Quick answer

Between integers the graph is flat; at each jump the output increases by 1.

Formula

  • y = ⌈x⌉
  • Constant on (n − 1, n] with height n

Introduction

Graphs make discrete jumps visible. They also explain why ceiling is not differentiable at integers.

Pair this with the definition article so the upward rule is fresh before you sketch.

Numeric checks on the calculator confirm each plotted point. For a table of inputs, open ceiling examples and plot the same pairs.

Step function view

Each step level equals the output integer. Between jumps the graph is flat.

Open circles versus filled points depend on your textbook convention at the jump. Follow your course notes for endpoint style.

The graph rises toward +∞ at each integer boundary, matching the ceiling rule on the number line.

Plotting tips

  • y = ⌈x⌉
  • Constant on (n − 1, n] with height n (convention-dependent endpoints)
  • Jump height increases by 1 at each integer break

Plot samples: 0.2 → 1, 1.2 → 2, −1.2 → −1.

Mark the interval form n − 1 < x ≤ n under each segment if your instructor wants analytic justification.

Compare with floor graphs mentally: floor steps move downward toward −∞ at breaks.

Reading the graph

  1. Pick sample x. Include a negative, a fraction, and an integer.
  2. Plot (x, ⌈x⌉). Draw points at integer heights. Label the coordinates.
  3. Connect steps. Horizontal lines between jumps. Do not slope the segments diagonally.
  4. Mark jumps. Show where the output increases by 1 when x crosses an integer from the left.

Interpretation

A flat segment means many inputs share one ceiling output until the next jump.

Near x = 2.9 the height is still 3; at x = 3 the height becomes 3 on the closed endpoint if your class uses the (n − 1, n] convention.

Large positive x and large negative x both produce wide flat sections, which is why the function looks like a staircase.