Can a graph of a rational function have no vertical asymptote?

Asked By: Ercilia Cal├žada | Last Updated: 11th March, 2020
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Can a graph of a rational function have no vertical asymptote? There is no vertical asymptote if the degree of the numerator of the function is greater than the degree of the denominator It is not possible. Rational functions always have vertical asymptotes.

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In respect to this, what function does not have a vertical asymptote?

Since the denominator has no zeroes, then there are no vertical asymptotes and the domain is "all x". Since the degree is greater in the denominator than in the numerator, the y-values will be dragged down to the x-axis and the horizontal asymptote is therefore "y = 0".

Furthermore, how do you find vertical asymptotes and holes? Set each factor in the denominator equal to zero and solve for the variable. If this factor does not appear in the numerator, then it is a vertical asymptote of the equation. If it does appear in the numerator, then it is a hole in the equation.

In this manner, how do you know if a rational function has a vertical asymptote or not?

Finding Vertical Asymptotes of Rational Functions. An asymptote is a line that the graph of a function approaches but never touches. The curves approach these asymptotes but never cross them. To find the vertical asymptote(s) of a rational function, simply set the denominator equal to 0 and solve for x.

How do you find the asymptotes of a function?

Finding Horizontal Asymptotes of Rational Functions

  1. If both polynomials are the same degree, divide the coefficients of the highest degree terms.
  2. If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.

38 Related Question Answers Found

What are vertical asymptotes?

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.)

How do you define Asymptotes?

An asymptote is a value that you get closer and closer to, but never quite reach. In mathematics, an asymptote is a horizontal, vertical, or slanted line that a graph approaches but never touches.

How do you tell if an equation has an asymptote?

Vertical asymptotes can be found by solving the equation n(x) = 0 where n(x) is the denominator of the function ( note: this only applies if the numerator t(x) is not zero for the same x value). Find the asymptotes for the function . The graph has a vertical asymptote with the equation x = 1.

When can there be no vertical asymptote?

here is no vertical asymptote if there are no x-intercepts. There is no vertical asymptote if the denominator of the function has only complex roots. There is no vertical asymptote if the degree of the numerator of the function is greater than the degree of the denominator It is not possible.

Do all functions have Asymptotes?


Does a linear function have any asymptotes? Surprisingly, this question does not have a simple answer. However, I hope to show you that while linear functions do not have any vertical asymptotes, they will have either a horizontal or oblique asymptote, depending on the slope of the line.

Will all rational functions have at least one vertical asymptote?

A rational function has at most one horizontal or oblique asymptote, and possibly many vertical asymptotes. Vertical asymptotes occur only when the denominator is zero. In other words, vertical asymptotes occur at singularities, or points at which the rational function is not defined.

What are the vertical and horizontal asymptotes of?

Solution. The vertical asymptotes will occur at those values of x for which the denominator is equal to zero: x − 1=0 x = 1 Thus, the graph will have a vertical asymptote at x = 1. To find the horizontal asymptote, we note that the degree of the numerator is two and the degree of the denominator is one.

How do you find the asymptotes of an exponential function?

Exponential Functions
A function of the form f(x) = a (bx) + c always has a horizontal asymptote at y = c. For example, the horizontal asymptote of y = 30e6x – 4 is: y = -4, and the horizontal asymptote of y = 5 (2x) is y = 0.

What are the rules for horizontal asymptotes?

The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m.
  • If n < m, the horizontal asymptote is y = 0.
  • If n = m, the horizontal asymptote is y = a/b.
  • If n > m, there is no horizontal asymptote.

What makes a function rational?


Rational function. In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K.

How do you graph asymptotes of a function?

Process for Graphing a Rational Function
  1. Find the intercepts, if there are any.
  2. Find the vertical asymptotes by setting the denominator equal to zero and solving.
  3. Find the horizontal asymptote, if it exists, using the fact above.
  4. The vertical asymptotes will divide the number line into regions.
  5. Sketch the graph.

What is the horizontal asymptote?

A horizontal asymptote is a y-value on a graph which a function approaches but does not actually reach. Here is a simple graphical example where the graphed function approaches, but never quite reaches, y=0 .

How do you find the horizontal asymptote of a graph?

Rule 1: If the degree of the numerator is less than the degree of the denominator, then there is a horizontal asymptote at y = 0 (the x-axis). Rule 2: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

How do you find slant asymptotes?

A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator. To find the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division. Examples: Find the slant (oblique) asymptote.

How do you graph a polynomial function?


Graphing Polynomial Functions
  1. Find the intercepts.
  2. Check for symmetry.
  3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts.
  4. Determine the end behavior by examining the leading term.
  5. Use the end behavior and the behavior at the intercepts to sketch the graph.

What is a simple rational function?

Rational Functions. A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1 . In other words, there must be a variable in the denominator. The general form of a rational function is p(x)q(x) , where p(x) and q(x) are polynomials and q(x)≠0 .

What makes a function not rational?

Non-Examples of Rational Functions
The function R(x) = (sqrt(x) + x^2) / (3x^2 - 9x + 2) is not a rational function since the numerator, sqrt(x) + x^2, is not a polynomial since the exponent of x is not an integer.