Question

How do you graph `f(x)=2|x-1|-3`?

Answer 1

Use the graph of the parent function `f(x)=absx` but change the slope to of each half of the graph to `+-2`, and shift the vertex right one unit and down three units.

Explanation:

The parent function of absolute value is `f(x)=abs(x)` and looks like
the graph below. Note the vertex is at `(0,0)` and the slope of the left branch of the graph is `-1` while the slope of the right branch is `+1`.
graph{abs(x) [-10, 10, -5, 5]}

Given the form, `f(x)=mabs(x-h)+k`,

`+-m` represents the slope of each branch of the graph. A positive value of `m` in the original equation means the graph has a "V" shape, while a negative value of `m` means the graph has an "upside-down V-shape".

`(h,k)` is the vertex. The vertex is shifted horizontally by h units and vertically by k units.

`f(x)=2abs(x-1)-3`

In this example, `m=2` is positive, and the graph will have a V-shape. The left branch of the V has a slope of `-2` and the right branch has a slope of `2`.

The vertex is `(1,-3)`, which represents a shift of one unit to the right and 3 units down.
graph{2abs(x-1)-3 [-10, 10, -5, 5]}