1. 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.

2. 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

3. Sec 4.5: Curve Sketching
F141

4. Sec 4.5: Curve Sketching
F092

5. 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

6. Sec 4.5: Curve Sketching
F101

7. Sec 4.5: Curve Sketching
F081

8. Sec 4.5: Curve Sketching
F101

9. 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

10. Sec 4.5: Curve Sketching
F091

11. Sec 4.5: Curve Sketching
F081

13. 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

14. 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

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

16. 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

17. 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

18. 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