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The "basic" hyperbola is the function

y = 1 x

This function has a graph which consists of two disjoint parts. Note that 0 is not in the domain . Note also that the function approaches 0 asymptotically as x grows infinitely large (or infinitely negative), and that it approaches infinity as x approaches 0 from the positive side, negative infinity as x approaches 0 from the negative side.

Math diagram

The center of the hyperbola is the point of rotational symmetry. In the above example, the center is ( 0 , 0 ) .

The graph of the rational function

y = a x h + k

is a hyperbola whose center is ( h , k ) . The constant a controls the "steepness" with which the graph approaches the asymptotes .

A hyperbola can also be defined as a conic section obtained by the intersection of a double cone with a plane that is intersects both pieces of the cone without intersecting the axis.

    Math diagram

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