Graphing Calculator: Plot, Trace and Analyze Functions

What it is and when to use it

The graphing calculator draws an expression as a curve on a coordinate plane and reads values back: coordinates under the cursor, zeros, minima and maxima, intersections, a table of samples, and the area between the curve and the x-axis. Five row types: y = f(x) curves, parametric pairs x(t), y(t), polar r(θ), sequences, and cobweb diagrams. Use it to check a function's shape before solving by hand or to see where two models cross.

Inputs and controls

Cartesian rowstake the right-hand side — x^2, sin(x), e^(-x^2) — not the y =. A row has a colour, a visibility toggle, an inequality setting that shades y ≥ f(x) or y ≤ f(x), and a derivative toggle that draws the slope as a dashed curve.
Other row types.Parametric rows take x(t) and y(t); polar rows take r(θ); sequence rows plot whole-number terms; cobweb rows iterate f(x) around the reference line y = x. Parameter rows start with the range 0 → 6.28 (≈ 2π).
Expression syntax.Operators are +, −, ×, ÷, with % for remainder, ! for factorial and ^ for powers; ^ is right-associative, so 2^3^2 is 512, and -x^2 means −(x²). Implicit multiplication works (2x, 3sin(x)), and 1e-5 is 0.00001. Functions include the trig set and their inverses, hyperbolics, ln, log (base 10), log2, sqrt, cbrt, abs, floor, ceil, round, min, max and hypot; constants include pi, π, e, tau and phi. Trigonometry is always in radians, so a degree angle needs sin(pi*x/180). The variable is x; xy is one undefined name, not x times y.
Sliders.Any single letter other than x, t, e or n becomes a slider running from −10 to 10 in steps of 0.05. Presets include sin(x)/x, e^(-x^2), floor(x) and 1/x.
Window and output.The view starts at −10 to 10 on both axes with grid lines every 2 units. The toolbar zooms by 70% or 140% per press, the wheel zooms around the pointer, dragging pans, and Reset restores the default. Export saves a PNG, and the share button copies a link to the same view.

Worked examples

A cubic with two turning points. Plot x^3 - 3*x. Zeros appear at x = −1.7321, 0.0000 and 1.7321; the extrema are labelled max (-1.000, 2.000) and min (1.000, -2.000). The integral from 0 to 2 reads −2.000000 (±2.8e-14): more area lies below the axis than above it there, and the sign is part of the answer.
Where two models meet. Plot x + 1 and x^2 - 1. The intersection markers land at (−1.000, 0.000) and (2.000, 3.000), the solutions of x + 1 = x² − 1.
An area, and its error bar. Enter sin(x), open the integral control, set the bounds to 0 and 3.14159265 — the fields take numbers, not π — and the readout gives 2.000000 (±2.2e-14). The ± is the quadrature's own error estimate, not physical uncertainty.

Background

René Descartes' 17th-century pairing of algebra with geometry turned equations into curves; polar coordinates use a distance and an angle, a natural fit for circles and spirals. Casio's fx-7000G, the first graphing calculator, arrived in 1985, and plotting later moved to screens like this one. Quadrature is the numerical route to a definite integral when no elementary antiderivative exists, and adaptive rules estimate their own error as they refine.

Reading values, and the limits of each readout

Readouts.: Hovering reports the cursor's x and every visible y = f(x) value to four decimals, and undefined where no finite value exists. Zeros are labelled x = … to four decimals; extrema and intersections are labelled to three, and at most 60 analysis markers are drawn. The table shows 21 rows from a chosen start and step at eight significant digits, with an em dash where a value is not finite. The integral covers the selected row between two bounds, reporting six decimals with a ± quadrature estimate; an integral that does not converge is labelled divergent.
Only the visible window.: The trap to remember: the analysis tools see only the visible window. In the default view sin(x) reports zeros at x = 0.0000, ±3.1416, ±6.2832 and ±9.4248 — seven markers for a function with infinitely many zeros. Zoom or pan to bring a root into view. A pole is not a zero, so tan(x) near ±π/2 is not marked; a flat stationary point is not an extremum, so x³ has no marker at the origin; and (x − 1)² is found at x = 1.
Window and aspect.: The window is square in data units while the plotting area usually is not, so one unit right may span more pixels than one unit up and a circle can look like an ellipse. Compare grid spacing before judging a shape, and read slopes from trace values.
Numeric reads, no symbolic algebra.: It does not solve an equation symbolically or return exact values; typed expressions with exact numbers belong to the scientific calculator.

Common mistakes

Typing y = into the function field, which expects the expression alone.
Using the radian default for a degree angle. sin(30) is about −0.9880316241; 30° is π/6.
Expecting 2^3^2 to be 64. The caret is right-associative and gives 512.
Trusting a flat segment. Sampling can hide a feature narrower than one screen column.

Where it is used

Checking a curve before solving. Plot x^3 - 3*x to see three zeros and two turning points, then solve by hand — the extrema appear as max (-1.000, 2.000) and min (1.000, -2.000).
Where two models cross. Draw x + 1 and x^2 - 1 and read the intersections at (−1.000, 0.000) and (2.000, 3.000), the solutions of x + 1 = x² − 1.
Area under a curve. Set the integral bounds with plain numbers — sin(x) from 0 to 3.14159265 reads 2.000000 (±2.2e-14).
Parameter sweeps. A single letter becomes a slider, so a family of curves can be redrawn live without re-typing the expression.

Privacy

Graphs and expressions stay in the browser; nothing is uploaded.

Sources

Wikipedia, Graphing calculator: https://en.wikipedia.org/wiki/Graphing_calculator （访问日期：2026-09-30）
Wikipedia, Cartesian coordinate system: https://en.wikipedia.org/wiki/Cartesian_coordinate_system （访问日期：2026-09-30）
Wikipedia, Polar coordinate system: https://en.wikipedia.org/wiki/Polar_coordinate_system （访问日期：2026-09-30）
Wikipedia, Numerical integration: https://en.wikipedia.org/wiki/Numerical_integration （访问日期：2026-09-30）
NIST DLMF, §3.5 Quadrature: https://dlmf.nist.gov/3.5 （访问日期：2026-09-30）

Sources & review

Reviewed by: CalcX Editorial Team
Updated: 2026-09-30