Quadratic function

Definition

A quadratic function is a function of the form:

$x\mapsto ax^{2}+bx+c$

where $a,b,c$ are real numbers and $a\neq 0$. In other words, a quadratic function is a polynomial function of degree two.

Key invariants

Expression	Name	Significance in the case $a,b,c\in \mathbb {R}$
$b^{2}-4ac$	(unnormalized) discriminant	The discriminant is positive (i.e., $b^{2}-4ac>0$) iff the quadratic has two distinct real roots The discriminant is zero (i.e., $b^{2}-4ac=0$) iff the quadratic has a real root of multiplicity two The discriminant is negative (i.e., $b^{2}-4ac<0$) iff the quadratic has no real roots
$a$	leading coefficient	Leading coefficient is positive (i.e., $a>0$) iff that the function approaches infinity as $x\to \infty$ and as $x\to -infty$ Leading coefficient is negative (i.e., $a<0$) iff that the function approaches infinity as $x\to \infty$ and as $x\to -infty$
$-b/a$	sum of roots	If the roots are $\alpha ,\beta$ (counted with multiplicity, and they need not be real roots), then the polynomial is $a(x-\alpha )(x-\beta )$ and the sum of roots $\alpha +\beta$ is $-b/a$.
$c/a$	product of rots	If the roots are $\alpha ,\beta$ (counted with multiplicity, and they need not be real roots), then the polynomial is $a(x-\alpha )(x-\beta )$ and the product of roots $\alpha \beta$ is $c/a$.
$b^{2}/(4a^{2})-c/a$	normalized discriminant	Similar observations as for the discriminant.