To find the period and write the equation of the sine function given the maximum and minimum points, we can follow these steps:
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Identify the points: The maximum is at \((\frac{\pi}{14}, 1)\) and the minimum is at \((\frac{3\pi}{14}, -1)\).
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Determine the period:
- The distance between a maximum and a minimum for a sine function is half the period.
- The distance between the maximum at \(\frac{\pi}{14}\) and the minimum at \(\frac{3\pi}{14}\) is: \[ \frac{3\pi}{14} - \frac{\pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7} \]
- Therefore, the full period \(P\) is twice this distance: \[ P = 2 \times \frac{\pi}{7} = \frac{2\pi}{7} \]
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Write the function:
- A sine function can be written as: \[ f(x) = A \sin(B(x - C)) + D \]
- Here, \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical shift.
- Since the maximum value is 1 and the minimum is -1, the amplitude \(A = 1\) and the vertical shift \(D = 0\). Thus, we can simplify: \[ f(x) = \sin(Bx) \]
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Determine \(B\):
- The relationship between the period \(P\) and \(B\) is: \[ P = \frac{2\pi}{B} \]
- Setting this equal to our calculated period: \[ \frac{2\pi}{B} = \frac{2\pi}{7} \]
- Solving for \(B\): \[ B = 7 \]
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Determine the phase shift:
- Since the maximum occurs at \(\frac{\pi}{14}\), we express the sine function in such a way that it peaks (maximum) at this point: \[ f(x) = \sin(7x - \frac{7\pi}{14}) = \sin(7x - \frac{\pi}{2}) \]
Taking all of this into account, our final results are:
- Period: \(\frac{2\pi}{7}\)
- Equation: \[ f(x) = \sin\left(7x - \frac{\pi}{2}\right) \]
Note: Alternatively, we can write it as \(f(x) = \cos(7x)\) since sine shifted by \(-\frac{\pi}{2}\) is a cosine function.
Final answers:
- Period: \(\frac{2\pi}{7}\)
- Function: \(f(x) = \cos(7x)\) or \(f(x) = \sin(7x - \frac{\pi}{2})\)