
We know that the absolute value of a number is always positive (or zero).
We can see this same result reflected in the graph of the absolute value parent function y =  x . All of the graph's yvalues will be positive (or zero).
The graph of the absolute value parent function is composed of two linear "pieces" joined together at a common vertex (the origin). The graph of such absolute value functions generally takes the shape of a V, or an upsidedown V. Notice that the graph is symmetric about the yaxis.
Linear "pieces" will appear in the equation of the absolute value function in the following manner:
y =  mx + b  + c
where the vertex is (b/m, c) and the axis of symmetry is x = b/m. 
Note that the slope of the linear "pieces" are +1 on the right side and 1 on the left side. Remember that when lines are perpendicular (form a right angle) their slopes are negative reciprocals. 
The absolute value function is one of the most recognized piecewise defined functions.

Features (of parent function):
• Domain: All Reals (∞,∞)
Unless domain is altered.
• Range: [0,∞)
• increasing (0, ∞)
• decreasing (∞,0)
• positive (∞, 0) U (0, ∞)
• absolute/relative min is 0
• no absolute max (graph → ∞)
• end behavior
f (x)
→ +∞, as x → +∞
f (x)
→ +∞, as x → ∞


Symmetric:
about x = 0
unless transformed
xintercept:
intersects xaxis at
(0, 0)
unless transformed
yintercept:
intersects yaxis at
(0, 0)
unless transformed
Vertex:
the point (0,0)
unless transformed


Range: When finding the range of an absolute value function, find the vertex (the turning point).
•
If the graph opens upwards, the range will be greater than or equal to the ycoordinates of the vertex.
• If the graph opens downward, the range will be less than or equal to the ycoordinate of the vertex.
Average rate of change:
is constant on each straight line section (ray) of the graph.


For help with absolute value graphs on your calculator,
Click Here! 




Absolute Value Function  Transformation Examples:
Translations 
Reflection 
Vertical Stretch/Shrink

General Form of Absolute Value Function: f (x) = a  x  h  + k
• the vertex is at (h,k)
• the axis of symmetry is x = h
• the graph has a vertical shift of k
• the graph opens up if a > 0, down if a < 0
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