Rational functions – Types of Asymptote

A line that approaches zero as one or both of the x and y coordinates approaches infinity are known as an asymptote of a curve in analytic geometry. An asymptote of a curve is a line that is tangent to the curve at infinity in projective geometry and related contexts.

In general, we will be given a rational function and asked to find the domain as well as any asymptotes. If there are any vertical asymptotes, we must first determine whether we have a horizontal or slant asymptote and what it is. We’ll need to know what steps to take and how to recognize different types of asymptotes to ensure we get the right (and complete) answer.

Types of Asymptote

A line approaching a curve but never touching it is called an asymptote. A line to which the graph of a function converges is known as an asymptote.

For a rational function, there are three types of asymptotes:

1. Horizontal Asymptotes.

2. Vertical Asymptotes.

3. Oblique Asymptote 

Horizontal Asymptotes

The degrees of the numerator and denominator can be used to determine the horizontal asymptote of a rational function. Horizontal asymptote at y = 0, numerator degree is less than denominator degree. The degree of the numerator is one higher than the degree of the denominator: Slant asymptote; no horizontal asymptote.

A rational function with a horizontal asymptote at y=2 is shown in the graph above. In general, horizontal asymptotes are represented by the equation y=a, where an is the value y of when x→±∞. These numbers also represent the maximum value that the function can achieve. 

Vertical Asymptotes

A vertical asymptote is a vertical line that guides but does not form part of the graph of a function. The graph will never cross it because it occurs at an x-value that is outside the function’s domain. There may be multiple vertical asymptotes for a function. 

Here’s an illustration of a graph with vertical asymptotes: x=-2 and x=2. This means that the function’s range is limited between -2 and 2. It’s worth noting that the graph’s curve never passes through the vertical asymptotes. This holds true for all functions with vertical asymptotes. 

Oblique Asymptote

Slanted asymptotes are another name for oblique asymptotes. This is due to its slanted form, which represents the linear function graph y=mx+b. Only when the numerator’s degree is exactly one degree higher than the denominator’s degree can a rational function have an oblique asymptote.

A slant asymptote is an imaginary oblique line that appears to touch a section of the graph. Only when the numerator’s degree is exactly one greater than the denominator’s degree does a rational function have a slant asymptote. 

Oblique asymptotes are linear functions that can be used to predict the end behaviour of rational functions, as shown in the example below.

 The f(x) oblique asymptote is represented by a dashed line in the graph, which guides the graph’s behaviour. Additionally, we can see that y=12x+1 is a linear function of the form y=mx+b.

 Conclusion

In this article we conclude that, A graph approaches an asymptote line without touching it. As the graph approaches positive infinity or negative infinity, it gets closer and closer to this line.  Asymptotes are useful guides for completing a function’s graph. An asymptote is a line along which the function’s curve approaches infinity or certain discontinuities. Vertical asymptotes, horizontal asymptotes, and oblique asymptotes are the three types of asymptotes.