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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 re-written as:

(4)

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 :

(5)

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 :

(6)

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:

  1. sign of

  2. -intercept

  3. -intercept

  4. asymptotes

Exercise 1:

Draw the graph of .

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

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

  3. What happens to the -values as the -values become very large?

  4. Give the equations of the asymptotes.

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

Draw the graph of .

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

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

To-do.

To-do.

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