Non Linear Relationships Functions and Graphs — NSW Year 10 Advanced Maths
Parabolas, circles, exponentials, hyperbolas and cubics, functions, rates of change and variation. Fourteen subtopics and 280 practice questions, aligned to NESA outcomes MA5-NLI-P-01, MA5-FNC-P-01 and MA5-RAT-P-01.
Non Linear Relationships Functions and Graphs in NSW Year 10 Advanced
This is the largest topic in Year 10 Advanced and the bridge to Year 11 functions. Students sketch parabolas, circles, exponential curves, hyperbolas and cubics and describe their transformations, use function notation, domain and range, rearrange formulas, read graphs that show change, and solve problems with direct and inverse variation such as \(y=\frac{k}{x}\).
The 14 subtopics below start with travel graphs, then cover exploring, sketching and applying parabolas, functions and notation, graphs of circles, exponential functions, hyperbolas and cubic equations, then further transformations, using graphs to describe change and literal equations, and finish with direct and inverse variation. It is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcomes MA5‑NLI‑C‑02, MA5‑NLI‑P‑01, MA5‑FNC‑P‑01, MA5‑RAT‑P‑01, MA5‑RAT‑P‑02, MA5‑EQU‑P‑01 and MA5‑EQU‑P‑02).
Subtopics
Non Linear Relationships Functions and Graphs: common questions
How do you find the vertex of a parabola?
Find the axis of symmetry \(x=-\frac{b}{2a}\), then substitute it. For \(y=-x^2+4x+5\), \(x=2\) and \(y=9\), so the vertex is \((2,9)\), a maximum because \(a<0\). Completing the square gives the same result.
What is the equation of a circle?
The circle centred at \((a,b)\) with radius \(r\) is \((x-a)^2+(y-b)^2=r^2\). For example, \((x+2)^2+(y-4)^2=25\) has centre \((-2,4)\) and radius 5.
What are the asymptotes of a hyperbola?
For \(y=\frac{k}{x-b}+c\), the asymptotes are the vertical line \(x=b\) and the horizontal line \(y=c\). So \(y=\frac{1}{x-1}+2\) has asymptotes \(x=1\) and \(y=2\).
What is the difference between direct and inverse variation?
In direct variation \(y=kx\): doubling \(x\) doubles \(y\), and the graph is a straight line through the origin. In inverse variation \(y=\frac{k}{x}\): doubling \(x\) halves \(y\), and the graph is a hyperbola.
Which NESA outcomes does this topic cover?
Parabolas, circles, exponentials, hyperbolas and cubics are MA5‑NLI‑P‑01, with real-life parabola problems in MA5‑NLI‑C‑02. Functions are MA5‑FNC‑P‑01, variation is MA5‑RAT‑P‑01, graphs of change are MA5‑RAT‑P‑02, cubic equations are MA5‑EQU‑P‑01 and changing the subject is MA5‑EQU‑P‑02, in Stage 5 of the NSW Mathematics K–10 Syllabus (2022).