Graph Plotter

• Explicit y = f(x) — the standard case. y = x², y = sin(x), y = e^x.
• Parametric (x(t), y(t)) — both coordinates as functions of a parameter. Curves that fail the "vertical line test" (circles, figure-eights, spirals) need this form.
• Polar r = f(θ) — radius as a function of angle. Roses (r = cos(2θ)), spirals (r = θ), cardioids (r = 1 - sin(θ)).
• Implicit f(x, y) = 0 — relations that cannot be solved for y alone. The unit circle x² + y² = 1 is implicit; ellipses, hyperbolas, transcendental curves.
• Vector fields — (u, v) at each (x, y). For ODEs and fluid dynamics intuition. Tool typically renders arrows on a grid.
• Sequences / discrete — scatter plot of (n, f(n)) for integer n. Fibonacci, recurrence relations, time series.

A grapher samples the function at N points across the x-range, then draws line segments connecting consecutive samples. If the function oscillates faster than the sampling rate, you get aliasing — the rendered curve does not match the true function. Plot y = sin(100x) at default sampling (100 points across [0, 2π]); you will see a slow wiggle that bears no relation to sin(100x).

Cure: more samples. The Nyquist criterion says you need at least 2 samples per period. For y = sin(kx), use at least 4k samples per 2π. Modern graphers adaptively increase sampling in regions of high curvature, but explicit control matters for high-frequency functions.

• Vertical asymptotes — at f(x) = 1/x near x=0, the function goes to ±∞. Default plotters draw a continuous line connecting +∞ and −∞ across the asymptote, which looks like a spike. Better: detect the sign change and break the curve at the asymptote.
• Holes — f(x) = (x²−1)/(x−1) is x+1 everywhere except at x=1 where it is 0/0 (undefined, removable). Graphers usually draw the line; the hole at x=1 is invisible unless explicitly marked.
• Discontinuities — step functions, sign(x), piecewise definitions. Default line-drawing falsely connects across the jump; use scatter mode or explicit discontinuity handling.
• Domain restrictions — sqrt(x) is undefined for x<0; log(x) is undefined for x≤0. The function "stops" at the boundary; do not extrapolate into the undefined region.

• You are studying calculus or precalculus and want to see what a function "looks like" before computing properties.
• You are doing physics or engineering homework where the relationship between variables is best understood graphically (projectile trajectories, RC circuit decay, harmonic oscillators).
• You are writing a slide or document and need a clean function plot without dragging out Mathematica or matplotlib.
• You are debugging "the formula in my code does not behave right" — plot it against the expected mathematical curve.

• It will not do symbolic computation. For symbolic derivatives, integrals, simplifications, use a CAS (SymPy, Maple, Wolfram Alpha). The plotter computes values numerically.
• It will not handle 3D surfaces. f(x, y) = ... needs a 3D plotter (Three.js based or Plotly). 2D-only here.
• It will not export interactive plots. Output is static PNG/SVG. For interactive embedding, use Plotly, Vega, or D3.
• It will not solve equations. To find where two curves intersect, plot both and read off graphically — for precision use a root-finder.