The topic Functions involves the use of both algebraic and graphical conventions and terminology to describe, interpret and model relationships of and between changing quantities. Knowledge of functions enables students to discover, recognise and generalise connections between algebraic and graphical representations of the same expression and to describe interactions through the use of both dependent and independent variables. The study of functions is important in developing students’ ability to find connections and patterns, communicate concisely and precisely, use algebraic techniques and manipulations, describe and solve problems, and predict future outcomes in areas such as finance, economics, data analysis, marketing and weather.
The topic of Trigonometric Functions involves the study of periodic functions in geometric, algebraic, numerical and graphical representations. A knowledge of trigonometric functions enables the solving of practical problems involving triangles or periodic graphs, such as waves and signals. The study of trigonometric functions is important in developing students’ understanding of periodic behaviour, a property not possessed by any previously studied functions. Utilising this property, mathematical models have been developed that describe the behaviour of many naturally occurring periodic phenomena, such as vibrations or waves, as well as oscillatory behaviour found in pendulums, electric currents and radio signals.
The topic of Calculus is concerned with how quantities change and provide a framework for developing quantitative models of change and deducing their consequences. The topic involves the development of the basic concepts upon which differential calculus is built, namely the connection between the gradient of the tangent to a curve and the instantaneous rate of change of a function, rates of change and derivatives of functions and the manipulative skills necessary for the effective use of differential calculus. The study of calculus is important in developing students’ ability to solve problems involving algebraic and graphical representations of functions and rates of change of a function with relevance to all quantitative fields of study including physics, chemistry, medicine, engineering, computing, statistics, business, finance, economics and the construction industry.
Exponential and Logarithmic Functions
The topic Exponential and Logarithmic Functions introduces exponential and logarithmic functions and develops their properties, including the manipulation of expressions involving them. The exponential function \( e^x \) is introduced by considering graphs of the derivative of exponential functions. A knowledge of exponential and logarithmic functions enables an understanding of practical applications, such as exponential growth and decay, as well as applications within the Calculus topic. The study of exponential and logarithmic functions is important in developing students’ ability to solve practical problems involving rates of change in contexts such as population growth and compound interest.
The topic of Statistical Analysis involves the exploration, display, analysis and interpretation of data to identify and communicate key information. Knowledge of statistical analysis enables careful interpretation of situations and awareness of the contributing factors when presented with information by third parties, including its possible misrepresentation. The study of statistical analysis is important in developing students’ ability to recognise, describe and apply statistical techniques in order to analyse current situations or predict future outcomes. It also develops an awareness of how conclusions drawn from data can be used to inform decisions made by groups such as scientific investigators, business people and policy-makers.
source – NSW Education Standards Authority