1. College Algebra Chapter 3 Polynomial and Rational Functions
Section 3.5
Rational Functions

2. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

3. Apply Notation Describing Infinite Behavior of a Function
Rational Function:
Let p(x) and q(x) be polynomials where q(x) ≠ 0.
A function f defined by
is called a rational function.

4. Apply Notation Describing Infinite Behavior of a Function
Meaning
x approaches c from the right
(but will not equal c).
x approaches c from the left
x approaches infinity
(x increases without bound).
x approaches negative infinity
(x decreases without bound).

5. Example 1:
The graph of is given.
Complete the statements.

7. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

8. Identify Vertical Asymptotes
The line x = c is a vertical asymptote of the graph of a function f if f (x) approaches infinity or negative infinity as x approaches c from either side.

9. Identify Vertical Asymptotes

10. Identify Vertical Asymptotes

11. Identify Vertical Asymptotes
Let where p(x) and q(x) have no common
factors other than 1. To locate the vertical asymptotes of
, determine the real numbers x where the denominator is zero, but the numerator is nonzero.

12. Example 2:
Identify the vertical asymptotes (if any).

13. Example 3:
Identify the vertical asymptotes (if any).

14. Example 4:
Identify the vertical asymptotes (if any).

16. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

17. Identify Horizontal Asymptotes
The line y = d is a horizontal asymptote of the graph of a function f if f (x) approaches d as x approaches infinity or negative infinity.

18. Identify Horizontal Asymptotes

19. Identify Horizontal Asymptotes

20. Identify Horizontal Asymptotes
Let f be a rational function defined by
where n is the degree of the numerator and m is the degree of the denominator.
1. If n > m, f has no horizontal asymptote.
2. If n < m, then the line y = 0 (the x-axis) is the horizontal asymptote of f.
3. If n = m, then the line is the horizontal asymptote of f.

21. Example 5:
Identify the horizontal asymptotes (if any).

22. Example 6:
Identify the horizontal asymptotes (if any).

23. Example 7:
Identify the horizontal asymptotes (if any).

25. Identify Horizontal Asymptotes
The graph of a rational function may not cross a vertical asymptote. The graph may cross a horizontal asymptote.

26. Example 8:
Determine if the function crosses its horizontal asymptote.

28. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

29. Identify Slant Asymptotes
A rational function will have a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator.
To find an equation of a slant asymptote, divide the numerator of the function by the denominator.
The quotient will be linear and the slant asymptote will be of the form y = quotient.

30. Example 9:
Determine the slant asymptote of

32. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

33. Graph Rational Functions
where p(x) and q(x) are polynomials with no common factors.
Intercepts: Determine the x- and y-intercepts
Asymptotes: Determine if the function has vertical asymptotes and graph them as dashed lines.
Determine if the function has a horizontal asymptote or a slant asymptote, and graph the asymptote as a dashed line.  
Determine if the function crosses the horizontal or slant asymptote.

34. Graph Rational Functions
Symmetry: f is symmetric to the y-axis if f (–x) = f (x).
f is symmetric to the origin if f (–x) = –f (x).
Plot points: Plot at least one point on the intervals defined by the x-intercepts, vertical asymptotes, and points where the function crosses a horizontal or slant asymptote.

35. Example 10:
Graph

36. Example 10 continued:

37. Example 11:
Graph

38. Example 11 continued:

39. Example 12:
Graph

40. Example 12 continued:

41. Example 13:
Use transformations to graph

47. Concepts
1. Apply Notation Describing Infinite Behavior of a Function
2. Identify Vertical Asymptotes
3. Identify Horizontal Asymptotes
4. Identify Slant Asymptotes
5. Graph Rational Functions
6. Use Rational Functions in Applications

48. Example 14:
Young's rule is a formula for calculating the approximate dosage for a child, given the adult dosage. For an adult dosage of d mg, the child's dosage C(x) may be calculated by
x is the child's age in years and

49. Example 14 continued:
a. What is the horizontal asymptote of the function?
What does it mean in the context of this problem?
b. What is the vertical asymptote of the function?

50. Example 14 continued:
c. For an adult dosage of 50 mg, what is the approximate dosage for a 9-year-old child?
Graph: