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The expression in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.
Origin of word 
The roots (zeros) of the quadratic function
are the values of x for which f(x) = 0.
where the discriminant is defined as
Forms of a quadratic function 
A quadratic function can be expressed in three formats:
- is called the standard form,
- is called the factored form, where and are the roots of the quadratic equation, it is used in logistic map
- is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.
To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots and . To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Regardless of the format, the graph of a quadratic function is a parabola (as shown above).
- If (or is a positive number), the parabola opens upward.
- If (or is a negative number), the parabola opens downward.
The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive a makes the function increase faster and the graph appear more closed.
The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex) which is at .
The coefficient b alone is the declivity of the parabola as y-axis intercepts.
The coefficient c controls the height of the parabola, more specifically, it is the point where the parabola intercept the y-axis.
The vertex of a parabola is the place where it turns, hence, it's also called the turning point. If the quadratic function is in vertex form, the vertex is . By the method of completing the square, one can turn the general form
so the vertex of the parabola in the vertex form is
If the quadratic function is in factored form
the average of the two roots, i.e.,
is the x-coordinate of the vertex, and hence the vertex is
The vertex is also the maximum point if or the minimum point if .
The vertical line
that passes through the vertex is also the axis of symmetry of the parabola.
Maximum and minimum points 
with the corresponding function value
so again the vertex point coordinates can be expressed as
The square root of a quadratic function 
The square root of a quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola. If then the equation describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola .
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 then the equation describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.
Given an , one cannot always deduce the analytic form of , which means the nth iteration of . (The superscript can be extended to negative number referring to the iteration of the inverse of if the inverse exists.) But there is one easier case, in which .
In such case, one has
- and .
So by induction,
can be obtained, where can be easily computed as
Finally, we have
in the case of .
Bivariate (two variable) quadratic function 
A bivariate quadratic function is a second-degree polynomial of the form
If the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.
If the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid.
The minimum or maximum of a bivariate quadratic function is obtained at where:
If and the function has no maximum or minimum, its graph forms a parabolic cylinder.
If and the function achieves the maximum/minimum at a line. Similarly, a minimum if A>0 and a maximum if A<0, its graph forms a parabolic cylinder.
See also 
- Quadratic form
- Quadratic equation
- Quadratic polynomial
- Matrix representation of conic sections
- Periodic points of complex quadratic mappings
- List of mathematical functions