Hyperbole: definition, function, formula, construction examples

In this publication, we will consider what a hyperbola is, give a formula by which its function is set, and use practical examples to analyze the algorithm for constructing this type of graph.

Definition and function of a hyperbola

Hyperbola is a graph of the inverse proportionality function, which is generally given by the following formula:

Here:

• x is an independent variable;
• k ≠ 0;
• with k > 0 the hyperbola is located in the I and III quarters of the coordinate plane;
• with k < 0 the schedule is in II and IV quarters.

The figure below shows an example of a hyperbola.

• The lines of the graph (in green) are called his by branches.
• The abscissa and ordinate axes (Ox и Oy) are asymptotes hyperbolas, i.e. the branches approach them endlessly, but never touch or cross them.
• Axis of symmetry (in blue) is a straight line:
  • y=x (at k > 0)
  • y = -x (at k < 0)

Asymptote bias

Let’s say we have a function given by the formula:

In this case:

• x = a is the vertical asymptote of the graph (at a ≠ 0) instead of wasps Oy;
• y=b is the horizontal asymptote (at b ≠ 0) instead of wasps Ox.

Canonical form of the hyperbola equation (coordinate axes coincide with the graph axes):

Algorithm for constructing a hyperbola

Example 1

Given a function y = ⁴/_x. Let’s plot it.

Solution

As k > 0, therefore, the hyperbola will be in the I and III coordinate quarters.

To build a graph, you first need to create a table of correspondence of values x и y. That is, we take a specific value x, substitute it into the formula of the function and get y.