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)
• 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 $$y = \frac{k}{x-a}$$ 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)
• 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 $$(x-a)^2 + (y-b)^2 = r^2$$ (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 $$(x-2)^2 + y^2 = 4$$,   $$y = (x-2)^2 - 4$$,   $$y = 4^x + 2$$,   $$y = x^2 + 2x - 4$$,   $$y = \frac{2}{x-4}$$
• 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 $$y = 2$$,   $$y = 2 -x$$,   $$y = (x-2)^2$$,   $$y = 2^x$$,   $$(x-2)^2 + (y-2)^2 = 4$$,   $$y = \frac{1}{x-2}$$,   $$y = 2x^2$$ (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(x-a)(x-b)(x-c), \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 ≥ 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 ≥ 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 ≥ 2 by using horizontal transformations

### 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.