Second-Degree Polynomial Function
Polynomial function whose general form is [latex]f(x) = \textrm{A}{x}^2 + \textrm{B} x + \textrm{C}[/latex], where A ≠ 0 and A, B, C ∈ [latex]\mathbb{R}[/latex].
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: [latex]x_{1} = \frac{−\textrm{B}\space + \space \sqrt{{\textrm{B}}^{2} − 4\textrm{AC}}}{2\textrm{A}}[/latex] and [latex]x_{2} = \frac{−\textrm{B}\space −\space \sqrt{{ \textrm{B}}^{2} − 4\textrm{AC}}}{2\textrm{A}}[/latex];
- If the function is of the form f(x) = A[latex]{x}^{2}[/latex] + B[latex]x[/latex], the zeros are : [latex]x_{1}[/latex] = 0 and [latex]x_{2}[/latex] = − [latex]\frac{B}{A}[/latex];
- If the function is of the form f(x) = A[latex]{x}^{2}[/latex] + C, the zeros are : [latex]x_{1}[/latex] = [latex]\sqrt{− \frac{C}{A}}[/latex] and [latex]x_{2}[/latex] = − [latex]\sqrt{− \frac{C}{A}}[/latex], where AC < 0;
- If the function is of the form f(x) = A[latex]{x}^{2}[/latex], the zeros are : [latex]x_{1}[/latex]= 0 and [latex]x_{2}[/latex]= 0.
Examples
- The graphical representation of a second-degree polynomial function defined by the relationship [latex]f(x) = {x}^{2}[/latex] is a basic parabola.
- The graphical representation of a second-degree polynomial function defined by the relationship [latex]f(x) = {(x − a)}^{2}[/latex] is a basic parabola translated horizontally.
- The graphical representation of the second-degree polynomial function defined by the relationship [latex]f(x) = {x}^{2} + k[/latex] is a basic parabola translated vertically.
- The graphical representation of the second-degree polynomial function defined by the relationship [latex]f(x) = a{(x − h)}^{2} + k[/latex] is a basic parabola translated horizontally and vertically.
This graph illustrates the function f defined by f(x) = [latex]\left ( x+3 \right )^{2} - 4[/latex]