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Article:Quadratic function
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[[Image:Polynomialdeg2.svg|thumb|right|<center><math>x^2 - x - 2\!</math></center>]]
A '''quadratic function''', in [[mathematics]], is a [[polynomial function]] of the form
:<math>f(x)=ax^2+bx+c,\quad a \ne 0.</math>
The [[graph of a function|graph]] of a quadratic function is a [[parabola]] whose axis of symmetry is parallel to the ''y''-axis.
The expression <math>ax^2+bx+c</math> in the definition of a quadratic function is a '''polynomial of [[Degree of a polynomial|degree]] 2''' or second order, or a '''2nd degree polynomial''', because the highest exponent of ''x'' is 2.
If the quadratic function is set equal to zero, then the result is a [[quadratic equation]]. The solutions to the equation are called the [[root of a function|root]]s of the equation.
==Origin of word==
The adjective ''quadratic'' comes from the [[Latin]] word ''[[wikt:en:quadratum#Latin|quadrātum]]'' (“[[square (geometry)|square]]”). A term like ''x''<sup>2</sup> is called a [[square (algebra)|square]] in algebra because it is the area of a ''square'' with side ''x''.
In general, a prefix [[quadr(i)-]] indicates the number [[4 (number)|4]]. Examples are quadrilateral and quadrant. ''Quadratum'' is the Latin word for square because a square has four sides.
{{Further|[[Quadratic equation]]}}
The [[root of a function|roots]] (zeros) of the quadratic function
: <math>f(x) = ax^2+bx+c\,</math>
are the values of ''x'' for which ''f''(''x'') = 0.
When the [[coefficient]]s ''a'', ''b'', and ''c'', are [[real numbers|real]] or [[complex numbers|complex]], the roots are
:<math>x=\frac{-b \pm \sqrt{\Delta}}{2 a}, </math>
where the [[discriminant]] is defined as
:<math>\Delta = b^2 - 4 a c \, . </math>
==Forms of a quadratic function==
==Forms of a quadratic function==
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