In this problem we will examine the short-run behavior of the following functions.

a) $f(x)=(x+3)^2 (x-2)(x-4)^2$
b) $g(x)=(x+1)(x-2)^3$
c) $m(x)=-(x+6)(x+3)(x-1)(x-2)$
d) $n(x)= (x-4)(x+2)^2 (x-3)^3$

Complete the following table using the formulas for each of the above functions and any additional information you can obtain from the graph of each function. You may want to sketch graphs on your own piece of paper. You are welcome to use an online graphing utility like www.desmos.com if you would like

Function features
$f(x)$
$g(x)$$2$
$m(x)$$4$
$n(x)$

b) In the order the roots appear in the factored forms of the functions provided, list the roots, the multiplicity of the roots, and if the function "bounces" off or "crosses" the $x$-axis at that root. If there are more blanks provided than needed for a given function, enter NONE in all remaining, unneeded blanks. Most of the information for function $f(x)$ has been completed as an example.

$f(x)$: $x = -3$, multiplicity: $2$,
$\hspace{2mm}$$x = 2$, multiplicity: $1$,
$\hspace{2mm}$$x =4$, multiplicity: $2$,
$\hspace{2mm}$$x =$NONE, multiplicity: NONE, NONE

$g(x)$: $x =$ , multiplicity: ,
$\hspace{2mm}$$x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,
$\hspace{2mm}$$x =$ , multiplicity: ,

$m(x)$, $x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,

$n(x)$, $x =$ , multiplicity: ,
$\hspace{2mm}$$x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,
$\hspace{2mm}$ $x =$ , multiplicity: ,