We propose a class of polynomially parameter-dependent quadratic (PPDQ) Lyapunov functions for assessing the stability of single-parameter-dependent linear, time-invariant, (s-PDLTI) systems in a non-conservative manner. It is shown that the stability of s-PDLTI systems is equivalent to the existence of a PPDQ Lyapunov function. A bound on the degree of the polynomial dependence of the Lyapunov function in the system parameter is given explicitly. The resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex, finite-dimensional optimisation problem. Extensions of the approach to state-feedback controller synthesis are also provided.