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.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