The Graphs of General Polynomial Functions

Compared with the graph of $f\left(x\right)=ax^n,\:$ the following also hold true.

The graph of $f\left(x\right)=-ax^n\:$ is reflected across the x-axis.
The graph of $f\left(x\right)=ax^n+k\:$ is translated (shifted) k units up if k>0 and $\mid k\mid\:$ units down if k<0.
The graph of $f\left(x\right)=a\left(x-h\right)^n\:$ is translated (shifted) h units to the right if h>0 and $\mid h\mid\:$ units to the left if h<0.
The graph of $f\left(x\right)=a\left(x-h\right)^n+k\:$ shows a combination of these translations.

Graph each polynomial function. Determine the largest open intervals of the domain over which each function is increasing or decreasing.

$f\left(x\right)=x^5-2$

The graph will be the same shape as that of $f\left(x\right)=x^5,\:$ but translated 2 units down. This function is increasing on its entire domain $\left(-\infty,\infty\right)$ .
$f\left(x\right)=\left(x+1\right)^6$
In $f\left(x\right)=\left(x+1\right)^6$ , the function has a graph with the same shape as $f\left(x\right)=x^6$ , but since $x+1=x-\left(-1\right)$ , it is translated 1 unit to the left. This function is decreasing on ( $-\infty,-1$ ] and increasing on [ $-1,\infty$ ).
$f\left(x\right)=-2\left(x-1\right)^3+3$
The negative sign in -2 causes the graph of the function to be reflected across the x-axis when compared with the graph of $f\left(x\right)=x^3$ . Because $\mid-2\mid>1$ , the graph is stretched vertically when compared to the graph of $f\left(x\right)=x^3$ . It is also translated 1 unit right and 3 units up. This function is increasing on its entire domain $\left(-\infty,\infty\right)$ .

Domain & Range of Polynomial Functions
Unless otherwise restricted, the domain of a polynomial function is the set of all real numbers. Polynomial functions are smooth, continuous curves on the interval $\left(-\infty,\infty\right)$ . The range of a polynomial function of odd degree is also the set of all real numbers.