Question

find the period and write the equation of the sine function that has a maximum at (pi/14,1) and a minimum at (3pi/14,-1)

Answer 1

To find the period and write the equation of the sine function based on the provided maxima and minima, we can follow these steps:

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

2. **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:
\[
f(x) = \sin(Bx)
\]

3. **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:
\[
\frac{2\pi}{B} = \frac{2\pi}{7} \Rightarrow B = 7
\]

4. **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 \( C = 0 \).

5. **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 \( \frac{2\pi}{7} \).