D. Math is Still Rational

1. D. Math is Still Rational Pre-Calculus 30

2. PC30.11 • Demonstrate understanding of radical and rational functions with restrictions on the domain.

3. Key Ideas • Rational Function • Vertical Asymptote • Point of Discontinuity • Discontinuity

5. 1. Transforming Rationals • PC30.11 • Demonstrate understanding of radical and rational functions with restrictions on the domain.

6. 1. Transforming Rationals • Recall that Rational Functions are functions of the form where p(x) and q(x) are polynomial expressions and

7. You can graph a rational function by creating a table of values and then graphing the points in the table. • To create a table of values: • Identify the non-permissible value(s) • Write the non-permissible value in the middle row of the table • Enter positive values above and negative values below the non-permissible value • Choose small and large values of x to give you a spread of values

8. The graph of a rational function represents a vertical stretch by a factor of “a” of the graph , because can be written as • Graphs of the rational functions of form have two parts that approach asymptotes at y=0 and x=0

9. Example 1

10. When it comes to more complicated rational functions we can use prior knowledge of rationals and our new knowledge of transformations to graph and analyze.

12. Some equations of rational functions can be manipulated algebraically into the form by creating a common factor in the numerator and the denominator.

13. Example 2

14. Example 3

15. Example 4

16. Example 5

17. Key Ideas p.441

18. Practice • Ex. 9.1 (p.442) #1-7odds in each, 8-16 evens #2-7odds in each, 8-20 evens

19. 2. Discontinuity • PC30.11 • Demonstrate understanding of radical and rational functions with restrictions on the domain.

20. 2. Discontinuity • Investigation p, 446 (use graphing calculators)

21. Graphs of rational functions can have a variety of shapes and features, for example vertical asymptotes • Vertical Asymptotes are not the only feature of a rational function that can occur at a non-permissible value as you saw in the Investigation. • This is called a Point of Discontinuity

22. The graph of a rational function may have a asymptote, a point of discontinuity, or both. • To establish these important characteristics of a graph, begin by factoring the numerator and denominator.

26. Example 1

27. Example 2

28. Example 3

29. Key Ideas p.451

30. Practice • Ex. 9.2 (p.451) #1-14 #3-21

31. 3. Solving Rationals • PC30.11 • Demonstrate understanding of radical and rational functions with restrictions on the domain.

32. 3. Solving Rationals • As we have seen in Pre-Calculus 20 we can solve a Rational Equation graphically and algebraically • When solving rational equations we must always remember to check for possible extraneous roots before we solve • Extraneous Roots are numbers that would make the original function undefined

33. Solving Algebraically: solving algebraically determines the exact solution and any extraneous roots. The steps are: • Factor the numerator and denominator fully • Multiply each term by the lowest common denominator to eliminate the fractions • Solve for x • Check solutions against restriction and in original equation For example:

34. Graphically: there are two methods for solving equations graphically

37. Example 1

38. Example 2

39. Example 3

40. Key Ideas p.465

41. Practice • Ex. 9.3 (p.465) #1-6 odds in each, 7-15 odds #4-6 odds in each, 7-17 odds