This paper deals with the frequency domain properties of an ellipsoidal family of rational functions, i.e. a family of rational functions whose coefficients depend affinely on an ellipsoidal parameter set. The considered problems are relevant to several recently developed techniques in the identification-for-control research area. A complete characterization of the frequency plots of such a family is provided and an efficient algorithm for computing the envelope of the Bode plots is devised. In particular, it is shown that the extremal values of the magnitude and phase of the family frequency response, which in general involve non-convex optimization problems, can be computed via a sequence of simple algebraic tests.