M115.  Syllabus.  Dr. Leisinger, Spring, 2014.
This file: Math_115_Syllabus.txt; Updated 1/24/2014, 6:10 pm.

Note: other information on the course is found elsewhere.

Prerequisites:
either
 Completion of Math-114
or
 a sufficient score on the Mathematics placement test A or B.

Calculators:
  In general, no calculator is needed, required, or permitted

=============== Topics covered: ======================================

Preliminaries
How to study for this course

Multiplication facts up to 9x9

Sets
  Set notation
  Element of, Union, Intersection, Complement
  Logical "and","or","not"
 
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
Order of operations

Working with fractions
 (add, subtract, multiply, divide, reduce, converting to and from decimal form)

Working with expressions
  Addition, Subtraction
  Multiplication
  Rational expressions

Working with inequalities
  Solving linear inequalities
  Absolute value
  Solving linear absolute value statements
  Notations:  set notation; graph notation; union-of-interval notation

Prime factorization of whole numbers
  Unique factorization

Equations and expressions

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:
    Concept of a parent function
    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
  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

Equations reducible to quadratic

Radical Equations
Fractional Equations

Formulae and problem solving

Systems of linear equations in 2 (or more) variables
  Solution methods:
    Substitution
    Gaussian Elimination
    Graphing
    Cramer's Rule (Method of Determinants)
    Matrix Inverse (2x2 case)

Graphing systems of linear inequalities

Word problems: certain types

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

Types of equations we can graph:
 Linear
 Linear systems
 Quadratic
 Rational [IF THERE IS TIME]