M130.  Syllabus.  Dr. Leisinger, Fall, 2013.
Updated 8/29/2013, 5:32 pm.

Note: other information on the course is found elsewhere.

Prerequisites:
Completion of Math-115, or a sufficient score on the Mathematics placement test.

Assumed for this class:
Multiplication facts up to 9x9
Working with fractions
 (add, subtract, multiply, divide, reduce, converting to and from decimal form)

Required for this class:
  A scientific calculator (Graphing calculators are not allowed)
    Examples of accepted calculators:
	Casio fx-260 solar (Available for less than $10 at Office Max)
	TI-30X.
    A calculator on your phone is not allowed.

Topics covered:

Preliminaries
How to study for this course

Properties of the number systems
  Natural numbers
  Whole numbers
  Integers
  Rational numbers
  Real numbers
  Complex numbers
 The square root of 2 is irrational

Field axioms

Working with inequalities

Prime factorization of whole numbers
  Unique factorization

Equations and expressions

Order of operations

Linear equations:
 Definition(s) of lines
 Forms of the linear equation (Standard, slope-intercept, point-slope, two-intercept, two-point)
 Converting between forms of the linear equation
 Graphing linear equations
[Note: solving systems of linear equations is covered in Math-115, and is assumed in this class]

The distance formula
Circles
 Definition of a circle
 Equation of a circle
 Graphing a circle
 Completing the square to change the polynomial form of a circle to its standard form
 Finding the center and radius of a circle in polynomial form

Polynomials
 Defintion of polynomial
 Definitions: monomial; term; degree; total degree; coefficient.
 Factoring polynomials over the integers into prime factors
   Descending order
   Common monomial factor
   Difference of two squares
   Sum or difference of two like odd powers
   Factoring trinomials
     Discriminant and its uses
     Factoring using the quadratic formula
   Factoring by grouping
   Polynomials reducible to quadratic
   Factor theorem; Rational Root theorem
   Factoring 3rd and 4th degree polynomials
   Factoring polynomials of two or more variables (some cases)
 Polynomial rings
   (definition and explanation)
   (unique factorization in the polynomial ring)

Exponents
  Definition
  Laws of exponents
  Negative exponents
  Fractional exponents and radicals


Functions and Relations
  What is a function?
  What is a relation?
  A function and its graph
  Vertical line test for a function
  Domain, Range, and Image for a function
  Function with domain specified
  Finding the domain of a function when the domain is not specified
  Making new functions from old:
    Shifting
    Mirroring on the x and y axis
    Even and Odd functions
      Definitions
      Examples
    Stretching and shrinking
    Combining functions by adding, subtracting, multiplying, dividing
  Function composition
  Function inverse
    Finding the function inverse by the "verbal string" method
    Finding the function inverse algebraically
    Testing a function inverse

Quadratic functions (a review)
  Polynomial form of the quadratic function
  Calculating the discriminant
  Using the discriminant
  Standard Form (or Vertex Form) of the quadratic function
  Finding roots of a quadratic function in polynomial form
    (a) by factoring
    (b) by the Quadratic Formula
  Graphing a quadratic function in Polynomial form
  Completing the square to transform Polynomial form to the Vertex form
  Finding roots from the Vertex form
  Graphing the vertex form
  Understanding the vertex form as a shifted/stretched form of y=x^2

Radical Equations
Fractional Equations

Exponential functions
  Definition
  Graphing exponential functions
  Growth and decay for exponential functions

Exponential Equations

Logarithmic functions
  The logarithm as the inverse of the exponential
  A logarithm is an exponent
  Properties of logarithms: know and use
  Change of Base formula <<<<
  Properties of logarithms: prove the properties
  Graphing logarithmic functions
  Shifting and mirroring a logarithmic function

Logarithmic Equations

Logs & Exponentials as mathematical Models
		

Trigonometry
  The Unit Circle
  Definitions of sine and cosine functions
  Coterminal angles
  Reference angle
  Special angles (0,30,45,60,90 degrees) and their trig functions
  Graphing the cosine function
  Graphing the general sinusoidal function:  y-C = A cos (B (x-D))
    Crest, trough, center line, amplitude, displacement, period
    The "Box Method"
    Determining the equation of a sinusoid from its graph
  Review sine and cosine definitions
  The six trig. functions
    Functions and Cofunctions
  Trigonometric identities (I)
    The fundamental Pythagorean Identity
    Other Pythagorean identities
    Reciprocal relations
  Inverse Trigonometric Functions (esp. Arcsin and Arccos)
  cos(A+B), sin(A+B) formulae: believing
  cos(A+B), sin(A+B) formulae: proving
  cos(A+B), sin(A+B) formulae: using
  Double angle formulae
  Half-angle formulae
  The area of a triangle
  Right Triangle Trigonometry
  The Law of Cosines
  The Law of Sines
  Solving the general oblique triangle
  Astronomy (an example)

Trigonometric Equations

=========== Review for Final Exam ===========
Types of equations we can solve:
 Linear Equations
 Polynomial Equations (by factoring)
 Quadratic Equations (two methods)
 Equations reducible to Quadratic
 Exponential Equations
 Radical Equations
 Fractional Equations
 Logarithmic Equations
 Trigonometric Equations

Relations and Function  (definition, identifying)
Operations on functions:
 +-x/; shift,mirror,stretch/shrink
 Function Composition
 Inverse Functions
  Calculating the inverse of a function (two methods)

Exponentials and Logarithms
 Inverses of each other
 Graphing both
 Laws of exponents
 Laws of logarithms
 Change of Base formula

Types of equations we can graph:
 Linear
 Quadratic
 Rational [IF THERE IS TIME]
 Exponential
 Logarithmic
 Trigonometric
 General Sinusoid
 Inverse Trigonometric

Review of Trigonometry