11 x1 t03 06 asymptotes (2012)
TRANSCRIPT
![Page 1: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/1.jpg)
Asymptotes
![Page 2: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/2.jpg)
AsymptotesCurves always bend towards the asymptotes
![Page 3: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/3.jpg)
AsymptotesCurves always bend towards the asymptotes
Curves never cross a vertical asymptote
![Page 4: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/4.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/5.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/6.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/7.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/8.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/9.jpg)
3 2e.g.
1 1x x
i yx x
![Page 10: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/10.jpg)
3 2e.g.
1 1x x
i yx x
2 21 6x x x
![Page 11: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/11.jpg)
3 2e.g.
1 1x x
i yx x
2 21 6x x x 1
2x 1
![Page 12: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/12.jpg)
3 2e.g.
1 1x x
i yx x
2 21 6x x x 1
2x 15x
![Page 13: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/13.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/14.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/15.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/16.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/17.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/18.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/19.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/20.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/21.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/22.jpg)
2 1 1e.g.
2 3x x x
i yx x
![Page 23: 11 x1 t03 06 asymptotes (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/23.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/24.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/25.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/26.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/27.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/28.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/29.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/30.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/31.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/32.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/33.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/34.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/35.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/36.jpg)
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 (2012)](https://reader033.vdocuments.site/reader033/viewer/2022042701/55a8dade1a28abb93e8b4708/html5/thumbnails/37.jpg)
Exercise 3G; 3, 6, 8ac, 16cf, 17a