Absolute value function

Definition

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.

5 Must Know Facts For Your Next Test

1. The standard form of an absolute value function is $f(x) = |x|$.
2. The graph of an absolute value function forms a 'V' shape with its vertex at the origin (0,0).
3. For $f(x) = |ax + b| + c$, the vertex is located at $(-\frac{b}{a}, c)$.
4. Absolute value functions are piecewise-defined: for $x \geq 0$, $f(x) = x$; for $x < 0$, $f(x) = -x$.
5. 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.