Asymptotes and centre of symmetry

To draw a homographic function only the two asymptotes and their meeting point are needed, all readable directly from the coefficients.

Property — Asymptotes of the homographic function

The function $y = \tfrac{ax+b}{cx+d}$ has:

• Vertical asymptote: $x = -\tfrac{d}{c}$ (the value of $x$ that makes the denominator vanish);
• Horizontal asymptote: $y = \tfrac{a}{c}$ (the limit of $y$ as $|x|\to\infty$);
• Centre of symmetry: $O'\left(-\tfrac{d}{c};\ \tfrac{a}{c}\right)$, the intersection of the two asymptotes.

The conditions $c\ne 0$ (otherwise it is a line) and $ad - bc \ne 0$ (otherwise the function is constant, because numerator and denominator would be proportional) are precisely what guarantees that the graph is indeed a hyperbola.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Horizontal asymptote · Vertical asymptote · Centre of symmetry · Homographic function
Functions: Homographic function
Methods: Canonical hyperbola
Skills: Analytic geometry · Sketching the graph