Question

Given the information:
Amplitude = 12
Period = 6/7
C = no phase shift
Midline = 12

Write a sine function

Answer 1

To write a sine function based on the provided information, we can use the general form of the sine function:

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

Where:
- \(A\) is the amplitude,
- \(B\) is related to the period,
- \(C\) is the horizontal phase shift,
- \(D\) is the vertical shift (midline).

Given your parameters:
- Amplitude (\(A\)) = 12
- Period = \(\frac{6}{7}\)
- Midline (\(D\)) = 12
- No phase shift (\(C\) = 0)

**Step 1: Calculate \(B\)**

The period of a sine function is given by the formula:

\[
\text{Period} = \frac{2\pi}{B}
\]

Given the period = \(\frac{6}{7}\), we can solve for \(B\):

\[
\frac{6}{7} = \frac{2\pi}{B}
\]

Cross-multiplying gives:

\[
6B = 14\pi
\]

Now, solving for \(B\):

\[
B = \frac{14\pi}{6} = \frac{7\pi}{3}
\]

**Step 2: Write the final function**

Now we can substitute \(A\), \(B\), \(C\), and \(D\) into the sine function formula. Since there's no phase shift (\(C = 0\)), our function simplifies to:

\[
f(x) = 12 \sin\left(\frac{7\pi}{3} x\right) + 12
\]

Thus, the sine function that meets the given specifications is:

\[
\boxed{f(x) = 12 \sin\left(\frac{7\pi}{3} x\right) + 12}
\]