Polynomial Functions
Rational Functions
2.6
Rational Functions
A rational function is a ratio (fraction) of polynomials;
f(x)=N(x)/D(x).
Domain:
x�R such that D(x) �= 0
Asymptotes
(V. A.) Vertical Asymptote – Vertical Line
x = a; where D(a) = 0
Find V.A. by looking for restrictions in the domain.
The function cannot touch nor cross a vertical asymptote.
Asymptotes
(H.A.) Horizontal Asymptotes – Horizontal Line
y = b; where lim f (x) = b
x→±∞
Find H.A. by looking at the end behavior of the function.
The function can touch or cross a horizontal asymptote.
If deg(N(x)) < deg(D(x)), then y=0 is the H.A.
If deg(N(x)) = deg(D(x)), then the H.A. is the ratio of leading
coefficients
If deg(N(x)) > deg(D(x)), then there is no H.A.
Asymptotes
(S. A.) Slant Asymptote
If y =
N (x)
D(x)
= ax + b + r(x)
And deg(N(x)) = deg(D(x)) + 1, then the rational function has
a slant asymptote; y = ax +b.
Find S.A. using long division.
A rational function NEVER has both a H.A. and S.A.
Sketching the Graph
  Identify domain.
  Simplify
  Find/Plot: vertical, horizontal, and slant asymptotes
  Find/Plot: x-intercept(s) and y-intercept
  Plot: At least one point on each side of the x-intercept(s) and
vertical asymptote(s)
  Fill in with smooth curve.