Quadratic function

The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.

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. However, where the "degree of a polynomial" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a power series.

A quadratic polynomial may involve a single variable x (the univariate case), or multiple variables such as x, y, and z (the multivariate case).

Any single-variable quadratic polynomial may be written as

${\displaystyle ax^{2}+bx+c,\,\!}$

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 ${\displaystyle ax^{2}+bx+c=0}$. 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

${\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f,\,\!}$

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, which are characterized by equating the expression for f (x, y) to zero. 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.

The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is (h, k). Using the method of completing the square, one can turn the standard form

${\displaystyle f(x)=ax^{2}+bx+c\,\!}$

into

${\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-h)^{2}+k\\&=a\left(x-{\frac {-b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right),\\\end{aligned}}}$

so the vertex, (h, k), of the parabola in standard form is

${\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}$

If the quadratic function is in factored form

${\displaystyle f(x)=a(x-r_{1})(x-r_{2})\,\!}$

the average of the two roots, i.e.,

${\displaystyle {\frac {r_{1}+r_{2}}{2}}\,\!}$

is the x-coordinate of the vertex, and hence the vertex (h, k) is

${\displaystyle \left({\frac {r_{1}+r_{2}}{2}},f\left({\frac {r_{1}+r_{2}}{2}}\right)\right).\!}$

The vertex is also the maximum point if a < 0, or the minimum point if a > 0.

The vertical line

${\displaystyle x=h=-{\frac {b}{2a}}}$

that passes through the vertex is also the axis of symmetry of the parabola.

Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:

${\displaystyle f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f'(x)=2ax+b\,\!.}$

x is a root of f '(x) if f '(x) = 0 resulting in

${\displaystyle x=-{\frac {b}{2a}}}$

with the corresponding function value

${\displaystyle f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}}\,\!,}$

so again the vertex point coordinates, (h, k), can be expressed as

${\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}$

The roots (or zeros), r₁ and r₂, of the univariate quadratic function

${\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-r_{1})(x-r_{2}),\\\end{aligned}}}$

are the values of x for which f(x) = 0.

When the coefficients a, b, and c, are real or complex, the roots are

${\displaystyle r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},}$

${\displaystyle r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}.}$

The modulus of the roots of a quadratic ${\displaystyle ax^{2}+bx+c\,}$ can be no greater than ${\displaystyle {\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,\,}$ where ${\displaystyle \phi }$ is the golden ratio ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}.}$[4]

If ${\displaystyle 4AB-E^{2}<0\,}$ the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.

If ${\displaystyle 4AB-E^{2}>0\,}$ the function has a minimum if both A > 0 and B > 0, and a maximum if both A < 0 and B < 0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at ${\displaystyle (x_{m},y_{m})\,}$ where:

${\displaystyle x_{m}=-{\frac {2BC-DE}{4AB-E^{2}}},}$

${\displaystyle y_{m}=-{\frac {2AD-CE}{4AB-E^{2}}}.}$

If ${\displaystyle 4AB-E^{2}=0\,}$ and ${\displaystyle DE-2CB=2AD-CE\neq 0\,}$ the function has no maximum or minimum; its graph forms a parabolic cylinder.

If ${\displaystyle 4AB-E^{2}=0\,}$ and ${\displaystyle DE-2CB=2AD-CE=0\,}$ the function achieves the maximum/minimum at a line—a minimum if A>0 and a maximum if A<0; its graph forms a parabolic cylinder.