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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

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3.6


Properties of Rational Functions and Their Graphs

Rational Function Vertical Asymptotes Horizontal Asymptotes Graphs of Rational Functions



Rational Function

A rational function is a quotient of two polynomials. Therefore a function is a rational function if and only if there is a polynomial and a non-zero polynomial such that

The domain of a rational function is the set of all real numbers for which is defined as a real number. Therefore, the domain of consists of all numbers x for which .



Vertical Asymptotes

Given the rational function

a zero of which is not a zero of represents a vertical asymptote, which is a vertical line , near which the absolute value of the function get larger and larger as x gets closer and closer to . To illustrate, let us look at the plot for the function . Now is a zero of which is not a zero of . Accordingly is a vertical asymptote.



Horizontal Asymptotes

We summarize how to find the horizontal asymptotes of a proper rational function as follows:The rational function

with has horizontal asymptote if if .



Slant Asymptote

If is an improper rational function such that degree of degree of then such a rational function does not have a horizontal asymptote. However, since we can write

with degree of degree of , then as the absolute value of gets larger gets close to . In case where is linear, it is called a slant or oblique asymptote.



Example 1Find the vertical and horizontal asymptotes of the rational function

has vertical asymptotes given by the solutions of

Then the vertical asymptotes are:

and has horizontal asymptote



Example 2Find the vertical and horizontal asymptotes and state if the graph of the rational function intersects the horizontal asymptote or not

has vertical asymptotes at and .

has horizontal asymptote

and since , has solution , the graph of intersects its horizontal asymptote at .



Graphs of Rational Functions

Guidelines for graphing rational functiona. The domain of b. The x and y-intercepts of (if any)c. The holes, vertical and horizontal asymptotes and

point(s) where the graph intersect the horizontal asymptote (if any)

d. Several points on the graph of as needed. Choose an x-value in each domain interval determined by vertical asymptotes and x-intercepts.



Example 3Sketch the graph of the rational function

The domain of consist of all the real numbers except the zeros of denominator. To determine those numbers we need to solve the equation:

Therefore the domain of is given by



Example 3 continue

Since is a rational expression, the x-intercepts are the zeros of numerator. Therefore, we need to solve the equation: The only x-intercept is . The other two numbers and are not in the domain of and therefore, do not represent x-intercepts.The y–intercept is .



Example 3 continue

By factoring the numerator and denominator of

For and the graph of has two holes at and . Furthermore, the graph has vertical asymptote and horizontal asymptote .



Example 3 continue

Finally to test if the graph crosses the horizontal asymptote, we need to solve the equation Which is a false statement. Therefore, the graph doesn’t cross its horizontal asymptote.

We finally obtain some points on the graph.



Example 3 continue

Interval Sign Position

negative below x-axis

positive above x-axis

negative below x-axis

Now we can give an accurate sketch of the graph of as:

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