Intermediate Algebra & Analytic Geometry

Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, systems involving quadratics, and determinants. Topics include determinants of higher order, application of Cramer's rule, second-order determinants, systems linear in quadratic terms, systems treatable by substitution, systems with a linear equation, and other systems treated by comparison. The manuscript ponders on trigonometric functions and equations, straight lines, and points, distances, and slopes, including intersection points of lines, perpendicular distances, angles between lines, positions of points, inverse trigonometric functions, and trigonometric equations. The publication is a valuable source of data for readers interested in intermediate algebra and analytic geometry.

Section One—Introduction

1 Preliminary Review

The Purpose of These Books

What is Advanced Mathematics?

Advanced Approaches to Mathematics

Recommended Sequence of Study

Recommended Pace of Study

Recommended Use of Practice Exercises

Assumed Background of the Reader

Note on Tables of Formulae

Review Table of Elementary Formulae

R-Series Formulae

Section Two—Intermediate Algebra

2 Linear Equations in Two Unknowns

Preliminary Definitions

Systems of Two Equations

Determinate and Indeterminate Systems

Consistent and Inconsistent Equations

Independent and Dependent Equations

Defective and Redundant Systems

Summary

3 Variables, Functions, and Graphs

Algebra and Geometry

Variables and Functions

Tables of Values

Explicit and Implicit Functions

First Quadrant Graphs

Rectangular Co-Ordinates

Graphic Interpretations

Summary

Note on Sequence of Study

4 Quadratic Equations in One Variable

A Different Type of Problem

Preliminary Definitions

Solution by Extracting Square Roots

Solution by Factoring

Solution by Completing the Square

Solutions by Formula

'Imaginary' Roots

Equations Equivalent to Quadratics

Graphic Interpretation of Quadratic Roots

Physical Interpretation of Quadratic Roots

Summary

5 Systems Involving Quadratics

Preliminary Definitions

Note on Sequence of Topics

Defective Systems

Systems with a Linear Equation

Systems Quadratic in Only One Variable

Other Systems Treated by Comparison

Systems Linear in Quadratic Terms

Systems with Only One Linear Term

Systems of Other Special Types

Systems Treatable by Substitution

Summary

Note on Sequence of Study

6 Linear Equations in N Variables

Systems in N Variables

Linear Dependence and Independence

Linear Consistency and Inconsistency

Geometric Interpretation in Schematic Form

Summary

7 Determinants

A Note on Sequence of Topics

Second-Order Determinants

Determinants of Higher Order

Expansion by Minors

Further Properties of Determinants

Applications of Cramer's Rule

Properties of Determinants, Continued

Exceptions to Cramer's Rule

Summary

Table of Determinant Formulae

8 Trigonometric Functions and Equations

Trigonometric Equations

Preliminary Algebraic Treatment

Inverse Trigonometric Functions

Angles of Any Magnitude

Radian Measure of Angles

Trigonometric Functions of Any Angle

Table: Typical Values of Trigonometric Functions

Applications of Generalized Trigonometric Functions

Complete Solution of Trigonometric Equations

Graphs of Trigonometric Functions

Variations of Trigonometric Functions

Summary

Section Three—Plane Analytic Geometry

9 Points, Distances, and Slopes

What is Analytic Geometry?

Positions of Points

Distances or Lengths

Slopes

Angles Between Lines

Problem-Solving Technique

Summary with Formulae

10 Straight Lines

Point-Slope Line Formulae

Two-Point Line Formulae

Points on Lines

Intersection Points of Lines

Parallel Lines

Perpendicular Lines

Perpendicular Distances

Problem-Solving Technique

Summary with Formulae

11 Conic Sections—Parabolas

Conic Sections as Loci

Parabolas: Definitions and Construction

Standard Equations for Parabolas

Applications of Parabola Formulae

12 Ellipses and Circles

Ellipses: Definitions and Construction of Ellipses

Eccentricity of Ellipses

The Standard Equation for an Ellipse

Applications of Ellipse Formulae

Circles: Formula and Applications

13 Hyperbolas

Definitions and Construction

Eccentricity of Hyperbolas

Standard Equation for a Hyperbola

Applications of Hyperbola Formulae

Summary (Chapters XI, XII, XIII) with Formulae

Section Four—Solid Analytic Geometry

14 Points and Directions in Space

Three-Dimensional Rectangular Co-Ordinates

Point and Length Formulae

Direction Numbers and Cosines

Parallel and Perpendicular Directions

15 Surfaces and Lines in Space

Normal Plane-Formulae

Parallel and Perpendicular Planes

Point Plane-Formulae

Lines as Intersections of Planes

Quadric Surfaces

Footnote on Hyper-Geometry

Summary (Chapters XIV and XV) with Formulae

Answers

Index