Example — studying a homographic function

Let us put the rules into practice by fully studying a homographic function.

Example — Study $y = \dfrac{2x+3}{x-1}$

Vertical asymptote: $x - 1 = 0 \iff x = 1$.

Horizontal asymptote: for $|x|\to\infty$, $y\to \tfrac{2x}{x} = 2$, so $y = 2$.

Centre of symmetry: $O'(1;2)$.

Intersections with the axes:

• with $Oy$ ($x=0$): $y = \tfrac{3}{-1} = -3$. Point $(0;-3)$.
• with $Ox$ ($y=0$): $\tfrac{2x+3}{x-1}=0 \iff 2x+3=0 \iff x = -\tfrac{3}{2}$. Point $\left(-\tfrac{3}{2};0\right)$.

Reading: the graph of $y=\tfrac{2x+3}{x-1}$ is a hyperbola with vertical asymptote $x=1$ (where the left branch and the right branch separate), horizontal asymptote $y=2$ and centre of symmetry $O'(1;2)$.

Links

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