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A rational function is acknowledged as a function that gets defined by some rational fractions. This is an algebraic function where both the denominator and the numerator happen to be polynomials.

The polynomials’ coefficients must not be rational numbers and you can take it as any field K. The value of a variable might be taken in a field L that contains K.

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What is Known as a Rational Function?

Rational functions are functions that you can write in the form of a ratio of a couple of polynomial functions. Here, neither the polynomials’ coefficients nor the values that the function takes are rational numbers. So, any function in a variable xx is known as a rational function only when you can write it as:


Here, PP and QQ are considered the polynomials functions of Q(x)≠0Q(x)≠0 and xx.

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The Method of Plotting a Graph of a Rational Function

For plotting a graph of rational function, you need to do the following:

Step one – Discover an intercept if there are present any.

Step two – You can find out the vertical asymptotes when you set the denominator equal to 0 and solve.

Step three – Find out the horizontal asymptote when there is present any, utilizing the fact mentioned above.

Step four – You must sketch the asymptote and also plot the x-intercepts and y-intercept on the graph.

Step five – In step five, you must sketch the graph.

You must use the steps mentioned above for plotting the graph of a rational function.

Functions of Graphing Rational

One highly vital concept to graph rational function is becoming aware of their asymptotes. Asymptotes are considered curves or lines that approach the curve stupidly. However, it never touches it.

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What are the Asymptotes of a Rational Function?

A rational function has got three kinds of asymptotes:

  • Vertical asymptotes
  • Horizontal asymptotes
  • Oblique asymptotes

Vertical asymptotes

R(x) will possess vertical asymptotes at the 0s of Q(x). It happens as the 0s of Q(x) and Q(x) are equal to 0. It means that towards the right and left of the 0 of Q(x) its value of Q would be a very small positive number and negative number and the value of R(x) would be largely positive and negative number respectively.

Horizontal asymptotes

Horizontal asymptotes are meant the asymptote is horizontal and it means it is parallel to the independent variable’s axis. For determining the asymptotes, you need to divide the denominator and numerator of R(x). After that, you must find the value of R(x) that approaches when x becomes a large value.

Oblique asymptote

R(x) will possess an oblique asymptote and when you can display it as T(x)+1Q(x). For quoting an instance, you need to take R(x)+x2+3x+3x+1. So the degree of P(x) happens to be greater in comparison to Q(x). Hence, it can’t possess a horizontal asymptote. However, it can possess a vertical asymptote at x=-1.

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