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".

Related Topics

Piecewise FunctionsFunction TransformationsLinear Equations

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.