Sec 4.5: Curve Sketching Asymptotes Horizontal Vertical 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes Slant or Oblique called a slant asymptote because the vertical distance between the curve and the line approaches 0.

Sec 4.5: 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

Sec 4.5: Curve Sketching F141

Sec 4.5: Curve Sketching F092

Deg(num)<Deg(den) Sec 4.5: Curve Sketching Asymptotes 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes Horizontal Special Case: (Rational function) Horizontal or Slant Degree Example Horizontal Slant Deg(num)<Deg(den) Deg(num)=Deg(den) Deg(num)=Deg(den)+1

Sec 4.5: Curve Sketching F101

Sec 4.5: Curve Sketching F081

Sec 4.5: Curve Sketching F101

Sec 4.5: 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

Sec 4.5: Curve Sketching F091

Sec 4.5: Curve Sketching F081

SKETCHING A RATIONAL FUNCTION Sec 4.5: Curve Sketching SKETCHING A RATIONAL FUNCTION Intercepts Asymptotes Intercepts even power odd power Vertical Asymptotes even power odd power Horizontal Asymptotes

Sec 4.5: Curve Sketching Intercepts Vertical Asymptot even power odd power reflect cross even power even power odd power odd power Vertical Asymptot Vertical Asymptot Same infinity different infinity even power odd power

SKETCHING A RATIONAL FUNCTION Sec 4.5: Curve Sketching SKETCHING A RATIONAL FUNCTION Intercepts Asymptotes even power odd power even power odd power

Sec 4.5: Curve Sketching common mistake Many students think that a graph cannot cross a slant or horizontal asymptote. This is wrong. A graph CAN cross slant and horizontal asymptotes (sometimes more than once). YES graph cannot cross a vertical asymptote

GUIDELINES FOR SKETCHING A CURVE Sec 4.5: Curve Sketching GUIDELINES FOR SKETCHING A CURVE Domain Intercepts Symmetry Asymptotes Intervals of Increase or Decrease Local Maximum and Minimum Values Concavity and Points of Inflection Sketch the Curve Symmetry symmetric about the y-axis symmetric about the origin

Sec 4.5: Curve Sketching Example Domain Intercepts Symmetry Asymptotes Intervals of Increase or Decrease Local Maximum and Minimum Values Concavity and Points of Inflection Sketch the Curve Domain: R-{1,-1} Intercepts : x=0 Symmetry: y-axis Asymptotes: V:x=1,-1 H:y=2 Intervals of Increase or Decrease: inc (-inf,-1) and (-1,0) dec (0,1) and (1,-inf) Local Maximum and Minimum Values: max at (0,0) Concavity and Points of Inflection down in (-1,1) UP in (-inf,-1) and (1,inf) Sketch the Curve