Graphing
Graphing is where algebra and calculus become visible. The tools here plot a function and its key features, place two functions on the same axes to find where they meet, draw parametric and polar curves, and shade the region defined by an inequality. Each result is rendered as a clean SVG and is paired with a numeric summary you can copy.
Every visual on this page is built from the same idea: sample the function at many x (or t, or θ) values, drop the non-finite points, and connect what remains. The function tools then run lightweight numeric algorithms on top — bisection to find zeros, sign changes in the derivative to find extrema, and bracketed root-finding on f(x) − g(x) to locate intersections.
Plot a single function
The function grapher draws y = f(x), then overlays the zeros it locates by bisection and the local maxima and minima it finds from sign changes in the numerical derivative. The y-intercept is drawn as an open circle when 0 falls inside the interval.
Compare two functions and find where they meet
Put f(x) and g(x) on the same axes and the calculator solves f(x) − g(x) = 0 numerically. Every crossing on the interval is reported and marked on the plot.
Parametric and polar curves
Parametric curves use a parameter t — both coordinates are functions of it. Polar curves describe a radius r at each angle θ; the calculator converts every (r, θ) sample to (r·cos θ, r·sin θ) before plotting.
Inequality regions and function-feature analysis
The inequality grapher shades the set of points above (or below) y = f(x), using a dashed boundary for strict inequalities. The feature analyser returns zeros, extrema, monotonicity intervals, vertical-asymptote candidates and end behavior without drawing anything — useful for homework summaries.
All solvers
Frequently asked questions
Can the calculator draw functions with vertical asymptotes?
Yes. When the function jumps by an outlier amount between two samples (as for 1/x or tan(x)), the polyline is broken so the asymptote is not drawn as a vertical line.
How precise are the zeros and intersections?
After locating a sign change between samples, the calculator runs ~50 bisection steps. That places the root within a tiny fraction of the sample spacing — far below the visible plot resolution.
How are extrema found?
By looking for sign changes in the numerical derivative between consecutive samples, then bisecting on f'(x) to refine the x-coordinate.
Why do polar and parametric plots look square?
Because they use an equal aspect ratio — one unit on x is one unit on y — so circles and other closed curves keep their shape instead of being squashed.