Absolute Value & Piecewise Functions

|x| answers exactly one question: how far is x from zero on the number line? Distance has no direction, so it is never negative — |5| = 5 and |−5| = 5, because both 5 and −5 sit exactly 5 units from zero, one to the right and one to the left.

That single idea — strip away the direction, keep the size — is all absolute value ever does. Everything else in this lesson, the piecewise rule and the V-shaped graph, is just that idea written out in different notations.

Plot y = |x| and the two piecewise branches show up as two straight rays: for x ≥ 0 the line y = x, and for x < 0 the line y = −x. They meet at the origin, forming a sharp V. The point where the two rays meet, (0, 0), is the vertex, and the vertical line x = 0 is the graph’s axis of symmetry — the left half is a mirror image of the right half.

This should look familiar — it is exactly the vertex-form pattern from transforming a parabola, applied to a V instead of a curve. The vertex sits at (h, k); h slides the V left or right, k slides it up or down. The number a controls the steepness — |a| > 1 makes the V narrower, |a| < 1 makes it wider — and if a is negative, the V flips upside down into a Λ shape, opening downward instead of upward.

|x| is the simplest example of a piecewise-defined function — a function built from different rules on different parts of the domain, like a shipping rate that charges one formula under 5 kg and a different formula at or above 5 kg. Reading a piecewise graph means checking, at each x-value, which piece of the domain you are in and applying only that branch’s rule; sketching one means drawing each branch only over its own stated interval and marking clearly whether each boundary point is included.