# How do you graph y= |x - 2| + 2?

Dec 26, 2017

you can see that:
when $x = 2$, $y \left(2\right) = \left\mid 2 - 2 \right\mid + 2 = 0 + 2 = 2$

when $x = 1$, $y \left(1\right) = \left\mid 1 - 2 \right\mid + 2 = 1 + 2 = 3$
when $x = 3$, $y \left(3\right) = \left\mid 3 - 2 \right\mid + 2 = 1 + 2 = 3$

when $x = 0$, $y \left(0\right) = \left\mid 0 - 2 \right\mid + 2 = 2 + 2 = 4$
when $x = 4$, $y \left(4\right) = \left\mid 4 - 2 \right\mid + 2 = 2 + 2 = 4$

when $x = - 1$, $y \left(- 1\right) = \left\mid - 1 - 2 \right\mid + 2 = 3 + 2 = 5$
when $x = 5$, $y \left(5\right) = \left\mid 5 - 2 \right\mid + 2 = 3 + 2 = 5$

and so on, in general:
when $x = 2 - a$, $y \left(2 - a\right) = \left\mid \cancel{2} - a \cancel{- 2} \right\mid + 2 = \left\mid - a \right\mid + 2 = a + 2$
when $x = 2 + a$, $y \left(2 + a\right) = \left\mid \cancel{2} + a \cancel{- 2} \right\mid + 2 = \left\mid + a \right\mid + 2 = a + 2$

We see this is a linear

so the graph will have a min when $x = 2$ and go linear in each side (down when $x < 2$ and up when $x > 2$)

graph{abs(x-2)+2 [-10, 10, -1, 9]}