Sec 4 5 Curve Sketching Asymptotes 1Vertical Asymptotes

Sec 4. 5: Curve Sketching Asymptotes 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes Horizontal Vertical 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 F 141

Sec 4. 5: Curve Sketching F 092

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)+1 5

Sec 4. 5: Curve Sketching F 101

Sec 4. 5: Curve Sketching F 081

Sec 4. 5: Curve Sketching F 101 8

Sec 4. 5: Curve Sketching How to find Slant Asymptotes Rational functions No t. R Find m and b so that the limit is zero atio nal long division

Sec 4. 5: Curve Sketching

Sec 4. 5: Curve Sketching We need the limit of this expression to be zero highest degree of the denominator is 1. So we force the degree of the numerator to be zero

Sec 4. 5: Curve Sketching

Sec 4. 5: Curve Sketching F 091

Sec 4. 5: Curve Sketching F 081

Sec 4. 5: Curve Sketching

Sec 4. 5: Curve Sketching SKETCHING A RATIONAL FUNCTION A. Intercepts B. Asymptotes Intercepts even power odd power Vertical Asymptotes even power odd power Horizontal Asymptotes

Sec 4. 5: Curve Sketching Intercepts even power odd power cross reflect even power Vertical Asymptot even power Same infinity even power odd power Vertical Asymptot odd power different infinity odd power

Sec 4. 5: Curve Sketching SKETCHING A RATIONAL FUNCTION A. Intercepts B. Asymptotes 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 (Rational function)

Sec 4. 5: Curve Sketching GUIDELINES FOR SKETCHING A CURVE A. B. C. D. E. F. G. H. 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 A. B. C. D. E. F. G. H. Domain Intercepts Symmetry Asymptotes Intervals of Increase or Decrease Local Maximum and Minimum Values Concavity and Points of Inflection Sketch the Curve A. B. C. D. E. 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) 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 21