Warm-Up: January 12, 2012

1. Warm-Up: January 12, 2012 • Find all zeros of

2. Homework Questions?

3. Rational Functionsand Their Graphs Section 2.6

4. Objectives • Find the domain of rational functions • Use arrow notation • Identify vertical asymptotes • Identify horizontal asymptotes • Graph rational functions • Identify slant asymptotes • Solve applied problems involving rational functions

5. Rational Functions • Rational Functions are quotients of polynomial functions • The domain of a rational function is all real numbers except those that cause the denominator to equal 0

6. Example 1 (like HW #1-8) • Find the domain of

7. You-Try #1 (like HW #1-8) • Find the domain of

8. Arrow Notation • As x  a+, f(x)  ∞ • “As x approaches a from the right, f(x) approaches infinity” • As x  a-, f(x)  -∞ • “As x approaches a from the left, f(x) approaches negative infinity” • As x  ∞, f(x)  0 • “As x approaches infinity, f(x) approaches zero”

9. Vertical Asymptotes • An asymptote is a line that the graph of f(x) approaches, but does not touch. • The line x=a is a vertical asymptote if f(x) increases or decreases without bound as x approaches a. • As x  a+, f(x)  ±∞ • As x  a-, f(x)  ±∞ • If “a” is a zero of q(x), but not a zero of p(x), then x=a is a vertical asymptote.

10. Example 2 (like HW #21-28) • Find the vertical asymptotes, if any, of

11. You-Try #2 (like HW #21-28) • Find the vertical asymptotes, if any, of

12. Holes • A hole is a point that is not part of the domain of a function, but does not cause an asymptote. • If “a” is a zero of q(x), and a zero of p(x), then there is a hole at x=a • Holes generally are not distinguishable on a graphing calculator graph

13. Example of a Hole

14. Horizontal Asymptotes • The line y=b is a horizontal asymptote if f(x) approaches “b” as x increases or decreases without bound • As x  ∞, f(x)  b • OR • As x  - ∞, f(x)  b

15. Identifying Horizontal Asymptotes • Only the highest degree term of the top and bottom matter • Let “n” equal the degree of p(x), the numerator • Let “m” equal the degree of q(x), the denominator • If n<m, then the x-axis (y=0) is the horizontal asymptote • If n=m, then the line is the horizontal asymptote • If n>m, then f(x) does not have a horizontal asymptote

16. Example 3 (like HW #29-33) • Find the horizontal asymptote, if any, of each function

17. Warm-Up: January 17, 2012 • Find the horizontal asymptotes, if any, of: • Find the vertical asymptotes, if any, of

18. Homework Questions?

19. You-Try #3 (like HW #29-33) • Find the horizontal asymptote, if any, of each function

20. Graphing Rational Functions • Find the zeros of p(x), the numerator • Find the zeros of q(x), the denominator • Identify any vertical asymptotes (numbers that are zeros of q(x) but not zeros of p(x)). Draw a dashed line. • Identify any holes (x-values are numbers that are zeros of both p(x) and q(x)) • Identify any horizontal asymptotes by examining the leading terms. Draw a dashed line. • Find f(-x) to determine if the graph of f(x) has symmetry: • If f(-x)=f(x), then there is y-axis symmetry • If f(-x)=-f(x), then there is origin symmetry

21. Graphing Rational Functions, cont. • Find the y-intercept by evaluating f(0) • Identify the x-intercepts (numbers that are zeros of p(x) but not q(x)) • Pick a few more points to plot • Draw a curve through the points, approaching but not touching the asymptotes. If there was a hole identified in step 4, put an open circle at that x-value. • Check your graph with a graphing calculator. Remember that it does not properly display asymptotes and holes.

22. Example 4 (like HW #37-58) • Graph

23. You-Try #4 (like HW #37-58) • Graph

24. Warm-Up: January 18, 2012 • Determine any and all asymptotes and holes of:

25. You-Try #5 (like HW #37-58) • Graph

26. Slant Asymptotes • A slant asymptote is a line of the form y=mx+b that the graph of a function approaches as x±∞ • The graph of f(x) has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator • Find the equation of the slant asymptote by division (synthetic or long), and ignore the remainder

27. Example 7 (like HW #59-66) • Find the slant asymptote and graph

28. You-Try #7 (like HW #59-66) • Find the slant asymptote and graph

29. Warm-Up: January 19, 2012 • Determine any and all asymptotes and holes of:

30. Applications of Rational Functions • The average cost of producing an item • Chemical concentrations over time • Used in numerous science and engineering fields to approximate or model complex equations

31. Example 8 (page 322 #70) The rational function describes the cost, C(x), in millions of dollars, to inoculate x% of the population against a particular strain of the flu. • Find and interpret C(20), C(40), C(60), C(80), and C(90) • What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function? • Graph the function

33. Assignment • Page 321 #1-39 odd, 59, 67