https://file.aiquickdraw.com/mathbot/file/1725779757978e256cln1.jpg

The given function is:

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

Let's break down and analyze this function:

### 1. **Amplitude**: 
   - The amplitude of the sine function is determined by the coefficient of \(\sin(\cdot)\). Here, the amplitude is \(2\), meaning the sine wave will oscillate between \(y = 3 - 2 = 1\) and \(y = 3 + 2 = 5\).

### 2. **Period**:
   - The period of a sine function \(\sin(bx)\) is given by \(\frac{2\pi}{b}\). In this case, \(b = 2\), so the period is:
   \[
   \text{Period} = \frac{2\pi}{2} = \pi
   \]
   - This means that the function completes one full cycle every \(\pi\) units along the x-axis.

### 3. **Phase Shift**:
   - The phase shift of the function is determined by the horizontal translation. For the function \(\sin(2x - \frac{\pi}{3})\), the phase shift is \(\frac{\pi}{3}\) to the right.

### 4. **Vertical Shift**:
   - The function is shifted vertically by \(+3\), meaning the midline of the sine wave is at \(y = 3\).

### Summary:
- **Amplitude**: 2
- **Period**: \(\pi\)
- **Phase Shift**: \(\frac{\pi}{3}\) to the right
- **Vertical Shift**: Up by 3 units

Would you like to explore any particular aspect of this function in more detail, or do you have any other questions?

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Here are 5 relative questions:
1. How do we calculate the maximum and minimum values of this function?
2. How does changing the coefficient of \(x\) inside the sine function affect its graph?
3. What happens if the vertical shift \(+3\) is changed to another value?
4. How does the phase shift \(\frac{\pi}{3}\) influence the graph's starting point?
5. Can you derive the general formula for the phase shift of any sine function?

**Tip:** When analyzing trigonometric functions, always identify the amplitude, period, phase shift, and vertical shift to fully understand the behavior of the graph.

Explore the sine function y = 2sin(2x - π/3) + 3 with amplitude, period, phase shift, and vertical shift analysis. Suitable for students in Grades 10-12, this problem helps understand trigonometric graph transformations and properties of sine functions.

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