Rational Function

Rational Function

1. Form: f(x)=N(x)/D(x); N(x) is the numerator, D(x) is the denominator, N(x) and D(x) are polynomials, D(x) cannot be 0

   • how to find all asymptote of f(x)
     • vertical: factor out the denominator, the real zeros of the denominator are the USE SIMPLIED FORM
       • The common factors of the top and bottom of f(x) gives the x values of the hole(use the SIMPLIED form of f(x)to find the y coordinates of the holes), the holes of the graph are not VA
     • USE SIMPLIED FORM horizontal: the greatest exponent of N(x) and D(x) are n and m, the leading coefficients of N(x) and D(x) are an and bm,
       • n<m: HA: f(x)=0, x approaches (+ or -) infinity
       • n=m: HA: f(x)=an/bm
       • n>m: HA: None
         • n>m: by exactly 1 degree, slant asymptote USE SYMPLIFIED FORM

2. how to graph a rational function

   • (1)simplify f(x) if possible: find the domain, holes and vertical asymptotes of f(x)
   • (2)all asymptote: see the category:how to find all asymptote of f(x)
   • (3)Y-intercept: set x=0, evaluate f(0), use the ORIGINAL f(x)
   • (4)X-intercepts: set the ORIGINAL f(x)=0, the numerator=0 find the factors of the numerator, find the real zeros of the numerator
   • (5)Table: VA,HA/Slant, all x and y intercepts, additional points in between those points
   • (6)Graph: plot the asymptotes on the graph and the points on the table