The FM-representation for the quadratic function Part 1

Abstract

Given 𝑏₀ ∈ R and 𝑐₀ ∈ R, let the families of functions 𝐹_a : R → R be given by 𝐹_a (𝑥) = 𝑎𝑥² +  𝑏₀x+ c₀ , 𝑎 ∈ R^*, and
𝑓_a : C → C be given by 𝑓_a (𝑧) = 𝑎𝑧² + 𝑏₀z+ c₀ , 𝑎 ∈ R^*. When we vary 𝑎, we obtain the F_aM-representation, which consists of the Cartesian plane in which the graph of 𝐹_a evolves and, right below, of the complex plane in which the paths of the 2 roots of 𝑓_a are traced. The importance of the F_aM-representation lies in the simultaneous analysis of what happens to 𝐹_a with what happens to the 2 roots of 𝑓_a. We show that the sequences of functions 𝐹_n ^{+ -1} : [0,+∞[→ [0,+∞[ given by 𝐹_n ^{+ -1}(𝑊) = (𝑊/𝑛)^1/2 and 𝐹_n ^{- -1} : [0,+∞[→] − ∞, 0] given by 𝐹_n ^{- -1}(𝑊) = - (𝑊/𝑛)^1/2, 𝑛 ∈ N = {1, 2, 3, ...}, converge pointwise, but not uniformly, to 0; while 𝐹_n ^{+ -1}|_[0,M]  and 𝐹_n ^{- -1}|_[0,M] converge uniformly to 0, for fixed 𝑀 > 0. For each 𝑎 ∈ R^*, we determine the curvature of 𝜆_a : R → R² given by 𝜆_a(𝑥) = (𝑥, a𝑥²) at a fixed 𝑥₀, and give examples.