Quadratic functions and polynomials

Summary

Given: a real-valued function of a real variable, \(f: R→R\)
We say that \(f\) is a quadratic function if there are constants \(a≠0,b,c\) such that

\[f\left( x \right)=ax^2+bx+c\]

for all \(x∈R\)
c is the intercept of the function.
The graph of a quadratic function is called a parabola . The shape is a "valley" if \(a>0\) and a "hill" if \(a<0\) .
A quadratic function can be written as

\[f\left( x \right)=a{\left( x+ \frac{b}{2a} \right)}^2+c- \frac{b^2}{4a}\]

\[\left( - \frac{b}{2a},c- \frac{b^2}{4a} \right)\]

is called the vertex of the parabola.
We say that \(f\) is a cubic function if there are constants \(a≠0,b,c,d\) such that

\[f\left( x \right)=ax^3+bx^2+cx+d\]

for all \(x∈R\)
We say that \(f\) is a polynomial of degree \(n\) if there are constants \(a_n≠0,a_{n-1},…,a_0\) such that

\[f\left( x \right)=a_nx^n+a_{n-1}x^{n-1}+…+a_1x+a_0\]