Question

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

Answer 1

Socratic graph is inserted. See the explanation.

Explanation:

As `8 - 4y = abs(x-2) >= 0, y <=2`.

Let me introduce the inverse operator `abs(...)^(-1)` for abs =

`abs(...)`.

If g(y) = `abs(f(x))`, then

f(x) =

`(abs)^(-1)(g(y)) = g(y)`, for `f(x) >= 0` and

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

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

`x -2 = 8 - 4y` or `x = 10 - 4y`, for `x-2 >= 0`

`x - 2= 4y - 8` or `x = 4y - 6`, for `x - 2 <= 0`.

The given equation is the combined equation, for these separate

piecewise equations

See the graph.

The vertex `(2, 2)` is the zenith of this pair.

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