ΣNODALUNE MATH
ESEN
ENGINE ACTIVE

FUNCTION LAB / 03

Write.
Graph. Understand.

Build any function with real mathematical notation, explore its graph and unfold the reasoning behind every result.

STRUCTURED EDITOR√ · a⁄b · xⁿ · logₐTOUCH · EDIT · ANALYZE

COMPOSER / 01

Your function, properly written

Symbols create editable structures: a fraction opens numerator and denominator; a root opens its content and index; Tab moves through every placeholder.

Preparing editor…

Click or tap any term to edit it. Use Tab to move through the numerator, denominator, index, exponent, and content.

TRY
AUTO WINDOW
Valid expression
LOCAL HISTORY

Recent functions

Functions you graph will appear here.

INTERACTIVE PLANE / 02

Universal function grapher

Drag to pan · use the wheel or buttons to zoom
Your browser cannot display the interactive Cartesian plane.
x [-3, 3] · y [-18.69, 18.69]
functionrootextremuminflection

The wheel zooms without scrolling the page. Drag to pan. Move the pointer near the curve to inspect a point. Enter selects the center point and Escape closes it. Detected 3 roots, 2 extreme points, 1 inflection, and 0 limits or discontinuities in the current window.

REPORT / 03

Complete analysis

Expand any result to see what it means, how it was obtained and how it was checked.

01Form and derivativesf′(x) = 3x^2-3exact＋

The derivative measures instantaneous slope; the second derivative describes curvature.

How it is obtained

  1. We simplify: x^3-3x
  2. Differentiate term by term: f′(x) = 3x^2-3
  3. Differentiate again: f″(x) = 6x
VERIFICATIONThe derivative was checked at a deterministic point using finite differences.
02Domain and continuity[-3, 3]estimated＋

The domain contains the x-values for which the expression produces a real number.

How it is obtained

  1. We find zeros of denominators, logarithm arguments, and even-root boundaries.
  2. We split the interval [-3, 3] at every critical point.
  3. We check an interior point and both endpoints of each segment.
VERIFICATIONContinuous on the analyzed window
03Roots and y-interceptx = -1.732051, x = 0, x = 1.732051exact＋

A root is a value where f(x)=0; the y-intercept is obtained by evaluating f(0).

How it is obtained

  1. We locate sign changes and tangent contacts with the x-axis.
  2. We refine every candidate by bisection or with the exact polynomial result.
  3. We evaluate f(0): 0.
VERIFICATIONMaximum residual |f(r)| = 0.
04Local extremamax. (-1, 2); min. (1, -2)verified＋

Local maxima and minima occur when the slope changes sign or at a cusp.

How it is obtained

  1. We solve f′(x)=0 numerically.
  2. We evaluate the function on both sides of every candidate.
  3. We classify −→+ as a minimum and +→− as a maximum.
VERIFICATIONEvery point was compared with independent neighboring values.
05Monotonicityincreasing on (-3, -1); decreasing on (-1, 1); increasing on (1, 3)verified＋

The sign of f′ indicates where the function increases or decreases.

How it is obtained

  1. We split the domain at extrema and discontinuities.
  2. We evaluate f′ at three interior points of every segment.
  3. We assign behavior only when the samples agree.
VERIFICATION3 segment(s) checked in the window.
06Concavity and inflection points(0, 0)verified＋

An inflection point requires continuity and a change in the sign of f″.

How it is obtained

  1. We find candidates where f″(x)=0.
  2. We discard discontinuities and points without a real function value.
  3. We compare concavity to the left and right.
VERIFICATIONconcave down on (-3, 0); concave up on (0, 3)
07Signnegative on (-3, -1.732051); positive on (-1.732051, 0); negative on (0, 1.732051); positive on (1.732051, 3)verified＋

This shows where the graph lies above or below the x-axis.

How it is obtained

  1. We order roots and discontinuities.
  2. We take three samples per interval.
  3. We verify that they keep the same sign.
VERIFICATION4 interval(s) classified.
08Symmetry and periodicityOdd about the originestimated＋

We compare f(x) with f(−x) and, for common periods, with f(x+T).

How it is obtained

  1. We take symmetric pairs of points.
  2. We measure the relative error of the even and odd identities.
  3. We test common trigonometric periods inside the window.
VERIFICATIONNo period was reported when the samples were inconclusive.
09Range and asymptotes≈ [-18, 18] for x ∈ [-3, 3]estimated＋

The displayed range applies to the selected window; asymptotes describe trends or divergence.

How it is obtained

  1. We sample 1,801 finite points to estimate the visible range.
  2. For vertical asymptotes, we approach from both sides.
  3. For behavior at infinity, we evaluate geometric scales.
VERIFICATIONNo asymptotes were detected with sufficient evidence.
✓
7/7 checks passed

Roots by residual, extrema by side comparison, inflections by concavity change and derivative by finite differences.

CONSISTENCY OK
Scope: Analysis of non-polynomial functions is limited to the selected window; it may miss extremely rapid oscillations. Estimated results are clearly identified and are never presented as exact.

TOOLS / 04

Quick calculation

The scientific calculator and quadratic solver use the same validation criteria.

01

Scientific calculator

Real-number evaluation

REAL NUMBERSRADIANS
Preparing editor…
RESULTEXACT
APPROXIMATE≈ 1

Real domain: divisor ≠ 0 · even root ≥ 0 · log > 0 · arcsin/arccos between −1 and 1

History

Your latest calculations will appear here.

02

Quadratic function

Stable formula and graph transfer

x²
+
x
+
= 0
DISCRIMINANTΔ = 4Two real roots
ROOTS1 · 3Verified with a numerically stable formula

ONLINE MATHEMATICS / 05

A calculator made for exploration, not just answers

Nodalune Math combines a free scientific calculator, a function graphing calculator and a quadratic equation solver. Every calculation runs in your browser.

01

Visual equation editor

Write fractions, roots, powers, logarithms and trigonometric functions in editable mathematical notation.

02

Interactive graphs

Pan, zoom and inspect points while discontinuous branches remain correctly separated.

03

Explained analysis

Study domain, roots, extrema, concavity, asymptotes and derivatives with steps and independent checks.

Frequently asked questions

Which functions can Nodalune Math graph?＋

Polynomials, rational functions, roots, absolute value, exponentials, logarithms and trigonometric functions within the selected analysis window.

Are the analysis results exact?＋

Symbolic results are labeled exact. Numerical findings are verified and labeled estimated whenever they depend on the interval or sampling.