Outcomes
A student:

 MA5.21WM
selects appropriate notations and conventions to communicate mathematical ideas and solutions

 MA5.23WM
constructs arguments to prove and justify results

 MA5.210NA
connects algebraic and graphical representations of simple nonlinear relationships
Content
 Students:
 Graph simple nonlinear relationships, with and without the use of digital technologies, and solve simple related equations (ACMNA296)
 graph parabolic relationships of the form \( y=ax^2, \,\,\, y=ax^2+c \), with and without the use of digital technologies
 identify parabolic shapes in the environment (Reasoning)
 describe the effect on the graph of \(\, y=x^2\, \) of multiplying \(x^2\) by different numbers (including negative numbers) or of adding different numbers (including negative numbers) to \(x^2\) (Communicating, Reasoning)
 determine the equation of a parabola, given a graph of the parabola with the main features clearly indicated (Reasoning)
 determine the \(x\)coordinate of a point on a parabola, given the \(y\)coordinate of the point
 sketch, compare and describe, with and without the use of digital technologies, the key features of simple exponential curves, eg sketch and describe similarities and differences of the graphs of \( y=2^x, \) \( \, y=2^x, \) \( \, y=2^{x}, \) \( \, y=2^{x}, \) \( \, y=2^x+1, \) \( \, y=2^x1 \)
 describe exponentials in terms of what happens to the \(y\)values as the \(x\)values become very large or very small, and the \(y\)value for \(x = 0\) (Communicating, Reasoning)
 use Pythagoras' theorem to establish the equation of a circle with centre the origin and radius of the circle \(r\)
 recognise and describe equations that represent circles with centre the origin and radius \(r\)
 sketch circles of the form \( x^2 + y^2 = r^2 \) where \(r\) is the radius of the circle
 Explore the connection between algebraic and graphical representations of relationships such as simple quadratics, circles and exponentials using digital technologies as appropriate (ACMNA239)
 identify graphs and equations of straight lines, parabolas, circles and exponentials
 match graphs of straight lines, parabolas, circles and exponentials to the appropriate equations
 sort and classify different types of graphs, match each graph to an equation, and justify each choice (Communicating, Reasoning)
Background Information
Various digital technologies with graphing capabilities facilitate the investigation of the shapes of curves and the effect of multiplying part(s) of the equation by different numbers (including negative numbers), or of adding different numbers (including negative numbers).
This substrand provides opportunities for mathematical modelling. For example, \( \, y=1.2^x \, \) for \( \, x \ge 0 \) models the growth of a quantity beginning at 1 and increasing 20% for each unit increase in \(x\).