Shape phase transitions in odd-A nuclei are investigated within the framework of the interacting boson-fermion model. The case of a single j = 9/2 fermion coupled to an even-even boson core is considered. This boson core transits from spherical to gamma-unstable shapes depending on the value of a control parameter in the boson Hamiltonian. The effect of the coupling of the odd particle to this core along the shape transition and, in particular, at the critical point is discussed. For that purpose, the ground-state energy surface in terms of the beta and gamma shape variables for the even core and odd-even energy surfaces for the different K states coming from j = 9/2 are constructed. The evolution of each individual coupled state along the transition from the spherical [U(5)] to the gamma-unstable [O(6)] situation is investigated. One finds that the core-fermion coupling gives rise to a smoother transition than in the even-core case.