Graphs for mod functions Can someone please teach me how to obtain graphs for the following  types of functions:


*

*$2+3|x-1|$

*$|x-1|+|x|+|x+1|$

*$|x-1|-|x|-|x+1|$

*$|x-1|^2$
Thanks.
 A: You should determine the critical points of the functions meanly values that make zero the absolute value. Then you get piecewise functions. For example for the first function $x=1$ is the critical point. Therefore your piecewise function will be as follows. 
$$f(x)=2+3|x-1|=\begin{cases}
2+3x-3 & \textrm{for}\: x\geq1\\
2+3-3x & \textrm{for}\: x<1
\end{cases}$$
Now it is easy to plot the function.
Edit:
For third one you have three critical points which are $-1,0,1$. So you have to investigate your function for intervals $(-\infty,-1]$, $(-1,0]$, $(0,1)$, $[1,+\infty)$,   after that you can get the following piecewise function.
$$f(x)=|x-1|-|x|-|x+1|=\begin{cases}
-x-2 & \textrm{for}\: x\geq1\\
-3x & \textrm{for}\:0<x<1\\
-x & \textrm{for}\:-1<x\leq0\\
x+2 & \textrm{for}\: x\leq-1
\end{cases}$$
A: Well first of all determine how $|x|$ looks. 


*

*$|x-d|$ is the same as $|x|$ but just shifted $d$ units in positive $x$-direction.

*$a|x|+b$ is the same as $|x|$ but shifted by $b$ in positive $y$-direction and streteched by factor $b$ in $y$ direction.


Using these 'rules' makes it way easier than 'stupidly' calculating values.
