Suppose $f$ is a real valued function with domain a subset of  real numbers, $D$.

Definition: $f$  is a quadratic function if there a numbers $A$, $B$, and $C$  with $A \ne 0$ so that for any $x \in D$,  $f(x) = Ax^2 + Bx +C$
Comment: The letters  $A$, $B$, and $C$ have been used to conform with the naming of these coefficients in quadratic equations, but there are alternatives. Here is an alternative:
$f$  is a quadratic function if there a numbers $a_2$, $a_1$, and $a_0$  with $a_2 \ne 0$ so that for any $x \in D$,  $f(x) =a_2x^2+ a_1 x +a_0$.

The number $A$ is sometimes referred to as  the "leading coefficient " of the quadratic function.
The number $B$ is sometimes referred to as  the "linear coefficient " of the quadratic function.
The number $C$ is sometimes referred to as "the $y$- intercept," "the initial value," or the "constant " of the quadratic function.