The typical path of instruction in high school algebra courses for the study of polynomial functions has been from linear functions, to quadratic functions, to polynomial functions of degree greater than two. This paper reports results of clinical interviews with an Algebra II student. The interviews were used to probe into the student's conceptual understanding of polynomial functions and the connections that the student made graphically and algebraically between the classes of polynomial functions. Results suggest that building polynomials from linear expressions may foster connections between the classes of polynomial functions and between the graphical and algebraic representations of these functions. (MKR)