vertical asymptote

Examples of vertical asymptote in the following topics:

• Asymptotes

  • Vertical asymptotes are vertical lines near which the function grows without bound.
  • An asymptote that is neither horizontal or vertical is an oblique (or slant) asymptote.
  • The $y$-axis is a vertical asymptote of the curve.
  • Vertical asymptotes occur only when the denominator is zero.
  • Therefore, a vertical asymptote exists at $x=1$.

• 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 $\displaystyle{x = -\frac{\pi}{2}}$.

• Parts of a Hyperbola

  • The vertices have coordinates $(h + a,k)$ and $(h-a,k)$.
  • The line connecting the vertices is called the transverse axis.
  • The asymptotes of the hyperbola are straight lines that are the diagonals of this 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.

• Introduction to Rational Functions

  • Note that there are vertical asymptotes at $x$-values of $2$ and $-2$.

• Rational Inequalities

  • For $x$ values that are zeros for the denominator polynomial, the rational function is undefined, with a vertical asymptote forming instead.

• Inverse Trigonometric Functions

  • The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next, instead of being divided into pieces by an asymptote.

• Standard Equations of Hyperbolas

  • 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.
  • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.
  • If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be rectangular or equilateral.
  • In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b.

• Types of Conic Sections

  • If the ellipse has a vertical major axis, the $a$ and $b$ labels will switch places.
  • Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
  • The general equation for a hyperbola with vertices on a horizontal line is:
  • It is the axis length connecting the two vertices.
  • The other degenerate case for a hyperbola is to become its two straight-line asymptotes.

• Applications of Hyperbolas

  • Sundials work by casting the shadow of a vertical marker, sometimes called a gnomon, over a clock face on the horizontal surface.
  • The parameters of the traced hyperbola, such as its asymptotes and its eccentricity, are related to the specific physical conditions that produced it, namely the angle between the sunlight and the ground, and the latitude at which the sundial exists.
  • So if we call this difference in distances $2a$, the hyperbola will have vertices separated by the same distance $2a$, and the foci of the hyperbola will be the two known points.