quadratic

Dictionary


  • an equation in which the highest power of an unknown quantity is a square a polynomial of the second degree of or relating to or resembling a square
  • "quadratic shapes" of or relating to the second power
  • "quadratic equation"

    • Wikipedia


    In mathematics, a quadratic function is a polynomial function (mathematics)function of the form !:f(x)=ax^2+bx+c< ;/math>,? where ''a is nonzero''. It takes its name from the Latin ''quadratus'' for square (geometry)square, because quadratic functions arise in the calculation of areas of squares. In the case where the domain of a functiondomain and codomain are R (the real numbers), the graph of a functiongraph of such a function is a parabola.If the quadratic function is set to be equal to zero, then the result is a quadratic equation.The square root of a quadratic function gives rise either to an ellipse or to a hyperbola. If ''a>0'' then the equation: y = \pm \sqrt describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola: y_p = a x^2 + b x + c. If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical.If ''a<0'' then the equation: y = \pm \sqrt describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola: y_p = a x^2 + b x + c is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an Empty setempty locus of points.A bivariate quadratic function is a second-degree polynomial of the form: f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F. Such a function describes a quadratic surface. Setting ''f(x,y)'' equal to zero describes the intersection of the surface with the plane ''z=0'', which is a locus (mathematics)locus of points equivalent to a conic section.

    Roots - The roots, or solutions to the quadratic function, for variable x, are: x = \frac } .This formula is called the quadratic formula. As for how the formula is derived, see quadratic equation.

    The Graph - The graph of a quadratic function :f(x) = a x^2 + b x + c or f(x) = a(x - h)^2 + k is called a parabola. The former is called the general form while the latter is the standard form. In either form, a is non-zero, and
  • If a > 0 , the parabola opens upward.
  • If a < 0 , the parabola opens downward. VertexThe place where the parabola turns is called the turning point or the vertex of the parabola. If the quadratic function is in standard form, the vertex is (''h'', ''k''). By the method of completing the square, one can turn the general form f(x) = a x^2 + b x + c to : f(x) = a\left(x + \frac \right)^2 + \frac , so that the vertex of the parabola in the general form will be: \left(-\frac , \frac \right). ''' Number of ''x''-intercepts''' The number of ''x''-intercepts is determined by the quantity b^2 - 4ac , which is called the discriminant.
  • If b^2 - 4ac > 0 and is a square number, the graph has two rational ''x''-intercepts since the quadratic formula yeilds two distinct real roots
  • If b^2 - 4ac > 0, and is ''not'' a square number, the graph has two irrational ''x''-intercepts since the quadratic formula yields two distinct real roots.
  • If b^2 - 4ac = 0, the graph has one ''x''-intercept since the quadratic formula yields one real root (or two equal real roots).
  • If b^2 - 4ac < 0, the graph has no ''x''-intercepts since the quadratic formula yields two imaginary roots.

    See also -
  • quadratic form
  • Matrix representation of conic sections
  • !quadricCategory:Polynomialsja: 二次関ਤ 8;nl:Kwadratische? functiepl:Funkcja kwadratowa DEBUG REDIRECT (quadratic function)

    • Websites


    Groups, Algorithms and Programming
    Group at the University of St. Andrews developing GAP, a free system for computational discrete algebra.
    http://www-gap.dcs.st-and.ac.uk/

    Eric Weisstein's World of Mathematics
    Glossary of terms. Material ranges from undergraduate to research level.
    http://mathworld.wolfram.com/

    Calculator City
    Includes tools to solve equations like Includes calculators for velocity, density, financial, geometry, algebra, calculus, astronomy, chemistry, physics and webpage color coders. Also, has trivia fact pages, such as state demographics and presidents.
    http://www.1728.com/
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