1. Rational Functions Algebra III, Sec. 2.6 Objective Determine the domains of rational functions, Find asymptotes of rational functions, and Sketch the graphs of rational functions.

2. Important Vocabulary  Rational Function – a quotient of polynomial functions  Vertical Asymptote – the line x = a if f(x)  ∞ or f(x)  -∞  Horizontal Asymptote – the line y = b if f(x)  b  Slant Asymptote – a diagonal asymptote when the degree of the numerator is exactly one more than the degree of the denominator

3. Introduction The domain of a rational function of x includes all real numbers except… To find the domain of a rational function of x, … x-values that make the denominator zero determine the excluded values (set the denominator =0 and solve for x)

4. Example 1 Find the domain of f(x) and discuss the behavior of f near any excluded x values. When is the function undefined? Use a table to determine the behavior of f near x=2 x1.51.91.9 9 1.99 9 22.00 1 2.0 1 2.12.5 f(x) As x approaches 2 from the left, f(x) decreases without bound (-∞) As x approaches 2 from the right, f(x) increases without bound (∞) So the domain is all real numbers except x=2

5. Example (on your handout) Find the domain of the function When is the function undefined? What is the domain?

6. Horizontal & Vertical Asymptotes The notation “f(x)  5 as x  ∞” means… Describe the end behavior of a rational function in relation to its horizontal asymptote. y approaches 5 as x approaches ∞ (increases without bound) the end behavior of a function approaches the horizontal asymptote

7. Horizontal & Vertical Asymptotes 1.The graph of f has vertical asymptotes at ____________________. 2.The graph of f has one or no horizontal asymptotes determined by _______________________________________. a)If n<m, the graph of f has ________________________________. b)If n=m, the graph of f has ________________________________. c)If n>m, the graph of f has ________________________________. the zeros of D(x) comparing the degrees of N(x) and D(x) horizontal asymptote y=0 horizontal asymptote y=a/b no horizontal asymptote leading coefficient

8. Example 2 Find the vertical and horizontal asymptotes of Vertical asymptotes? Horizontal asymptotes? Set the denominator =0 and solve… Compare the degrees…

9. Example (on your handout) Find the vertical and horizontal asymptotes of Vertical asymptotes? Horizontal asymptotes? Set the denominator =0 and solve… Compare the degrees…

10. Analyzing Graphs of Rational Fns To sketch the graph of the rational function where N(x) and D(x) are polynomials with no common factors… 1.Simplify f if possible 2.Find & plot the y-intercept (find f(0)) 3.Find the x-intercepts (set N(x)=0 and solve) 4.Find the vertical asymptotes (set D(x)=0 & solve) 5.Find the horizontal asymptotes 6.Plot at least one point point between each x- values (intercepts & asymptotes) 7.Use smooth curves

11. Example 3 Sketch the graph of f(x) and state its domain. Simplify? y-intercept? x-intercept? vertical asymptotes? horizontal asymptotes?

12. Example (on your handout) Sketch the graph of f(x) and state its domain. Simplify? y-intercept? x-intercept? vertical asymptotes? horizontal asymptotes?

13. Example 4 Sketch the graph of f(x) and state its domain. Simplify? y-intercept? x-intercept? vertical asymptotes? horizontal asymptotes?

14. Example 5 Sketch the graph of f(x). Simplify? y-intercept? x-intercept? vertical asymptotes? horizontal asymptotes?

15. Example 6 Sketch the graph of f(x). Simplify? y-intercept? x-intercept? vertical asymptotes? horizontal asymptotes?

16. Slant Asymptotes To find the equation of a slant asymptote… Use long division or synthetic division

17. Example (on your handout) Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote. a)

18. Example (on your handout) Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote. b)

19. Example 7 Sketch the graph of X-intercepts? Y-intercepts? Vertical asymptote? Horizontal asymptote? Slant asymptote? Additional Points

20. Applications of Rational Fns Give an example of asymptotic behavior that occurs in real life.

21. Example 8 The cost C (in millions of dollars) fore removing p% of the industrial pollutants discharged into a river is given by for 0 ≤ p < 100. A proposed new law would require companies to remove 80% of their pollutants discharged into the river. The current law requires 45% removal. How much additional cost would the companies incur as a result of this law?

23. Example 9 A rectangular page is designed to contain 57 square inches of print. The margins on each side, as well as the top and bottom, are all 1 inch deep. What should the dimensions of the page be so that the least amount of paper is used?