Business Mathematics Module

CONTENTS

1. Review of High School Topics1.1

2. Working With Functions

   • Converting Rules to Single Formulas

   • Converting Rules to Multiple Formulas

   • Graphs of Functions

   • Functions as Tables

3. Applications of Linear Functions

   • Linear Depreciation

   • Linear Interpolation

   • Linear Extrapolation

   • Linear Regression

4. Linear Inequalities

   • Manipulating Inequalities

   • Graphical Interpretations

   • Combining Inequalities

   • Linear Programming

5. Logarithms

   • Solving Index Equations for the Base

   • Solving Selected Index Equations for the Index

   • Definition of a Logarithm

   • Properties of Logarithms

   • Solving an Equation for the Index

6. Exponential Functions

   • Graphs of Exponential Functions

   • Business Applications

   • Using a Common Base

7. Boolean Algebra

   • Truth and Falsity

   • Negation

   • The "AND" Conjunction

   • The "OR" Conjunction

8. Counting Principles

   • The Multiplication Principle of Counting

   • Applications of the Counting Principle

9. Marginal Functions

   • Marginal Values and Functions

   • Marginal Profits

   • Marginal Taxation Rates

   • Interpretation of Marginal Functions as Rates of Change

INTRODUCTION

• This module covers essential mathematics techniques for Business and Property Studies students.

• Content from high school, particularly grade 12 major syllabuses, has been reviewed for clarity.

• The appendices provide questions and exercises on all topics in the syllabus. Students who struggled with grade 12 math should focus on understanding these foundational techniques.

Calculators

• Familiarity with a scientific calculator is necessary as not all calculators work the same.

• Essential functions to know:

  • Make corrections to calculations

  • Recall previous calculations

  • Use calculator memory

  • Arithmetic functions including root, square, power, exponential, logarithm, and percentage

  • Recognize and enter scientific notation

• Many resources, including recommended texts such as "The Power of Mathematics – Applications to Business" by Whipkey and Conway, are available in libraries.

REVIEW OF HIGH SCHOOL TOPICS

ORDER OF OPERATIONS

• Mathematical operations have different precedence.

• Perform multiplications and divisions before additions and subtractions.

• Evaluate functions (e.g., square, square root, power) before multiplications/divisions. Use brackets to change the order.

INDICES

• Shorthand index notation is often used in algebra.

• Index expressions: x^n, where x is a base and n is an index.

• Special rules regarding indices include:

  • Multiplication: a^m * a^n = a^(m+n)

  • Division: a^m / a^n = a^(m-n)

  • Powers: (a^m)^n = a^(m*n)

  • Roots: n√(a^m) = a^(m/n) with restrictions depending on n’s parity.

MANIPULATING FORMULAS AND EQUATIONS

• To rearrange a formula, apply the same operation to both sides.

• Highest Common Factors (HCF) and Lowest Common Multiples (LCM) are pivotal for simplifying expressions.

• Techniques like removing brackets are necessary to simplify expressions (product of factors into a sum/difference).

• Factoring and manipulating equations involve switching sides while maintaining equality.

FUNCTIONS AND GRAPHS OF FUNCTIONS

• Functions express relationships between variables.

• Example: Simple Interest (SI = P x R x T).

• Represent functions using graphs showing independent (x-axis) and dependent variables (y-axis).

• Notably, no two points on a vertical line should coincide within a function.

LINEAR FUNCTIONS

LINEAR FUNCTIONS AND GRAPHS

• Linear functions fit many real-life situations with a straight-line graph format.

• Standard form: y = mx + c (where m=slope, c=y-intercept).

QUADRATIC FUNCTIONS

• Quadratic functions feature parabolic graphs.

• Synopsis of intercept points and turning points enhances comprehension of their behavior.

SIMULTANEOUS EQUATIONS

• To find solutions satisfying multiple equations simultaneously, combine them to reduce the number of variables.

• Critical for solving real-world problems through algebraic modeling.

WORKING WITH FUNCTIONS

CONVERTING RULES TO FORMULAS

• To transform mathematical descriptions into formulas, carefully define variables and express relationships.

BOOLEAN ALGEBRA

TRUTH AND FALSITY

• Focus on the evaluation of statements based upon truth/falsity criteria.

• Essential for data retrieval in database contexts.

COUNTING PRINCIPLES

THE MULTIPLICATION PRINCIPLE

• Offers a systematic way of counting combinations possible within defined parameters without manual enumeration.

MARGINAL FUNCTIONS

MARGINAL VALUES AND FUNCTIONS

• Define marginal functions in economics to encapsulate the effects of incremental changes in various business scenarios (e.g., marginal cost, marginal profit).

MARGINAL TAXATION RATES

• Understand how marginal tax rates apply to changes in income levels impacting tax paid for additional income.