A core negative power function has the form $P_n(x) = x^n = \frac 1 {x^{-n}}$ where $n < 0$ is a negative integer.
For example, $P_{-3}(x) = x^{-3} = \frac 1 {x^3}$.

A core negative power function is the simplest of rational functions .

Since "division by $0$" doesn't make sense,  $0$ is not in the domain of $P_n$ when $n<0$, the line $x=0$ has no point in common with the graph of $P_n$ and $x=0$ is a "pole" for $P_n$.

A core negative power function can understood as the composition of a core positive power function with the core negative power function $P_{-1}$.
For example $P_{-5} = P_{-1} \circ P_5$.

The "Vertex" Form:

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

$f(x) = a_n (x-h)^n + k = \frac {a_n} {(x-h)^{-n}} +k$  where $n <0$.
When $n$ is an odd number then $f$ has very large positive values on one side close to the pole $x=h$ and very large negative values close to the other side of the pole $x=h$.

When $n$ is an even number
with $a_n > 0$ then $f$ has very large positive values close to the pole $x=h$ while with $a_n < 0$ then $f$ has very large negative values close to the pole $x=h$.

When $x$ is a very large positive or negative number, then $f(x)$ is very close to $k$, which is described on the graph by saying $y = k$ is a horizontal asymptote  for the graph of $f$.

We explore how mapping diagrams for core negative power functions help us visualize and understand further the meaning of the shape of the graph of $P_n$ when $n<0$..

Example OAF.CNPF.1 : $P_{-3}(x) = x^{-3} = \frac 1 {x^3}$.

Example OAF.CNPF.2   $P_{-4}(x) = x^{-4} = \frac 1 {x^4}$.

Example OAF.CNPF.3 :  $P_{-5}(x) = x^{-5} = \frac 1 {x^5}$

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

Example OAF.DCNPF.0 : Dynamic Visualization of Core Negative Power Function: Graphs and Mapping Diagrams

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