Transcript USACAS_withScreenShots - Michael Buescher`s Home Page

CAS in Algebra 2 and
Precalculus
Michael Buescher
Hathaway Brown School
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
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.
The Distributive Property
 Type the following into your TI-89 and
write down its response.
– a. 3  2  5  3  2  3  5
– b. x   y  z   x  y  x  z
– c. 2  52  22  52
– d. x  y 2  x 2  y 2
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
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
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
Rationalizing Denominators?
a)
1
3
4
b)
8 5
c)
d)
4
x
x
x 1
[examples from UCSMP Advanced Algebra, supplemental materials, Lesson Master 8.6B]
Powers and Roots
Show that
6 2

4
1 3
2
2
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.
9
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
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)
Polynomials, Early On
x  2 x  5x  6  x  2x  1x  3
3
2
15
.786,8.209
10
5
2,0
-5
-4
-3
-2
1,0
-1
1
-5
-10
-15
2
3,0
3
4
2.120, 4.061
5
More Polynomials
 Expand the understanding of factors and
graphs, through …
– Irrational zeros
– Non-real zeros
– And finally, the Fundamental Theorem of
Algebra
Irrational Zeros
UCSMP Advanced Algebra, Example 3, page 707
px   x 2  5
 Find the zeros using the quadratic formula.
 Consider
 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):
Non-Real Zeros
UCSMP Advanced Algebra, Example 4, page 708
2


p
x

x
 x 1
 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).
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?
x  3x  2x  2  i x  2  i 
A Test Question: Polynomials
 Sketch a graph of f x   x5  7 x 3  15x 2  7
 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:
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.
Xscl = 1; Yscl = 1;
all intercepts are integers.
Rational Functions: The Old Rule
 Let f be the rational function
N x  an x n  an 1 x n 1    a1 x  a0
f x  

D x  bm x m  bm1 x m1    b1 x  b0
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 y  an b is a horizontal asymptote.
m
– If n > m, the graph of f has no horizontal asymptote.
 Oblique (slant) asymptotes are treated separately.
Rational Functions
 Expanded Form:
2 x 2  13x  18
f x  
x3
 Factored Form:

x  2 2 x  9 
f x  
x  3
 Quotient-Remainder Form:
3
f x   2 x  7 
x3
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.
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.
x  3x  4 x  6
f x  
x 1
3
 Analyze
2
is just as easy as any other rational function.
Rational Functions
 Analyze
x 3  3x 2  4 x  6
f x  
x 1
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
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.
Rational Functions
 Analyze
x 2  3x
f x  
x
Factored form:
wait … what?
Quotient-Remainder form:
still very odd ...
What do the  and the  have to say?
Thank You!
Michael Buescher
Hathaway Brown School
[email protected]