Polynomial Graphing Calculator

A polynomial function combines constant coefficients with nonnegative whole-number powers of x. Examples range from a constant or straight line to quadratics, cubics, quartics, and higher-degree curves. Unlike functions with variables in denominators or radicals, a polynomial is defined for every real x-value and draws a smooth, continuous graph without gaps or vertical asymptotes. This calculator plots that graph over a chosen horizontal interval and summarizes several important algebraic features. The graph is generated by evaluating the polynomial at many evenly spaced x-values. Horner's method processes the coefficients efficiently, repeatedly multiplying the accumulated value by x and adding the next coefficient. The sampled y-values are scaled to fit the graph panel, allowing both small and large polynomial values to remain visible. Changing X-Min and X-Max lets you zoom into a region or inspect wider end behavior. A real root is an x-value where f(x) equals zero, which is where the graph crosses or touches the x-axis. The calculator scans the selected interval for sign changes and refines each crossing with bisection. Roots outside the chosen range are not reported. Even-multiplicity roots can touch the axis without changing sign, so numerical graph scanning may not identify every tangent root; an algebraic factoring or dedicated root solver is preferable when complete root multiplicity is required. The y-intercept is f(0), equal to the polynomial's constant term. It marks where the graph meets the vertical axis. The derivative describes the curve's instantaneous slope. Each term ax^n differentiates to nax^(n-1), while a constant differentiates to zero. Critical points occur where this derivative is zero, helping locate local maxima and minima. Degree and leading coefficient govern end behavior. An even-degree polynomial points in the same vertical direction at both ends, while an odd-degree polynomial points in opposite directions. A positive leading coefficient rises on the right; a negative one falls. A degree-n polynomial can have at most n real roots and at most n - 1 turning points, though it may have fewer of either. Polynomial graphs appear throughout algebra, calculus, physics, economics, interpolation, and engineering models. Use the plotted curve to build intuition, then use the listed roots, intercept, and derivative to confirm symbolic work. Enter terms with ^ for exponents; multiplication signs are optional, so both 3*x^2 and 3x^2 are accepted.