QUADRATIC FUNCTION

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A 'quadratic function', in mathematics, is a polynomial function of the form $f(x)=ax^2+bx+c\; ,!$, where $a,\; b,\; c\; ,!$ are real numbers and $a$
e 0 ,!. If variable and coefficients are complex number one gets complex quadratic polynomial.

It takes its name from the Latin ''quadratus'' for square, because quadratic functions arise in the calculation of areas of squares. Because the (highest) exponent of ''x'' is 2, a quadratic function is sometimes referred as a 'degree 2 polynomial' or a '2nd degree polynomial'. The graph of such a function is a parabola.
If the quadratic function is set to be 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.

Contents

Origin of word

Roots

Forms of a quadratic function

Graph

Number of ''x''-intercepts

Vertex

The square root of a quadratic function

Bivariate quadratic function

Minimum/Maximum

See also

External links

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 $x^2$, the prefix is also sometimes used in words involving the number 2.

Roots

The roots of the quadratic equation $0=ax^2+bx+c,!$, where $a$
e 0 ,! are

This formula is called the quadratic formula. To see how the formula is derived, see quadratic equation.
To transform them to integers ( multiplying a nonzero constant to an equation will not change the roots nor their nature). In the case where ''a'', ''b'' and ''c'' are real, the following does not always apply.

★ say '$Delta\; =\; b^2-4ac\; ,$'

★ If '$Delta\; >\; 0,!$' and $Delta$ is a square number, then there are two distinct rational roots since $sqrt\{Delta\}$ is rational.

★ If '$Delta\; >\; 0,!$' and $Delta$ is 'not' a square number, then there are two distinct irrational roots since $sqrt\{Delta\}$ is irrational.

★ If '$Delta\; =\; 0,!$,' then there are two equal (a.k.a. double) roots since $sqrt\{Delta\}$ is zero.

★ If ',' then there are two distinct complex roots since $sqrt\{Delta\}$ is imaginary.
By letting and or vice versa, one can factor $a\; x^2\; +\; b\; x\; +\; c\; ,!$ as $a(x\; -\; r\_1)(x\; -\; r\_2),!$.

Forms of a quadratic function

A quadratic function can be expressed in three formats:

★ $f(x)\; =\; a\; x^2\; +\; b\; x\; +\; c\; ,!$ is called the 'general form or polynomial form',

★ $f(x)\; =\; a(x\; -\; r\_1)(x\; -\; r\_2),!$ is called the 'factored form', where $r\_1$ and $r\_2$ are the roots of the quadratic equation, and

★ $f(x)\; =\; a(x\; -\; h)^2\; +\; k\; ,!$ 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 $r\_1$ and $r\_2$. 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.

Graph

Regardless of the format, the graph of a quadratic function is a parabola (as shown above).

★ If $a\; >\; 0\; ,!$, 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 height of the parabola, 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 $Delta\; >\; 0,!,$ then there are two ''x''-intercepts because the two real roots are distinct.

★ If $Delta\; =\; 0,!,$ 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).

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 $(h,\; -k),!$. By the method of completing the square, one can turn the general form: $f(x)\; =\; a\; x^2\; +\; b\; x\; +\; c\; ,!$ to
:
ight)^2 - rac{b^2-4ac}{4 a} ,
so the vertex of the parabola in the general form will be
:
ight).
If the quadratic function is in factored form $f(x)\; =\; a(x\; -\; r\_1)(x\; -\; r\_2)\; ,!$
the average of the two roots, i.e.,
:
is the ''x''-coordinate of the vertex, and hence the vertex is
:
ight).!
The vertex is also the maximum point if 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'
: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 $f(x)\; =\; ax^2\; +\; bx\; +\; c\; ,!$ as sample quadratic equation, to find its maximum or minimum points (which depends on $a\; ,!$, if $a\; >\; 0\; ,!$, it has a minimum point, if , it has a maximum point) we have to first, take its derivative:
::$f(x)=ax^2+bx+c\; Leftrightarrow\; ,!$$f\text{'}(x)=2ax+b\; ,!$
:Then, we find the roots of $f\text{'}(x),!$:
::$2ax+b=0\; Rightarrow\; ,!$ $2ax=-b\; Rightarrow,!$
:So, is the $x,!$ value of $f(x),!$. Now, to find the $y,!$ value, we substitute on $f(x),!$:
::
ight)^2+b left (-rac{b}{2a}
ight)+cRightarrow y= rac{ab^2}{4a^2} - rac{b^2}{2a} + c Rightarrow y= rac{b^2}{4a} - rac{b^2}{2a} + c Rightarrow
::
:Thus, the maximum or minimum point coordinates are:
::
ight).

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 $a>0,!$ then the equation$y\; =\; pm\; sqrt\{a\; x^2\; +\; b\; x\; +\; c\}$describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola$y\_p\; =\; a\; x^2\; +\; b\; x\; +\; c\; ,!$
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 $y\; =\; pm\; sqrt\{a\; x^2\; +\; b\; x\; +\; c\}$ describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola
$y\_p\; =\; a\; x^2\; +\; b\; x\; +\; c\; ,!$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

Bivariate quadratic function

A 'bivariate quadratic function' is a second-degree polynomial of the form
:$f(x,y)\; =\; A\; x^2\; +\; B\; y^2\; +\; C\; x\; +\; D\; y\; +\; E\; x\; y\; +\; F\; ,!$
Such a function describes a quadratic surface. Setting $f(x,y),!$ equal to zero describes the intersection of the surface with the plane $z=0,!$, which is a locus of points equivalent to a conic section.

Minimum/Maximum

If the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.
If $4AB-E^2\; >0\; ,$ 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 $(x\_m,\; y\_m)\; ,$ where:
:
:
If $4AB-\; E^2\; =0\; ,$ and $DE-2CB=2AD-CE$
e 0 , the function has no maximum or minimum, its graph forms a parabolic cylinder.
If $4AB-\; E^2\; =0\; ,$ and $DE-2CB=2AD-CE\; =0\; ,$ 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

External links

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