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

May 17, 2017

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 $\left(x - 1\right)$ for $| x - 1 |$ and the domain restriction $x - 1 \ge 0$
y = 2( x - 1 ) - 3; x-1>=0

Substitute $\left(1 - x\right)$ 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 $\left(1 , - 3\right)$; the graph is two rays extending from this point.

Another point on the line with positive slope is $\left(2 , - 1\right)$
Another point on the line with negative slope is $\left(0 , - 1\right)$

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