Polynomial function whose general form is \(f(x) = \textrm{A}{x}^2 + \textrm{B} x + \textrm{C}\), where A ≠ 0 and A, B, C ∈ \(\mathbb{R}\).

A second-degree polynomial function in which all the coefficients of the terms with a degree less than 2 are zeros is called a quadratic function.

### Properties

- The graph of a second-degree polynomial function has its vertex at the origin of the Cartesian plane.
- The zeros of a second-degree polynomial function are given by the following :
- If (B
^{2}– 4AC) ≥ 0, the zeros are real numbers: \(x_{1} = \frac{−\textrm{B}\space + \space \sqrt{{\textrm{B}}^{2} − 4\textrm{AC}}}{2\textrm{A}}\) and \(x_{2} = \frac{−\textrm{B}\space −\space \sqrt{{ \textrm{B}}^{2} − 4\textrm{AC}}}{2\textrm{A}}\); - If the function is of the form
*f*(*x*) = A\({x}^{2}\) + B\(x\), the zeros are : \(x_{1}\) = 0 and \(x_{2}\) = − \(\frac{B}{A}\); - If the function is of the form
*f*(*x*) = A\({x}^{2}\) + C, the zeros are : \(x_{1}\) = \(\sqrt{− \frac{C}{A}}\) and \(x_{2}\) = − \(\sqrt{− \frac{C}{A}}\), where AC < 0; - If the function is of the form
*f*(*x*) = A\({x}^{2}\), the zeros are : \(x_{1}\)= 0 and \(x_{2}\)= 0.

- If (B

### Examples

- The graphical representation of a second-degree polynomial function defined by the relationship \(f(x) = {x}^{2}\) is a basic parabola.

- The graphical representation of a second-degree polynomial function defined by the relationship \(f(x) = {(x − a)}^{2}\) is a basic parabola translated horizontally.

- The graphical representation of the second-degree polynomial function defined by the relationship \(f(x) = {x}^{2} + k\) is a basic parabola translated vertically.

- The graphical representation of the second-degree polynomial function defined by the relationship \(f(x) = a{(x − h)}^{2} + k\) is a basic parabola translated horizontally and vertically.

This graph illustrates the function *f* defined by *f*(*x*) = \(\left ( x+3 \right )^{2} – 4\)