Section+2.6

__2.6 Graphs of Rational Functions__

Ex. Find the domain of f(x)=1/(x-2) D: R - {2} lim f(x)= -infinity lim f(x)= infinity x--->2- x--->2+
 * 1. Rational Functions**: r(x)=p(x)/q(x) where q(x) does not equal 0 and p(x) and q(x) are polynomial functions.
 * 1) Domain: All real numbers except for the values that make the denominators zero.

//lim f(x)=±b lim f(x)= x---> a-
 * 2. Vertical Asymptote**: x=a is a vertical asymptote of the graph of f if:

or

lim f(x)=±infinity x--->a+//

//lim f(x)=b x---> -infinity
 * 3. Horizontal Asymptote**: y=b is a horizontal asymptote of the graph of f if:

or

lim f(x)=b x---> infinity//

1. Any vertical asymptote will occur at the zeros of the denominator. 2. If the degree of p(x) is less than q(x), then y=0 is a horizontal asymptote. 3. If the dgree of p(x)=q(x), then the ratio of the leading coefficients is a horizontal asymptote. 4. If the degree of p(x)>q(x), then the quotient of the function (ignoring the remainder) is an asymptote. -If the degree of p(x) is exactly one more than q(x), then we get a __slant__ asymptote. Ex. Find the asymptotes and intercepts of f(x)=x^3/x^2-9 Vertical: x^2-9=0 sqrt(x^2)=sqrt(9) //**x=±3**// Horizontal: (x^3+0x^2+0x+0)/(x^2+0x-9) y-intercept: (0,0) x-intercept: (0,0)
 * 4. Asymptote Rules**: Let r(x)=p(x)/q(x) be a rational function in lowest terms, then
 * x**+9x/(x^2-9)
 * //y=x//**

Ex. Find the asymptotes, intercepts, and graph f(x)=(x-1)/(x^2-x-6) Vertical: (factor the denominator) Horizontal: (use Asymptote Rules #2)
 * //x=3, x=-2//**
 * //y=0//**