For example: $P_3(x) = x^3$.

A general polynomial is a weighted sum of core positive power
functions together with the constant function $P_0(x) = 1$ .

Simple modifications of $P$ can involve compositions with the core linear functions, giving functions in a "vertex" form similar to that for quadratic polynomials:

$f(x) = a_n (x-h)^n + k$.

When $n$ is an odd number, then $f$ always has a unique root, $c$
where $f(c)=0$.When $n$ is an even number, then $f$ may or may not have one, two or no roots, $c$ where $f(c)=0$. This will depend on the relation of $a_n$ and $k$.

If $a_n*k > 0$ then $f$ will have no real roots.

If $a_n*k < 0$ then $f$ will have exactly two distinct roots.

If $k = 0$ then $f$ will have exactly one root: $x = h$.

We explore how mapping diagrams for core positive polynomial functions help us visualize and understand further the meaning of the shape of the graph of $P_n$.