Since the function starts at it maximum value, let's use a cosine function to represent it:
[tex]f(x)=A+B\cos(C(x+D))[/tex]
Since the midline of the periodic function is y = 2, we have A = 2.
The period of the function is 4pi/3, so we have:
[tex]\begin{gathered} T=\frac{2\pi}{C}\\ \\ \frac{4\pi}{3}=\frac{2\pi}{C}\\ \\ \frac{2}{3}=\frac{1}{C}\\ \\ C=\frac{3}{2} \end{gathered}[/tex]
Since the function already starts at its maximum value, there is no horizontal phase shift, so D = 0.
The amplitude is 1 (it goes up and down 1 unit from the midline), so we have B = 1.
Therefore the function is:
[tex]f(x)=2+\cos(\frac{3}{2}x)[/tex]
