To find the period of the sine function based on the given maximum and minimum points, let's first identify the properties of the sine function.
The standard sine function, \( f(x) = A \sin(B(x - C)) + D \), has:
- A vertical shift determined by \( D \),
- Amplitude given by \( A \),
- A horizontal shift determined by \( C \),
- A period given by \( \frac{2\pi}{B} \).
Step 1: Determine the period.
You provided a maximum at \( \left(\frac{\pi}{14}, 1\right) \) and a minimum at \( \left(\frac{3\pi}{14}, -1\right) \).
- The distance between the maximum and minimum points is:
\[ \text{Distance} = \frac{3\pi}{14} - \frac{\pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7} \]
In a sine function, the distance from a maximum to the next minimum is a quarter of the period (\( \frac{P}{4} \)), thus, we can setup the following equation for the period \( P \):
\[ \frac{P}{4} = \frac{\pi}{7} \]
Step 2: Solve for the period \( P \):
\[ P = 4 \cdot \frac{\pi}{7} = \frac{4\pi}{7} \]
Step 3: Write the equation of the sine function.
The maximum value is 1, which suggests the amplitude \( A = 1 \) (since sine ranges from -1 to 1), and the vertical shift \( D = 0 \). Since we need the function to have a maximum at \( x = \frac{\pi}{14} \), we need to adjust for the horizontal shift.
Considering a standard sine function has its first maximum at \( x = \frac{\pi}{2} \), we would typically need to shift it to the left to have it peak earlier.
Using the formula for the function, we have:
\[ f(x) = A \sin\left(B(x - C)\right) + D \]
Substituting our values \( A = 1 \), \( D = 0 \), \( P = \frac{4\pi}{7} \) gives us \( B \):
\[ B = \frac{2\pi}{P} = \frac{2\pi}{\frac{4\pi}{7}} = \frac{7}{2} \]
We need to find \( C \). Since the maximum occurs at \( x = \frac{\pi}{14} \), we can express \( C \) as follows:
\[ \text{Set } B\left(x - C\right) = \frac{\pi}{2} \text{ when } x = \frac{\pi}{14} \]
Thus,
\[ \frac{7}{2}\left(\frac{\pi}{14} - C\right) = \frac{\pi}{2} \]
Solving for \( C \):
\[ \frac{\pi}{14} - C = \frac{\pi}{7} \cdot \frac{2}{7} = \frac{\pi}{7} \cdot \frac{1}{4} = \frac{\pi}{2} \cdot \frac{1}{4} = \frac{\pi}{28} \] \[ C = \frac{\pi}{14} - \frac{\pi}{28} = \frac{2\pi}{28} - \frac{\pi}{28} = \frac{\pi}{28} \]
So, plugging it all into our equation, we get:
\[ f(x) = \sin\left(\frac{7}{2}\left(x - \frac{\pi}{28}\right)\right) \]
In conclusion, the function can be written as:
\[ f(x) = \sin\left(\frac{7}{2}x - \frac{7\pi}{56}\right) \]
This final equation represents the sine function with the specified maximum and minimum values.