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The expression ax2 + bx + c in the definition of a univariate quadratic function is a polynomial of degree 2, or a 2nd degree polynomial, because the highest exponent of x is 2. This expression is also called a quadratic polynomial or quadratic.
A quadratic function can also be multivariate (having more than one variable). The bivariate case in terms of variables x and y has the form
In general there can be an arbitrarily large number of variables, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
- 1 Origin of word
- 2 Terminology
- 3 Forms of a quadratic function
- 4 Graph
- 5 Roots
- 6 The square root of a quadratic function
- 7 Iteration
- 8 Bivariate (two variable) quadratic function
- 9 See also
- 10 References
- 11 External links
Origin of word
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.
A quadratic polynomial may involve a single variable x (the univariate case), or multiple variables such as x, y, and z (the multivariate case).
The one-variable case
Any single-variable quadratic polynomial may be written as
where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation . The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Any quadratic polynomial with two variables may be written as
where x and y are the variables and a, b, c, d, e, and f are the coefficients. Such polynomials are fundamental to the study of conic sections. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.
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 standard form to factored form, one needs only the quadratic formula to determine the two roots x1 and x2. 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 a > 0, (or is a positive number), the parabola opens upward.
- If a < 0, (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, greater positive a values makes the function increase faster and the graph appears 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 (h, k). 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 a < 0, or the minimum point if a > 0.
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 roots (zeros) of the quadratic function
are the values of x for which f(x) = 0.
where the discriminant is defined 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
where A, B, C, D, and E are fixed coefficients and F is the constant term. 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 hyperbolic paraboloid.
If the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs 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.
- Quadratic form
- Quadratic equation
- Matrix representation of conic sections
- Periodic points of complex quadratic mappings
- List of mathematical functions