To analyze the function \( y = \frac{4}{3} \sin\left(\frac{\pi}{3} x\right) \), let's break it down step by step:

1. **Identify the function type:** This is a sine function of the form \( y = A \sin(Bx + C) + D \).

2. **Amplitude (A):** The amplitude of the sine function is the coefficient of the sine, which in this case is \(\frac{4}{3}\).

3. **Frequency (B):** The coefficient of \( x \) inside the sine function is \(\frac{\pi}{3}\). This determines the period of the sine function.
   - The period \( P \) of the sine function is given by \( P = \frac{2\pi}{B} \).
   - Substituting \( B = \frac{\pi}{3} \), we get \( P = \frac{2\pi}{\frac{\pi}{3}} = 6 \).

4. **Phase Shift (C):** There is no phase shift in this function since there is no constant added inside the sine function.

5. **Vertical Shift (D):** There is no vertical shift in this function since there is no constant added outside the sine function.

So, the characteristics of the function \( y = \frac{4}{3} \sin\left(\frac{\pi}{3} x\right) \) are:

- **Amplitude:** \(\frac{4}{3}\)
- **Period:** \(6\)
- **Phase Shift:** None
- **Vertical Shift:** None

### Graphing the Function

To graph \( y = \frac{4}{3} \sin\left(\frac{\pi}{3} x\right) \):

1. The sine function starts at 0, reaches its maximum value of \(\frac{4}{3}\) at \(\frac{P}{4} = 1.5\), returns to 0 at \(\frac{P}{2} = 3\), reaches its minimum value of \(-\frac{4}{3}\) at \(\frac{3P}{4} = 4.5\), and returns to 0 at \(P = 6\).
2. This pattern repeats every 6 units along the x-axis.

Would you like a detailed graph of this function or further explanation on any particular aspect?

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Here are five numbered questions for you to ask next:

1. How do you determine the amplitude and period of a trigonometric function?
2. Can you explain the significance of phase shift and vertical shift in trigonometric functions?
3. What are some real-world applications of sine and cosine functions?
4. How do you transform a sine function to match specific amplitude and period values?
5. Can you demonstrate how to graph a sine function with a given period and amplitude?

**Tip:** Remember, the period of a sine or cosine function is inversely proportional to the coefficient of \( x \) inside the function.

Explore the function y = (4/3) sin(pi/3 x) step-by-step. Learn about its amplitude, period, and graph characteristics. Suitable for students in Grades 10-12.

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