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

2. Homework Questions?

3. Rational Functions and Their Graphs Section 2.6

4. Objectives 1. Find the domain of rational functions 2. Use arrow notation 3. Identify vertical asymptotes 4. Identify horizontal asymptotes 5. Graph rational functions 6. Identify slant asymptotes 7. 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 1. Find the zeros of p(x), the numerator 2. Find the zeros of q(x), the denominator 3. Identify any vertical asymptotes (numbers that are zeros of q(x) but not zeros of p(x)). Draw a dashed line. 4. Identify any holes (x-values are numbers that are zeros of both p(x) and q(x)) 5. Identify any horizontal asymptotes by examining the leading terms. Draw a dashed line. 6. 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. 7. Find the y-intercept by evaluating f(0) 8. Identify the x-intercepts (numbers that are zeros of p(x) but not q(x)) 9. Pick a few more points to plot 10. 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. 11. 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. a) Find and interpret C(20), C(40), C(60), C(80), and C(90) b) What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function? c) Graph the function

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