Graph of a function with one absolute value

Text

Sketch the function $y=1-|x-2|+x$.

Solution

Breakpoint $x=2$.

• For $x\ge2$: $y=1-(x-2)+x=3$ (horizontal line);
• for $x<2$: $y=1-(2-x)+x=2x-1$ (slope $2$).

The two pieces join at $(2;3)$: the function rises along $y=2x-1$ up to $(2;3)$, then stays constant $y=3$.

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