Absolute value graph: y = |x|

Q: What does the absolute value graph look like?

A V-shape with its vertex at the origin, made of the lines y = x (for x ≥ 0) and y = −x (for x < 0).

Q: How do I find the vertex of y = a|x − h| + k?

The vertex is (h, k). The sign of a tells you whether the V opens up (a > 0) or down (a < 0).

Q: Is |x| differentiable?

Everywhere except x = 0, where the graph has a corner. For x ≠ 0 the derivative is the sign of x.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The absolute value graph is a V. For x ≥ 0 it is the line y = x, for x < 0 it is the line y = −x, and the two halves meet at a sharp corner at the origin.

Open in calculator

The graph of

y=|x|

for −6 ≤ x ≤ 6. Dots mark its vertex.

Key features

Domain: All real numbers
Range: y ≥ 0
Vertex: (0, 0), a minimum
Slopes: −1 for x < 0, +1 for x > 0
Piecewise form: y = x if x ≥ 0, y = −x if x < 0
Symmetry: Even: |−x| = |x|
Differentiability: Not differentiable at x = 0 (a corner)
y-intercept: None
Derivative: sign(x)
Antiderivative: x|x|2+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of |x| looks like

The absolute value measures distance from zero, so it is never negative: |−3| = |3| = 3. The graph keeps the right half of the line y = x and reflects the left half up above the x-axis.

The corner at the origin is where the slope jumps from −1 to +1. There is no single tangent line there, which is why |x| is the standard example of a continuous function that is not differentiable.

Absolute value graphs appear wherever a quantity is a distance: |x − 3| is the distance from x to 3, and its graph is the same V moved so its vertex sits at x = 3.

Points of interest

Found numerically to about twelve digits in the window −6 ≤ x ≤ 6, then shown in exact form where one exists.

x-intercepts (roots): x = 0
Local maxima: None in this window
Local minima: None in this window
Inflection points: (−6, 6), (−5.997, 5.997), (−5.994, 5.994), (−5.991, 5.991), (−5.988, 5.988), (−5.985, 5.985) … (4001 in view)

Table of values

A few points to plot, with exact values where they are simple.

Values of |x|
x	y = \|x\|
−3	3
−2	2
−1	1
0	0
1	1
2	2
3	3

How to graph |x| by hand

1
Plot the vertex (0, 0).
2
Plot (1, 1), (2, 2), (3, 3) to the right and (−1, 1), (−2, 2), (−3, 3) to the left.
3
Join each side with a straight ray, meeting in a sharp point at the origin.

Transformations

How the graph moves when you change the formula. The original is |x| in blue.

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y = abs(x - 2) + 1 Vertex moved to (2, 1).
y = -2abs(x) Flipped upside down and twice as steep.
y = abs(x^2 - 4) Absolute value of a parabola: the part below the axis is reflected up, making a W.