Quadratic functions are usually the first we encounter that have curved or nonlinear graphs. The graph of a quadratic function is a specific kind of curve called a **parabola** (all parabolas are symmetric with respect to a line called the axis of symmetry). In algebra, **quadratic functions** are any form of the equation `f(x) = ax^2 + bx + c`, where `a` is not equal to `0`.

The vertex of a parabola is the point at the bottom of the "U" shape (or the top, if the parabola opens downward). The equation for a parabola can also be written in "vertex form": `f(x)=a(x−h)^2+k`. In this equation, the vertex of the parabola is the point `(h,k)`.

Created with GeoGebra by Vitor Nunes

There are two "families" of quadratic functions. Try changing the parameters of each of the functions.

Let `f` be the functions of type:

`a(x-h)^2 + k`

Let `g` be the functions of type:

`ax^2 + bx + c`

Quadratic functions can be highly useful when trying to solve any number of problems that involve measurements or quantities with unknown variables. You may find these equations useless. But, if you understand how to use these relatively simple equations to determine a range of results, you can easily solve problems that involve unknown amounts and factors. The parabola appears in physics as the path described by a ball thrown at an angle to the horizontal. The vertex of the parabola gives information regarding maximum height and combined with the symmetry of the curve also tells us how to find the horizontal range.