Pages

Saturday, May 12, 2012

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 .

Sunday, May 6, 2012

Java Programming



The methods inside the constructor of the frame class, that are called by the frame1 object does the following: