We consider a one-dimensional p-wave superconducting quantum wire with the modulated chemical potential, which is described by $\hat{H}= \displaystyle\sum\nolimits_{i}\left[ \left( -t\hat{c}_{i}^{\dagger }\hat{c}_{i+1}+\Delta \hat{c}_{i}\hat{c}_{i+1}+ h.c.\right) +V_{i}\hat{n}_{i}\right]$, $V_{i}=V\dfrac{\cos \left( 2{\text{π}} i\alpha + \delta \right) }{1-b\cos \left( 2{\text{π}} i\alpha+\delta \right) }$ and can be solved by the Bogoliubov-de Gennes method. When $b=0$, $\alpha$ is a rational number, the system undergoes a transition from topologically nontrivial phase to topologically trivial phase which is accompanied by the disappearance of the Majorana fermions and the changing of the $Z_2$ topological invariant of the bulk system. We find the phase transition strongly depends on the strength of potential V and the phase shift $\delta$. For some certain special parameters $\alpha$ and $\delta$, the critical strength of the phase transition is infinity. For the incommensurate case, i.e. $\alpha=(\sqrt{5}-1)/2$, the phase diagram is identified by analyzing the low-energy spectrum, the amplitudes of the lowest excitation states, the $Z_2$ topological invariant and the inverse participation ratio (IPR) which characterizes the localization of the wave functions. Three phases emerge in such case for $\delta=0$, topologically nontrivial superconductor, topologically trivial superconductor and topologically trivial Anderson insulator. For a topologically nontrivial superconductor, it displays zero-energy Majorana fermions with a $Z_2$ topological invariant. By calculating the IPR, we find the lowest excitation states of the topologically trivial superconductor and topologically trivial Anderson insulator show different scaling features. For a topologically trivial superconductor, the IPR of the lowest excitation state tends to zero with the increase of the size, while it keeps a finite value for different sizes in the trivial Anderson localization phase.