Answer:
\(y=2.14sin(\frac{ \pi}{6}(x-4))+6.05\)
Step-by-step explanation:
Use a sinusoidal function of the form
\(y=Asin(B(x-C))+D\)
A is the amplitude, obtain the difference between the highest and the lowest value, and then divide by two:
\(A=\frac{ 8.19-3.91}{2}=\frac{4.28}{2}=2.14\)
D is the vertical shift, is equal to the lowest value plus the amplitude:
\(D=3.91+2.14=6.05\)
B is the frequency, being 12 months for each cycle (period T=12) and the relation between the period and the frequency is:
\(B=\frac{2\pi}{T}\)
\(B=\frac{2\pi}{12}=\frac{\pi}{6}\)
Finally, for C the phase shift, the highest value in a sine function the highest point is in the first quartes of the period if the period is 12, then this maximum is in x=3. In the data the highest is in x=7, so the phase shift is to the right and equal in value to 4 (C=4).
So:
\(y=Asin(B(x-C))+D\)
\(y=2.14sin(\frac{\pi}{6}(x-4))+6.05\)
For the form \(y=Asin(Bx-E)+D\)
multiply C by B to find E (\(4*\frac{\pi}{6}=\frac{4\pi}{6} =\frac{2\pi}{3}\)):
\(y=2.14sin(\frac{\pi}{6}x-\frac{2\pi}{3})+6.05\)
