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To determine the equation of the sine curve from the graph, we need to identify the **amplitude**, **period**, and **vertical shift**. The general form of a sine function is:

\[
y = A \sin(Bx + C) + D
\]
Where:
- \( A \) is the amplitude (vertical stretch/shrink),
- \( B \) determines the period \( T = \frac{2\pi}{B} \),
- \( C \) is the phase shift (horizontal translation),
- \( D \) is the vertical shift.

---

### Step 1: Identify the Amplitude \( A \)
The amplitude is half the distance between the maximum and minimum values of the graph.

- The maximum value is \( y = 6 \),
- The minimum value is \( y = -2 \).

The amplitude is:
\[
A = \frac{\text{Max} - \text{Min}}{2} = \frac{6 - (-2)}{2} = \frac{8}{2} = 4.
\]

---

### Step 2: Identify the Vertical Shift \( D \)
The vertical shift is the average of the maximum and minimum values.

\[
D = \frac{\text{Max} + \text{Min}}{2} = \frac{6 + (-2)}{2} = \frac{4}{2} = 2.
\]

---

### Step 3: Identify the Period and \( B \)
The graph completes one full cycle from \( x = -2\pi \) to \( x = 2\pi \), so the period \( T \) is:
\[
T = 4\pi.
\]
The relationship between \( B \) and the period is:
\[
T = \frac{2\pi}{B}.
\]
Substitute \( T = 4\pi \):
\[
4\pi = \frac{2\pi}{B}.
\]
Solve for \( B \):
\[
B = \frac{2\pi}{4\pi} = \frac{1}{2}.
\]

---

### Step 4: Identify the Phase Shift \( C \)
There is no phase shift visible because the sine graph appears symmetric around the origin and follows the standard sine curve's pattern. Therefore, \( C = 0 \).

---

### Final Equation
The equation of the graph is:
\[
y = 4 \sin\left( \frac{1}{2}x \right) + 2.
\]

---

### Summary:
- Amplitude: \( 4 \),
- Period: \( 4\pi \) (hence \( B = \frac{1}{2} \)),
- Vertical shift: \( 2 \),
- Phase shift: None.

Would you like more clarification on any of the steps? Let me know!

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### Related Questions:
1. How do you determine the amplitude and period of a sine function?
2. What is the difference between sine and cosine graphs?
3. How would the graph look if the amplitude were doubled?
4. How do you find the phase shift of a sine function?
5. What happens if you add a negative sign to the amplitude?

**Tip:** To identify the period of a trigonometric function, observe the horizontal distance required for the curve to repeat its pattern.

The following graph is a sine curve. Determine the equation of the graph.

Learn how to determine the equation of a sine curve from a graph. This example identifies the amplitude, vertical shift, and period to form the sine function equation: y = 4sin(1/2x) + 2. Suitable for high school students studying trigonometry.

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