To find the period and equation of the sine function based on the given information about its maximum and minimum points, we can follow these steps:
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Identifying the period: The sine function has a maximum at \( ( \frac{\pi}{14}, 1 ) \) and a minimum at \( ( \frac{3\pi}{14}, -1 ) \). The distance between the maximum and minimum points is: \[ x_{min} - x_{max} = \frac{3\pi}{14} - \frac{\pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7} \] The distance between a maximum point and the next maximum point is half the period. Thus, the full period \( P \) can be calculated as follows: \[ P = 2 \times \frac{\pi}{7} = \frac{2\pi}{7} \]
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Writing the function: The general form of a sine function is: \[ f(x) = A \sin(B(x - C)) + D \]
- Amplitude \( A = 1 \) (since it reaches a maximum of 1 and minimum of -1).
- Midline \( D = 0 \) (the average of the maximum and minimum, \( \frac{1 + (-1)}{2} = 0 \)).
- Period \( P = \frac{2\pi}{B} \). Since we found \( P = \frac{2\pi}{7} \), we can solve for \( B \): \[ B = \frac{2\pi}{P} = \frac{2\pi}{\frac{2\pi}{7}} = 7 \]
- The sine function reaches its maximum at \( \frac{\pi}{14} \), which means we need a phase shift \( C \). The general sine function reaches a maximum at \( \frac{\pi}{2} \), so we can set: \[ B \left( x - C \right) = \frac{\pi}{2} \quad \text{when} \quad x = \frac{\pi}{14} \] Substitute \( B = 7 \): \[ 7 \left( \frac{\pi}{14} - C \right) = \frac{\pi}{2} \] Solve for \( C \): \[ \frac{7\pi}{14} - 7C = \frac{\pi}{2} \ \frac{\pi}{2} - 7C = \frac{\pi}{2} \ 7C = 0 \ C = 0 \] Thus, no phase shift is needed.
Putting it all together, we have: \[ f(x) = \sin(7x) \]
Finally, the answers are:
- Period: \(\frac{2\pi}{7}\)
- Function: \(f(x) = \sin(7x)\)
So to summarize:
- period = \(\frac{2\pi}{7}\)
- \(f(x) = \sin(7x)\)