The single defining feature of quadratic functions is that they are of the second order, or of degree two. This means that in all quadratic functions, the highest exponent of $x$ in a non-zero term is equal to two.  A quadratic function is of the general form:

$f(x)=ax^2+bx+c$

where $a$, $b$, and $c$ are constants and $x$ is the independent variable.  The constants $b$and $c$ can take any finite value, and $a$ can take any finite value other than $0$.  

A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:

$ax^2+bx+c=0$

When all constants are known, a quadratic equation can be solved as to find a solution of $x$.  Such solutions are known as zeros.  There are several ways of finding $x$, but these methods will be discussed later.

Differences Between Quadratics and Linear Functions

Quadratic equations are different than linear functions in a few key ways. 

• Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope). All quadratic functions both increase and decrease.
• With a linear function, each input has an individual, unique output (assuming the output is not a constant).  With a quadratic function, pairs of unique independent variables will produce the same dependent variable, with only one exception (the vertex) for a given quadratic function.
• The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

Forms of Quadratic Functions

Quadratic functions can be expressed in many different forms. The form written above is called standard form. Additionally

$f(x)=a(x-x_1)(x-x_2)$

is known as factored form, where $x_1$ and $x_2$ are the zeros, or roots, of the equation. These are $x$ values at which the function crosses the y-axis (and thus where $y$ equals zero). 

The vertex form is displayed as:

$f(x)=a(x-h)^2+k$

where $h$ and $k$ are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if $a$ is negative) or minimum (if $a$ is positive).