Editing Parabola (section)

== Generalizations ==
If one replaces the real numbers by an arbitrary [[Field (mathematics)|field]], many geometric properties of the parabola <math> y = x^2</math> are still valid:
# A line intersects in at most two points. 
# At any point <math>(x_0, x_0^2)</math> the line <math>y = 2 x_0 x - x_0^2</math> is the tangent.
Essentially new phenomena arise, if the field has characteristic 2 (that is, <math>1 + 1 = 0</math>): the tangents are all parallel.

In [[algebraic geometry]], the parabola is generalized by the [[rational normal curve]]s, which have coordinates {{math|(''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ..., ''x<sup>n</sup>'')}}; the standard parabola is the case {{math|1=''n'' = 2}}, and the case {{math|1=''n'' = 3}} is known as the [[twisted cubic]]. A further generalization is given by the [[Veronese variety]], when there is more than one input variable.

In the theory of [[quadratic form]]s, the parabola is the graph of the quadratic form {{math|''x''<sup>2</sup>}} (or other scalings), while the [[elliptic paraboloid]] is the graph of the [[Definite bilinear form|positive-definite]] quadratic form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} (or scalings), and the [[hyperbolic paraboloid]] is the graph of the [[indefinite quadratic form]] {{math|''x''<sup>2</sup> − ''y''<sup>2</sup>}}. Generalizations to more variables yield further such objects.

The curves {{math|1=''y'' = ''x''{{isup|''p''}}}} for other values of {{mvar|p}} are traditionally referred to as the '''higher parabolas''' and were originally treated implicitly, in the form {{math|1=''x''{{isup|''p''}} = ''ky''{{isup|''q''}}}} for {{mvar|p}} and {{mvar|q}} both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula {{math|1=''y'' = ''x''{{isup|''p''/''q''}}}} for a positive fractional power of {{mvar|x}}. Negative fractional powers correspond to the implicit equation {{math|1=''x''{{isup|''p''}} ''y''{{isup|''q''}} = ''k''}} and are traditionally referred to as '''higher hyperbolas'''. Analytically, {{mvar|x}} can also be raised to an irrational power (for positive values of {{mvar|x}}); the analytic properties are analogous to when {{mvar|x}} is raised to rational powers, but the resulting curve is no longer algebraic and cannot be analyzed by algebraic geometry.