An absolute value function is a function that contains an algebraic expression within absolute value symbols. The output of the absolute value function is always non-negative.
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The standard form of an absolute value function is $f(x) = |x|$.
The graph of an absolute value function forms a 'V' shape with its vertex at the origin (0,0).
For $f(x) = |ax + b| + c$, the vertex is located at $(-\frac{b}{a}, c)$.
Absolute value functions are piecewise-defined: for $x \geq 0$, $f(x) = x$; for $x < 0$, $f(x) = -x$.
Shifts and reflections can be applied to the basic absolute value graph by modifying the equation, such as horizontal shifts ($f(x-h)$), vertical shifts ($f(x)+k$), and reflections ($-f(x)$).
Review Questions
What is the shape of the graph of an absolute value function?
How do you determine the vertex of the function $f(x) = |ax + b| + c$?
Write the piecewise definition for $f(x) = |x|$.
Related terms
Piecewise Function: A function composed of multiple sub-functions, each defined on a specific interval.
Vertex: The point where a parabola or 'V'-shaped graph changes direction; in an absolute value graph, it is where the two linear pieces meet.
Transformation: Operations that alter the form of a figure or graph, including translations, reflections, stretches, and compressions.