Absolute value graph sketching: $||x-1|-1|$

1) $y=|x|$

2) $y=|x-1|$

3) $y=|x-1|-1$

4) $y=||x-1|-1|$

If you know the graph of $f(x)$, then the graph of $|f(x)|$ is just the result of “folding” $f$ about to the $x$ axis. Hence, plotting $f(x) = \bigl\lvert \lvert x - 1 \rvert - 1 \bigr\rvert$ goes like this.

1. Plot $y = x - 1$ and fold the portion of it that lies below the $x$ axis over the $x$ axis. The result is the graph of $\lvert x - 1 \rvert$.

2. Then shift this graph downward by $1$ and do the same folding again. Then you will get $\bigl\lvert \lvert x - 1 \rvert - 1 \bigr\rvert$.

Start by drawing $y = |x|$. Now move it right one and you get the graph of $y = |x - 1|$. Next yank it down one and get the graph of $y = |x-1|-1$. Finally, reflect up the stuff below the $x$-axis and you are done.