Function Analyzer

The Function Analyzer performs the complete analysis a curve-sketching problem asks for. Domain and range. x and y intercepts. Symmetry. Vertical, horizontal, and slant asymptotes with the limits that establish them, and holes where a factor cancels. First derivative: critical points, intervals of increase and decrease, local maxima and minima by the first or second derivative test. Second derivative: concavity and inflection points. End behaviour. It then assembles a sign chart and describes the graph section by section so you can draw it. Transformations of a parent function and piecewise functions are handled too. Parametric and polar curves are analysed for the same features where they apply, and a function given as a table of values or a described graph is handled by working from the data given.

Type the function or photograph it: "f(x) = (x^2 - 1)/(x - 2)", "analyze g(x) = x e^(-x)", "sketch y = ln(x^2 + 1)". Ask for specific parts if you do not need everything: "just the asymptotes", "find the inflection points". Restrict the domain if the problem does. Set Answer form to exact for critical values with radicals or decimal for rounded coordinates. For a piecewise function, write each piece with its interval. For an applied problem that reduces to optimising a function, paste the whole problem and the solver builds the function, states its domain from the context, and then analyses it. Ask for the graph of the derivative if the question wants it.

Output: a summary table of the features first, then each derived: the domain computation, the intercept equations, the limits for asymptotes, the derivatives with their zeros, the sign charts, and the classification of each critical point. A final paragraph describes the shape of the curve from left to right. The working is shown so you can check it. For the derivative on its own with the rule named, use the Derivative Calculator. For roots of the function only, use the Equation Solver. A follow-up can ask for a table of points to plot, the analysis restricted to a smaller interval, or the same function after a transformation. Sign charts are printed as tables with the test points used, so each conclusion about increase or concavity can be reproduced.