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
$f(x) = a_n (x-h)^n + k$.
When $n$ is an odd number, then $f$ always has a unique root, $c$
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 $P_n$.
Dynamic Visualization of Core
Positive Power Function: Graphs and Mapping Diagrams
Dynamic Visualization of Vertex Form
for a Positive Power Function: Graphs and Mapping Diagrams