|x|

Absolute Value Functions

The V-shape — reflections and shifts in one clean fold.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualFold the line y=x in half along the y-axis — every negative output flips positive.

Algebra $y = |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$ . Vertex at origin.

y = |x|

VisualDrag the red vertex — the V pivots around (h,k). a controls the opening steepness and direction.

Algebra $y = a|x-h| + k$ . Vertex at $(h,k)$ . Opens up if $a>0$ , down if $a<0$ .

y = a|x-h| + k

VisualTwo candidate solutions — one from each arm of the V. Always verify in the original.

Algebra $|x-h|=d \iff x-h = \pm d$ .

|A| = d \implies A = \pm d

Absolute value parent

y = |x|

Vertex at origin

Vertex form

y = a|x-h|+k

Vertex (h,k)

Split rule

|A|=d \Rightarrow A=\pm d

d ≥ 0 required

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easyabsolute value

Find the vertex of

y=|x-2|+3

easyabsolute value

Evaluate

|-7| + |4|

mediumabsolute value

Solve

|2x-3| = 5

mediumabsolute value

Write

y=|x|+2

as a piecewise function.

hardabsolute value

Solve

|x-3| < 2

. Express as an interval.