Asymptote

Sketching graph involve finding :

1. domain
2. x and y intercepts
3. Asymptotes (for rational Functions)
4. relative extremum , Critical Numbers, increasing or decreasing
5. concavity(concave up/ concave down), Inflection Points
6. Table
7. Sketch the Graph

3 types Asymptotes

1. Vertical Asymptotes
2. Slant/oblique Aymptotes
3. horizontal Asymptotes

if y = p(x)/q(x)

Vertical Asymptotes

to find vertical asymptotes, factorize p(x) and q(x) completely and let denominator equals to ZERO. 

e.g : y = x^2 - x - 6 / x -  4

x - 4 = 0

x= 4, vertical Asymptotes.

Slant / Oblique Aymptotes

If degree of p(x) > degree of q(x) by 1, there is an oblique asymptotes.

use LONG DIVISION.

e.g : y = x^2 + 1/ x- 1

         x + 1     

x-1 | x^2 + 1

    - ( x  - x )

              x + 1

              x + 1

                    0

Horizontal Asymptotes

1. If degree of P(x) <  degree  of Q(x), HORIZONTAL Asymptotes is y=0.
2. If degree of P(x) =  degree  of Q(x), horizontal Asymptotes

e.g :  y = 2x^2 + 3 / 5x^2 +4 

horizontal Asymptotes = 2/5 .