CAS in Algebra 2 and Precalculus

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CAS in Algebra 2 and Precalculus

• Michael Buescher
• Hathaway Brown School

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Where Im 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

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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.

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The Distributive Property

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

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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

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What Distributes Where?

• Exponents (including roots) distribute over
  Multiplication and Division but NOT Addition and
  Subtraction.
• Multiplication and Division distribute 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

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Rationalizing Denominators?
examples from UCSMP Advanced Algebra,
supplemental materials, Lesson Master 8.6B
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Powers and Roots

• Show that

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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.
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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

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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)

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Polynomials, Early On
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More Polynomials

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

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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 (x2 5)
• factor (x2 5) use ??
• factor (x2 5, x)

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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).

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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?

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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

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A Test Question Polynomials

• The function f (x) -x3 5x2 kx 3 is
  graphed below, where k is some integer. Use the
  graph and your knowledge of polynomials to find
  k.

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Rational Functions The Old Rule

• Let f be the rational function
• where N(x) and D(x) have no common factors.
• If n lt m, the line y 0 (the x-axis) is a
  horizontal asymptote.
• If n m, the line is a horizontal
  asymptote.
• If n gt m, the graph of f has no horizontal
  asymptote.
• Oblique (slant) asymptotes are treated separately.

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Rational Functions

• Expanded Form
• Factored Form
• Quotient-Remainder Form

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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.

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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.

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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
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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.

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Rational Functions

• Analyze

Factored form wait what?
Quotient-Remainder form still very odd ...
What do the ? and the ? have to say?
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Thank You!

• Michael Buescher
• Hathaway Brown School
• mbuescher_at_hb.edu