Polynomial graphing calculator

If the range start is not strictly less than the range end, the task fails with invalid_input instead of inventing a reversed or empty plot.

What a sampled polynomial graph is and how Horner evaluation works

A quadratic such as coefficients [1, 0, 0] on [−2, 2] with five samples yields the familiar parabola points (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4). Constants produce a horizontal line; a linear binomial produces a straight segment. Degree is reported from the first non-zero leading coefficient; the zero polynomial is treated as degree zero and graphs as the x-axis. Nothing here finds roots, sketches asymptotes of rationals, or emits an image: the contract is the numeric point list a graphing client already knows how to draw. A univariate polynomial is a finite sum p(x) equals a_n x to the n plus a_{n-1} x to the n minus one, down to the constant a_0. Graphing that function on a closed interval is not the same job as finding its roots, dividing it, or integrating it: a plot needs a list of sampled points (x_i, p(x_i)) that a renderer can connect. This calculator builds that list by walking a uniform grid from x_start to x_end inclusive and evaluating p at each abscissa with Horner's method. Horner rewrites the polynomial as a nested multiply-add, p(x) equals ((a_n x + a_{n-1}) x + … ) x + a_0, so each sample costs one multiply and one add per coefficient and stays numerically stabler than summing independent powers of x. The grid has samples points (default eleven), so the step is (x_end minus x_start) divided by (samples minus one). Both endpoints are included: the first point is always (x_start, p(x_start)) and the last point is pinned to x_end so accumulated floating-point drift cannot miss the right edge of the window you asked for.

Input contract, range check, response shape, and errors

Missing fields, empty coefficient arrays, non-finite entries, and a degree above sixty-four fail the same way, naming the field. Values are rounded to twelve decimal places so goldens compare byte-for-byte across runtimes. POST a JSON body with required coefficients, required x_start, and required x_end. Coefficients is a non-empty array of finite numbers in highest-degree-first order: [a_n, a_{n-1}, …, a_0] for p(x) equals a_n x^n plus … plus a_0. The aliases coeffs and poly are accepted. x_start (aliases start, x_min) is the left endpoint of the closed window; x_end (aliases end, x_max) is the right endpoint. Both must be finite numbers, and x_start must be strictly less than x_end. Optional samples (aliases n, count) is an integer from two to one thousand and one inclusive; omit it and the default of eleven points is used. On success the body echoes the cleaned coefficients, reports degree and a compact polynomial display string, repeats x_start, x_end, samples, and the uniform step, and returns points as an ordered array of objects each with x and y, plus point_count. Pair the list with your own charting library; this endpoint does not rasterize a picture or invent SVG.

Where a polynomial graphing API earns its keep

Engineering and science notebooks often need a quick polyline of a fitted polynomial, a characteristic polynomial, or a low-order Taylor piece before they hand the same coefficients to a root finder or a Riemann-sum tool; calling a tiny parse API avoids pulling a full plotting stack for a coefficient vector and two numbers.