Quadratic function: Difference between revisions

Revision as of 10:18, 27 November 2022

Revision as of 11:16, 27 November 2022

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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]].

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Any quadratic polynomial with two variables may be written as

:<math> f(x,y) = a x^2 + b y^2 + cxy + dx+ e y + 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, 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 [[hypersurface]]s. In [[linear algebra]], quadratic polynomials can be generalized to the notion of a [[quadratic form]] on a [[vector space]].

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</ref>

* <math>f(x) = a x^2 + b x + c \,\!</math> is called the '''standard form''',

* <math>f(x) = a(x - r_1)(x - r_2)\,\!</math> is called the '''factored form''', where {{math|''r''₁}} and {{math|''r''₂}} are the roots of the quadratic function and the solutions of the corresponding quadratic equation.

* <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.

The coefficient {{math|''a''}} is the same value in all three forms. To convert the '''standard form''' to '''factored form''', one needs only the [[quadratic formula]] to determine the two roots {{math|''r''₁}} and {{math|''r''₂}}. 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.

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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 {{math|(''h'', ''k'')}}. Using the method of completing the square, one can turn the standard form

:<math>f(x) = a x^2 + b x + c \,\!</math>

into

: <math>\begin{align}

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: <math> \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right). </math>{{Citation needed|date=October 2022}}

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 {{math|(''h'', ''k'')}} 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}}.

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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>

{{math|''x''}} is a root of {{math|''f'' '(''x'')}} if {{math|''f'' '(''x'') {{=}} 0}}

resulting in

:<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 = c-\frac{b^2}{4a} \,\!,</math>

so again the vertex point coordinates, {{math|(''h'', ''k'')}}, can be expressed as

:<math> \left (-\frac {b}{2a}, c-\frac {b^2}{4a} \right). </math>

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===Upper bound on the magnitude of the roots===

The [[absolute value|modulus]] of the roots of a quadratic <math>ax^2+bx+c\,</math> can be no greater than <math>\frac{\max(|a|, |b|, |c|)}{|a|}\times \phi,\, </math> where <math>\phi</math> is the [[golden ratio]] <math>\frac{1+\sqrt{5}}{2}.</math><ref>Lord, Nick, "Golden bounds for the roots of quadratic equations", ''Mathematical Gazette'' 91, November 2007, 549.</ref>{{importance inline|<!--Formula doesn't scale under scale of ''x''; a realistic formula should scale by α when b ↦ bα and c ↦cα²-->}}

==The square root of a univariate quadratic function==

The [[square root]] of a univariate 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, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the [[ordinate]] of the [[minimum]] point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!.</math>. If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.

If <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either a circle or other 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==

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one has

:<math>f(x)=a(x-c)^2+c=h^{(-1)}(g(h(x))),\,\!</math>

where

:<math>g(x)=ax^2\,\!</math> and <math>h(x)=x-c.\,\!</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-c)^{2^n}+c\,\!</math>

as the solution.

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{{Further|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>

where ''A, B, C, D'', and ''E'' are fixed [[coefficient]]s and ''F'' is the constant term.

Such a function describes a quadratic [[Surface (mathematics)|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 a hyperbolic [[paraboloid]].

If <math> 4AB-E^2 >0 \,</math> the function has a minimum if both {{nowrap|''A'' > 0}} and {{nowrap|''B'' > 0}}, and a maximum if both {{nowrap|''A'' < 0}} and {{nowrap|''B'' < 0}}; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at <math> (x_m, y_m) \,</math> where:

:<math>x_m = -\frac{2BC-DE}{4AB-E^2},</math>

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:<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—a minimum if ''A''>0 and a maximum if ''A''<0; its graph forms a parabolic cylinder.

==See also==