Floor Function Graph Calculator

Configure Floor Function

Function form

y₁ = ⌊ a·x + b ⌋

Analyze classic stepwise behavior with precise control of parameters.

a (slope) y₁

b (shift) y₁

Domain start (x)

Domain end (x)

Step

Enable comparison with second floor function y₂ = ⌊ a₂·x + b₂ ⌋

a₂ (slope) y₂

b₂ (shift) y₂

Overlay continuous line y₁ = a·x + b (pre-floor).

Highlight jump discontinuities on the graph.

Show horizontal guide lines at integer output levels.

Quick presets

Floor Function Graph

Steps display constant integer outputs. Options overlay continuous lines, jump markers, and dual floor functions for deeper comparisons.

Export & Analysis Tools

Export sampled values for one or both functions. Ideal for worksheets, lectures, proofs, and numerical experiments requiring exact stepwise outputs.

Includes parameters, domain, and step size configuration.
Compatible with spreadsheets and CAS tools.
Great for documenting discontinuities and integer intervals.

Calculated Values Table

#	x	a·x + b	⌊a·x + b⌋ (y₁)
1	-5	-5	-5
2	-4.75	-4.75	-5
3	-4.5	-4.5	-5
4	-4.25	-4.25	-5
5	-4	-4	-4
6	-3.75	-3.75	-4
7	-3.5	-3.5	-4
8	-3.25	-3.25	-4
9	-3	-3	-3
10	-2.75	-2.75	-3
11	-2.5	-2.5	-3
12	-2.25	-2.25	-3
13	-2	-2	-2
14	-1.75	-1.75	-2
15	-1.5	-1.5	-2
16	-1.25	-1.25	-2
17	-1	-1	-1
18	-0.75	-0.75	-1
19	-0.5	-0.5	-1
20	-0.25	-0.25	-1
21	0	0	0
22	0.25	0.25	0
23	0.5	0.5	0
24	0.75	0.75	0
25	1	1	1
26	1.25	1.25	1
27	1.5	1.5	1
28	1.75	1.75	1
29	2	2	2
30	2.25	2.25	2
31	2.5	2.5	2
32	2.75	2.75	2
33	3	3	3
34	3.25	3.25	3
35	3.5	3.5	3
36	3.75	3.75	3
37	4	4	4
38	4.25	4.25	4
39	4.5	4.5	4
40	4.75	4.75	4
41	5	5	5

Compare outputs row-by-row to study parameter effects and discrete jumps.

Example: y₁ = ⌊x⌋ from -2 to 3

x	⌊x⌋
-2.0	-2
-1.5	-2
-0.2	-1
0.0	0
0.7	0
1.2	1
2.9	2
3.0	3

Observe constant values between integers and jumps at integer boundaries.

Formula Used

The primary function is y₁ = ⌊a·x + b⌋. Optionally, a second function y₂ = ⌊a₂·x + b₂⌋ is plotted.

For any real number t, ⌊t⌋ is the greatest integer less than or equal to t.
Each horizontal step satisfies k ≤ a·x + b < k + 1 for integer k.
Jumps occur where a·x + b or a₂·x + b₂ equals an integer.

Overlaying y₁ = a·x + b clarifies how flooring transforms linear growth into a staircase with discrete integer outputs.

How to Use This Calculator

Set a, b, domain range, and step for y₁.
Enable and configure y₂ to compare two floor functions visually.
Toggle continuous line, jump markers, and integer guides as needed.
Use quick presets for instant demonstrations in class or notes.
Review the live table to read exact values at each x.
Export CSV for spreadsheets or programming verification tasks.
Export PDF to archive configurations, examples, and sample outputs.

Key Features of the Floor Function Graph Tool

Interactive plotting of y₁ and y₂ floor functions on a clean grid.
Configurable domain, resolution, and parameters to match lesson objectives.
Visual overlays for continuous lines, jumps, and integer guide levels.
Instant numeric table synchronized with plotted sample points.

Common Use Cases and Learning Scenarios

Teaching step functions, discontinuities, and rounding behavior in calculus or algebra.
Comparing transformations like vertical shifts, reflections, and scaling of floor graphs.
Preparing assignment solutions, handouts, and answer keys with exportable data.
Testing analytical inequalities involving floor expressions across chosen intervals.

Data Outputs Generated by This Calculator

Ordered pairs (x, ⌊a·x + b⌋) for the chosen sampling step.
Optional pairs for y₂ to compare alternative floor transformations.
Intermediate linear values a·x + b and a₂·x + b₂ for context.
Structured CSV ready for spreadsheets, plotting libraries, or coding tasks.

Graph Interpretation and Discontinuity Insights

Horizontal segments represent intervals where the floor value remains constant.
Jump markers highlight boundary points where integer transitions occur.
Integer guide lines emphasize alignment of outputs to discrete levels.
Dual graphs reveal how small parameter changes shift entire step structures.

Frequently Asked Questions

1. What does the floor function represent on this graph?

The floor function maps each real x to the greatest integer less than or equal to a·x + b, producing a stepwise staircase instead of a smooth curve.

2. How should I choose the step size for accurate graphs?

Use smaller step sizes to capture jumps precisely and show detailed steps. Larger steps generate fewer sample points, suitable for quick overviews or broad conceptual demonstrations.

3. Why enable the continuous line alongside the floor graph?

The continuous line shows the original linear expression before flooring. Comparing both helps students see how rounding down at each point creates the staircase shape.

4. When is the second floor function useful?

The second function helps compare two different parameter sets directly. It is useful for exploring shifts, reflections, scaling effects, and validating algebraic inequalities involving multiple floor expressions.

5. What can I do with the CSV export option?

CSV export lets you import all computed values into spreadsheets or code. You can build custom charts, verify formulas, or integrate the dataset into lessons and technical documentation.

6. What information is included in the PDF summary?

The PDF summary includes chosen parameters, domain, step size, and sample rows from the calculation table, providing a compact record suitable for assignments, reports, or lecture notes.

7. Can this tool help verify solutions with floor expressions?

Yes. Adjust parameters and intervals, then inspect tables and graphs to confirm where inequalities hold, where jumps occur, and how expressions behave across different ranges.