Functions
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Absolute Value Functions

The V-shape that measures distance from zero

The absolute value of a number is its distance from zero — always positive. The function y = |x| creates a distinctive V-shape: it goes down-left like y = −x, then bounces up-right like y = x.

You can write it as a piecewise function: when x < 0, y = −x; when x ≥ 0, y = x. Shifts like y = |x − 3| + 2 move the V right 3 and up 2. Solving |x − 3| = 5 means finding two points where the V-shape hits y = 5.

Try the sliders to shift and stretch the V — or ask the AI "Solve |x − 3| = 5".

Graph

FAQ

What does absolute value mean?
Absolute value measures distance from zero on the number line. |−7| = 7 and |7| = 7 because both are 7 units from zero. It strips away the sign.
Why is the graph a V-shape?
For positive x, |x| = x (a line going up-right). For negative x, |x| = −x (a line going up-left). These two lines meet at the origin, forming a V.
How do I shift the absolute value graph?
y = |x − h| + k shifts the vertex to (h, k). So y = |x − 3| + 2 moves the V to vertex (3, 2). The h shifts horizontally (opposite sign!), and k shifts vertically.
How do I solve an absolute value equation?
An equation like |x − 3| = 5 has two solutions because distance can go in two directions. Set x − 3 = 5 (giving x = 8) and x − 3 = −5 (giving x = −2). On the graph, these are the two points where the V-shape intersects y = 5.

Related Topics

Piecewise FunctionsFunction TransformationsLinear Equations