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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 an equation of the form \( y = A \sin(B(x - C)) + D \) based on the sine wave from the graph, we need to determine the following parameters:

1. **Amplitude (A)**: The vertical distance from the midline to the maximum or minimum point of the wave.
2. **Period (T)**: The horizontal distance required for the wave to complete one full cycle.
3. **Phase Shift (C)**: The horizontal shift of the wave.
4. **Vertical Shift (D)**: The vertical displacement of the wave's midline from the \(x\)-axis.

### Observations from the graph:

1. **Amplitude (A)**:
   - The maximum value is \(3\), and the minimum value is \(-1\).
   - The midline is halfway between \(3\) and \(-1\), which is \(1\).
   - The amplitude is the distance from the midline to the maximum, so \(A = 3 - 1 = 2\).

2. **Vertical Shift (D)**:
   - The midline is at \(y = 1\), so \(D = 1\).

3. **Period (T)**:
   - From the graph, the sine wave completes one cycle between \(x = 0\) and \(x = 2\pi\).
   - The period \(T = 2\pi\).
   - The formula for period in a sine function is \(T = \frac{2\pi}{B}\), so \(B = 1\).

4. **Phase Shift (C)**:
   - The graph seems to have a phase shift to the right.
   - The first peak occurs at \(x = \frac{\pi}{3}\), indicating a phase shift of \(C = \frac{\pi}{3}\).

### Equation:

Based on the observations, the equation of the sine wave is:

\[
y = 2 \sin\left( x - \frac{\pi}{3} \right) + 1
\]

Let me know if you need further details or explanations!

Here are 5 related questions to deepen your understanding:

1. How does the amplitude of a sine wave affect its shape?
2. What role does the period play in defining the frequency of the sine wave?
3. How would you adjust the equation if the graph was reflected across the x-axis?
4. What happens to the sine wave if we change the vertical shift (D)?
5. How can you determine the phase shift directly from a sine wave's graph?

**Tip**: Always check where the peaks and troughs of the sine wave occur to confirm the phase shift.

Learn how to write an equation of the form y = A sin(B(x - C)) + D based on the provided sine wave graph. This covers trigonometry concepts like amplitude, period, phase shift, and vertical shift. Ideal for Grades 10-12.

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