a x + b y = ----------- with c not 0 c x + dis 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 = ----------- xThe 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 = ----- xThe 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 + dEach 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).