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.

Unless otherwise specified, we consider quadratic functions where the inputs, outputs, and coefficients are all real numbers.

Key data

Item	Value
Default domain	all real numbers, i.e., all of $\mathbb {R}$
range	Case $a>0$: $\left[c-{\frac {b^{2}}{4a}},\infty \right)$ Case $a<0$: $\left(-\infty ,c-{\frac {b^{2}}{4a}}\right]$

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.