Second-Degree Polynomial Function

Second-Degree Polynomial Function

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 (B2 – 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.

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) = $$\left ( x+3 \right )^{2} – 4$$