Quick answer

y = |x| falls with slope −1 for x < 0 and rises with slope 1 for x ≥ 0.

Formula

  • Vertex of y = |x − h| + k is (h, k)
  • y = |ax + b| is a V with corner where ax + b = 0

Introduction

Plotting |x| helps you see why equations can have two solutions and why inequalities look like intervals. The graph makes the piecewise rule visible.

The basic graph is a V with vertex at the origin. Coefficients and shifts move the corner and change steepness without flipping the upward opening.

The Absolute Value Calculator supplies numeric values for table building. Link graph ideas to absolute value equations and absolute value inequalities when you analyze intersections.

Shape and symmetry

The left and right arms are mirror images across the vertical line through the vertex. The graph is never below the x-axis because |x| ≥ 0.

A horizontal line y = k crosses twice when k > 0, once when k = 0, and never when k < 0. That picture matches the solution count for |x| = k.

For y = |x − h| + k, the vertex moves to (h, k). Inside the bars shifts horizontally; outside shifts vertically.

Slopes on y = |x| are −1 for x < 0 and 1 for x ≥ 0. The corner is sharp, not smooth, which matters in later calculus topics.

Key features

  • y = |x|: vertex (0, 0)
  • y = |x − 3|: vertex (3, 0)
  • y = |x| + 2: vertex (0, 2)
  • y = |ax + b|: corner where ax + b = 0

Make a small table with x values on both sides of the corner, then plot points and draw two rays.

Strict inequalities correspond to segments below a horizontal line on the graph; greater-than forms use the outer arms.

For transformations inside algebra courses, connect each shift to the vertex coordinates before you plot.

Graphing steps

  1. Make a table. Choose x values on both sides of the corner point. Include the x that makes the inside zero.
  2. Compute y = |expression|. Use the definition or the calculator for each row.
  3. Plot points. Connect with two straight rays meeting at the vertex.
  4. Mark the vertex. Label the lowest point on y = |x| or the shifted corner for other forms.

Quick sketches

For y = |x − 1|, plot (0, 1), (1, 0), and (2, 1). The V opens upward with vertex (1, 0).

For y = |x| and y = 3, intersections are at (−3, 3) and (3, 3), matching |x| = 3.

For y = |2x|, the corner stays at (0, 0) but the arms are steeper because x changes faster inside the bars.