Answer:
[tex]y=2.14sin(\frac{ \pi}{6}(x-4))+6.05[/tex]
Step-by-step explanation:
Use a sinusoidal function of the form
[tex]y=Asin(B(x-C))+D[/tex]
A is the amplitude, obtain the difference between the highest and the lowest value, and then divide by two:
[tex]A=\frac{ 8.19-3.91}{2}=\frac{4.28}{2}=2.14[/tex]
D is the vertical shift, is equal to the lowest value plus the amplitude:
[tex]D=3.91+2.14=6.05[/tex]
B is the frequency, being 12 months for each cycle (period T=12) and the relation between the period and the frequency is:
[tex]B=\frac{2\pi}{T}[/tex]
[tex]B=\frac{2\pi}{12}=\frac{\pi}{6}[/tex]
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:
[tex]y=Asin(B(x-C))+D[/tex]
[tex]y=2.14sin(\frac{\pi}{6}(x-4))+6.05[/tex]
For the form [tex]y=Asin(Bx-E)+D[/tex]
multiply C by B to find E ([tex]4*\frac{\pi}{6}=\frac{4\pi}{6} =\frac{2\pi}{3}[/tex]):
[tex]y=2.14sin(\frac{\pi}{6}x-\frac{2\pi}{3})+6.05[/tex]
