1. 2.6 Rational Functions and Asymptotes

2. Rational Function Rational function can be written in the form where N(x) and D(x) are polynomials and D(x) is not 0 Domain of a rational function is all real numbers except the x-values that make D(x) = 0

3. Finding Domain/Range Find the domain and the range of the function x-.5-.1-.001-.0001 f(x)-2-10-100-1000 x.001.01.1.51 f(x)10001001021

4. Horizontal and Vertical Asymptotes 1)The line x = a is a vertical asymptotes of the graph f if f(x)  ∞ or f(x)  -∞ as x  a, either from the right or from the left 2) The line y = b is a horizontal asymptote of the graph of f if f(x)  b as x  ∞ or x  -∞

5. Asymptotes of a Rational Function Let f be the rational function where N(x) and D(x) have no common factors. To Find A Vertical Asymptote: The graph has a vertical asymptotes at the zeros of D(x)

6. Asymptotes of a Rational Function Let f be the rational function where N(x) and D(x) have no common factors. To Find A Horizontal Asymptote: The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x) n = the degree of N(x) and d = the degree of D(x) 1.If n < d, then y = 0 2.If n = d, then y = a n /b n (leading coefficients) 3.If n > d, then there is no horizontal asymptote

7. Finding the Asymptotes 1) 2)

8. Finding Horizontal and Vertical Asymptotes

9. Find the a) domain, b) vertical asymptotes, and c) horizontal asymptotes

10. Two Horizontal Asymptotes A function that is not rational can have two horizontal asymptotes. One to the right and one to the left.

11. Word Problems A utility company burns coal to generate electricity. The cost C (in $) of removing p% of the smokestack pollutants is given by C=80,000/(100-p) for 0 < p < 100. Use your calculator to graph the function. You are a member of a state legislature that is considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would there be to the utility company because of the new law?