Rational Functions

6 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros 6)Slant Asymptotes

Domain In general, the domain of a rational function includes all real numbers except those x-values that make the denominator zero

Horizontal Asymptote Describes the end behavior of the graph as x approaches If the degree of p(x) < the degree of q(x), there is a horizontal asymptote at y = 0 If the degree of p(x) > the degree of q(x), there is NO horizontal asymptote If the degree of p(x) = the degree of q(x), there is a horizontal asymptote at

Vertical Asymptote Shows excluded values for which the function, f(x) is not defined for x The graph of f has vertical asymptotes at the unique zeros of q(x) (cannot be from a factor that is also in p(x)

Holes in the graph Do not occur unless there are factors in p(x) that are the same as factors in q(x) Occur at any zeros of factors that cancel out of p(x) and q(x)

Zeros of f(x) X-intercepts of the graph Occur at the zeros of p(x) –q(x) will not affect the zeros unless factors cancel out

Slant Asymptotes If the degree of p(x) is exactly one more than the degree of q(x), the graph of f(x) will have a slant asymptote To determine the equation of the slant asymptote, you must divide p(x) by q(x) and find the quotient

Ex1) Give the domain, asymptotes, holes and zeros of the following function. Then, graph it without using a calculator.

Ex2) Find the domain, asymptotes, holes and zeros. Then, graph f(x) without using a calculator

Ex3) Find the domain, asymptotes, holes and zeros. Then, graph f(x) without using a calculator

Ex4) Find the domain, asymptotes, holes and zeros. Then, graph f(x) without using a calculator

Practice Pg. 304 (13 – 18, 49 – 55odd)