Why is the absolute value graph V-shaped?

For positive x values, |x| = x, so the right side follows the line y = x. For negative x values, |x| = −x, so the left side follows y = −x (a line with slope −1). These two lines meet at the origin, forming a V.

Is the absolute value function differentiable everywhere?

No. f(x) = |x| is not differentiable at x = 0 because the graph has a sharp corner (cusp) there. From the right, the slope is +1; from the left, the slope is −1. Since these one-sided derivatives are different, the derivative does not exist at x = 0.

Where is the absolute value function used in real life?

Absolute value is used to measure distances (GPS, navigation), calculate errors in data analysis, define norms and distances in vector spaces, represent the magnitude of quantities in physics (speed vs. velocity), and in computer programming to compare or constrain values.

VIZMath · Function Guide

Absolute Value Function

f(x) = |x|

The V-shaped function that measures distance from zero — always returning a non-negative value.

Absolute Value Function

The absolute value function f(x) = |x| returns the non-negative magnitude of any real number x. For positive x, it returns x unchanged; for negative x, it flips the sign and returns -x (making it positive); and |0| = 0.

Its graph is a distinctive V-shape with the vertex (corner point) at the origin. The right branch follows y = x (rising at 45°) and the left branch follows y = -x (also rising at 45° from the other side).

The absolute value is essentially a distance function — |x| measures how far x is from zero on the number line.

Standard Form

f(x) = a|x - h| + k

Vertical stretch/compression and reflection (a < 0 flips it downward)

Horizontal shift — moves the vertex left or right

Vertical shift — moves the vertex up or down

Key Properties

Domain

All real numbers: (-∞, +∞)

Range

[0, +∞) — always non-negative

Vertex

(0, 0) for basic form; (h, k) for transformed form

Symmetry

Even function: |−x| = |x| — symmetric about the y-axis

Slope

+1 for x > 0 and −1 for x < 0 (not differentiable at x = 0)

Shape

V-shaped — two straight rays meeting at the vertex

Examples

f(x) = |x|

Basic V-shape: vertex at origin, slopes of ±1.

f(x) = |x - 2| + 1

Vertex shifted to (2, 1).

f(x) = -2|x|

Inverted and stretched — opens downward with slope ±2.

Visualize the Absolute Value Function

Shift and stretch the V-shape and see the vertex transform in real time

Try in VIZMath Pro →

FAQ

Absolute value means the distance from zero, always giving a non-negative result. |5| = 5, |−5| = 5, and |0| = 0. It strips away the sign and tells you only the size (magnitude) of a number.

Related Functions

Square Root Function Quadratic Function