Graphing calculator

Write your functions as on paper, with real fractions, exponents and roots, and see their graph instantly. Tap a function to see its roots, maxima and minima, y-intercept and intersections with the other curves; drag along it to trace it.

What you can plot

Functions y = f(x) and x = g(y), and your own functions: f(x) = x³ − 3x.
Implicit curves like x² + y² = 9 and shaded inequalities like y < sin x.
Parametric curves (cos t, sin t) and polar curves r = 1 + cos θ.
Points, animated sliders for parameters and domain restrictions {x > 0}.
Angles in radians or degrees, zoom with the wheel or two fingers.

Guide: how to analyze a function with its graph

A good graph answers many questions at a glance: where it crosses the axes, where it rises or falls, where it has maxima and minima and where it meets another curve. This guide reviews what to look for in a graph, how to confirm it with calculus and what kinds of curves the tool can draw.

What to look for in the graph of a function

Domain: where it is defined. √x requires x ≥ 0; ln x, x > 0; 1/x, x ≠ 0.
Roots or zeros: the x-intercepts, where f(x) = 0.
y-intercept: the point (0, f(0)), if 0 is in the domain.
Local maxima and minima: where the function switches from rising to falling or vice versa. If f is differentiable, f′(x) = 0 there.
Asymptotes: vertical (the function grows without bound near a point, like 1/x at 0) and horizontal or slant (behavior as x → ±∞).
Intersections with another curve: the solutions of f(x) = g(x).

Worked example: f(x) = x³ − 3x

Roots: x³ − 3x = x(x² − 3) = 0, so x = 0 and x = ±√3 ≈ ±1.732.
y-intercept: f(0) = 0.
Extrema: f′(x) = 3x² − 3 = 0 gives x = ±1. Since f″(x) = 6x, there is a local maximum at x = −1, f(−1) = 2, and a local minimum at x = 1, f(1) = −2.
Symmetry: f(−x) = −f(x): the function is odd and its graph is symmetric about the origin.
Intersection with g(x) = x: x³ − 3x = x leads to x³ − 4x = x(x² − 4) = 0, so x = 0 and x = ±2; the points are (−2, −2), (0, 0) and (2, 2).
Area between the two curves from x = 0 to x = 2: g is on top there, so A = ∫₀² (x − (x³ − 3x)) dx = ∫₀² (4x − x³) dx = [2x² − x⁴/4]₀² = 8 − 4 = 4.

The tool marks the same roots, the maximum, the minimum and the three intersections with y = x.

Why the tool does not make up points

A graphing calculator draws by joining points, and that can mislead: joining both sides of an asymptote, skipping a fast oscillation or taking a spike next to an asymptote for a maximum. This tool bounds the function on each piece (interval arithmetic) to know whether two points can safely be joined by a line, breaks the line at jumps and asymptotes, and only marks a maximum or minimum if the function is continuous around it and the slope changes sign.

What can be drawn

Functions y = f(x), and also x = f(y).
Implicit equations, such as x² + y² = 25.
Inequalities, such as y < x² or x² + y² ≤ 9, with the region shaded.
Parametric curves (x(t), y(t)) and polar curves r(θ).
Points, user-defined functions and constants with a slider to see how a family of curves changes.

How to use the calculator

Type each expression in a row, with the equation editor. Each row's color is changed with the circle on its left.
Select a row to see its roots, extrema, y-intercept and intersections with the other rows.
Use the row tools: table of values, tangent line that follows the cursor and area under the curve.
Export the graph as PNG or SVG.

Common mistakes

Trusting the default window: a maximum or a root can be out of view. Zoom out before concluding.
Calling a point a root where the function only approaches 0 (like eˣ as x → −∞).
Confusing a local maximum with the absolute maximum.
Working in degrees when trigonometric functions are used in calculus, which assumes radians.

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