## Asmptotes of Hyperbola From Algebraic Fraction Form

Given a hyperbola written as an algebraic fraction,
$y=\frac{ax+b}{cx+d}$
, we can find the equations of the asymptotes using long division and rearranging into the form
$(x-x_0)(y-y_0=K$
. The asymptotes are then
$x=x_0, \; y=y_0$
. If
$K \gt 0$
the hyperbola is in the 1st and 3rd quadrant and if
$K \lt 0$
the hyperbola is in the 2nd and 4th quadrant. Example:
$y=\frac{3x-2}{x+1}$
.
Long division gives
$y=3- \frac{5}{x+1}$
.
Subtract 3 to give
$y-3=- \frac{5}{x+1}$
.
Multiply by
$(x+1)$
to give
$(x+1)(y-3)=-5$
.
The asymptotes are
$x=-1, \; y=3$
and the hyperbola is in the 2nd and 4th quadrant.