A core positive power function has the form $P_n(x) = x^n$ where $n > 0$ is a natural number.
For example: $P_3(x) = x^3$.

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

The "leading coefficient" of $P_n$ is $1$ and the value of $P_n(0) = 0$  so $(0,0)$ is always a point on the graph of $P_n$.

The "Vertex" Form:

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 $P_n$.

Example OAF.CPPF.1 : $P_3(x) = x^3$.

Example OAF.CPPF.2 $P_4(x) =x^4$.

Example OAF.CPPF.3 : $P(x) =x^5$.

Example OAF.CPPF.4 : $P(x) =\frac 1 2 (x-1)^3 + 2$.

Example OAF.DCPPF.0 : Dynamic Visualization of Core Positive Power Function: Graphs and Mapping Diagrams

Example OAF.DVPPF.0 : Dynamic Visualization of Vertex Form for a Positive Power Function: Graphs and Mapping Diagrams