1. 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

2. 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.

3. 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

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

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

6. 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)

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

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

9. 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.

10. 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

11. 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.

12. 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

13. Oblique asymptote: y = x + 6