E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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3.6
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Properties of Rational Functions and Their Graphs
Rational Function Vertical Asymptotes Horizontal Asymptotes Graphs of Rational Functions
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
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KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
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KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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: