Quadratic function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a quadratic function of a single <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a> is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of the form

f
        (
        x
        )
        =
        a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
        ,
        
        a
        ≠
        0
        ,
      
    
    {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,}

where ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠ is its variable, and ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, ⁠
 
 
 
 b
 
 
 {\displaystyle b}
 
⁠, and ⁠
 
 
 
 c
 
 
 {\displaystyle c}
 
⁠ are <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a>. The <a href="/facts/Mathematical_expression/MPrMlYbE">expression</a> ⁠
 
 
 
 
 a
 
 x
 
 2
 
 
 +
 b
 x
 +
 c
 
 
 
 {\displaystyle \textstyle ax^{2}+bx+c}
 
⁠, especially when treated as an <a href="/facts/Mathematical_object/Juhyow9M">object</a> in itself rather than as a function, is a quadratic polynomial, a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> of degree two. In <a href="/facts/Elementary_mathematics/J0Jt18pb">elementary mathematics</a> a polynomial and its associated <a href="/facts/Polynomial_function/Lzak8VVx">polynomial function</a> are rarely distinguished and the terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic.

The <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a <a href="/facts/Function_of_a_real_variable/bbNG3DnN">real</a> single-variable quadratic function is a <a href="/facts/Parabola/Cweo3wgw">parabola</a>. If a quadratic function is <a href="/facts/Equation/pxFzLAVR">equated</a> with zero, then the result is a <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equation</a>. The solutions of a quadratic equation are the <a href="/facts/Zero_of_a_function/mlK3dhoT">zeros</a> (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the <a href="/facts/Quadratic_formula/j5ftMowG">quadratic formula</a>.
A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠ and ⁠
 
 
 
 y
 
 
 {\displaystyle y}
 
⁠ has the form

f
        (
        x
        ,
        y
        )
        =
        a
        
          x
          
            2
          
        
        +
        b
        x
        y
        +
        c
        
          y
          
            2
          
        
        +
        d
        x
        +
        e
        y
        +
        f
        ,
      
    
    {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,}

with at least one of ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, ⁠
 
 
 
 b
 
 
 {\displaystyle b}
 
⁠, and ⁠
 
 
 
 c
 
 
 {\displaystyle c}
 
⁠ not equal to zero. In general the zeros of such a quadratic function describe a <a href="/facts/Conic_section/teqgLptX">conic section</a> (a <a href="/facts/Circle/9fy5ggKn">circle</a> or other <a href="/facts/Ellipse/1wUDnGxY">ellipse</a>, a <a href="/facts/Parabola/Cweo3wgw">parabola</a>, or a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a>) in the ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠–⁠
 
 
 
 y
 
 
 {\displaystyle y}
 
⁠ plane. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a <a href="/facts/Quadric/L3cTApd4">quadric</a>, which is a <a href="/facts/Surface_(geometry)/whF08jvy">surface</a> in the case of three variables and a <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> in general case.

Quadratic function open-in-new

Quadratic function