Slant Asymptote

Definition

A slant asymptote is a line that a rational function's graph approaches as the independent variable approaches positive or negative infinity. It represents the oblique line that the graph of the rational function tends to parallel as it moves further from the origin.

5 Must Know Facts For Your Next Test

1. The equation of a slant asymptote has the form $y = mx + b$, where $m$ is the slope of the asymptote and $b$ is the $y$-intercept.
2. Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator in a rational function.
3. To find the equation of a slant asymptote, one can use polynomial long division to divide the numerator by the denominator and obtain the slope and $y$-intercept.
4. Slant asymptotes provide information about the behavior of a rational function's graph as it approaches positive or negative infinity.
5. The presence of a slant asymptote indicates that the rational function has a removable discontinuity, which can be identified and addressed.

Review Questions

• Explain the relationship between the degrees of the numerator and denominator of a rational function and the existence of a slant asymptote.
  • The existence of a slant asymptote for a rational function is determined by the relationship between the degrees of the numerator and denominator polynomials. Specifically, if the degree of the numerator is one more than the degree of the denominator, then the rational function will have a slant asymptote. This is because the leading term of the numerator will dominate the behavior of the function as the independent variable approaches positive or negative infinity, resulting in the graph of the function approaching a line with a specific slope and $y$-intercept.
• Describe the process of finding the equation of a slant asymptote for a rational function.
  • To find the equation of a slant asymptote for a rational function, one can use the method of polynomial long division. By dividing the numerator polynomial by the denominator polynomial, the result will be a quotient and a remainder. The quotient represents the linear equation of the slant asymptote, with the slope being the coefficient of the linear term and the $y$-intercept being the constant term. The remainder, if any, indicates the presence of a horizontal asymptote in addition to the slant asymptote.
• Explain how the presence of a slant asymptote provides information about the behavior of a rational function's graph.
  • The presence of a slant asymptote for a rational function indicates that the graph of the function will approach and parallel the slant asymptote as the independent variable approaches positive or negative infinity. This information is valuable in understanding the overall behavior and characteristics of the rational function, as the slant asymptote represents the oblique line that the graph will tend to follow at its extremes. Additionally, the equation of the slant asymptote can be used to make predictions about the function's behavior and to identify any removable discontinuities that may be present.

Related terms

Rational Function:

A rational function is a function that can be expressed as the ratio of two polynomial functions.

Horizontal Asymptote:

A horizontal asymptote is a horizontal line that a rational function's graph approaches as the independent variable approaches positive or negative infinity.

Vertical Asymptote:

A vertical asymptote is a vertical line that a rational function's graph approaches but never touches as the independent variable approaches a specific value.