f(x) = x² - x - 2

A quadratic function, in mathematics, is a polynomial function of the form , where .
It takes its name from the Latin quadratus for square, because quadratic functions arise in the calculation of the area of a square. A quadratic function is also referred to as a degree 2 polynomial or a 2nd degree polynomial, because the highest exponent of x is 2. The graph of such a function is a parabola. Image File history File links Polynomial of degree 2 : f(x) = x2 - x - 2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... For other uses, see Latin (disambiguation). ... For other uses, see Square. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

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 or the zeros of the function. In mathematics, a quadratic equation is a polynomial equation of the second degree. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

Origin of word

The prefix quadri- is used to indicate the number 4. Examples are quadrilateral and quadrant. However, because it is in the Latin word for square (since a square has 4 sides), and the area of a square with side length x is x², the prefix is also sometimes used in words involving the number 2. A numerical prefix is a prefix that denotes a number, which is usually a multiplier for the thing being prefixed. ... This article discusses the number Four. ... This article is about the geometric shape. ... Look up Quadrant on Wiktionary, the free dictionary Quadrant can mean: HMS Quadrant (G11), a WW-II British/Australian warship. ... This article does not cite any references or sources. ...

Roots

The two roots of the quadratic equation , where are

This formula is called the quadratic formula. To see how the formula is derived, see quadratic equation. In mathematics, a quadratic equation is a polynomial equation of the second degree. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

• Let
• If and Δ is a perfect square, then there are two distinct rational roots since is rational.
• If and Δ is not a perfect square, then there are two distinct irrational roots since is irrational.
• If , then the two roots are equal, since is zero.
• If , then the two roots are complex conjugates, since is imaginary.

By letting and or vice versa, one can factor as . —a complex is different from complicated or composed, the complex is more than the sum of its parts Look up complex in Wiktionary, the free dictionary. ...

Forms of a quadratic function

A quadratic function can be expressed in three formats:

• is called the general form or polynomial form,
• is called the factored form, where r₁ and r₂ are the roots of the quadratic equation, and
• is called the standard form or vertex form.

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 factors. Completing the square is an algebra technique, also used in many types of calculus. ...

Graph

Regardless of the format, the graph of a quadratic function is a parabola (as shown above). A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

• If , the parabola opens upward.
• If , 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).

The coefficient b alone is the declivity of the parabola as it crosses the y-axis.

The coefficient c controls the y-intercept; more specifically, it is the point were the parabola crosses the y-axis.

Number of x-intercepts

The number of x-intercepts can be determined by the discriminant too.

• If then there are two x-intercepts because the two real roots are distinct.
• If then there is exactly one x-intercept because of the two real roots are equal. In this case, the parabola is tangent to the x-axis.
• If the graph has no x-intercepts because the two roots are imaginary. In this case, the parabola is either completely above the x-axis (if a > 0) or completely below the x-axis (if a < 0).

For other uses, see tangent (disambiguation). ...

Vertex

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 standard form, the vertex is . By the method of completing the square, one can turn the general form: to

so the vertex of the parabola in the general form will be

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

The maximum or minimum of the function is always obtained at the vertex, the following method is another derivation of the same fact using calculus, the advantage of this method is that it works for more general functions.

Taking as sample quadratic equation, to find its maximum or minimum points (which depends on , if , it has a minimum point, if , it has a maximum point) we have to first, take its derivative:

Then, we find the roots of :

So, is the value of . Now, to find the value, we substitute on :

Thus, the maximum or minimum point coordinates are:

For other uses, see Calculus (disambiguation). ... A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the... For a non-technical overview of the subject, see Calculus. ...

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 equationdescribes 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. In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... For other uses, see Ellipse (disambiguation). ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... Ordinate means the y coordinate on an (x, y) graph; the plotted output of a mathematical function. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... The empty set is the set containing no elements. ...

Bivariate 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. An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

Minimum/Maximum

If the function has no maximum or minimum, its graph forms an hyperbolic paraboloid. Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ...

If the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid. Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic 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
• Matrix representation of conic sections
• Quadric
• Periodic points of complex quadratic mappings

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... This article on periodic points of complex quadratic mappings describes periodic points of some quadratic polynomial mappings on the complex numbers. ...

External links

• Eric W. Weisstein, Quadratic at MathWorld.