-An absolute value function has the form of y= |f(x)| where f(x) is a function.
-The x-intercept of the graph of f(x) is a critical point. The graph of y=|f(x)| changes direction at this point.
-The domain of y=|f(x)| is . The range is , which means that the graph always stays in quadrant 1 and/or 2. Anything supposed to be in quadrant 3 and\or 4 will be reflected through the x-axis.
-The absolute value of a number is often defined using piecewise notation:
For example: f(x)= |x|

*To write an absolute value function in piecewise notation:
-Start by identifying when the expression in the absolute value symbols is positive, zero, or negative.
-Identifying x-intercept (critical point).
-Deciding the restriction for x, which create definitions for different subsets of the domain.
Example 1: y = |2x-1|. X-intercept is
When , the expression (2x-1) is positive or zero.
So, |2x-1| = 2x-1 for .
When x< , the expression (2x-1) is negative. So, |2x-1|= -(2x-1) for x< . Example 2: . X-intecepts are x=-3 and x=0. When , the expression is positive or zero. So, for . When x<-3 or x>0, the expression is negative.
So, for x<-3 or x>0.
*Graphing:
Base on piecewise notation, graph accordingly. Remember, all the part in quadrant 3 and 4 will be reflected through x-axis.
-Example 1: y=|3x+6|. X-intercept (critical point) is x=-2
When , the expression (3x+6) is positive or zero.
So, |3x+6| = 3x+6 for .
When x<-2, the expression (3x+6) is negative. So, |3x+6| = -(3x+6) for x< -2, which means the part which x< -2 will be reflected through x-axis.

The resulting graph will be:

-Example 2: . X-intercepts are x=-3 and x=-1.
When , the expression (-2(x+2)^2 +2) is positive or zero.
So, |-2(x+2)^2 +2| = -2(x+2)^2 +2 for .
When x <-3 or x>-1, the expression (-2(x+2)^2 +2) is negative.
So, |-2(x+2)^2 +2| = -(-2(x+2)^2 +2) for x <-3 or x>-1.

The final result will be: