vertical

Vertical1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes

Asymptotes

1)(

2

3

x

xxf

Horizontal

)(lim

),(lim

xf

xf

study

x

x

Slant or Oblique

0)()(lim

bmxxfstudyx

called a slant asymptote because the vertical distance between the curve and the line approaches 0.

For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following

:Example

Sec 4.4: Curve Sketching

1)(

2

3

x

xxf

Slant or Oblique

0)()(lim

bmxxfstudyx

called a slant asymptote because the vertical distance between the curve and the line approaches 0

For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following

:Example

Sec 4.4: Curve Sketching

3

1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes

Asymptotes

Degree Example Horizontal Slant

Deg(num)<Deg(den)

Deg(num)=Deg(den)

Deg(num)=Deg(den)+1

0y NO

n

n

x

xy

of coeff

of coeff NO

NO

Special Case: (Rational function) Horizontal or Slant

onLongDivisi

4

2

4

1

x

xy

2

2

24

31

x

xy

1

12

3

xx

xy

Horizontal

Sec 4.4: Curve Sketching

F091

Sec 4.4: Curve Sketching

F101

Sec 4.4: Curve Sketching

F081

Sec 4.4: Curve Sketching

F092

Sec 4.4: Curve Sketching

1)(

2

3

x

xxf

Slant or Oblique

0)()(lim

bmxxfstudyx

called a slant asymptote because the vertical distance between the curve and the line approaches 0

For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following

:Example

xexf x )(

:Example

Sec 4.4: Curve Sketching

9

F101

Sec 4.4: Curve Sketching

A. InterceptsB. Asymptotes

SKETCHING A RATIONAL FUNCTION

)2()2(3

)4()(

2

2

xx

xxxf

:Example

Sec 4.4: Curve Sketching

A. DomainB. InterceptsC. SymmetryD. AsymptotesE. Intervals of Increase or DecreaseF. Local Maximum and Minimum ValuesG. Concavity and Points of InflectionH. Sketch the Curve

GUIDELINES FOR SKETCHING A CURVE

Symmetry

)()( :functioneven xfxf

)()( :function odd xfxf

symmetric aboutthe y-axis

symmetric aboutthe origin

Sec 4.4: Curve Sketching

12

Example

1

22

2

x

xy

A. DomainB. InterceptsC. SymmetryD. AsymptotesE. Intervals of Increase or DecreaseF. Local Maximum and Minimum ValuesG. Concavity and Points of InflectionH. Sketch the Curve

A. Domain: R-{1,-1}B. Intercepts : x=0C. Symmetry: y-axisD. Asymptotes: V:x=1,-1 H:y=2E. Intervals of Increase or Decrease: inc (-

inf,-1) and (-1,0) dec (0,1) and (1,-inf)F. Local Maximum and Minimum Values:

max at (0,0)G. Concavity and Points of Inflection down

in (-1,1) UP in (-inf,-1) and (1,inf)H. Sketch the Curve

Sec 4.4: Curve Sketching

F081

Sec 4.4: Curve Sketching

Easy to sketch:

2)( xxf

xxf )(

21 xy

2410 xy

Study the limit at inf

criticals all find 1)

lim lim :study 2)xx

asymptotes verticalall find 3)

Sec 4.4: Curve Sketching

Study the limit at inf

criticals all find 1)

lim lim :study 2)xx

asymptotes verticalall find 3)

Sec 4.4: Curve Sketching

Study the limit at inf

degeven poly with 1)

or )(lim x

xf

deg oddpoly with 2)

second the is one)(lim x

xf

Sec 4.4: Curve Sketching

F083

Sec 4.4: Curve Sketching

18

F091

Sec 4.4: Curve Sketching

19

F091

Sec 4.4: Curve Sketching