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[[Image:Polynomialdeg2.svg|thumb|right|<center><math>x^2 - x - 2\!</math></center>]]
A '''quadratic function''', in [[mathematics]], is a [[polynomial function]] of the form
 
:<math>f(x)=ax^2+bx+c,\quad a \ne 0.</math><ref name="wolfram">{{cite web | url=http://mathworld.wolfram.com/QuadraticEquation.html | title=Quadratic Equation -- from Wolfram MathWorld | accessdate=January 6, 2013}}</ref>
 
The [[graph of a function|graph]] of a quadratic function is a [[parabola]] whose axis of symmetry is parallel to the {{math|''y''}}-axis.
 
The expression {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}} in the definition of a quadratic function is a '''polynomial of [[Degree of a polynomial|degree]] 2''' or second order, or a '''2nd degree polynomial''', because the highest exponent of {{math|''x''}} is 2.  This expression is also called a '''quadratic polynomial''' or '''quadratic'''.
 
If the quadratic function is set equal to zero, then the result is a [[quadratic equation]]. The solutions to the equation are called the [[root of a function|root]]s of the equation.
 
==Origin of word==
 
The adjective ''quadratic'' comes from the [[Latin]] word ''[[wikt:en:quadratum#Latin|quadrātum]]'' ("[[square (geometry)|square]]"). A term like {{math|''x''<sup>2</sup>}} is called a [[square (algebra)|square]] in algebra because it is the area of a ''square'' with side {{math|''x''}}.
 
In general, a prefix [[quadr(i)-]] indicates the number {{math|[[4 (number)|4]]}}. Examples are quadrilateral and quadrant. ''Quadratum'' is the Latin word for square because a square has four sides.
 
==Roots==
{{Further2|[[Quadratic equation]]}}
 
The [[root of a function|roots]] (zeros) of the quadratic function
: <math>f(x) = ax^2+bx+c\,</math>
are the values of {{math|''x''}} for which {{math|''f''(''x'') {{=}} 0}}.
 
When the [[coefficient]]s {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, are [[real numbers|real]] or [[complex numbers|complex]], the roots are
:<math>x=\frac{-b \pm \sqrt{\Delta}}{2 a}, </math>
where the [[discriminant]] is defined as
:<math>\Delta = b^2 - 4 a c \, . </math>
 
==Forms of a quadratic function==
A quadratic function can be expressed in three formats:<ref>{{citation
|title=College Algebra
|first1=Deborah
|last1=Hughes-Hallett
|first2=Eric
|last2=Connally
|first3=William G.
|last3=McCallum
|publisher=John Wiley & Sons Inc
|year=2007
|isbn=0-471-27175-6, 9780471271758
|page=205
|url=http://books.google.be/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22}},  [http://books.google.be/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22 Search result]
</ref>
 
* <math>f(x) = a x^2 + b x + c \,\!</math> is called the '''standard form''',
* <math>f(x) = a(x - x_1)(x - x_2)\,\!</math> is called the '''factored form''', where {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} are the roots of the quadratic equation, it is used in [[logistic map]]
* <math>f(x) = a(x - h)^2 + k \,\!</math> is called the '''vertex form''', where {{math|''h''}} and {{math|''k''}} are the {{math|''x''}} and {{math|''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 {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}. 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.
 
==Graph==
[[Image:Function ax^2.svg|thumb|350px|<math>f(x) = ax^2 |_{a=\{0.1,0.3,1,3\}} \!</math>]]
[[Image:Function x^2+bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{1,2,3,4\}} \!</math>]]
[[Image:Function x^2-bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{-1,-2,-3,-4\}} \!</math>]]
Regardless of the format, the graph of a quadratic function is a [[parabola]] (as shown above).
* If {{math|''a'' &gt; 0}}, (or is a positive number), the parabola opens upward.
* If {{math|''a'' &lt; 0}}, (or is a negative number), the parabola opens downward.
 
The coefficient {{math|''a''}} controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive {{math|''a''}} makes the function increase faster and the graph appear more closed.
 
The coefficients {{math|''b''}} and {{math|''a''}} together control the axis of symmetry of the parabola (also the {{math|''x''}}-coordinate of the vertex) which is at <math>x = -\frac{b}{2a}</math>.
 
The coefficient {{math|''b''}} alone is the declivity of the parabola as {{math|''y''}}-axis intercepts.
 
The coefficient {{math|''c''}} controls the height of the parabola, more specifically, it is the point where the parabola intercept the {{math|''y''}}-axis.
 
===Vertex===<!-- This section is linked from [[Quadratic equation]] -->
 
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 {{math|(''h'', ''k'')}}. By the method of completing the square, one can turn the general form
:<math>f(x) = a x^2 + b x + c \,\!</math>
into
: <math> f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,</math>
so the vertex of the parabola in the vertex form is
: <math> \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right). </math>
If the quadratic function is in factored form
:<math>f(x) = a(x - r_1)(x - r_2) \,\!</math>
the average of the two roots, i.e.,
: <math>\frac{r_1 + r_2}{2} \,\!</math>
is the {{math|''x''}}-coordinate of the vertex, and hence the vertex is
: <math> \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!</math>
 
The vertex is also the maximum point if {{math|''a'' &lt; 0}}, or the minimum point if {{math|''a'' &gt; 0}}.
 
The vertical line
 
: <math> x=h=-\frac{b}{2a} </math>
 
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 [[minima and maxima|maximum or minimum]] of the function, can be obtained by finding the roots of the [[derivative]]:
:<math>f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!,</math>
giving
:<math>x=-\frac{b}{2a}</math>
with the corresponding function value
:<math>f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,</math>
so again the vertex point coordinates can be expressed as
:<math> \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right). </math>
 
==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 <math>a>0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes a hyperbola. The axis of the hyperbola is determined by the [[ordinate]] of the [[minimum]] point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!</math>.<br>If the ordinate is negative, then the hyperbola's axis is horizontal.  If the ordinate is positive, then the hyperbola's axis is vertical.<br>If <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either an ellipse or nothing at all.  If the ordinate of the [[maximum]] point of the corresponding parabola
<math> y_p = a x^2 + b x + c \,\!</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an [[Empty set|empty]] locus of points.
 
==Iteration==
Given an <math>f(x)=ax^2+bx+c</math>, one cannot always deduce the analytic form of <math>f^{(n)}(x)</math>, which means the ''nth'' iteration of <math>f(x)</math>. (The superscript can be extended to negative number referring to the iteration of the inverse of <math>f(x)</math> if the inverse exists.) But there is one easier case, in which <math>f(x)=a(x-x_0)^2+x_0</math>.
 
In such case, one has
:<math>f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x)))\,\!</math>,
where
:<math>g(x)=ax^2\,\!</math> and <math>h(x)=x-x_0\,\!</math>.
So by induction,
:<math>f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!</math>
can be obtained, where <math>g^{(n)}(x)</math> can be easily computed as
:<math>g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}\,\!</math>.
Finally, we have
:<math>f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\,\!</math>,
in the case of <math>f(x)=a(x-x_0)^2+x_0</math>.
 
See [[Topological conjugacy]] for more detail about such relationship between ''f'' and ''g''. And see [[Complex quadratic polynomial]] for the chaotic behavior in the general iteration.
 
==Bivariate (two variable) quadratic function==
{{see|Quadric|Quadratic form}}
A '''bivariate quadratic function''' is a second-degree polynomial of the form
:<math> f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!</math>
Such a function describes a quadratic [[surface]].  Setting <math>f(x,y)\,\!</math> equal to zero describes the intersection of the surface with the plane <math>z=0\,\!</math>, which is a [[locus (mathematics)|locus]] of points equivalent to a [[conic section]].
 
===Minimum/maximum===
 
If <math> 4AB-E^2 <0 \,</math> the function has no maximum or minimum, its graph forms an hyperbolic [[paraboloid]].
 
If <math> 4AB-E^2 >0 \,</math> 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 <math> (x_m, y_m) \,</math> where:
 
:<math>x_m = -\frac{2BC-DE}{4AB-E^2}</math>
 
:<math>y_m = -\frac{2AD-CE}{4AB-E^2}</math>
 
If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE \ne 0 \,</math> the function has no maximum or minimum, its graph forms a parabolic [[cylinder (geometry)|cylinder]].
 
If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE =0 \,</math> 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 polynomial==
In mathematics, a quadratic polynomial or quadratic is a [[polynomial]] of [[degree of a polynomial|degree]] two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For example, <math>x^2 - 4x + 7</math> is a quadratic polynomial, while <math>x^3 - 4x + 7</math> is not.
 
===Coefficients===
The [[coefficients]] of a polynomial are often  taken to be real or [[Complex quadratic polynomial|complex number]]s, but in fact, a polynomial may be defined over any [[ring (mathematics)|ring]].
 
===Degree===
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 "[[Degeneracy (mathematics)|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.
 
===Variables===
 
A quadratic polynomial may involve a single [[Variable (mathematics)|variable]] ''x'', or multiple variables such as ''x'', ''y'', and ''z''.
 
====The one-variable case====
 
Any single-variable quadratic polynomial may be written as
:<math>ax^2 + bx + c,\,\!</math>
where ''x'' is the variable, and ''a'', ''b'', and ''c'' represent the [[coefficient]]s. In [[elementary algebra]], such polynomials often arise in the form of a [[quadratic equation]] <math>ax^2 + bx + c = 0</math>.  The solutions to this equation are called the [[Root of a function|roots]] of the quadratic polynomial, and may be found through [[factorization]], [[completing the square]], [[Graph of a function|graphing]], [[Newton's method]], or through the use of the [[quadratic formula]].  Each quadratic polynomial has an associated quadratic function, whose [[graph of a function|graph]] is a [[parabola]].
 
If the polynomial is a polynomial in one [[Variable (mathematics)|variable]], it determines a quadratic function in one variable. An example is given by ''f''(''x'')&nbsp;=&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''x''&nbsp;−&nbsp;2;. The [[Graph of a function|graph]] of such a [[Function (mathematics)|function]] is a [[parabola]] (in degenerate cases a [[line (mathematics)|line]]), and its [[Root of a function|zero]]es can be found by solving the [[quadratic equation]] ''f''(''x'')&nbsp;=&nbsp;0.
 
There are three main '''forms''' :
* general form,  <math> f(x) =  a x^2 + b x + c \,</math>.
* [[logistic map|logistic form]], <math>f_r(x) = r x ( 1-x ) \,</math>, used to study [[Euclidean space|1D]] [[Dynamical_system#Maps|discrete dynamics]],
* [[Complex quadratic polynomial|monic and centered form]], <math>f_c(x) = x^2 +c\,</math>, used to study [[complex dynamics]].
 
====Two variables case====
 
Any quadratic polynomial with two variables may be written as
:<math>ax^2 + bxy + cy^2 + dx + ey + f,\,\!</math>
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 section]]s. 
Similarly, quadratic polynomials with three or more variables correspond to [[quadric]] surfaces and [[hypersurface]]s. In [[linear algebra]], quadratic polynomials can be generalized to the notion of a [[quadratic form]] on a [[vector space]].
 
====N variables case====
In the general case, a quadratic polynomial in ''n'' variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> can be written in the form
:<math>
\sum_{i, j = 1}^{n} Q_{i,j}  x_i  x_j + \sum_{i = 1}^{n} P_i  x_i + R
</math>
where ''Q'' is a symmetric ''n''-dimensional [[matrix (mathematics)|matrix]], ''P'' is an ''n''-dimensional [[Vector (geometric)|vector]], and ''R'' a constant.
 
==See also==
* [[Quadratic form]]
* [[Quadratic equation]]
* [[Matrix representation of conic sections]]
* [[Quadric]]
* [[Periodic points of complex quadratic mappings]]
* [[List of mathematical functions]]
 
==References==
{{reflist}}
*Algebra 1, Glencoe, ISBN 0-07-825083-8
*Algebra 2, Saxon, ISBN 0-939798-62-X
 
==External links==
* {{MathWorld|title=Quadratic|urlname=Quadratic}}
 
{{Polynomials}}
 
{{DEFAULTSORT:Quadratic Function}}
[[Category:Polynomials]]
[[Category:Parabolas]]

Revision as of 11:31, 17 February 2014

If you really desire to lose weight plus feel better about oneself it's significant to make a dedication to change the way you eat. There are a lot of diets out there which work because they restrict what we eat. Someone once mentioned that they're certainly just controlled starvation, plus folks can't stick to that lengthy term plus end up gaining all weight back.

diet plans really have the above revealed criteria and let the dieter to eat what they want at certain certain periods. This would confirm that the dieter does not receive into binge eating plus refuses to drop away from the diet.

There are three major points a individual must remember on her diet plans losing fat. One is that calories are enemies, 2nd is the fact that fiber is a good friend and 3rd is that water will fill 1 up.

If you should lose weight rapidly (like 48 hours), this diet may interest you. For 48 hours all we are enabled to consume is a unique fruit juice. It is supposed to have all that your body demands for which time. Many persons have reported noticeable fat reduction in those 48 hours. The question is, what happens in the next 48?

Just these diets should not tell you to eat less and starve yourselves. They must not tell you to eat less carbs, no protein, no greens. They can't dramatically lower your daily calorie intake plus they cannot market you on some magic pills and do nothing.

The Cabbage Diet. Lose weight fast with the cabbage diet. This diet consists of creating a soup with cabbage, that you'll eat during the 7-day duration of the diet. The diet is low in fat, and high in fiber which might aid you lose fat quickly. During this diet you'll rotate between eating cabbage soup and fruit with eating greens and meat on each alternative day. This diet might help we lose weight considering it really is low inside calories. Some people find it hard to keep this diet due to the limited food choices it allows.

But we personally choose to lose we excess pounds, you must wish To create it result. If you go at your fat reduction half-hearted, we will lose the battle of the bulge! Carefully plan the route you need to take and be willing, or prepare to fail! There are numerous lucrative diets which function fast out there. Focus on 1 proven system and stick with it.