Question

Find the period and write the equation of the sine function that has a maximum at (π/14,1) and a minimum at (3π/14,-1).
period = ____
f(x)=sin (_____)

Answer 1

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:

1. **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:
\[
P = 2 \times \frac{\pi}{7} = \frac{2\pi}{7}
\]

2. **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 = \(\frac{2\pi}{7}\)
- \(f(x) = \sin(7x)\)