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    Precalc Notes

Functions, Vocabulary     Quadratic Functions Exponent & Log Laws
Orientation, ... Higher Order Polynomials Trig Functions, "Before Unit Circle"
Calculator Resources Rationals Trig Functions, "Unit Circle & On"
Lines Log & Exponential Functions     Trig Cheat Sheet

Higher Order Polynomial Functions

Definition & Notation
Simple definition (for algebra I and geometry):   polynomial
 
Precalc definition:
an algebraic expression or function, named by its highest degree, produced by the MULTIPLICATION of constants, monomials, binomials, ..., with the use of one variable x,
 
anxn + an-1xn-1 + an-2xn-2 + ... +a2x2 + a1x1 + a0x0,
 
in which the constants an, an-1, an-2, ... , a2, a1, a0 are real,
and serve as coefficients or the constant term,
and in which all exponent of x are counting numbers.

Notes
Polynomial Functions: Notes
      Definition & Notation
      Polynomial Computation
      Possible Roots
      Quadratic, Cubic, Quartic Formulas
      Complex Roots
Polynomial Functions
      How to sketch a polynomial function
      Examples of sketching polynomial functions
      Problems to sketch and their solution
Intercepts/Roots/Factors Vocabulary & Images
Games and Manipulatives
Who Has? Polynomial F(x) Game
 

Rational Functions

 
Definition & Notation
Precalc definition:
an algebraic expression or function, produced by the DIVISION of polynomials, with the use of one variable x -- fraction expressions or fraction functions
 
anxn + an-1xn-1 + an-2xn-2 + ... +a2x2 + a1x1 + a0x0
__________________________________________
bnxn + bn-1xn-1 + bn-2xn-2 + ... +b2x2 + b1x1 + b0x0
 
in which the constants an, an-1, an-2, ... , a2, a1, a0, bn, bn-1, bn-2, ... , b2, b1, b0 are real,
and serve as coefficients or the constant term,
and in which all exponent of x are counting numbers.

Notes
Rational Functions
      a list of things to do when sketching a rational function
      two examples of sketching rational functions
      a problems to try and their solution
Features of a Rational Function
Sketch f(x) = [ (x2 - 4)(x + 3)] / [(x+2)(x-1)]
Synthetic division & long division
      to find zeros
      to rewrite a rational function to find horizontal and other asymptotes and
      to determine the end behavior.
 
Games
Who Has? Rat'l F(x)
 
sketchpad sketchpad
Visit the Interactive Sketch Pad Material On This Topic
    Polynomial and Rational Functions
  • parabola   -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,
        * both generated by equations defined by parameters.
        * WITH SHOW/HIDE ACTION BUTTON
        * graphs of 2 parabolas: 1 in general form, 1 in standard forms,
        * 1 parabola, set a, then generat parabola w/2 points.
        * 1 parabola controlled by 3 points, used for determinant analysis
        * 1 vertical axis of symmetry, 1 horizontal "axis of symmetry"
  • TracePoly   -- drag point f(x) to state and show new ( x, f(x))
  • recipfx   -- graph of parameter-defined reciprocal function,
        * with slide-able tangent line,
        * VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER!


spread sheet notes
Spread Sheets -- Dynamic Computation
  • polynomial spreadsheet --
        graph a polynomial function written in factored form
        complete synthetic division
  • synthetic division spreadsheet -- just complete synthetic division of a polynomial or of a quadratic by a linear binomial
  • rational function spread sheet
        Graph and table of (Axm)/(Bxn)
        Watch the power of the top or bottom equal or dominate the other.
        Examine hoirzontal asymptotes or infinite increase or decrease.
        Think endbehavior and explore.
        Graph and table of rational function, polynomial A(x) divided by B(x).
        Examine asymptotes.
        Examine the role of factors A(x) and 1/B(x).
        Graph and table of rational functions written as A(x) / B(x) + C(x)
        Really play with C(x), the asymptote
        Complete synthetic division and rewrite of quotient and remainder.


Functions, Vocabulary     Quadratic Functions Exponent & Log Laws
Orientation, ... Higher Order Polynomials Trig Functions, "Before Unit Circle"
Calculator Resources Rationals Trig Functions, "Unit Circle & On"
Lines Log & Exponential Functions     Trig Cheat Sheet
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