To graph the function \( y = 4\sin(3x) \), we'll break down the transformations step by step.

### 1. **Start with the basic sine function \( y = \sin(x) \):**
   - The basic sine function oscillates between -1 and 1, with a period of \( 2\pi \). 
   - Its graph is a smooth wave that starts at \( (0, 0) \), reaches a maximum of 1 at \( \frac{\pi}{2} \), returns to 0 at \( \pi \), reaches a minimum of -1 at \( \frac{3\pi}{2} \), and returns to 0 at \( 2\pi \).

### 2. **Amplitude change:**
   - The coefficient 4 in \( y = 4\sin(3x) \) affects the amplitude.
   - The amplitude (the height from the centerline to the peak) is now 4 instead of 1.
   - The graph will oscillate between -4 and 4 instead of -1 and 1.

### 3. **Period change:**
   - The coefficient 3 inside the sine function affects the period.
   - The period of \( y = \sin(x) \) is \( 2\pi \), but for \( y = \sin(3x) \), the period becomes \( \frac{2\pi}{3} \).
   - This means the sine wave completes one full cycle more quickly, within an interval of \( \frac{2\pi}{3} \).

### 4. **Graphing the function:**
   - **Key points to plot:**
     1. Start at \( x = 0 \), \( y = 0 \).
     2. The next peak is at \( x = \frac{\pi}{6} \), \( y = 4 \).
     3. The graph returns to \( y = 0 \) at \( x = \frac{\pi}{3} \).
     4. The minimum point is at \( x = \frac{\pi}{2} \), \( y = -4 \).
     5. The graph returns to \( y = 0 \) at \( x = \frac{2\pi}{3} \).
   - Continue this pattern to complete one full cycle of the wave.
  
### Final Graph Characteristics:
- **Amplitude:** 4
- **Period:** \( \frac{2\pi}{3} \)
- **Wavelength:** The graph repeats every \( \frac{2\pi}{3} \) units along the x-axis.

Would you like to explore how to graph this function further, or do you have other questions in mind?

### Related Questions:
1. How does the graph of \( y = 2\sin(x) \) differ from \( y = 4\sin(3x) \)?
2. What happens to the graph of \( y = 4\sin(3x) \) if we add a phase shift, such as \( y = 4\sin(3x - \frac{\pi}{2}) \)?
3. How do you determine the phase shift of a sine function?
4. What is the impact of adding a vertical shift, such as in \( y = 4\sin(3x) + 2 \)?
5. How do you graph \( y = \cos(3x) \) and compare it with \( y = \sin(3x) \)?
6. What is the significance of the amplitude in trigonometric functions?
7. How does the graph change if we use \( y = 4\sin\left(\frac{x}{2}\right) \)?
8. How would you graph \( y = 4\sin(3x) \) for negative values of \( x \)?

**Tip:** When dealing with trigonometric functions, always note the amplitude, period, phase shift, and vertical shift to accurately graph the function.

Learn how to graph the function y = 4sin(3x) step-by-step. Understand the impact of amplitude and period changes on the sine function graph. Suitable for high school students studying trigonometry.

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