Answer
$y=3\cos \left(\frac{1}{2}x-\frac{\pi}{4}\right)+2$
Work Step by Step
The cosine function is:
$$y=a\cos (bx-c)+d$$
Rewriting the equation:
$$y=a\cos b\left(x-\frac{c}{b}\right)+d$$
Finding $b$:
$$period=\frac{2\pi}{b}$$ $$b=\frac{2\pi}{period}=\frac{2\pi}{4\pi}=\frac{1}{2}$$
With right phase shift of $\frac{\pi}{2}$ or phase shift of positive $\frac{\pi}{2}$:
$$\frac{c}{b}=\frac{\pi}{2}$$ $$c=\frac{\pi}{2}b=\frac{\pi}{2}\left(\frac{1}{2}\right)=\frac{\pi}{4}$$
Substituting $a=3,~b=\frac{1}{2},~c=\frac{\pi}{4}$ and $d=2$, the function is:
$$y=3\cos \left(\frac{1}{2}x-\frac{\pi}{4}\right)+2$$
