Horizontal Asymptote

Horizontal asymptote is a straight horizontal line that continually proposes a given curve, but fails to meet it at any fixed distance. In other words, it is the usual behaviour of the horizontal line at the very edges of the graph.

It's important to understand the difference between horizontal and vertical asymptotes:

• It is hard to access horizontal asymptote, whereas touching a vertical asymptote is not a difficult task even done unknowingly.
• Whereas vertical asymptotes get attracted towards the origin, horizontal asymptotes repel the sides of the graph.  

These simple examples give detailed idea of the horizontal asymptotes.

Horizontal asymptotes and its operation:

Horizontal asymptotes work for the functions where the numerators as well as the denominator both are polynomials.

Now, the question arises what are polynomials?

Well, polynomials are the mathematical expressions, including a sum of terms, each term having a variable raised to a power and multiplied by a coefficient.  The simplest polynomials consist of one variable. This is how polynomials look like :

The graphs of polynomials act like "roller coaster", the constant ups and downs.

These functions are named as rational expressions. Let's give a look how a horizontal asymptote looks like.

So, the functional area is the fraction of two polynomials.

Our horizontal asymptote is y = 0. We can observe how the function's graph gets closer and closer to that line as it moves towards the ends of the graph.

Horizontal Asymptotes Rules:

Horizontal asymptotes follow three rules depending on the degree of the polynomials of the rational expression.

Let's understand it this way:

Our function is having a polynomial of degree "N" on top and a polynomial of degree "M" on the bottom. Horizontal asymptote rules work according to this degree.

• When n is less than m, the horizontal asymptote is y = 0 or the x-axis.
• When n is equal to m, then the horizontal asymptote is equal to y = a/b or we can simply divide the coefficients of the terms.
• When n is greater than m, (n>m) there is no horizontal asymptote.

The degrees of the polynomials in the function determine if there is a horizontal asymptote and where it actually lies. While working on these problems, go through all these steps in order, so you can easily remember them. They're not so hard to crack once you get the nerve of them, so just practice the exercises and be an expert in it.

Let's solve a few simple problems of Horizontal asymptotes:

1. Find the horizontal asymptote of y = 1/x

Clue:

There is one horizontal and one vertical asymptote.

Answer:

There is a horizontal asymptote at x = 0, and a vertical one at y = 0.

2. Let's find the horizontal asymptote for (x) = (4x²- 3x + 1) / (2x²- 1)

If we go with the long division:

(2x²- 1) / (4x²- 3x + 1)

Which gives 4x²- 2

Subtracting gives -3x

We got this 2 by looking at 4x²/ 2x²= 2

So, the horizontal asymptote is the line

y = 2

We could also look at

4x²/ 2x²

and it's done!...

So, that's how horizontal asymptotes work with a wide variety of functions. This lesson was all about graphing rational functions. Graphing program proves to be a great way to understand the working of rational functions. Function graphs strike to play with your imagination and make you quick interpreter.

Some Helpful Links:

1. This link can help you master the chapter with the practice sheets.
2. StackExchange too will provide you the practice sheets with solutions.
3. You will enjoy learning horizontal asymptotes in this link with very easy methods.
4. You can study the horizontal asymptotes in this link vividly with graphical presentations.