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?

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.