Dinâmica de Funções Quadráticas: uma abordagem no Ensino Médio

Abstract

In this work, we present a detailed exposition about the dynamics of a family of quadratic functions indexed by a parameter µ. More specifically, our object of study is the family of functions fµ(x) = µx(1 − x) with µ ∈ (1, 3) ∪ (4, +∞). We will cover basic concepts of Dynamic Systems Theory, such as: fixed point, attractor or repulsor, periodic points and orbit of a point. The study will be divided into three cases, according to the variation of parameter µ: (Case l) 1 < µ ≤ 2, (Case ll) 2 < µ < 3 and (Case lll) µ > 4. We will see the influence of the parameter on the behavior of the orbits that remain in the range [0.1]. This will lead us to conclude how rich and complex quadratic functions can be from the perspective of Dynamic Systems. Finally, we bring as a pedagogical proposal a Notebook of Activities, aimed at the student of first year of High School, which aims to approach this family of quadratic functions in an investigative way from the perspective of dynamics.