The purpose of this study was to explore how individual students understand various aspects of quadratic functions such as quadratic growth, quadratic correspondence, quadratic graphs, vertex points, x-intercepts, y-intercept, line of symmetry, parameters of general quadratic functions, and quadratic equations, in order to provide detailed characterizations of the scope and depth of students' understandings of these concepts. To this end, a qualitative multiple case study methodology was used. Semi-structured, video recorded, in-depth interviews with three university students and one high school student, who either recently completed a formal pre-calculus course or were currently enrolled in a pre-calculus course, constituted the study's primary data source. Students were given a twelve problem task instrument and their problem solving activities were analyzed using cognitive constructivist theories in which the participants' acts of understanding, bases of understanding, and cognitive structures were explicated and modeled. The first case, pseudo named Ken, yielded an understanding of quadratic function as a unique type of equation where one "solves for y." The analysis of the second case, Sarah, led to the emergence of an understanding of quadratic function as a unique type of graph where every value of x has only one y value on the parabola shaped graph. And, three of all four cases suggested a way of understanding quadratic functions as a collection of things that are compartmentalized in multiple ways. In addition, all four cases confirmed some of the major findings in the literature on students' understandings of functions. All four cases were compatible with both the action view of functions and the compartmentalization of function knowledge. They thus added to the existing findings in the literature by providing holistic fabrics of common ways of understanding quadratic functions. These findings emerged through several cross analyses between and among the multiple cases of the study. The design of the study allowed this multiple layers of analyses, while yielding rich descriptions and explanations throughout.
