The framework of linear parameter-varying (LPV) systems has shown to be a powerful tool for the design of controllers for complex nonlinear systems using linear tools. In this work, we derive novel methods that allow us to synthesize LPV state-feedback controllers directly from only a single sequence of data and guarantee stability and performance of the closed-loop system. We show that if the measured open-loop data from the system satisfies a persistency of excitation condition, then the full open-loop and closed-loop input-scheduling-state behavior can be represented using only the data. With this representation, we formulate data-driven analysis and synthesis problems, where the latter yields controllers that guarantee stability and performance in terms of infinite horizon quadratic cost, generalization of the H2-norm, and l2-gain of the closed-loop system. The controllers are synthesized by solving a semi-definite program. Additionally, we provide a synthesis method to handle noisy measurement data. Competitive performance of the proposed data-driven synthesis methods is demonstrated w.r.t. model-based synthesis in multiple simulation studies, including a nonlinear unbalanced disc system.