Linear and Quadratic Functions

Linear and quadratic functions make lines and parabola, respectively, when graphed. They are one of the simplest functional forms.

Linear Function

In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:

$f(x + y) = f(x) + f(y) \\ f(ax) = af(x)$

Linear functions may be confused with affine functions. One variable affine functions can be written as $f(x)=mx+b$. Although affine functions make lines when graphed, they do not satisfy the properties of linearity. However, the term "linear function" is quite often loosely used to include affine functions of the form $f(x)=mx+b$. Linear functions form the basis of linear algebra.

Affine Function

An affine transformation (from the Latin, affinis, "connected with") is a transformation which preserves straight lines (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances between points lying on a straight line (e.g., the midpoint of a line segment remains the midpoint after transformation). It does not necessarily preserve angles or lengths, but does have the property that sets of parallel lines will remain parallel to each other after an affine transformation.

Quadratic Function

A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis . The expression $ax^2+bx+c$ in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of $x$ is 2. If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.