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NSW Syllabuses

Mathematics K–10 - Stage 5.3 - Number and Algebra Non-Linear Relationships §

Outcomes

A student:

  • MA5.3-1WM

    uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

  • MA5.3-3WM

    uses deductive reasoning in presenting arguments and formal proofs

  • MA5.3-9NA

    sketches and interprets a variety of non-linear relationships

Content

  • find x- and y-intercepts, where appropriate, for the graph of \(y=ax^2+bx+c\), given aand c
  • graph a variety of parabolas, including where the equation is given in the form \(y=ax^2+bx+c\), for various values of aand 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) ICTCCT
  • describe features of a parabola by examining its equation (Communicating) CCT
  • determine the equation of the axis of symmetry of a parabola using: CCT
  • 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: CCT
  • 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
    Two rows with headings x and y. X row shows 0, 1, 2, 3, 4, 5. Y row shows 1, 2, 5, 10, 17, 26.
  • 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) CCT
  • 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) LCCT
  • explain why it may be useful to choose both small and large numbers when constructing a table of values for a hyperbola (Communicating, Reasoning) CCT
  • graph a variety of hyperbolic curves, including where the equation is given in the form \( y = \frac{k}{x} + c \) or \( y = \frac{k}{x-a} \) 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 \( y = \frac{k}{x-a} \) (Problem Solving) CCT
  • 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) CCT
  • 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) CCT
  • recognise and describe equations that represent circles with centre \( (a,b) \) and radius \(r\) CCT
  • establish the equation of the circle with centre \((a,b)\) and radius \(r\), and graph equations of the form \( (x-a)^2 + (y-b)^2 = r^2 \) (Communicating, Reasoning) CCT
  • determine whether a particular point is inside, on, or outside a given circle (Reasoning) CCT
  • 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 \( (x-2)^2 + y^2 = 4 \),   \( y = (x-2)^2 - 4 \),   \( y = 4^x + 2 \),   \( y = x^2 + 2x - 4 \),   \( y = \frac{2}{x-4} \) CCT
  • determine how to sketch a particular curve by determining its features from its equation (Problem Solving) CCT
  • identify equations whose graph is symmetrical about the y-axis (Communicating, Reasoning) CCT
  • determine a possible equation from a given graph and check using digital technologies ICT
  • compare and contrast different types of graphs and determine possible equations from the key features, eg \( y = 2 \),   \( y = 2 -x \),   \( y = (x-2)^2 \),   \( y = 2^x \),   \( (x-2)^2 + (y-2)^2 = 4 \),   \( y = \frac{1}{x-2} \),   \( y = 2x^2 \) (Communicating, Reasoning) CCTICT
  • determine the points of intersection of a line with a parabola, hyperbola or circle, graphically and algebraically ICT
  • compare methods of finding points of intersection of curves and justify choice of method for a particular pair of curves (Communicating, Reasoning) CCT
  • Describe, interpret and sketch cubics, other curves and their transformations
  • graph and compare features of the graphs of cubic equations of the forms 
    \( \begin{align} y &= kx^3 \\ y &= kx^3 + c \\ y &= k(x-a)(x-b)(x-c), \end{align} \)
    describing the effect on the graph of different values of a, b, c and k LCCT
  • graph a variety of equations of the form \(y=kx^n\) for n an integer, n ≥ 2, describing the effect of n being odd or even on the shape of the curve CCT
  • graph curves of the form \(y=kx^n + c\) from curves of the form \(y = kx^n\) for n an integer, n ≥ 2 by using vertical transformations CCT
  • graph curves of the form \(y = k(x - a)^n\) from curves of the form \(y = kx^n\) for n an integer, n ≥ 2 by using horizontal transformations CCT

Background Information

This substrand links to other learning areas and real-life 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.