https://file.aiquickdraw.com/mathbot/file/17277969508658fm1iw27.jpg

The problem asks you to graph the function \( y = \sin\left(\frac{1}{4}x\right) \) and identify the correct graph.

### Key points:
1. **General Form of Sine Function**: The standard form of a sine function is \( y = \sin(bx) \), where \( b \) affects the **period** of the function.
2. **Period Calculation**: The period of a sine function is given by \( \frac{2\pi}{b} \). In this case, \( b = \frac{1}{4} \), so the period is:
   \[
   \text{Period} = \frac{2\pi}{\frac{1}{4}} = 8\pi
   \]
   This means the sine wave completes one full cycle over an interval of \( 8\pi \).

### Checking the options:
- **Option A and B**: These show graphs with a smooth wave completing one cycle by \( 8\pi \). This aligns with the period we calculated.
- **Option C and D**: These show graphs with much more rapid oscillations, completing several cycles within \( 8\pi \), which would indicate a different period (likely for a function like \( \sin(4x) \)).

Since the function completes one full cycle over \( 8\pi \), the correct graph is either **A** or **B**. Both show the correct period, but:

- **Option A**: Starts at \( (0, 0) \) and goes upwards first, which matches the behavior of \( \sin\left(\frac{1}{4}x\right) \), as \( \sin(0) = 0 \) and the sine function first increases.
- **Option B**: Shows a sine wave starting downwards, which does not match the sine function’s typical behavior starting from 0.

### Conclusion:
The correct graph is **Option A**.

Would you like more details on how the sine function works, or how to identify different transformations?

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### Follow-up questions:
1. What does the amplitude of a sine function represent?
2. How does the value of \( b \) affect the period of a sine function?
3. What would the graph look like if the function were \( y = \sin(4x) \)?
4. How can you determine phase shifts in a sine function?
5. How do you graph a cosine function compared to a sine function?

### Tip:
To remember the effect of \( b \) on the period, use the formula \( \text{Period} = \frac{2\pi}{b} \), where smaller values of \( b \) stretch the graph horizontally, and larger values compress it.

Graph the function y = sin(1/4 * x). Give the period. Choose the correct graph of the function.

Understand how to graph the sine function y = sin(1/4 * x) and determine its period. This problem involves trigonometric concepts, including calculating the period using 2π / b. Suitable for high school students learning trigonometry.

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