Question

How do you graph `y = 2 | x - 1 | - 3`?

Answer 1

Substitute the definition: `|A|={(A;A>=0),(-A;A<0):}`
Simplify the domain restrictions and the equation.
Start at the common point plot and plot the two rays extending from that point.

Explanation:

Given: `y = 2 | x - 1 | - 3`

Substitute `(x-1)` for `|x-1|` and the domain restriction `x-1>=0`
`y = 2( x - 1 ) - 3; x-1>=0`

Substitute `(1-x)` for `|x-1|` and the domain restriction `x-1<0`
`y = 2( 1-x ) - 3; x-1<0`

Simplify the domain restrictions:

`y = 2( x - 1 ) - 3; x>=1`
`y = 2( 1-x ) - 3; x<1`

Simplify the equations:

`y = 2x - 5; x>=1`
`y = -2x - 1; x<1`

Please observe that the common point is `(1,-3)`; the graph is two rays extending from this point.

Another point on the line with positive slope is `(2,-1)`
Another point on the line with negative slope is `(0,-1)`

The following is a graph of the function:
graph{y=2|x-1|-3 [-14.24, 14.24, -7.12, 7.12]}