- Equation of the homographic function

- Simplifying the equation by means of translations

- Back to the general equation

- Highly recommended

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'.

The horizontal asymptote does not change. The equation of the translated graph has the form

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:

- For b" > 0.

The graph of y = b"/x arises when we multiply all the images of previous graph by b". This graph lies in the first and the third quadrant. The asymptotes do not change. - Voor b" < 0.

The graph of y = b"/x arises when we multiply all the images of previous graph by the negative number b". Now, the graph lies in the second and fourth quadrant. The asymptotes do not change.

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).

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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