Rational Functions and Asymptotes

Rational Functions and Asymptotes:

The rational function is a function which can be written as the ratio of two polynomials where the denominator is not zero.

f(x) = p(x)/q(x)


The domain of a rational function is all the real values apart from where the denominator, q(x) = 0.


The roots, solutions, zeros, x-intercepts (whatever you wish to call them) of the rational function will be the positions where p(x) = 0. That is, totally ignore the denominator. Whatever makes the numerator 0 (zero) will be the roots of rational function, just similar to they were the roots of polynomial function former.

When you can write it in the factored form, then you can tell whether it will touch or cross the x-axis at each and every x-intercept by whether the multiplicity on the factor is even or odd.

Vertical Asymptotes:

The asymptote is a line from which the curve approaches however does not cross. The equations of vertical asymptotes can be found by determining the roots of q(x). Completely avoid the numerator whenever looking for vertical asymptotes, only in the denominator matters.

When you can write it in the factored form, then you can tell whether the graph will be asymptotic in similar direction or in different directions by whether the multiplicity is odd or even.

Asymptotic in similar direction signifies that the curve will go up and down on both left and right sides of the vertical asymptote. Asymptotic in various directions signifies that the one side of curve will go down and other side of the curve will go up at vertical asymptote.

Horizontal Asymptotes:

The horizontal line is an asymptote merely to the far left and the far right of graph. ‘Far’ left or ‘far’ right is stated as anything past the vertical asymptotes or x-intercepts. The Horizontal asymptotes are not asymptotic in middle. It is okay to cross the horizontal asymptote in middle.

The place of the horizontal asymptote is recognized by looking at the degrees of numerator (n) and denominator (m).

a) When n<m, the x-axis, y = 0 is a horizontal asymptote.

b) When n = m, then y = an/bm is a horizontal asymptote. That is, the ratio of leading coefficients.

c) When n>m, then there is no horizontal asymptote. Though, if n = m+1, then there is an oblique or slant asymptote.


Sometimes, a factor will show in the numerator and in denominator. Let us suppose the factor (x-k) is in the numerator and denominator. Since the factor is in denominator, x = k will not be in the domain of function. This signifies that one of two things can occur. There will either a vertical asymptote at x = k or there will be a hole at x = k.

Let's look at what will occur in each of such cases:

a) There are more (x-k) factors in denominator. Subsequent to dividing out all the duplicate factors, the (x-k) is still in denominator. Factors in the denominator outcome in vertical asymptotes. Thus, there will be a vertical asymptote at x = k.

b) There are more (x-k) factors in numerator. After dividing out all duplicate factors, the (x-k) is still in the numerator. The factors in numerator outcome in x-intercepts. However since you cannot use x = k, then there will be a hole in the graph on x-axis.

c) There are equivalent numbers of (x-k) factors in numerator and denominator. After dividing out all factors (since there are equivalent amounts), there is no (x-k) left at all. Since there is no (x-k) in denominator, there is no vertical asymptote at x = k. Since there is no (x-k) in the numerator, there is no x-intercept at x = k. There is simply a hole in the graph, somewhere other than on x-axis. To determine the precise place, plug in x = k to the reduced function (you cannot plug it to the original, it is undefined, there), and see what y-value you obtain.

Oblique Asymptotes:

Whenever the degree of numerator is precisely one more than the degree of denominator, the graph of the rational function will contain an oblique asymptote. The other name for an oblique asymptote is the slant asymptote.

To determine the equation of oblique asymptote, execute long division (that is, synthetic if it will work) by dividing the denominator to the numerator. Since x gets very big (this is the far left or far right), the remainder part becomes much small, almost zero. Therefore to find the equation of oblique asymptote, execute the long division and reject the remainder.

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