This article describes an activity for secondary school students that may constitute an appropriate opportunity to discuss with them the idea of proof, particularly in an algebraic context. During the activity the students may experience and understand some of the roles played by proof in mathematics in addition to verification of truth: explanation, discovery, intellectual challenge and self-fulfilment. They may learn that proof is not always a necessary condition for conviction since the work with several examples may provide confidence in the truth of a conjecture. The activity promotes the investigation of several families of quadratic functions and several families of cubic functions. The most important results are discussed and proven in the paper.