function from a graph?
2. What effect does changing the amplitude have on the graph of a sine function?
3. How do phase shifts impact the position of sine and cosine graphs?
4. Can you derive the formula for the period of a sine function from its general form?
5. What are the differences between horizontal and vertical shifts in trigonometric functions?

### Tip:
For a sine function \( y = a \sin(bx - c) + d \), remember that \( a \) affects the amplitude, \( b \) affects the period (\( \frac{2\pi}{b} \)), \( c \) affects the horizontal shift, and \( d \) adjusts the vertical shift.

Write a sine function in the form y = a sin(bx - c) + d for the three graphs below.

Learn how to write the equation of a sine function in the form y = a sin(bx - c) + d based on given graphs. This problem involves determining amplitude, period, phase shift, and vertical shift for each graph. Suitable for high school students studying trigonometric functions.

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