Let's say we have the expression $f(x) = \left|\left|x-1\right|\ -3\left|x+1\right|\right|$.
To draw it's graph, I would normally find the critical points ($x = 1, x = -1, x = \frac{-1}{2}$), then find the corresponding values of $y$ for each f(x), then join the points making the graph.
This would give us a graph looking like this. However, it turns out this graph is actually for $f(x) = \left|\left|x-1\right|\ +3\left|x+1\right|\right|$.
Normally, for the $3|x+1|$ expression, I would multiply the expression with the number outside it. But when that number is negative, the graph I was making was wrong, and my logic is incorrect.
The actual graph for $f(x) = \left|\left|x-1\right|\ -3\left|x+1\right|\right|$ is here.
So here is my question. How do you draw the graph of a modulus function where 2 modulus expressions are subtracted, and how do you go about multiplying a modulus expression (like $|x+1|$ with $-3$) in $f(x)$ with a negative number?
Best Answer
Draw the graph of $|x-1|-3|x+1|$ then reflect the parts of the graph which are below the $x$ axis with respect the $x$ axis. Draw the graph of the above function $f(x)=|x-1|-3|x+1|$ for the interval $(-\infty, -1]$, where $f(x)=-(x-1)+3(x+1)$. Then for the interval $[-1,1]$, and finally for $[1,+\infty)$.
This is the way.