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Abstract
Present research on the teaching and learning of asymptotes at the high school level is limited to applications to higher level mathematics courses or rational function guides for secondary teachers, leaving a gap in the research on how students perceive asymptotes and productive ways to construct student thinking around asymptotes. In this thesis, I engaged three high school students in semi-structured task-based interviews to give insight into how they conceive and construct understandings of asymptotes, specifically within rational functions, as well as how they develop definitions of approaching using variation in quantities. I discuss the models of each student’s thinking that resulted from analysis of these interviews. Finally, I present suggestions for future research to help fill this gap in research.