Algebra Brain Summary

Malcolm E. Hays 28 October 2002

 

This is a list of all the thoughts located in the Algebra Brain. Each thought is followed by a statement indicating the content associated by that thought. Thought followed by (jump) are jump thoughts, which are located on the left hand side of the central thought in the Brain matrix. Jump thoughts often take the user to entirely different sections of the Brain or provide reference information about the central thought.

 

This compilation of thoughts only includes those thoughts listed under Content in the Algebra Brain.

 

Ordering Real Numbers How real numbers relate to each other with respect to the origin

 

Functions - Introduction to one of the fundamental concepts in Algebra

Function Terminology (jump) A list of common terms associated with functions

Inverse Functions - Interchanging the domain and range of a function

Composition of Functions (jump) A function of a function

Finding the Inverse - How to find the inverse of a function

Horizontal Line Test - The test which determines if a function has an inverse

Polynomial Functions - Functions involving a sum of powers of x

Fundamental Theorem of Algebra (jump) - The number of roots of a polynomial function equals the highest power of x

Complex Numbers - Numbers the complex plane, of which the real numbers are a subset

Polynomial Operations - How to manipulate polynomials

Adding/Subtracting - How to add and subtract two (or more) polynomials

Multiplying Polynomials - How to multiply two (or more) polynomials

Special Product Patterns (jump) - A list of commonly occurring products that occur in Algebra

Polynomial Division - How to divide two polynomials

Synthetic Division - Technique used to divide polynomials of a particular form

Factor Theorem - A polynomial f(x) has a factor (x k) if and only if f(k) = 0

Remainder Theorem - If a polynomial f(x) is divided by (x k), then the remainder is r = f(k)

Real Zeros - The real solutions of a polynomial equations (as opposed to the complex solutions)

Descartes's Rule of Signs - How to determine the number of positive and negative roots of a polynomial equation

Rational Zero Test - How to find possible roots of a polynomial function with integer coefficients

Special Factoring Patterns - A list of commonly occurring factoring patterns that occur in Algebra

Special Product Patterns - A list of commonly occurring products that occur in Algebra

Multiplying Polynomials (jump) - How to multiply two (or more) polynomials

Rational Functions - One polynomial function divided by another

Conic Sections (jump) A look at some practical examples of rational functions

Polynomial Division (jump) Intimately tied with rational functions

Properties of Fractions (jump) Necessary for understanding how rational functions work

Asymptotes What they are and how they are used to help sketch rational functions

Partial Fractions - How to convert a rational function into a sum of two fractions

Distinct Linear Factors - Denominator of rational function is composed of distinct linear factors

Distinct Quadratic Factors - Denominator of rational function is composed of irreducible quadratic factors

Mixed Factors Denominator of rational function contains both linear and quadratic factors (distinct and/or repeating)

Repeated Linear Factors - Linear factors that repeat themselves in the denominator of a rational function

Repeated Quadratic Factors - Irreducible repeating quadratic factors in the denominator of a rational function

Sketching Rational Functions - How to sketch the graph of a rational function

Transcendental Functions - A category of functions that includes both exponential and logarithmic functions

Exponential Functions - Basic definition of the exponential function

Properties of Exponents (jump) The various rules governing the behavior of exponents

Exponential / Logarithmic Equations How to use logarithms to solve mathematical equations

Logarithmic/Exponential Models - Applications of logarithmic/exponential functions

Exponential Growth and Decay - Very common application of logs

Gaussian - Model which produces the bell-shaped curve used in statistical analysis

Logarithmic - Used in a wide variety of applications including earthquakes, sound, and time of death

Logistics Growth - Model used to accurately represent population growth in an environment

Logarithmic Functions The inverse of an exponential function

Properties of Logarithms (jump) - Various rules governing behavior of logarithmic functions

Exponential / Logarithmic Equations How to use logarithms to solve mathematical equations

Logarithmic/Exponential Models - Applications of logarithmic/exponential functions

Exponential Growth and Decay - Very common application of logs

Gaussian - Model which produces the bell-shaped curve used in statistical analysis

Logarithmic - Used in a wide variety of applications including earthquakes, sound, and time of death

Logistics Growth - Model used to accurately represent population growth in an environment

Natural Base e - Definition of logarithm of base e

Translations and Combinations - Moving graphs around the coordinate plane

Arithmetic Combinations - How to add, subtract, multiply, and divide two functions

Composition of Functions - A function of a function

Reflections in Coordinate Axes - How to reflect a function around either the vertical or horizontal axis

Vertical and Horizontal Shifts - How to move a function vertically and horizontally around a graph

 

Quadratics - Second-degree polynomial functions

Conic Sections - Classification of conic sections by the discriminant of a quadratic function

Circles - Definition of a circle and its related equation

Ellipses - Definition of ellipse and its related equation

Hyperbolas - Definition of hyperbola and its related equation

Parabolas - Definition of a parabola and its related equation

Translations of Conics - Moving the graph of a conic section around in the plane

Circle Translation - How to move a circle around in the plane

Ellipse Translation - How to move an ellipse around in the plane

Hyperbola Translation - How to move a hyperbola around in the plane

Parabola Translation - How to move a parabola around in the plane

Quadratic Equations - One of the most important applications of polynomial functions

1. Factoring - Factor the quadratic, then set each factor equal to zero to find the roots

2. Extracting Square Roots - Extracting a square root from a quadratic equation

3. Completing the Square - Transforming a quadratic into a perfect square

4. Quadratic Formula - The granddaddy of all methods of solving a quadratic when nothing else works

 

Linear Equations Equations that involve variables raised to the first power only

Linear Inequalities - How to solve an inequality instead of an equation

Absolute Value Inequalities - Inequalities involving absolute value operator

Polynomial Inequalities - How to find test intervals for solving polynomial inequalities

Rational Inequalities Solving inequalities that involve one polynomial divided by another

Properties of Inequalities (jump) - A list of common properties that are obeyed by inequalities

Linear Systems - How to solve more than one linear equation at a time, often involving two or more variables

Linear Systems in a Matrix (jump) Relationship between linear equations and matrices

Graphical Approach - How to solve a linear system from a graph of the equations

Graphical Interpretation of Systems (jump) What linear systems have to do with the real world

Method of Elimination - How to solve a linear system of equations by eliminating a variable

Multivariable Linear Systems - Transformation of a linear system into row-echelon form

Substitution Method - Summary of how to solve a system of linear equations using back-substitution

Systems of Inequalities Solving and graphing systems of linear inequalities

Lines and Slope - Various formulas involving lines and their slopes

Equations of Lines - Several different ways of writing equations of lines

Parallel and Perpendicular Lines - Basic definitions

Point-Slope Form of a Line - One form of an equation representing a line

Slope-Intercept Form of a Line - One form of an equation representing a line

 

Matrices and Determinants - Basic definition of a matrix

Determinants - A special operation performed on a matrix

Properties of Determinants (jump) A list of properties that we can use to simplify the process of finding determinants

Applications of Determinants - Several useful ways to apply matrices

Area of Triangle - How to find the area of a triangle with the given vertices using determinants

Cramer's Rule - Solving a system of linear equations using determinants

Lines in Plane - How to tell if three points lie on the same line and a method for finding the equation of a line

Triangular Matrix - A handy form of matrix used to quickly solve linear systems of equations

Inverse of a Square Matrix - Analogous to the inverse of a function

Finding Matrix Inverse - How to find the inverse of a matrix

Inverse of a 2 x 2 Matrix - Basic quick formula for finding the inverse of a simple square matrix

Linear Systems in a Matrix Using a matrix to represent a linear system of equations

Elementary Row Operations - One method of solving systems of linear equations using matrices

Linear Systems (jump) - How to solve more than one linear equation at a time

Gaussian Elimination Matrices Using ERO to find the solution of a system of linear equations

Gauss-Jordan Elimination Similar to Gaussian elimination but it goes a step further

Matrix Operations - List of ways of representing matrices

Identity Matrix - A matrix consisting of 1's along its diagonal and zeros everywhere else

Matrix Addition - How to add two matrices together

Matrix Multiplication - How to multiply two matrices together

Properties of Matrix Operations - Properties of matrix addition, matrix multiplication and scalar multiplication

Scalar Multiplication - Multiplication of a matrix by a constant (scalar)

 

Sequences and Probability - Introductory page about sequences, series, and probability

Binomial Theorem - Expanding a binomial raised to an integer power

Pascal's Triangle - A more visual way to find binomial coefficients

Counting Principle - Method of finding total number of ways a particular event can occur

Combinations Number of ways a group of objects can be arranged irregardless of order

Permutations Number of ways a group of objects can be arranged, with order being important

Factorial A special type of function used to multiply integers together in a particular fashion

Probability - Terminology used throughout probability problems

Independent Events - An event that has no bearing on any previous or subsequent events

Union of Two Events - Also defined as the probability of events A or B occurring

Complementary Events - Probability of a complement is sort of the left over of the probability of an event

Sequences An arrangement of numbers in a particular order based on a relationship between those numbers

Arithmetic Sequence A sequence whose terms all have a common difference between consecutive terms

Geometric Sequence A sequence whose consecutive terms have a common ratio

Summation Notation A shortcut method of representing the sum of a large sequence of numbers

 


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