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The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
The constant c translates the graph vertically up c units when c > 0 or down when c < 0. The asymptotes of a truncus are found at x = -b (for the vertical asymptote) and y = c (for the horizontal asymptote). This function is more commonly known as a reciprocal squared function, particularly the basic example /. [1]
An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation =, y becomes arbitrarily small in magnitude as x increases.
The asymptotic directions are the same as the asymptotes of the hyperbola of the Dupin indicatrix through a hyperbolic point, or the unique asymptote through a parabolic point. [1] An asymptotic direction is a direction along which the normal curvature is zero: take the plane spanned by the direction and the surface's normal at that point. The ...
Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve. [1] Equate first and second derivatives to 0 to find the stationary points and inflection points respectively. If the equation of the curve cannot be solved explicitly for x or y, finding these derivatives requires implicit ...
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function , which is defined by the formula: [ 1 ]
The function admits a horizontal asymptote. The curve is symmetrical with respect to the y-axis. The curvature radius is r = a cot x / y . A great implication that the tractrix had was the study of its surface of revolution about its asymptote: the pseudosphere.
The Cartesian equation is = / (+). The curve resembles the Folium of Descartes [1] and the line x = –a is an asymptote to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by =.