Introduction

Non-Linear Relationships are an exciting area of mathematics that provide rich opportunities for exploring real life patterns through algebra.

These relationships are defined by two variables, usually referred to as x and y, which do no interact in a linear way. The exponent of one of the variables is always more than one, for example in the relationship y = x^{2 .} This topic opens doors to model many real life situations, such as the motion of falling objects, compounding rates of pay, breeding of species such as rabbits, and our increase in height as we age. We tend to express these relationships in two ways; as equations, or through sketches, and use tables of values to relate between the two. Parabolas, exponentials, circles and hyperbolas are all examples of non-linear graphs.

Non-linear relationships rely strongly on understandings within Linear Relationships and Coordinate Geometry. Units of work exploring these concepts have been developed by hard-working students, and can be found here:

• Understanding what an equation is.
• Linear Relationships-The Cartesian Plane: Individual lessons not yet available, however these topics are most poignant:
  • Axes
  • Quadrants
  • Coordinates
  • Identifying and Point Plotting on the Cartesian Plane
  • Origin
  • Number patterns
• Substitution to complete a table of values.
• Finding points and connecting into lines using a table of values
• Find x and y intercepts using substitution.
• Students are assumed to have a basic grasp of the graphing software Desmos.

Syllabus Outcomes:

• Graph simple non-linear relations, with and without the use of digital technologies (ACMNA296)
  • complete tables of values to graph simple non-linear relationships and compare these with graphs drawn using digital technologies, eg y=x^2, y=x^2 + 2, y=2^x
• Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technologies as appropriate (ACMNA239)
  • use digital technologies to graph simple quadratics, exponentials and circles, eg y=-x^2
    • describe and compare a variety of simple non-linear relationships (Communicating, Reasoning)
    • connect the shape of a non-linear graph with the distinguishing features of its equation (Communicating, Reasoning)

(NESA, K-10 Syllabus)