A quadratic function, in mathematics, is a polynomial function of the form
The graph of a quadratic function is a parabola whose major axis is parallel to the y-
The expression ax + bx + c 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.
If the quadratic function is set equal to zero, then the result is a quadratic equation.
The solutions to the equation are called the roots of the equation.
Origin of word
The adjective quadratic comes from the Latin word quadratum for square. A term
like x2 is called a square in algebra because it is the area of a square with side x.
In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and
quadrant. Quadratum is the Latin word for square because a square has four sides.
The roots (zeros) of the quadratic function
are the values of x for which f(x) = 0.
When the coefficients a, b, and c, are real or complex, the roots are
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 x1 and x2
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 general form to factored form, one needs only the quadratic formula to
determine the two roots r1 and r2. To convert the general form to standard form, one
needs a process called completing the square. To convert the factored form (or
standard form) to general form, one needs to multiply, expand and/or distribute the
Regardless of the format, the graph of a quadratic function is a parabola (as shown
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 x = -b/2a.
The coefficient b alone is the declivity of the parabola as it crosses the y-axis.
The coefficient c controls the height of the parabola, more specifically, it is the point
where the parabola crosses the y-axis.
Inspection of the factored form shows that the x-intercepts of the graph are given by
the roots of the quadratic function.
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 general 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
Using calculus, the vertex point, being a maximum or minimum of the function, can
be obtained by finding the roots of the derivative:
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 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 f(x) = ax + bx + c, one cannot always deduce the analytic form of
f(n)(x), which means the nth iteration of f(x). (The superscript can be extended to
negative number referring to the iteration of the inverse of f(x) if the inverse exists.)
But there is one easier case, in which f(x) = a(x − x0) + x0.
In such case, one has
So by induction,
can be obtained, where g (x) can be easily computed as
Finally, we have
in the case of f(x) = a(x − x0)2 + x0.
See Topological conjugacy for more detail about such relationship between f and g.
And see Complex quadratic polynomial for the chaotic bahavior in the general
Bivariate (two variable) quadratic function
A bivariate quadratic function is a second-degree polynomial of the form
Such a function describes a quadratic surface. Setting equal to zero describes
the intersection of the surface with the plane , which is a locus of points
equivalent to a conic section.
If the function has no maximum or minimum, its graph forms an
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
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.