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). (1 point) perio d=□ f(x)=sin (□ )

Answer 1

To find the period and the equation of the sine function given the maximum and minimum points, we start with the provided points: a maximum at \(\left(\frac{\pi}{14}, 1\right)\) and a minimum at \(\left(\frac{3\pi}{14}, -1\right)\).

### Step 1: Determine the period
The distance between the maximum and the minimum points is half of the period of the sine function. The distance between these two points is:

\[
\text{Distance} = \left(3\frac{\pi}{14} - \frac{\pi}{14}\right) = \frac{2\pi}{14} = \frac{\pi}{7}
\]

Since this distance represents half of the period, we multiply by 2 to find the full period \(P\):

\[
P = 2 \cdot \frac{\pi}{7} = \frac{2\pi}{7}
\]

### Step 2: Write the equation of the sine function
The general form of a sine function is given by:

\[
f(x) = A \sin(B(x - C)) + D
\]

where:
- \(A\) is the amplitude,
- \(B\) determines the period \(P\) (related by \(P = \frac{2\pi}{B}\)),
- \(C\) is the horizontal shift,
- \(D\) is the vertical shift.

### Step 3: Determine the amplitude and vertical shift
From the provided maximum and minimum points:
- The maximum value is \(1\) and the minimum value is \(-1\), which means the amplitude \(A = 1\).
- The vertical shift \(D\) (the midline) is given by:

\[
D = \frac{1 + (-1)}{2} = 0
\]

### Step 4: Determine \(B\)
We have found the period \(P\):

\[
P = \frac{2\pi}{B}
\]

Setting the known period equal to \(\frac{2\pi}{7}\):

\[
\frac{2\pi}{B} = \frac{2\pi}{7}
\]

This gives:

\[
B = 7
\]

### Step 5: Determine the phase shift \(C\)
Since there is a maximum at \(x = \frac{\pi}{14}\), we can find \(C\). The sine function has a maximum at \( \frac{\pi}{2} \) plus any multiple of the period. The sine function has a maximum at the point \( \frac{\pi}{2} \) when \(B = 1\). Since we have \(B = 7\), we need to adjust for that.

To relate the maximum to our sine function's phase shift:

\[
B\left(x - C\right) = \frac{\pi}{2}
\]

Plugging in the maximum point \(x = \frac{\pi}{14}\):

\[
7\left(\frac{\pi}{14} - C\right) = \frac{\pi}{2}
\]

### Solving for \(C\):

We simplify:

\[
\frac{7\pi}{14} - 7C = \frac{\pi}{2}
\]
\[
\frac{\pi}{2} - 7C = \frac{\pi}{2}
\]
\[
7C = 0 \Rightarrow C = 0
\]

### Final equation
Now we can write the function:

\[
f(x) = \sin(7x)
\]

Thus, the results are:
- Period: \( \frac{2\pi}{7} \)
- Function: \( f(x) = \sin(7x) \)

To summarize:

\[
\text{period} = \frac{2\pi}{7}, \quad f(x) = \sin(7x)
\]