Definition of the homographic function

The inverse proportionality $y=\tfrac{k}{x}$ is the simplest case of a broader family of functions.

Definition — Homographic function

A homographic function is a function of the form $y = \frac{ax+b}{cx+d}, \qquad c \ne 0,\ ad - bc \ne 0.$ Its graph is a rectangular hyperbola with asymptotes parallel to the Cartesian axes (that is, a “translated rectangular hyperbola”).

The two conditions on the coefficients are essential: if $c=0$ we would have a line, and if $ad-bc=0$ the function would be constant.

Links

Topics: Homographic hyperbola
Concepts: Homographic function · Rectangular hyperbola
Functions: Homographic function · Hyperbola