37
Asymptotes

11 x1 t03 06 asymptotes (2013)

Embed Size (px)

Citation preview

Page 1: 11 x1 t03 06 asymptotes (2013)

Asymptotes

Page 2: 11 x1 t03 06 asymptotes (2013)

AsymptotesCurves always bend towards the asymptotes

Page 3: 11 x1 t03 06 asymptotes (2013)

AsymptotesCurves always bend towards the asymptotes

Curves never cross a vertical asymptote

Page 4: 11 x1 t03 06 asymptotes (2013)

Asymptotes

x

Curves always bend towards the asymptotes

Curves never cross a vertical asymptote

Curves approach horizontal and oblique asymptotes as

Page 5: 11 x1 t03 06 asymptotes (2013)

Asymptotes

x

Curves always bend towards the asymptotes

Curves never cross a vertical asymptote

Curves approach horizontal and oblique asymptotes as

xAxRxQ

xAxPy

Page 6: 11 x1 t03 06 asymptotes (2013)

Asymptotes

x

Curves always bend towards the asymptotes

Curves never cross a vertical asymptote

Curves approach horizontal and oblique asymptotes as

xAxRxQ

xAxPy

solve A(x) = 0 to find vertical asymptotes

Page 7: 11 x1 t03 06 asymptotes (2013)

Asymptotes

x

Curves always bend towards the asymptotes

Curves never cross a vertical asymptote

Curves approach horizontal and oblique asymptotes as

xAxRxQ

xAxPy

solve A(x) = 0 to find vertical asymptotes

y = Q(x) is the horizontal/oblique

asymptote

Page 8: 11 x1 t03 06 asymptotes (2013)

Asymptotes

x

Curves always bend towards the asymptotes

Curves never cross a vertical asymptote

Curves approach horizontal and oblique asymptotes as

xAxRxQ

xAxPy

solve A(x) = 0 to find vertical asymptotes

y = Q(x) is the horizontal/oblique

asymptote

solve R(x) = 0 to find where (if anywhere) the curve cuts

the horizontal/oblique asymptote

Page 9: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

Page 10: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

2 21 6x x x

Page 11: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

2 21 6x x x 1

2x 1

Page 12: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

2 21 6x x x 1

2x 15x

Page 13: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

Page 14: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2

Page 15: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

Page 16: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

Page 17: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

horizontal asymptote: 1y

1

Page 18: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

horizontal asymptote: 1y

1

cuts horizontal asymptote at 5x

(5,1)

Page 19: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

horizontal asymptote: 1y

1

cuts horizontal asymptote at 5x

(5,1)

Page 20: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

horizontal asymptote: 1y

1

cuts horizontal asymptote at 5x

(5,1)

Page 21: 11 x1 t03 06 asymptotes (2013)

3 2e.g.

1 1x x

i yx x

y

x

2 21 6x x x 1

2x 15x

51

1 1xy

x x

x intercepts: (–3,0) , (2,0) –3 2y intercept: (0,6)

6

vertical asymptotes: 1x

–1 1

horizontal asymptote: 1y

1

cuts horizontal asymptote at 5x

(5,1)

Page 22: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

Page 23: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x

Page 24: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x x

3 2 6x x x

Page 25: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

Page 26: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

1

2 6x x

Page 27: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

1

2 6x x 4 4x

Page 28: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

1

2 6x x 4 4x

Page 29: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) 2–1 1

1

2 6x x 4 4x

Page 30: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) 2–1 1

1

2 6x x 4 4x

1 intercept: 0,3

y

13

Page 31: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

Page 32: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

oblique asymptote: 1y x

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

1y x

Page 33: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

oblique asymptote: 1y x

cuts horizontal asymptote at 1x

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

1y x

Page 34: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

oblique asymptote: 1y x

cuts horizontal asymptote at 1x

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

1y x

Page 35: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

oblique asymptote: 1y x

cuts horizontal asymptote at 1x

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

1y x

Page 36: 11 x1 t03 06 asymptotes (2013)

2 1 1e.g.

2 3x x x

i yx x

y

x

2 3 26 2 2x x x x x x

3 2 6x x x 2 5 2x x

4 412 3xy x

x x

x intercepts: (–1,0), (1,0), (2,0) –2 2

vertical asymptotes: 2,3x

–1 1

oblique asymptote: 1y x

cuts horizontal asymptote at 1x

1

2 6x x 4 4x

1 intercept: 0,3

y

13

3

1y x

Page 37: 11 x1 t03 06 asymptotes (2013)

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