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Write an equation of the form y=A sin(B(x-c))+D whose graph is the given sine wave. 

To write the equation of the sine wave in the form \( y = A \sin(B(x - C)) + D \), let's break down each component based on the graph:

1. **Amplitude (\(A\))**: The amplitude is the distance from the middle of the wave (midline) to the maximum or minimum value. In the graph, the maximum is 3, and the midline is at \( y = 1 \). Therefore, the amplitude \( A \) is:
   \[
   A = 3 - 1 = 2
   \]

2. **Vertical shift (\(D\))**: The midline of the wave is the horizontal line where the wave oscillates around. In this case, the midline is \( y = 1 \), so the vertical shift \( D \) is:
   \[
   D = 1
   \]

3. **Period and \(B\)**: The period of a sine wave is the distance between two consecutive peaks or troughs. Here, one full cycle of the wave occurs between \( -\pi \) and \( \pi \), so the period is \( 2\pi \). The formula for the period is \( \frac{2\pi}{B} \), so we can find \( B \):
   \[
   2\pi = \frac{2\pi}{B} \implies B = 1
   \]

4. **Horizontal shift (\(C\))**: The wave appears to have a horizontal shift to the right. The maximum point is at \( x = 0 \) (instead of \( x = -\frac{\pi}{2} \) for a standard sine wave). Since the sine wave reaches its maximum at \( x = 0 \), there is no horizontal shift, so \( C = 0 \).

Now, combining all these values into the sine equation:

\[
y = 2 \sin(x) + 1
\]

This is the equation of the sine wave.

Would you like further details or clarifications?

Here are 5 related questions:
1. How does changing the amplitude \(A\) affect the graph of the sine wave?
2. What happens to the graph if the vertical shift \(D\) is changed?
3. Can you explain how to calculate the period of a trigonometric function?
4. What effect does the horizontal shift \(C\) have on the graph?
5. How would the graph look if \(B\) were greater than 1?

**Tip**: When identifying the amplitude of a sine wave, always check the vertical distance from the midline to the highest point on the graph.

Write an equation of the form y=A sin(B(x-C))+D whose graph is the given sine wave.

Learn how to write the equation of a sine wave in the form y = A sin(B(x - C)) + D based on a graph. This problem covers key concepts in trigonometry, including amplitude, period, and phase shift, with step-by-step calculations for understanding graph transformations.

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