You probably made a mistake with the calculation of the period. The amplitude of this graph and $\displaystyle{y}={4}{\cos{{\left(\frac{{1}}{{2}}{x}\right)}}}\usepackage{cmbright}\Huge\$ is 4. However, this graph has 4 cycles in the interval $\displaystyle{\left[{0},{4}\pi\right]}\usepackage{cmbright}\Huge\$ , so it has a period of $\displaystyle\pi\usepackage{cmbright}\Huge\$ .

But $\displaystyle{y}={4}{\cos{{\left(\frac{{1}}{{2}}{x}\right)}}}\usepackage{cmbright}\Huge\$ has a period of $\displaystyle\frac{{{2}\pi}}{{\frac{{1}}{{2}}}}=\frac{{{2}\pi}}{{1}}\cdot\frac{{2}}{{1}}={4}\pi\usepackage{cmbright}\Huge\$ .