a x + b
y = ----------- with c not 0
c x + d
is called a homographic function.
Since c is different from zero, we can divide the numerator and the denominator of the functional rule by c. Then the equation has the form
a' x + b'
y = -----------
x + d'
We calculate the asymptotes of this function and we find a horizontal asymptote y = a'
and a vertical asymptote x = -d'.
a x + b"
y = -----------
x
The horizontal asymptote is still y = a.
Now we translate the last graph a units downwards. Then the horizontal asymptote is on de x-axis. The new equation of the function is
b"
y = -----
x
The shape of the curve has not changed during the translations. De graph of the last equation is very simple. For b = 1, the graph looks like this:
a x + b
y = ----------- with c not 0
c x + d
Each graph of a homographic function is called an orthogonal hyperbola
with a horizontal and vertical asymptote.
Each bisector of the asymptotes is an axis of symmetry of the hyperbola and the intersection point of the asymptotes is a point of symmetry.
The asymptotes of this general equation are y = a/c and x= -d/c. The symmetry point is S(-d/c,a/c).