We are given a general cosine function, that is:
[tex]y=A\cos (B(x-C))+D[/tex]Where:
[tex]A=\text{ amplitude}[/tex]The amplitude "A" is half the distance between the maximum and minimum values of the function, in this case the maximum value is 3.7 and the minimum value is 0.3, therefore, the amplitude is:
[tex]A=\frac{3.7-0.3}{2}=1.7[/tex]Therefore, A = 1.7
[tex]B=frequency[/tex]The value "B" is the number of cycles it takes the function to go from 0 to 2pi
The number B is usually expressed as:
[tex]T=\frac{2\pi}{B}[/tex]"T" is called the period and is how much it takes the curve to do one cycle, in this case the period is:
[tex]T=3.9-2.3=1.6[/tex]Therefore, replacing we get:
[tex]\begin{gathered} 1.6=\frac{2\pi}{B} \\ B=\frac{2\pi}{1.6} \end{gathered}[/tex]The number C is called the phase and refers to how much is the function is shifted horizontally, in this case, the function is shifted by 2.3 in the positive direction, therefore C = 2.3.
The number D is how much the graph is shifted vertically, in this case, it is shifted by 0.3 plus the amplitude, that is:
[tex]D=0.3+1.7=2[/tex]Replacing the values we get:
[tex]y=1.7\cos (\frac{2\pi}{1.6}(x-2.3))+2[/tex]