We know that the absolute value of a number is always positive.

We can see this same result reflected in the graph of the absolute value parent function y = | x |. All of the graph's y-values 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 up-side-down V. Notice that the graph is symmetric about the y-axis.

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.

abs12
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.

ABsoluteFeatures
The absolute value function is one of the most recognized piecewise defined functions.

CRgraph2ABS1a
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

x-intercept:
intersects x-axis at (0, 0)
unless transformed

y-intercept:
intersects y-axis at (0, 0)
unless transformed

Vertex:
the point (0,0)
unless transformed


Table: Y1: y = | x | CRgraph1
Read more about Absolute Value.
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 y-coordinates of the vertex.
• If the graph opens downward, the range will be less than or equal to the y-coordinate of the vertex.

Average rate of change:
is constant on each straight line section (ray) of the graph.

ti84c
For help with absolute value graphs on your calculator,
Click Here!

Absolute Value Function - Transformation Examples:
ABS3
Translations
abs2
Reflection
ABS4
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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