The rates of enzyme reactions that are activated or inhibited by added modifiers can in some cases be expressed as a rational function of the first degree, v = (alpha 0 + alpha 1[Q] )/(beta 0 + beta 1 [Q] ) where [Q] is the concentration of the modifier and alpha 0, alpha 1, beta 0, and beta 1 are functions of rate constants and sometimes of the enzyme and substrate concentrations; the behaviour is then said to be linear. Three simple mechanisms that give rise to linear kinetics are examined, and the conditions under which there is activation or inhibition are determined. Sometimes there is a transition from activation to inhibition as the substrate concentration is varied. Definitions of competitive, uncompetitive, and noncompetitive activation are suggested, by analogy with the generally accepted definitions for inhibition. In second-degree activation or inhibition the rate can be expressed as the ratio of two quadratic polynomials with positive coefficients. Ten patterns are then possible for plots of v against [Q], and they may be classified with respect to (i) overall activation or inhibition, (ii) initial (at [Q] leads to 0) activation or inhibition, (iii) terminal (at [Q] leads to oo) activation or inhibition, and (iv) whether there is an initial inflexion. The general case of an n:n rational function is also discussed.