Quadratic Function
A Quadratic Function is a polynomial function with a function degree of two.
- AKA: Second-Order Polynomal.
- Context:
- It can range from being a Univariate Second-Order Polynomal Function to being a Multivariate Second-Order Polynomal Function.
- It can be referenced by a Quadratic Equation.
- Example(s):
- [math]\displaystyle{ f(x)=ax^2+bx+c, \quad a \ne 0 }[/math].
- [math]\displaystyle{ f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f, \quad a \ne 0, \quad b \ne 0 }[/math].
- …
- Counter-Example(s):
- See: Quadratically Constrained Quadratic Program, Graph of a Function, Parabola, Root of a Function, Conic Section, Circle, Ellipse, Hyperbola.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/quadratic_function Retrieved:2015-11-7.
- In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: : [math]\displaystyle{ f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j, }[/math] with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.
A univariate (single-variable) quadratic function has the form : [math]\displaystyle{ f(x)=ax^2+bx+c,\quad a \ne 0 }[/math] in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
The bivariate case in terms of variables x and y has the form : [math]\displaystyle{ f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\! }[/math] with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
In general there can be an arbitrarily large number of variables, in which case the resulting surface is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
- In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: : [math]\displaystyle{ f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j, }[/math] with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.