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
x5
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
x5
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
x5
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
x5
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
x5
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
x5
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
x5
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
x5
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
x5
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
