vertical 1)vertical asymptotes 2) horizontal asymptotes 3) slant asymptotes asymptotes sec 4.5...
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Vertical1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes
Asymptotes
Sec 4.5 SUMMARY OF CURVE SKETCHING
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 4.5 SUMMARY OF 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
3
1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes
Asymptotes
Sec 4.5 SUMMARY OF CURVE SKETCHING
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
F091
Sec 4.5 SUMMARY OF CURVE SKETCHING
F101
Sec 4.5 SUMMARY OF CURVE SKETCHING
Sec 4.5 SUMMARY OF CURVE SKETCHING
F081
F092
Sec 4.5 SUMMARY OF CURVE SKETCHING
Sec 4.5 SUMMARY OF 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
9
F101
Sec 4.5 SUMMARY OF CURVE SKETCHING
A. InterceptsB. Asymptotes
SKETCHING A RATIONAL FUNCTION
Sec 4.5 SUMMARY OF CURVE SKETCHING
)2()2(3
)4()(
2
2
xx
xxxf
:Example
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
Sec 4.5 SUMMARY OF CURVE SKETCHING
Symmetry
)()( :functioneven xfxf
)()( :function odd xfxf
symmetric aboutthe y-axis
symmetric aboutthe origin
12
Example
Sec 4.5 SUMMARY OF CURVE SKETCHING
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 4.5 SUMMARY OF CURVE SKETCHING
F081
Absolute Maximum and Minimum
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)
Absolute Maximum and Minimum
Study the limit at inf
criticals all find 1)
lim lim :study 2)xx
asymptotes verticalall find 3)
Absolute Maximum and Minimum
Study the limit at inf
degeven poly with 1)
or )(lim x
xf
deg oddpoly with 2)
second the is one)(lim x
xf
F083
18
F091
19
F091