# Section 5.2 Properties of Rational Functions

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Section 5.2 Properties of Rational Functions
Objectives Find the Domain of a Rational Function Determine the Vertical Asymptotes of a Rational Function Determine the Horizontal or Oblique Asymptotes of a Rational Function

A rational function is a function of the form
where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator q is 0.

Find the domain of the following rational functions.
All real numbers x except -6 and -2. All real numbers x except -4 and 4. All Real Numbers

Recall that the graph of is
(1,1) (-1,-1)

Graph the function using transformations
(1,1) (-1,-1) (3,1) (1,-1) (2,0) (3,2) (1,0) (2,0) (0,1)

If, as x or as x , the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R. If, as x approaches some number c, the values |R(x)| , then the line x = c is a vertical asymptote of the graph of R. In the previous example, there was a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. (3,2) (1,0) (2,0) (0,1)

Examples of Horizontal Asymptotes
y = L y = R(x) y x y = L y = R(x) y x

Examples of Vertical Asymptotes:
x = c y x = c y x x

Theorem: Locating Vertical Asymptotes
If an asymptote is neither horizontal nor vertical it is called oblique. y x Theorem: Locating Vertical Asymptotes A rational function R(x) = p(x) / q(x), in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

Vertical asymptotes: x = -1 and x = 1
Example: Find the vertical asymptotes, if any, of the graph of each rational function. Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes Vertical asymptote: x = -4

Consider the rational function
in which the degree of the numerator is n and the degree of the denominator is m. 1. If n < m, then y = 0 is a horizontal asymptote of the graph of R. 2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R. 3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division. 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.

Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3
Example: Find the horizontal or oblique asymptotes, if any, of the graph of Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3

Oblique asymptote: y = x + 6