# 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 $x\geq 2$. Apr 29, 2016 at 18:50

1) $y=|x|$

2) $y=|x-1|$

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

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

• plus 1, because it is always nice to see visually
– nz_
Apr 29, 2016 at 19:23
• +1; Also, there's a vertical bar missing in the formula for the fourth picture. Apr 29, 2016 at 20:03

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. Apr 29, 2016 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. Apr 29, 2016 at 23:08
• @FixedPoint fixed. Apr 30, 2016 at 4:16
• You still have the word "reflection" in the very first line. Apr 30, 2016 at 6:45

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.