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what is the equation for the following sine function

To determine the equation of the sine function from the graph, we need to identify several key characteristics:

1. **Amplitude (A)**: The maximum displacement from the center line (y = 0). Here, the maximum y-value is 2, and the minimum y-value is -2. Therefore, the amplitude \(A\) is 2.

2. **Period (T)**: The length of one complete cycle. From the graph, one full cycle occurs from \(x = 0\) to \(x = 180\). Hence, the period \(T\) is 180.

3. **Frequency (f)**: The number of cycles per unit of time, which is the reciprocal of the period. Here, \( f = \frac{1}{T} = \frac{1}{180} \).

4. **Horizontal shift (phase shift)**: There appears to be no horizontal shift since the graph starts at the origin (0, 0).

5. **Vertical shift (D)**: There appears to be no vertical shift since the graph oscillates around the x-axis (y = 0).

Given these observations, the general form of the sine function is:
\[ y = A \sin(B(x - C)) + D \]

Here:
- \( A = 2 \)
- \( B = \frac{2\pi}{T} = \frac{2\pi}{180} = \frac{\pi}{90} \)
- \( C = 0 \) (no phase shift)
- \( D = 0 \) (no vertical shift)

So, the equation of the sine function is:
\[ y = 2 \sin\left(\frac{\pi}{90} x\right) \]

Would you like more details or have any questions?

Here are some additional questions you can ask:
1. How do you determine the amplitude of a sine function?
2. What is the general form of a trigonometric function?
3. How does the period of a function relate to its frequency?
4. What is the significance of phase shifts in trigonometric functions?
5. How can you identify vertical shifts in sine and cosine graphs?

**Tip:** When writing the equation of a sine or cosine function, always check the graph for amplitude, period, phase shift, and vertical shift.

Learn how to determine the equation of a sine function from its graph. Understand key concepts such as amplitude, period, frequency, and phase shift. Suitable for high school students studying trigonometry.

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