Transcript
Asymptotes
AsymptotesCurves always bend towards the asymptotes
AsymptotesCurves always bend towards the asymptotes
Curves never cross a vertical asymptote
Asymptotes
x
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as
Asymptotes
x
Curves always bend towards the asymptotes
Curves never cross a vertical asymptote
Curves approach horizontal and oblique asymptotes as
xAxRxQ
xAxPy
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
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
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
3 2e.g.
1 1x x
i yx x
3 2e.g.
1 1x x
i yx x
2 21 6x x x
3 2e.g.
1 1x x
i yx x
2 21 6x x x 1
2x 1
3 2e.g.
1 1x x
i yx x
2 21 6x x x 1
2x 15x
3 2e.g.
1 1x x
i yx x
2 21 6x x x 1
2x 15x
51
1 1xy
x x
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
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
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
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
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)
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)
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)
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)
2 1 1e.g.
2 3x x x
i yx x
2 1 1e.g.
2 3x x x
i yx x
2 3 26 2 2x x x x x
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 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
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
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
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
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
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
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
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
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
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
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
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
Exercise 3G; 3, 6, 8ac, 16cf, 17a
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