Horizontal Asymptote

Definition

A horizontal asymptote is a horizontal line that a graph approaches as the input value (x) either increases or decreases without bound. This concept helps identify the behavior of functions at the extremes, indicating how they behave as they approach positive or negative infinity. Understanding horizontal asymptotes is crucial for sketching curves accurately and analyzing limits at infinity.

5 Must Know Facts For Your Next Test

1. Horizontal asymptotes can be found by evaluating the limit of a function as x approaches positive or negative infinity.
2. If the degree of the numerator is less than the degree of the denominator in a rational function, the horizontal asymptote is at y=0.
3. If the degrees of the numerator and denominator are equal, the horizontal asymptote is at y equal to the ratio of their leading coefficients.
4. If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote, but there may be an oblique or slant asymptote instead.
5. Horizontal asymptotes indicate long-term behavior but do not affect how a graph behaves near vertical asymptotes or intercepts.

Review Questions

• How do you determine the horizontal asymptote of a rational function, and what does it tell you about the function's behavior?
  • To determine the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than that of the denominator, the asymptote is at y=0. If they are equal, it is at y equal to the ratio of their leading coefficients. This information helps describe how the function behaves as x approaches positive or negative infinity, indicating where the graph levels off.
• Explain why understanding horizontal asymptotes is important for curve sketching and how it relates to limits at infinity.
  • Understanding horizontal asymptotes is vital for curve sketching because they reveal how a function behaves as it extends toward infinity. This knowledge helps in accurately depicting end behavior on graphs. By evaluating limits at infinity, we can determine these asymptotes, which guide us in predicting whether the graph will rise or fall infinitely or settle at a particular value. Thus, limits at infinity and horizontal asymptotes work together to provide a complete picture of function behavior.
• Analyze how horizontal asymptotes influence the overall shape of a graph and contrast this with vertical asymptotes.
  • Horizontal asymptotes influence a graph's shape by indicating where it levels off as x approaches positive or negative infinity. This contrasts with vertical asymptotes, where the graph tends to approach infinite values and can indicate points where functions are undefined. While horizontal asymptotes help describe long-term behavior, vertical asymptotes often signify critical points that lead to significant changes in graph shape. Together, they provide insight into both end behavior and local characteristics of functions.

Related terms

Vertical Asymptote: A vertical asymptote is a vertical line that a graph approaches but never touches, typically occurring at values of x where a function becomes undefined.

End Behavior: End behavior describes how a function behaves as the input approaches positive or negative infinity, often determined by its leading terms.

Limit at Infinity: A limit at infinity refers to the value that a function approaches as the input grows larger and larger in the positive or negative direction.