The study investigated learners' mathematical discourse on the hyperbola from a commognitive perspective, and focused on algebraic, graphic, and numeric representations of the hyperbola. Task-based interviews were conducted with five Grade 10 learners from a township school. Learners' mathematical discourse was analysed by means of the Discourse Profile of the Hyperbola, which was adapted from the Arithmetic Discourse Profile of Ben-Yahuda et al. We found that, while learners' graphs were mainly mathematically correct, their word use was colloquial and their interpretation of the mediators and their narratives were generally visual. Taken together, this suggests that their routines were ritualised. Learners' tables of values did not show asymptotic behaviour yet they drew graphs showing two asymptotes. However, four learners tended to talk as if there were only one asymptote and they had difficulty in seeing the axes as asymptotes.