11 X1 T03 06 asymptotes (2010)

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11 X1 T03 06 asymptotes (2010)

  1. 1. Asymptotes
  2. 2. Asymptotes Curves always bend towards the asymptotes
  3. 3. Asymptotes Curves always bend towards the asymptotes Curves never cross a vertical asymptote
  4. 4. Asymptotes Curves always bend towards the asymptotes Curves never cross a vertical asymptote Curves approach horizontal and oblique asymptotes as x  
  5. 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. 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. 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. 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. 9. e.g.  i  y   x  3 x  2   x  1 x  1
  10. 10. e.g.  i  y   x  3 x  2   x  1 x  1 x2 1 x2  x  6
  11. 11. e.g.  i  y   x  3 x  2   x  1 x  1 1 x2 1 x2  x  6 x2 1
  12. 12. e.g.  i  y   x  3 x  2   x  1 x  1 1 x2 1 x2  x  6 x2 1 x 5
  13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 22. e.g.  i  y   x  2  x  1 x  1  x  2  x  3
  23. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 37. Exercise 3G; 3, 6, 8ac, 16cf, 17a

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