Graphs for mod functions

Question

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

1. $2+3|x-1|$

2. $|x-1|+|x|+|x+1|$

3. $|x-1|-|x|-|x+1|$

4. $|x-1|^2$

Thanks.

Answer 1

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. $$f(x)=|x-1|-|x|-|x+1|=\begin{cases} -x-2 & \textrm{for}\: x\geq1\\ -3x & \textrm{for}\:0<x<1\\ -x & \textrm{for}\:-1<x\leq0\\ x+2 & \textrm{for}\: x\leq-1 \end{cases}$$

Answer 2

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.