To find the period of the sine function that has a maximum at \((\frac{\pi}{14}, 1)\) and a minimum at \((\frac{3\pi}{14}, -1)\), we can start by recognizing the properties of the sine wave.
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Finding the Period:
- The maximum occurs at \(\frac{\pi}{14}\).
- The minimum occurs at \(\frac{3\pi}{14}\).
The distance between the maximum and the minimum is: \[ \frac{3\pi}{14} - \frac{\pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7}. \]
From a maximum to a minimum is \(\frac{1}{2}\) of a full cycle (since the sine function goes from maximum to minimum). Therefore, the full period \(P\) is: \[ P = 2 \times \frac{\pi}{7} = \frac{2\pi}{7}. \]
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Writing the Equation: The general form of a sine function is: \[ y = A \sin(B(x - C)) + D, \] where:
- \(A\) is the amplitude (1 in this case, as the max is 1 and min is -1),
- \(B\) is related to the period by \(B = \frac{2\pi}{P}\),
- \(C\) is the phase shift,
- \(D\) is the vertical shift.
Given \(P = \frac{2\pi}{7}\): \[ B = \frac{2\pi}{P} = \frac{2\pi}{\frac{2\pi}{7}} = 7. \]
Since the sine function has a maximum at \((\frac{\pi}{14}, 1)\), we consider a cosine function starting at maximum for simplicity (as \(\cos(x)\) starts at its maximum). A sine function can be converted by phase shifting: \[ y = A \cos(B(x - C)) + D. \] Here,
- \(A = 1\) (the amplitude),
- \(B = 7\),
- There is no vertical shift (\(D = 0\)),
- The maximum at \(\frac{\pi}{14}\) indicates \(C = \frac{\pi}{14}\).
Thus, the equation becomes: \[ y = \cos\left(7x - \frac{\pi}{2}\right). \]
We can also convert back to sine if needed: \[ y = \sin\left(7x - \frac{3\pi}{2}\right) \quad \text{(equivalent to a cosine function due to phase shift)}. \]
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Final Answer: The equation of the sine function is: \[ y = \sin\left(7x - \frac{3\pi}{2}\right), \] or, equivalently, \[ y = \cos(7x). \] The period of the function is \(\frac{2\pi}{7}\).