Quadratic function
In elementary algebra, a quadratic function is a function containing a quadratic expression, a polynomial where the degree (the highest exponent it has) is 2. The single-variable standard form of a quadratic function isː [math]\displaystyle{ f(x)=ax^2+bx+c }[/math] where [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math] and [math]\displaystyle{ c }[/math] are all constants and [math]\displaystyle{ a\ne 0 }[/math].
When such a function gets plotted on a graph where [math]\displaystyle{ f(x)=y }[/math], a curve that extends infinitely called a parabola will appear.
When a quadratic function is set to some value, it makes a quadratic equation. When the value is zero, the equation is said to be in standard form, and its solutions are the places where the function crosses the [math]\displaystyle{ x }[/math]-axis.
Properties
Quadratic functions have a single extremum. This point, which is a minimum if [math]\displaystyle{ a\gt 0 }[/math] and a maximum if [math]\displaystyle{ a\lt 0 }[/math], is called the vertex of the parabola.
The derivative of a quadratic function is a linear function.
Etymology
The word quadratic comes from the Latin word quadrātum ("square"). The highest degree term, [math]\displaystyle{ x^2 }[/math], is the area of a square with side length [math]\displaystyle{ x }[/math]. The word "quadratic" is applied to many things in mathematics that involve this [math]\displaystyle{ x^{2} }[/math] term. A similar etymology is shared with cubic functions, which have an [math]\displaystyle{ x^{3} }[/math] term that is the volume of the cube of side length [math]\displaystyle{ x }[/math]. Higher degrees like quartic functions and up take their name from the degree directly using numeric prefixes.
Quadratic Function Media
A quadratic polynomial with two real roots (crossings of the x axis).
Function f(x)=ax2{\displaystyle f(x)=ax^{2}}
Function f(x)=x2+bx{\displaystyle f(x)=x^{2}+bx}
Function f(x)=x2+bx{\displaystyle f(x)=x^{2}+bx}
