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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 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. |
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.
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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 → -∞
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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
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Table:
Y1: y = | x | 
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Range: When finding the range of an absolute value function, find the vertex (the turning point).
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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.
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Transformations on Absolute Value |
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We know that transformations have the ability to move functions by sliding, reflecting, rotating, stretching, and shrinking them. Let's see how these changes will affect the absolute value function:
Absolute Value Function - Transformation Examples:
Translation
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Translations:
Vertical Shift: f (x) + k
Horizontal Shift: f (x + h)
Reflections:
-f (x) over x-axis
f (-x) over y-axis
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Reflection
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Vertical Stretch/Compress
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Vertical Stretch/Compress
c • f (x) stretch ( |c| > 1)
c • f (x) compress (0 < |c| < 1)
Horizontal Stretch/Compress
f (c • x) stretch (0 < |c| < 1)
f (c • x) compress ( |c| > 1) |
Horizontal Stretch/Compress
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Special Note: When working with absolute value, a vertical stretch can equal a horizontal compression. Consider: | 3x | = 3 | x |. This is true since a property of absolute value states that "the absolute value of a product is the product of the absolute value."
| a • b | = | a | • | b |
y = 3 | x | is a vertical stretch y = | 3x | is a horizontal compression.
 For an absolute value function, a vertical stretch is equivalent to a horizontal compression.
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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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For help with absolute value
graphs
on your
calculator.
click here.
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For help with absolute value
plus
on your
calculator,
click here. |
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