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