# QA – Chapter 1 : Mathematical Techniques

**1. Mathematical Techniques**

**Functions**

Functions are a fundamental concept in mathematics that play a crucial role in various business problems. They represent a relationship between two variables, where one variable (the dependent variable) is determined by the other variable (the independent variable). Functions can be classified into several types based on their properties and behavior.

**1.1 Linear Functions**

Linear functions have a constant rate of change, meaning that the change in the dependent variable is directly proportional to the change in the independent variable. They are represented by the formula y = mx + b, where m is the slope, and b is the y-intercept. Linear functions are commonly used in business problems involving simple cost-profit analysis, break-even analysis, and linear programming.

**1.2 Quadratic Functions**

Quadratic functions have a parabolic shape and are defined by the formula y = ax^2 + bx + c, where a, b, and c are constants. These functions have a unique maximum or minimum point called the vertex, which occurs when x = -b/(2a). Quadratic functions are used in business problems related to revenue, total cost, and inventory management.

**1.3 Cubic Functions**

Cubic functions have a more complex shape than linear or quadratic functions and are defined by the formula y = ax^3 + bx^2 + cx + d. These functions have up to three turning points, which can be found using the method of completing the square. Cubic functions are used in business problems involving advanced cost analysis, production planning, and project management.

**1.4 Exponential Functions**

Exponential functions grow or decay at a constant percentage rate and are represented by the formula y = ab^x, where a and b are constants. They are commonly used in business problems related to population growth, compound interest, and inventory valuation.

**1.5 Logarithmic Functions**

Logarithmic functions have an inverse relationship between the independent and dependent variables, and they are defined by the formula y = log_b(x), where b is the base of the logarithm. Logarithmic functions are used in business problems involving elasticity of demand, discounted cash flow analysis, and depreciation.

**2. Mathematical Techniques: Equations and Graphs**

Equations and graphs are essential tools for solving business problems using mathematical techniques. They help visualize the relationship between variables, identify critical points, and find solutions.

**2.1 Linear Equations**

Linear equations are equations in which the dependent variable is a linear function of the independent variable. They can be solved using algebraic techniques, such as substitution and elimination methods, or graphically by plotting points and drawing a line that represents the equation.

**2.2 Quadratic Equations**

Quadratic equations are equations in which the dependent variable is a quadratic function of the independent variable. They can be solved using algebraic techniques, such as factoring, completing the square, or the quadratic formula.

**2.3 Systems of Linear Equations**

Systems of linear equations consist of two or more linear equations with two or more variables. They can be solved using algebraic techniques, such as substitution, elimination, and matrix methods, or graphically by plotting points and drawing lines that represent the equations.

**2.4 Systems of Non-Linear Equations**

Systems of non-linear equations consist of two or more non-linear equations with two or more variables. They can be solved using algebraic techniques, such as substitution and elimination methods, or numerically using iterative methods like the Newton-Raphson method.

**3. Matrix Algebra**

Matrix algebra is a powerful technique for solving complex problems in business and other fields. Matrices are rectangular arrays of numbers, and matrix operations include addition, subtraction, multiplication, transposition, and inversion.

**3.1 Matrix Operations**

Matrix operations involve performing arithmetic operations on matrices. Addition and subtraction involve adding or subtracting corresponding elements, while multiplication involves performing a series of elementary row and column operations. Transposition involves interchanging the rows and columns of a matrix. Inversion involves finding the inverse of a matrix, which is the matrix that, when multiplied by the original matrix, yields the identity matrix.

**3.2 Applications of Matrix Algebra**

Matrix algebra has various applications in business, including statistical modeling, Markov analysis, input-output analysis, and general problem-solving. Statistical modeling uses matrices to represent and analyze data, while Markov analysis helps in understanding the transition probabilities between states in a stochastic process. Input-output analysis is used to study the interdependence between different sectors of an economy, and matrix algebra can be applied to solve linear programming problems and other optimization problems.

In conclusion, the CPA Quantitative Analysis unit 1 Mathematical techniques subtopics, including functions, equations and graphs, and matrix algebra, are essential tools for solving business problems. They provide a solid foundation for understanding and analyzing complex situations in the business world.