asymptote

(noun)

A line which a curved function or shape approaches but never touches.

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

(noun)

A straight line which a curve approaches arbitrarily closely as it goes to infinity. 

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

(noun)

A line that a curve approaches arbitrarily closely, as it extends toward infinity.

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

(noun)

A line that a curve approaches arbitrarily closely. Asymptotes can be horizontal, vertical or oblique.

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

(noun)

A line that a curve approaches arbitrarily closely. An asymptote may be vertical, oblique or horizontal. Horizontal asymptotes correspond to the value the curve approaches as $x$ gets very large or very small.

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

(noun)

A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity".

Related Terms

  • vertical asymptotes
  • logarithmic function
  • logarithm
  • exponential function
  • tangent
  • periodic function
  • odd function
  • period
  • directrix
  • nappe
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • conic section
  • ellipse
  • hyperbola
  • rational function
  • vertical asymptote
  • oblique
  • exponential growth
  • vertex
  • parabola
  • locus

Examples of asymptote in the following topics:

  • Asymptotes

    • A rational function can have at most one horizontal or oblique asymptote, and many possible vertical asymptotes; these can be calculated.
    • An asymptote that is neither horizontal or vertical is an oblique (or slant) asymptote.
    • A rational function has at most one horizontal or oblique asymptote, and possibly many vertical asymptotes.
    • If $n>m$, then there is no horizontal asymptote (However, if $n = m+1$, then there exists a slant asymptote).
    • Hence, horizontal asymptote is given by:
  • Standard Equations of Hyperbolas

    • At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center.
    • A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them.
    • Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote.
    • The distance b (not shown in below) is the length of the perpendicular segment from either vertex to the asymptotes.
    • If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be rectangular or equilateral.
  • Parts of a Hyperbola

    • The asymptotes of the hyperbola are straight lines that are the diagonals of this rectangle.
    • Then draw in the asymptotes as extended lines that are also the diagonals of the rectangle.
    • Finally, draw the curve of the hyperbola by following the asymptote inwards, curving in to touch the vertex on the rectangle, and then following the other asymptote out.
    • The asymptotes of a rectangular hyperbola are the $x$- and $y$-axes.
  • Polynomial and Rational Functions as Models

    • They can take on only a limited number of shapes and are particularly ill-suited to modeling asymptotes.
    • For lots of datasets, their are no asymptotes and data is more or less bounded.
    • To deal with the asymptotic problems of polynomials, we also use rational functions:
    • Rational functions are a little more complex in form than polynomial functions, but they have an advantage in that they can take on a much greater range of shapes and can effectively model asymptotes.
    • However, rational functions sometimes include undesirable asymptotes that can disrupt an otherwise smooth trend line.
  • Limited Growth

    • Graphically, the logistic function resembles an exponential function followed by a logarithmic function that approaches a horizontal asymptote.
    • This horizontal asymptote represents the carrying capacity.
    • That is, $y=c$ is a horizontal asymptote of the graph.
    • Additionally, $y=o$ is also a horizontal asymptote.
    • Logistic functions have an "s" shape, where the function starts from a certain point, increases, and then approaches an upper asymptote.
  • Basics of Graphing Exponential Functions

    • That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis is a horizontal asymptote of the function.
    • That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis a horizontal asymptote of the function.
    • The function $y=b^x$ has the $x$-axis as a horizontal asymptote because the curve will always approach the $x$-axis as $x$ approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.
    • The $x$-axis is a horizontal asymptote of the function.
  • Graphs of Logarithmic Functions

    • The $y$-axis is a vertical asymptote of the graph.
    • This means that the $y$-axis is a vertical asymptote of the function.
    • However, the logarithmic function has a vertical asymptote descending towards $-\infty$ as $x$ approaches $0$, whereas the square root reaches a minimum $y$-value of $0$.
    • The graph of the square root function resembles the graph of the logarithmic function, but does not have a vertical asymptote.
  • Tangent as a Function

    • At these values, the graph of the tangent has vertical asymptotes.
    • The tangent function has vertical asymptotes at $\displaystyle{x = \frac{\pi}{2}}$ and $\displaystyle{x = -\frac{\pi}{2}}$.
  • Types of Conic Sections

    • Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
    • The other degenerate case for a hyperbola is to become its two straight-line asymptotes.
  • Graphs of Exponential Functions, Base e

    • The graph always lies above the $x$-axis, but gets arbitrarily close to it for negative $x$; thus, the $x$-axis is a horizontal asymptote.
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