1. Asymptotes

2. Asymptotes
Curves always bend towards the asymptotes

3. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote

4. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as x  

5. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as x  
P x  R x 
y  Q x  
A x  A x 

6. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as x  
P x  R x 
y  Q x  
A x  A x 
solve A(x) = 0 to find
vertical asymptotes

7. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as x  
P x  R x 
y  Q x  
A x  A x 
y = Q(x) is the solve A(x) = 0 to find
horizontal/oblique vertical asymptotes
asymptote

8. Asymptotes
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as x  
solve R(x) = 0 to find where
(if anywhere) the curve cuts
the horizontal/oblique
asymptote
P x  R x 
y  Q x  
A x  A x 
y = Q(x) is the solve A(x) = 0 to find
horizontal/oblique vertical asymptotes
asymptote

9. e.g.  i  y 
 x  3 x  2 
 x  1 x  1

10. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
x2 1 x2  x  6

11. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6
x2 1

12. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6
x2 1
x 5

13. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6
x2 1
x 5
x5
y  1
 x  1 x  1

14. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1
x 5
x5
y  1
 x  1 x  1
x intercepts: (–3,0) , (2,0)
–3 2 x

15. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
x intercepts: (–3,0) , (2,0)
–3 2 x
y intercept: (0,6)

16. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
x intercepts: (–3,0) , (2,0)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1

17. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
1
x intercepts: (–3,0) , (2,0)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1
horizontal asymptote: y  1

18. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
1
x intercepts: (–3,0) , (2,0) (5,1)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1
horizontal asymptote: y  1
cuts horizontal
asymptote at x  5

19. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
1
x intercepts: (–3,0) , (2,0) (5,1)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1
horizontal asymptote: y  1
cuts horizontal
asymptote at x  5

20. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
1
x intercepts: (–3,0) , (2,0) (5,1)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1
horizontal asymptote: y  1
cuts horizontal
asymptote at x  5

21. e.g.  i  y 
 x  3 x  2 
 x  1 x  1
1
x2 1 x2  x  6 y
x2 1 6
x 5
x5
y  1
 x  1 x  1
1
x intercepts: (–3,0) , (2,0) (5,1)
–3 –1 1 2 x
y intercept: (0,6)
vertical asymptotes: x  1
horizontal asymptote: y  1
cuts horizontal
asymptote at x  5

22. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3

23. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 2  x  6 x3  2 x 2  x  2

24. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x
x 2  x  6 x3  2 x 2  x  2
x3  x 2  6 x

25. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x
x 2  x  6 x3  2 x 2  x  2
x3  x 2  6 x
 x2  5x  2

26. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2
x3  x 2  6 x
 x2  5 x  2
2
x  x  6

27. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2
x3  x 2  6 x
 x2  5 x  2
2
x  x  6
4x  4

28. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2
x3  x 2  6 x
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3

29. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –1 1 2 x

30. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –1 1 1 2 x

y intercept:  0,  
1
  3
 3

31. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3

32. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
y  x 1
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3
oblique asymptote: y  x  1

33. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
y  x 1
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3
oblique asymptote: y  x  1
cuts horizontal
asymptote at x  1

34. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
y  x 1
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3
oblique asymptote: y  x  1
cuts horizontal
asymptote at x  1

35. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
y  x 1
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3
oblique asymptote: y  x  1
cuts horizontal
asymptote at x  1

36. e.g.  i  y 
 x  2  x  1 x  1
 x  2  x  3
x 1
x 2  x  6 x3  2 x 2  x  2 y
x3  x 2  6 x
y  x 1
 x2  5 x  2
2
x  x  6
4x  4
4x  4
y  x 1
 x  2  x  3
x intercepts: (–1,0), (1,0), (2,0) –2 –1 1 1 2 3 x

y intercept:  0,  
1
  3
 3
vertical asymptotes: x  2,3
oblique asymptote: y  x  1
cuts horizontal
asymptote at x  1

37. Exercise 3G; 3, 6, 8ac, 16cf, 17a