Hyperbolic functions
Functions of the form
Functions of the general form are called hyperbolic functions.
Functions of the form
The effect of
The effect of is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).

For , the graph of is shifted vertically upwards by units.

For , the graph of is shifted vertically downwards by units.
The horizontal asymptote is the line and the vertical asymptote is always the axis, the line .
The effect of
The sign of determines the shape of the graph.

If , the graph of lies in the first and third quadrants.
For , the graph of will be further away from the axes than .
For , as a tends to 0, the graph moves closer to the axes than .

If , the graph of lies in the second and fourth quadrants.
For , the graph of will be further away from the axes than .
For , as a tends to 0, the graph moves closer to the axes than .
Discovering the characteristics
The standard form of a hyperbola is the equation .
Domain and range
For , the function is undefined for . The domain is therefore .
We see that can be rewritten as:
This shows that the function is undefined only at .
Therefore the range is .
Intercepts
The intercept:
Every point on the axis has an coordinate of 0, therefore to calculate the intercept, let .
For example, the intercept of is given by setting :
which is undefined, therefore there is no intercept.
The intercept:
Every point on the axis has a coordinate of 0, therefore to calculate the intercept, let .
For example, the intercept of is given by setting :
This gives the point .
Asymptotes
There are two asymptotes for functions of the form .
The horizontal asymptote is the line and the vertical asymptote is always the axis, the line .
Axes of symmetry
There are two lines about which a hyperbola is symmetrical: and .
Sketching graphs of the form
In order to sketch graphs of functions of the form, , we need to determine four characteristics:

sign of

intercept

intercept

asymptotes
Exercise 1:
Draw the graph of .

Does the point lie on the graph? Give a reason for your answer.

If the value of a point on the drawn graph is 0,25 what is the corresponding value?

What happens to the values as the values become very large?

Give the equations of the asymptotes.

With the line as line of symmetry, what is the point symmetrical to ?
Draw the graph of .

How would the graph compare with that of ? Explain your answer fully.

Draw the graph of on the same set of axes, showing asymptotes, axes of symmetry and the coordinates of one point on the graph.
Todo.
Todo.