To find the period and write the equation of the sine function based on the provided maxima and minima, we can follow these steps:
-
Identify the Period:
- The maximum occurs at \( x = \frac{\pi}{14} \) and the minimum occurs at \( x = \frac{3\pi}{14} \).
- The distance between the maximum and minimum is: \[ \frac{3\pi}{14} - \frac{\pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7} \]
- This distance represents half of the period of the sine function (since a sine wave goes from maximum to minimum in half a cycle).
- Therefore, the full period \( P \) is: \[ P = 2 \times \frac{\pi}{7} = \frac{2\pi}{7} \]
-
Formulate the Function:
- The general sine function can be expressed as: \[ f(x) = A \sin(B(x - C)) + D \]
- From the information provided, we know that the amplitude \( A = 1 \) (since the maximum is 1 and the minimum is -1) and the vertical shift \( D = 0 \).
- Thus, our function simplifies to: \[ f(x) = \sin(Bx) \]
-
Determine B:
- The relationship between the period \( P \) and \( B \) is given by: \[ P = \frac{2\pi}{B} \]
- We previously found that \( P = \frac{2\pi}{7} \), so we can set up the equation: \[ \frac{2\pi}{B} = \frac{2\pi}{7} \Rightarrow B = 7 \]
-
Determine the Phase Shift \( C \):
- Since the maximum occurs at \( x = \frac{\pi}{14} \), we can set up the phase shift. The sine function normally has a maximum at \( \frac{\pi}{2} \), so we need to shift it: \[ Bx - C = \frac{\pi}{2} ,, \text{when} ,, x = \frac{\pi}{14} \]
- Substitute \( B = 7 \) and \( x = \frac{\pi}{14} \): \[ 7\left(\frac{\pi}{14}\right) - C = \frac{\pi}{2} \] \[ \frac{\pi}{2} - C = \frac{\pi}{2} \]
- This gives us \( C = 0 \).
-
Write the Final Equation:
- Substituting these values into the sine function: \[ f(x) = \sin(7x) \]
Thus, the equation of the sine function that has a maximum at \( \left(\frac{\pi}{14}, 1\right) \) and a minimum at \( \left(\frac{3\pi}{14}, -1\right) \) is:
\[ f(x) = \sin(7x) \]
The period of this function is \( \frac{2\pi}{7} \).