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To calculate the parameters for the wave function in the form of \( X(t) = A \sin(\omega t + \varphi) + y_0 \), we need to estimate the following:

1. **Amplitude (A)**: This is half the difference between the maximum and minimum values of \( X(t) \).
2. **Angular frequency (\( \omega \))**: This depends on the period \( T \) of the wave, where \( \omega = \frac{2\pi}{T} \).
3. **Phase shift (\( \varphi \))**: This can be determined by analyzing where the wave crosses the horizontal axis or using known points on the graph.
4. **Vertical shift (\( y_0 \))**: This is the average value of \( X(t) \), which can be approximated by averaging the maximum and minimum values.

### Steps:
1. I'll calculate the minimum, maximum, and average values for \( X(t) \) to estimate the amplitude and vertical shift.
2. I'll calculate the period using the data in the table to estimate \( \omega \).
3. Finally, I'll estimate the phase shift \( \varphi \).

Let's begin with the first step, analyzing the data from the table.

Here are the key values derived from the dataHere are the key values derived from the 

1. **Maximum value of \( X(t) \)**: 31
2. **Minimum value of \( X(t) \)**: 24
3. **Vertical shift (\( y_0 \))**: 27.5 (average of max and min)
4. **Amplitude (\( A \))**: 3.5 (half of the range)
5. **Average period (\( T \))**: 0.265
6. **Angular frequency (\( \omega \))**: 23.71 (calculated as \( \omega = \frac{2\pi}{T} \))

Next, we can estimate the phase shift \( \varphi \) based on the starting point of the wave and its comparison to the sine function. If you'd like to proceed with that or need more details, feel free to ask.

Would you like me to calculate the phase shift \( \varphi \), or provide more details on these calculations?

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Here are 5 relevant follow-up questions:
1. Would you like to calculate the exact phase shift (\( \varphi \)) for the wave?
2. Should we attempt to model this data using a different wave equation (e.g., cosine)?
3. Do you want to verify how well the sine wave fits this data by plotting it?
4. Is there a specific point on the graph you'd like to focus on for analysis?
5. Would you like to see a breakdown of how the period and angular frequency were estimated?

**Tip**: When analyzing waveforms, always verify that the estimated period matches the observed oscillations closely, as small changes can significantly affect the function.

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This solution guides you through calculating parameters like amplitude, angular frequency, and phase shift for wave functions using data. Learn how to fit a sine function to the given time series.

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