11 X1 T03 06 asymptotes (2010)

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