Thread: Graphing Absolute value function with 2 absolute value components.

1. Graphing Absolute value function with 2 absolute value components.

Hi,

Im trying to solve a problem from my algebra 2 book having to do with absolute value function graphing. The equation is:

y=|x+1|+|x|

here is the graph:
Graph 1.pdf

What I have done is find both of the slanted lines algebraically, and i found those to be:
y=2x+1 and y=-2x-1.

I did that by splitting the original equation into 2 different positive and negative equations, then simplifying:
y=x+1+x=2x+1
y=-(x+1+1)=-2x+1

I have the feeling i did that prematurely, because i didn't find the domain of each equation.

Basically my biggest question stems from the conceptual perspective. What do you do when there are two absolute value terms in an function and how do you find the individual functions and their respective domains?

Thanks!!

2. Re: Graphing Absolute value function with 2 absolute value components.

Originally Posted by bass1989
Hi,

Im trying to solve a problem from my algebra 2 book having to do with absolute value function graphing. The equation is:

y=|x+1|+|x|

here is the graph:
Graph 1.pdf

What I have done is find both of the slanted lines algebraically, and i found those to be:
y=2x+1 and y=-2x-1.

I did that by splitting the original equation into 2 different positive and negative equations, then simplifying:
y=x+1+x=2x+1
y=-(x+1+1)=-2x+1

I have the feeling i did that prematurely, because i didn't find the domain of each equation.

Basically my biggest question stems from the conceptual perspective. What do you do when there are two absolute value terms in an function and how do you find the individual functions and their respective domains?

Thanks!!
There's no magic going on.

$|x+1|=\begin{cases}-(x+1) &x\leq -1 \\(x+1) &x \geq -1\end{cases}$

$|x|=\begin{cases}-x &x\leq 0\\x &x\geq 0\end{cases}$

so there are 3 regions of interest

$x \leq -1$

$-1 < x \leq 0$

$0 < x$

and looking carefully you see

$|x+1|+|x|=\begin{cases} -(x+1)+(-x) & x \leq -1 \\ (x+1)+(-x)&-1 < x \leq 0 \\ (x+1)+x&0 < x \end{cases}$

$|x+1|+|x|=\begin{cases} -2x-1 & x \leq -1 \\ 1&-1 < x \leq 0 \\ 2x+1&0 < x \end{cases}$

3. Re: Graphing Absolute value function with 2 absolute value components.

Thank you very much for your reply! Upon working on the problem a bit more I figured out the last portion, where you find the equations for each region of interest, but I could still not figure out how to get the domains of each equation.

Thanks!

4. Re: Graphing Absolute value function with 2 absolute value components.

I can't give you any general recipe for dealing with absolute values, other than using the definition. The definition of course is |x| = -x when x < 0, and |x| = x when x is greater than or equal to 0. Here's your specific problem:

5. Re: Graphing Absolute value function with 2 absolute value components.

Originally Posted by bass1989
Thank you very much for your reply! Upon working on the problem a bit more I figured out the last portion, where you find the equations for each region of interest, but I could still not figure out how to get the domains of each equation.

Thanks!
from the definition of $|x|=\begin{cases}x &x\geq 0\\ -x &x<0\end{cases}$

on what domain is (x+1) > 0?

on what domain is (x+1) < 0?