Course

# VCE Further Mathematics Units 3 and 4 – Graphs and Relations

6.1 Straight Lines
6.2 Simultaneous Equations
6.3 Graphs
6.4 Inequalities

30 Lessons

This module covers the use of linear relations, including piecewise-defined relations, and non-linear relations to model a range of practical situations and solve related problems, including optimisation problems by linear programming.

Construction and interpretation of graphs, including:

• straight-line graphs, line segment graphs and step graphs and their use to model and analyse practical situations
• simultaneous linear equations in two unknowns and their use to model and analyse practical situations including break-even analysis, where cost and revenue functions are linear
• non-linear graphs and their use to model and analyse practical and familiar situations including the practical significance and interpretation of intercepts, slope, maximum/minimum points and the average rate of change when interpreting the graph
• non-linear graphs, either constructed from a table of data or given, the use of interpolation and extrapolation to predict values, estimation of maximum/minimum values and location; and coordinates of points of intersection for applications such as break-even analysis with non-linear cost and revenue functions
• graphical representation of relations of the form of $y = kx^n$ for $x \ge 0$, where $n ∈ {–2, –1, 1, 2, 3}$, and their use in modelling practical situations including the determination of the constant of proportionality $k$ by substitution of known values or by plotting $y$ against $x^n$ to linearise a given set of data, and the use of linearisation to test the validity of a proposed model.

Linear programming, including:

• review of linear inequalities in one and two variables and their graphical representation
• graphs of systems of linear inequalities (no more than five including those involving only one variable) and the use of shading-in to identify a feasible region
• linear programming and its purpose
• formulation of a linear programming problem including the identification of the decision variables, the construction of a system of linear inequalities to represent the constraints, and the expression of the quantity to be optimised (the objective function) in terms of the decision variables
• use of the graphical method to solve simple linear programming problems with two decision variables, and the sliding-line method and the corner-point principle as alternative methods for identifying optimal solutions
• extension of the linear programming method to include problems where integer solutions are required (for feasible regions containing only a small number of possible integer solutions only).

source – VCE Mathematics Study Design