# How do you graph f(x)= -1/4|x-2|+2?

Dec 22, 2016

Socratic graph is inserted. See the explanation.

#### Explanation:

As $8 - 4 y = \left\mid x - 2 \right\mid \ge 0 , y \le 2$.

Let me introduce the inverse operator ${\left\mid \ldots \right\mid}^{- 1}$ for abs =

$\left\mid \ldots \right\mid$.

If g(y) = $\left\mid f \left(x\right) \right\mid$, then

f(x) =

(abs)^(-1)(g(y)) = g(y), for $f \left(x\right) \ge 0$ and

$$                          = - g(y), for f(x) <= 0.


Here, f(x) = x - 2 and g(y) = 8 - 4y. And so,

$x - 2 = 8 - 4 y$ or $x = 10 - 4 y$, for $x - 2 \ge 0$

$x - 2 = 4 y - 8$ or $x = 4 y - 6$, for $x - 2 \le 0$.

The given equation is the combined equation, for these separate

piecewise equations

See the graph.

The vertex $\left(2 , 2\right)$ is the zenith of this pair.

graph{y+1/4|x-2|-2=0 [-20, 20, -10, 10]}