The main purpose of the paper is to study the effect of a quadratic action on some classes of natural exponential families (NEFs) and to use it for deciding on the existence of certain NEFs whose variance functions have the form of Pick functions. Section 2 considers the group \(\text{SL}(2,{\mathbf R})\) of the \(2\times 2\) (invertible) real matrices with determinant 1, and defines a new quadratic action of \(\text{SL}(2,{\mathbf R})\) by the Moebius transforms \(h(x)= (ax+ b)/(cx+d)\) on the NEFs on \({\mathbf R}\). If \(V_F\) denotes the variance function of the NEF \(R\), then a quadratic action on \(F\) is proved to have a variance function of the form \((cm+d)^2V_F(h(m))\). Section 3 is devoted to a brief review of the properties of Pick functions. Pick functions are special analytic functions \(f\) on \(H^+=\{z\mid \text{Im} (z)>0\}\) such that \(f(H^+)\subset H^+\), and represent a basic tool used in the present work. Section 4 provides two classes of NEFs which are closed to the quadratic action of \(\text{SL}(2,{\mathbf R})\) on those NEFs, and whose images of their elements are easily described. Namely, when \(k'\) is the mean function of the NEF \(F\), and \(z\mapsto k'(z)\) and \(z\mapsto ak'(a \log z)\), with \(a\neq 0\), are Pick functions, then the new mean function \(h(k')\) associated to the quadratic action can be explicitly described. Section 5 proves that certain cubic NEFs belong to the NEF classes pointed out in Section 4. This fact entails in Section 6 a classification of the variance functions \(P(m)/m\), where the polynomial \(P\) has degree less than 3 and does not have complex zeroes.