The homographic function

Equation of the homographic function

Each function, different from 0, with an equation of the form
     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'.

Simplifying the equation by means of translations

The vertical asymptote is on the y-axis, if we translate the graph of this last function d' units to the right.
The horizontal asymptote does not change. The equation of the translated graph has the form
     a x + b"
y = -----------
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

y = -----
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:

Back to the general equation

     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).

Highly recommended

Choose random values for a,b,c,d.
Plot the graph with a plotting tool.
Check if the results of the previous theory are met.
Repeat this procedure a few times.

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