Graph of a sum of absolute values

Text

Sketch the graph of the function $y=|x|+|x-2|-x+3$.

Solution

Breakpoints $x=0$ and $x=2$; remove the absolute values interval by interval.

• $x<0$: $y=-x+(2-x)-x+3=-3x+5$;
• $0\le x<2$: $y=x+(2-x)-x+3=5-x$;
• $x\ge2$: $y=x+(x-2)-x+3=x+1$.

The graph is piecewise linear: slope $-3$ up to $(0;5)$, then $-1$ up to $(2;3)$, then $+1$. The minimum point is $(2;3)$.

$\boxed{y=\begin{cases}-3x+5 & x<0\ 5-x & 0\le x<2\ x+1 & x\ge2\end{cases}}$