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Write an expression defining y as a function of x which best describes this graph.


The graph shows a sinusoidal wave that appears to oscillate in a regular pattern. From the graph, the following observations can be made:

1. **Amplitude**: The wave peaks at 10 and troughs at -10, suggesting an amplitude of 10.
2. **Period**: The wave completes a cycle approximately every 6 units along the x-axis.
3. **Vertical Shift**: There is no visible vertical shift, as the wave oscillates symmetrically around the x-axis.
4. **Phase Shift**: The wave appears to start from zero at x = 0, indicating there may not be a phase shift.

Based on these observations, the function \( y(x) \) can be expressed as:
\[
y(x) = 10 \sin\left(\frac{\pi}{3}x\right)
\]

### Explanation:
- The amplitude is 10, as the wave reaches 10 and -10.
- The period \( T \) is approximately 6. The period of a sine function is given by \( T = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. Solving for \( b \) gives \( b = \frac{2\pi}{6} = \frac{\pi}{3} \).
  
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### Related Questions:
1. What is the general form of a sinusoidal function?
2. How do we determine the period and amplitude of a trigonometric function from a graph?
3. Can we write this function using cosine instead of sine?
4. What effect does changing the amplitude have on the graph of a trigonometric function?
5. How do phase shifts affect the appearance of a sine or cosine graph?

### Tip:
To identify a trigonometric function from a graph, always check for amplitude, period, vertical shift, and phase shift.

Write an expression defining y as a function of x which best describes this graph.

Discover how to express y as a function of x using sinusoidal functions. This guide provides insights into determining amplitude, period, and phase shift from a graph, exemplified by y(x) = 10 sin(π/3 x). Suitable for high school students in Grades 10-12.

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