1. Properties of Rational Functions 1

2. Learning Objectives 2 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function 3. Find the horizontal or oblique asymptotes of a rational function

3. Rational Function

4. To Find the Domain Domain The domain of a rational function is all real values except where the denominator, q(x) = 0

5. Example

8. Points Not in The Domain

9. Holes and Vertical Asymptotes

10. Examples Find holes and vertical asymptotes

11. Examples Find holes and vertical asymptotes

12. Example Find holes and vertical asymptotes

13. Example

14. More on Holes 14

15. Holes and Vertical Asymptotes Holes and vertical asymptotes are discontinuities, but they are very different vertical asymptotes are non-removable discontinuities but holes are removable discontinuities, and by the addition of a point, we can create a function continuous at that point 15

16. Example Hole at x = 2 Holes do not appear on the graph, but are clearly indicated on the table X-2 evenly divides both the numerator and the denominator 16

17. Example Vertical Asymptote at x = 2 Holes do appear on the graph and are clearly indicated on the table 17

18. Example x  0 -  f(x)  ∞ x  0 +  f(x)  ∞ x  0  f(x)  ∞

19. Horizontal and Oblique Asymptotes

20. End Behavior A function will not have both an oblique and a horizontal asymptote

21. A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle. Horizontal Asymptote

22. Example Find equation for horizontal asymptote 22

23. Example Find equation for horizontal asymptote 23

24. Example Find x-value(s) where f(x) crosses horizontal asymptote 24

25. Example Find equation for horizontal asymptote 25

26. Example Find equation for horizontal asymptote 26

27. Find equation for horizontal asymptote Example 27

28. Example 28

29. 29 Example This means for very large values of R 2 the total resistance approaches 10 ohms.

30. Oblique Asymptotes When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. Another name for an oblique asymptote is a slant asymptote. To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator and discarding the remainder 30

31. Example Oblique Asymptote y = x+2 Y2 is the end behavior of y1 X-2 divides the numerator with a remainder 31

32. Finding Oblique Asymptotes 32

33. Example 33

34. Example 34

35. Example 35

36. Example 36

37. Example Using synthetic division 37

38. Our rational function Our rational function and OA Example 38

39. Example 39

40. Example Using synthetic division 40

41. Our rational function Our rational function and OA Example 41