# Graphing Functions of the form ${x}^{n}$

A
**
polynomial function
**
is a
function
in which
$f\left(x\right)$
is a polynomial in
$x$
.

A polynomial function of degree $n$ is written as $f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$ .

The graph of a
**
polynomial function of degree
$0$
**
, which is of the form
$f\left(x\right)=a$
is a
horizontal line
.

The graph of a
**
polynomial function of degree
$1$
**
, which is of the form
$f\left(x\right)=ax+b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\ne 0$
, is a non-horizontal line. Read more about graphing lines
here
.

The graph of a
**
polynomial function of degree
$2$
**
, also called a
quadratic equation
, which is of the form
$f\left(x\right)=a{x}^{2}+bx+c,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\ne 0$
, is a parabola. Read more about graphing parabolas
here
.

Higher-degree polynomials can be somewhat more difficult to graph. Some easy cases are the parent graphs $f\left(x\right)={x}^{3},f\left(x\right)={x}^{4},f\left(x\right)={x}^{5}$ , etc. These graphs have a predictable shape depending on whether the exponent is even or odd, as shown below.

Other higher-degree polynomials with more terms require other techniques. One method is to factor the polynomials to find $x$ -intercepts.

You can also use calculus to find the critical points of a polynomial function.