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

Where would you start if you were told to plot:

$$||x-1|-1|$$

I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then: $\pm(\pm(x-1)-1)$, but how would I know which regions which equation would work for?

• Carefully think about it, for example if $x\geq 2$, there is no need for the absolute value, hence $||x-1|-1|=x-2$ for $x\geq 2$. – Mathematician 42 Apr 29 '16 at 18:50

## 3 Answers

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 $\bigl\lvert \lvert x - 1 \rvert - 1 \bigr\rvert$.

• It's more like folding it across the $x$ axis, not reflecting it. Reflecting it would be negation; the positive parts would become negative as well as the negative parts becoming positive. – user2357112 Apr 29 '16 at 19:42
• Please listen to user2357112 and don't use the word "reflection" because "reflection" has a specific meaning in math which is different from what you want it to mean in this answer. Somebody at the level of the OP could become terribly confused. – Fixed Point Apr 29 '16 at 23:08
• @FixedPoint fixed. – dezdichado Apr 30 '16 at 4:16
• You still have the word "reflection" in the very first line. – Fixed Point Apr 30 '16 at 6:45
• plus 1, because it is always nice to see visually – i squared - Keep it Real Apr 29 '16 at 19:23
• +1; Also, there's a vertical bar missing in the formula for the fourth picture. – Daan Michiels Apr 29 '16 at 20:03

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.