Editing Quadratic function (section)

==Graph of the univariate function==
[[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 univariate quadratic function <math>f(x) = ax^2 + bx + c</math> is a [[parabola]] (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation <math>y = ax^2 + bx + c</math>.
* If {{math|''a'' &gt; 0}}, the parabola opens upwards.
* If {{math|''a'' &lt; 0}}, the parabola opens downwards.

The coefficient {{math|''a''}} controls the degree of curvature of the graph; a larger magnitude of  {{math|''a''}} gives the graph a more closed (sharply curved) appearance.

The coefficients {{math|''b''}} and {{math|''a''}} together control the location of the axis of symmetry of the parabola (also the {{math|''x''}}-coordinate of the vertex and the ''h'' parameter in the vertex form) which is at
:<math>x = -\frac{b}{2a}.</math>

The coefficient {{math|''c''}} controls the height of the parabola; more specifically, it is the height of the parabola  where it intercepts 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 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}
  f(x) &= a x^2 + b x + c \\
       &= a (x - h)^2 + k \\
       &= a\left(x - \frac{-b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right),\\
\end{align}</math>
so the vertex, {{math|(''h'', ''k'')}}, of the parabola in standard form is
: <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}}.

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>
{{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>