The polynomial division trick

There is a quick way to recognise the structure of a homographic function without applying the formulae from memory: carry out the division between the two polynomials.

Remark — Trick: polynomial division

To grasp quickly the structure of the homographic function one can do the (polynomial) division of $ax+b$ by $cx+d$. In the previous example: $\frac{2x+3}{x-1} = 2 + \frac{5}{x-1}.$ In this form the horizontal asymptote $y=2$ and the vertical one $x=1$ are read off directly, and one recognises that the curve is the rectangular hyperbola $y = \tfrac{5}{t}$ translated by $(+1;+2)$.

Writing the homographic function as “constant $+$ translated inverse proportionality” makes both sketching the graph and the connection with the already-known function $y=\tfrac{k}{x}$ immediate.

Links

Topics: Homographic hyperbola
Concepts: Polynomial division · Homographic function · Inverse proportionality · Translation
Functions: Homographic function
Skills: Using formulae