CAS in Algebra 2 and Precalculus

1. CAS in Algebra 2 and Precalculus Michael Buescher Hathaway Brown School

2. Where I’m Coming From • Using CAS in Algebra 2 and Precalculus classes for four years • TI-89 for all, Mathematica for me • Traditional curriculum, heavily influenced by College Board AP Calculus

3. The Basics • Pedagogical Use #1: What I Already Know is True • Verify Distributive Property (and deny some fallacies!) -- Day One of calculator use in Algebra 2.

4. The Distributive Property • Type the following into your TI-89 and write down its response. • a. • b. • c. • d.

5. The Distributive Property • Use the answers you got above to answer the following True or False: • Multiplication distributes over addition and subtraction • Division distributes over addition and subtraction • Exponents distribute over addition and subtraction • Roots distribute over addition and subtraction

6. What Distributes Where? • Exponents(including roots) distribute over Multiplication and Division but NOT Addition and Subtraction. • Multiplication and Divisiondistribute over Addition and Subtraction. • P • E • MD • AS

7. Powers and Roots • Pedagogical Use #2: There seem to be some more truths out there. • Rationalize denominators. • When should denominators be rationalized? • Why should denominators be rationalized? • Imaginary and complex numbers

8. Rationalizing Denominators? [examples from UCSMP Advanced Algebra, supplemental materials, Lesson Master 8.6B]

9. Powers and Roots Show that

10. Is there something else out there? What are the two things you have to look out for when determining the domain of a function? What does your calculator reply when you ask it the following? a. 9 ÷ 0 b.

11. Powers and Roots • Pedagogical Use #3: Different forms of an expression highlight different information • Polynomials: • Standard form vs. factored form • Rational Functions: • Numerator-denominator vs. quotient-remainder

12. Polynomials, Early On • Take an equation and put it on the board: • Standard form • Factored form • Sketch the graph • Identify all intercepts • Find all turning points (max/min)

13. Polynomials, Early On

14. More Polynomials • Expand the understanding of factors and graphs, through … • Irrational zeros • Non-real zeros • And finally, the Fundamental Theorem of Algebra

15. Irrational Zeros UCSMP Advanced Algebra, Example 3, page 707 • Consider • Find the zeros using the quadratic formula. • Find the x-intercepts using the graph on your calculator. • On your calculator: • factor (x^2 – 5): • factor (x^2 – 5) [use ]: • factor (x^2 – 5, x):

16. Non-Real Zeros UCSMP Advanced Algebra, Example 4, page 708 • Consider • Sketch a graph and find the x-intercepts. • Use the quadratic formula to solve p(x) = 0. • Check your answer with cSolve. • Use the zeros to factor p (x).

17. Approaching the Fundamental Theorem of Algebra Ask your calculator to cfactor f (x) = x4 – 5x3 + 3x2 + 19x – 30. Use the factored form to find all four complex number solutions. How many x-intercepts will the graph have?

18. A Test Question: Polynomials • Sketch a graph of • Label the x- and y-intercepts. • How many complex zeros does the function have? • How many of those solutions are real numbers? Find them. • How many of them are non-real numbers? Find them:

19. Xscl = 1; Yscl = 1; all intercepts are integers. A Test Question: Polynomials The function f (x) = -x3 + 5x2 + k∙x + 3 is graphed below, where k is some integer. Use the graph and your knowledge of polynomials to find k.

20. Rational Functions: The Old Rule • Let f be the rational function where N(x) and D(x) have no common factors. • If n < m, the line y = 0 (the x-axis) is a horizontal asymptote. • If n = m, the line is a horizontal asymptote. • If n > m, the graph of f has no horizontal asymptote. • Oblique (slant) asymptotes are treated separately.

21. Rational Functions • Expanded Form: • Factored Form: • Quotient-Remainder Form:

22. Rational Functions: The New Rule • Given a rational function f (x), • Find the quotient and remainder. • The quotient is the “macro” picture. • The remainder is the “micro” picture -- it gives details near specific points.

23. Rational Functions • No need to artificially limit ourselves to expressions where the degree of the numerator is at most one more than the degree of the denominator. • Analyze is just as easy as any other rational function.

24. Rational Functions • Analyze • Expanded form: • y-intercept is (0, 6) • vertical asymptote x = -1 • Factored form: • x-intercept at (1, 0) • Quotient-Remainder form: • Approaches f (x) = x2 - 4x

25. Rational Functions: Test Question • Find the equation of a rational function that meets the following conditions: • Vertical asymptote x = 2 • Slant (oblique) asymptote y = 3x – 1 • y-intercept (0, 4) • Show all of your work, of course, and graph your final answer. Label at least four points other than the • y-intercept with integer or simple rational coordinates.

26. Rational Functions • Analyze • Factored form: • wait … what? • Quotient-Remainder form: • still very odd ... • What do the  and the  have to say?

27. Thank You! Michael Buescher Hathaway Brown School mbuescher@hb.edu