Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.39NA
sketches and interprets a variety of nonlinear relationships
Content
 Students:
 Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267)
 find \(x\) and \(y\)intercepts, where appropriate, for the graph of \(y=ax^2+bx+c\), given \(a\), \(b\) and \(c\)
 graph a variety of parabolas, including where the equation is given in the form \(y=ax^2+bx+c\), for various values of \(a\), \(b\) and \(c\)

use digital technologies to investigate and describe features of the graphs of parabolas given in the following forms for both positive and negative values of \(a\) and \(k\), eg
\( \begin{align} y& = ax^2 \\ y& = ax^2 + k \\ y& = (x+a)^2 \\ y& = (x+a)^2 + k \end{align} \)
(Communicating, Reasoning)  describe features of a parabola by examining its equation (Communicating)
 determine the equation of the axis of symmetry of a parabola using:
 the midpoint of the interval joining the points at which the parabola cuts the \(x\)axis
 the formula \( x = \frac{b}{2a} \)
 find the coordinates of the vertex of a parabola by:
 using the midpoint of the interval joining the points at which the parabola cuts the \(x\)axis and substituting to obtain the \(y\)coordinate of the vertex
 using the formula for the axis of symmetry to obtain the \(x\)coordinate and substituting to obtain the \(y\)coordinate of the vertex
 completing the square on \(x\) in the equation of the parabola
 identify and use features of parabolas and their equations to assist in sketching quadratic relationships, eg identify and use the \(x\) and \(y\)intercepts, vertex, axis of symmetry and concavity

determine quadratic expressions to describe particular number patterns, eg generate the equation \(y=x^2+1\) for the table
 graph hyperbolic relationships of the form \( y = \frac{k}{x} \) for integer values of \(k\)
 describe the effect on the graph of \( y = \frac{1}{x} \) of multiplying \(\frac{1}{x} \) by different constants (Communicating)
 explain what happens to the \(y\)values of the points on the hyperbola \( y = \frac{k}{x} \) as the \(x\)values become very large or closer to zero (Communicating)
 explain why it may be useful to choose both small and large numbers when constructing a table of values for a hyperbola (Communicating, Reasoning)
 graph a variety of hyperbolic curves, including where the equation is given in the form \( y = \frac{k}{x} + c \) or \( y = \frac{k}{xa} \) for integer values of \(k\), \(a\) and \(c\)
 determine the equations of the asymptotes of a hyperbola in the form \( y = \frac{k}{x} + c \) or \( y = \frac{k}{xa} \) (Problem Solving)
 identify features of hyperbolas from their equations to assist in sketching their graphs, eg identify asymptotes, orientation, \(x\) and/or \(y\)intercepts where they exist (Problem Solving, Reasoning)
 describe hyperbolas in terms of what happens to the \(y\)values of the points on the hyperbola as \(x\) becomes very large or very small, whether there is a \(y\)value for every \(x\)value, and what occurs near or at \(x = 0\) (Communicating, Reasoning)
 recognise and describe equations that represent circles with centre \( (a,b) \) and radius \(r\)
 establish the equation of the circle with centre \((a,b)\) and radius \(r\), and graph equations of the form \( (xa)^2 + (yb)^2 = r^2 \) (Communicating, Reasoning)
 determine whether a particular point is inside, on, or outside a given circle (Reasoning)
 find the centre and radius of a circle whose equation is in the form \( x^2 + y^2 + ax + by + c = 0 \) by completing the square (Problem Solving)
 identify and name different types of graphs from their equations, eg \( (x2)^2 + y^2 = 4 \), \( y = (x2)^2  4 \), \( y = 4^x + 2 \), \( y = x^2 + 2x  4 \), \( y = \frac{2}{x4} \)
 determine how to sketch a particular curve by determining its features from its equation (Problem Solving)
 identify equations whose graph is symmetrical about the \(y\)axis (Communicating, Reasoning)
 determine a possible equation from a given graph and check using digital technologies
 compare and contrast different types of graphs and determine possible equations from the key features, eg \( y = 2 \), \( y = 2 x \), \( y = (x2)^2 \), \( y = 2^x \), \( (x2)^2 + (y2)^2 = 4 \), \( y = \frac{1}{x2} \), \( y = 2x^2 \) (Communicating, Reasoning)
 determine the points of intersection of a line with a parabola, hyperbola or circle, graphically and algebraically
 compare methods of finding points of intersection of curves and justify choice of method for a particular pair of curves (Communicating, Reasoning)
 Describe, interpret and sketch cubics, other curves and their transformations

graph and compare features of the graphs of cubic equations of the forms
\( \begin{align} y &= kx^3 \\ y &= kx^3 + c \\ y &= k(xa)(xb)(xc), \end{align} \)
describing the effect on the graph of different values of \(a\), \(b\), \(c\) and \(k\)  graph a variety of equations of the form \(y=kx^n\) for \(n\) an integer, \(n\ge 2\), describing the effect of \(n\) being odd or even on the shape of the curve
 graph curves of the form \(y=kx^n + c\) from curves of the form \(y = kx^n\) for \(n\) an integer, \(n \ge 2\) by using vertical transformations
 graph curves of the form \(y = k(x  a)^n\) from curves of the form \(y = kx^n\) for \(n\) an integer, \(n \ge 2\) by using horizontal transformations
Background Information
This substrand links to other learning areas and reallife examples of graphs, eg exponential graphs used for population growth in demographics, radioactive decay, town planning, etc.
The substrand could provide opportunities for modelling, eg the hyperbola \( y = \frac{k}{x} \) for \(x > 0\) models sharing a prize of $\(k\) between \(x\) people or the length of a rectangle, given area \(k\) and breadth \(x\).