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The problem shown asks to graph the function \( y = \sin \left( \frac{1}{6}x \right) \) over a two-period interval and find its period.

### Step-by-Step Solution:

1. **General Sine Function Form**: The general form of the sine function is:
   \[
   y = \sin(bx)
   \]
   where the period \( P \) is given by:
   \[
   P = \frac{2\pi}{|b|}
   \]

2. **Find the Period**: For the given function \( y = \sin \left( \frac{1}{6}x \right) \), the coefficient of \( x \), \( b \), is \( \frac{1}{6} \). Thus, the period is:
   \[
   P = \frac{2\pi}{\frac{1}{6}} = 12\pi
   \]

3. **Two-Period Interval**: Since the period is \( 12\pi \), two periods would be \( 2 \times 12\pi = 24\pi \). Therefore, the graph should span from \( 0 \) to \( 24\pi \).

4. **Correct Graph**: The graph should complete one full sine wave in the interval \( [0, 12\pi] \) and another in \( [12\pi, 24\pi] \). This means the wave should have a lower frequency (stretched horizontally compared to the standard sine wave).

### Choosing the Correct Graph:
Now, looking at the options:
- The correct graph will show a sine wave with a much longer wavelength (period) than the standard sine wave. This corresponds to a graph that extends over \( 12\pi \) for one full wave.

Thus, the correct answer should be **option C**, as it correctly shows a graph with two full cycles over a \( 24\pi \)-interval, which matches the characteristics of the sine function with a period of \( 12\pi \).

Would you like more details on any part of this explanation?

### 5 Relative Questions:
1. What is the relationship between the period and the frequency of a sine function?
2. How does the graph of \( y = \sin(bx) \) change as \( b \) increases or decreases?
3. How would the graph change if the function was \( y = \cos \left( \frac{1}{6}x \right) \)?
4. What is the amplitude of the function \( y = \sin \left( \frac{1}{6}x \right) \)?
5. What effect does multiplying \( x \) by a constant inside the sine function have on the graph?

### Tip:
Remember, the period of a sine or cosine function is always \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \). This helps quickly identify how the graph stretches or shrinks horizontally.

Graph the following function over a two-period interval. Give the period: y = sin(1/6 * x). Which of the following is the correct graph of the function?

Understand how to graph the sine function y = sin(1/6 * x) over a two-period interval, and determine the period. This solution includes identifying the correct graph from multiple options and calculating the period of the function.

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