# Quadratics When graphed on the Cartesian plane, a quadratic is represented with a [[Conic Section]] called a parabola, which is produced when intersecting a cone with a plane that is parallel to a line that is on the cone's surface. All points on a parabola are equidistant from a point called the focus point and a line called the directrix. Both can be derived from the [[#Vertex Form]] of the quadratic. Given $y=a(x-m)^2+n$, the focus point is located at $(m, n+ \frac 1 {4a})$, with the directrix being the line $y=-n+\frac 1 {4a}$. An algebraic definition of a parabola is a second degree [[Polynomials|polynomial]]. See [[The Quadratic Formula]] for a formula for solving any quadratic in its standard form $f(x) = ax^2 + bx + c$. ![[The Quadratic Formula#^l9kegs]] Note also the simplified version: ![[The Quadratic Formula#^5vr6d5]] Solving and [[Factoring Quadratics]] are two ways of representing the same problem: that of finding roots of a quadratic. ## Non-standard forms ### Factored form $y = a(x-k)(k-j)$ in which $k$ and $j$ represent the zeros (or roots) of the quadratic and $a$ equals the standard form's leading coefficient. ### Vertex Form $y=a(x-m)^2+n$ in which $a$ equals the standard form's leading coefficient and $m$ and $n$ represent the $x,y$ coordinates of the vertex. The coordinates of the vertex can be calculated from the standard form $m=-\frac{b}{2a}$, $n= -\frac {b^2}{4a} + c$ [[Proof for the Vertex Form of Quadratics]]