Quadratic Function: a specific case of the Parabola as a conic section

Abstract

This article presents a constructivist approach for the graph of the quadratic function and its properties, suitable for students in their first contact with the subject. Different from traditional approaches, a sequence inspired by Analytic Geometry is proposed, starting from the analysis of the function $g(x)=ax^2$ and using horizontal and vertical translations to construct the canonical form geometrically, motivating its algebraic deduction. The formalization that the graph of the quadratic function is a parabola was done in a simple way; to solidify the concept, two activities were proposed in \textit{GeoGebra}, the first for visualizing the geometric definition and the second with the objective to verify, in a specific case, that the function graph coincides with the geometric locus (parabola). The main objective was the conceptual enrichment for the student, contemplating from intuition to rigorous formalization, focusing on clarifying and justifying the properties of the quadratic function graph, contributing to more meaningful learning.