Vertical 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes Asymptotes Sec 4.5 SUMMARY OF CURVE SKETCHING Horizontal Slant or Oblique 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

Sec 4.5 SUMMARY OF CURVE SKETCHING Slant or Oblique 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

3 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes Asymptotes Sec 4.5 SUMMARY OF CURVE SKETCHING DegreeExampleHorizontalSlant Deg(num)<Deg(den) Deg(num)=Deg(den) Deg(num)=Deg(den)+1 Special Case: (Rational function) Horizontal or Slant Horizontal

F091 Sec 4.5 SUMMARY OF CURVE SKETCHING

F101 Sec 4.5 SUMMARY OF CURVE SKETCHING

F081

F092 Sec 4.5 SUMMARY OF CURVE SKETCHING

Slant or Oblique 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

9 F101 Sec 4.5 SUMMARY OF CURVE SKETCHING

A.Intercepts B.Asymptotes SKETCHING A RATIONAL FUNCTION Sec 4.5 SUMMARY OF CURVE SKETCHING

A.Domain B.Intercepts C.Symmetry D.Asymptotes E.Intervals of Increase or Decrease F.Local Maximum and Minimum Values G.Concavity and Points of Inflection H.Sketch the Curve GUIDELINES FOR SKETCHING A CURVE Sec 4.5 SUMMARY OF CURVE SKETCHING Symmetry symmetric about the y-axis symmetric about the origin

12 Example Sec 4.5 SUMMARY OF CURVE SKETCHING A.Domain B.Intercepts C.Symmetry D.Asymptotes E.Intervals of Increase or Decrease F.Local Maximum and Minimum Values G.Concavity and Points of Inflection H.Sketch the Curve A.Domain: R-{1,-1} B.Intercepts : x=0 C.Symmetry: y-axis D.Asymptotes: V:x=1,-1 H:y=2 E.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: Study the limit at inf

Absolute Maximum and Minimum Study the limit at inf

Absolute Maximum and Minimum Study the limit at inf

F083

18 F091

19 F091