VCE Further Mathematics Units 3 and 4 Graphs and Relations
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

STRAIGHT LINES

Gradient of Lines

Intercepts

Parallel Lines

Perpendicular Lines

Application of Perpendicular Lines

Vertical Lines and Horizontal Lines

Finding Equations of Lines

Finding Equations of Lines involving Angles

General Form of a Line

Finding Gradient from an Equation

Finding a Linear Equation from a Graph

Sketching Linear Graphs

Finding Intersection of Lines Graphically

SIMULTANEOUS EQUATIONS

Linear Simultaneous Equations - Elimination Method

Linear Simultaneous Equations - Substitution Method

Numerical Application of Linear Simultaneous Equations

Practical Application of Linear Simultaneous Equations

Undefined or All Real Values of Simultaneous Equations

GRAPHS

Further Graphs of Basic Power Functions

Polynomial Graphs

Interpolation and Extrapolation

Predictions

INEQUALITIES

Linear Inequalities involving Fractions

Simultaneous Linear Inequalities

Regions of Linear Inequalities

Problem Solving with Regions of Linear Inequalities