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This article focuses on first-year university students’ understanding of concepts related to third-degree polynomial and trigonometric functions that they encountered during their study of Grade 12 mathematics. In this study three online questions from two firstyear mathematics quizzes at the University of KwaZulu-Natal were analysed. The first question focused on the characteristics of a polynomial function, while the second and third questions focused on the characteristics of trigonometric functions. The characteristics included calculus-related concepts, for example, intervals of increase or decrease, concavity and local extrema. It was found that about a fifth of the participants (n = 557), science students who studied the core mathematics module Introduction to Calculus, had difficulties in answering those calculus-related questions compared to determining the general non-calculus characteristics, for example, the range and domain. The study also found that about a quarter of the participants lacked relational understanding with regard to calculus-related concepts. This study recommends that lecturers need to spend more time on calculus-related concepts of a function that focus on relational understanding.