To graph \( y = 4\sin x \), it's important to understand the key characteristics of the sine function and how it changes with a coefficient.

### Characteristics of \( y = 4\sin x \):
1. **Amplitude**: The amplitude of the function \( y = 4\sin x \) is 4. This means the graph will oscillate between \( -4 \) and \( 4 \).
   
2. **Period**: The period of \( \sin x \) is \( 2\pi \). This remains unchanged since there is no coefficient affecting the \( x \)-variable.
   
3. **Frequency**: The frequency is also the same as \( \sin x \) because the coefficient inside the function (which affects frequency) is 1.

4. **Vertical Shift**: There is no vertical shift since no constant is added or subtracted from \( y \).

5. **Horizontal Shift**: There is no horizontal shift as there is no constant added or subtracted within the \( x \)-variable.

### Key Points of the Graph:
- **At \( x = 0 \)**: \( y = 4\sin(0) = 0 \)
- **At \( x = \frac{\pi}{2} \)**: \( y = 4\sin\left(\frac{\pi}{2}\right) = 4 \)
- **At \( x = \pi \)**: \( y = 4\sin(\pi) = 0 \)
- **At \( x = \frac{3\pi}{2} \)**: \( y = 4\sin\left(\frac{3\pi}{2}\right) = -4 \)
- **At \( x = 2\pi \)**: \( y = 4\sin(2\pi) = 0 \)

The graph oscillates between 4 and -4 with a period of \( 2\pi \).

### Graph Representation:
If you want to visualize the graph, it will be a standard sine curve but with a stretched amplitude reaching a maximum of 4 and a minimum of -4.

Would you like me to provide a visual representation of this graph?

### Related Questions:
1. How does the graph of \( y = 4\sin x \) compare to \( y = \sin x \)?
2. What happens to the graph if we change \( y = 4\sin x \) to \( y = 4\sin(2x) \)?
3. What is the period of \( y = 4\sin\left(\frac{1}{2}x\right) \)?
4. How would you find the maximum and minimum values of \( y = 4\sin x \)?
5. How does changing the amplitude affect the shape of a sine wave?

**Tip:** When sketching trigonometric graphs, always identify key features like amplitude, period, and shifts first for a more accurate plot.

Learn how to graph the function y = 4sinx by understanding its amplitude, period, and key points. This solution provides a detailed breakdown of how the sine graph oscillates between -4 and 4 with a period of 2π. Ideal for students in Grades 9-12 studying trigonometry.

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