Sketch the graph of the following sine functions. Give the period and amplitude of each function. 1. y sin 3x

Let's analyze and sketch the graph of the function \( y = \sin(3x) \).

### Step 1: Determine the Amplitude and Period

1. **Amplitude**: The amplitude of a sine function \( y = a \sin(bx) \) is the absolute value of \( a \). In this case, since \( a = 1 \) (the coefficient in front of the sine function), the amplitude is:
   \[
   \text{Amplitude} = |1| = 1
   \]

2. **Period**: The period of a sine function \( y = \sin(bx) \) is given by \( \frac{2\pi}{|b|} \). Here, \( b = 3 \), so the period is:
   \[
   \text{Period} = \frac{2\pi}{3}
   \]

### Step 2: Sketching the Graph

To sketch the graph of \( y = \sin(3x) \):

- Start by marking the amplitude (1 and -1) on the y-axis.
- Mark the period, \( \frac{2\pi}{3} \), on the x-axis.
- Divide the period into four equal parts to plot key points: \( 0 \), \( \frac{\pi}{6} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and \( \frac{2\pi}{3} \).
- Use the sine pattern (starting from 0, going to the max at \( \frac{\pi}{6} \), back to 0 at \( \frac{\pi}{3} \), to min at \( \frac{\pi}{2} \), and returning to 0 at \( \frac{2\pi}{3} \)).

Would you like a detailed sketch, or do you have further questions about the graph?

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### Related Questions

1. How would the graph change if we used \( y = 2\sin(3x) \)?
2. What would be the period of \( y = \sin(4x) \)?
3. How does the amplitude affect the shape of the sine wave?
4. What happens to the graph if \( y = \sin\left(\frac{x}{3}\right) \) is used?
5. How would you graph \( y = \sin(3x) + 1 \)?

### Tip

Remember, the period of \( y = \sin(bx) \) is always \( \frac{2\pi}{|b|} \), while the amplitude is the absolute value of the coefficient in front of the sine term.

Sketch the graph of the following sine functions. Give the period and amplitude of each function. 1. y = sin(3x)

Learn how to graph the sine function y = sin(3x) with a detailed analysis of its amplitude and period. The amplitude is 1, and the period is 2π/3. This guide is perfect for students in Grades 9-11 studying trigonometric functions.

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