Australian Curriculum Mathematics/Mathematical Methods/Functions and Graphs

Australian Curriculum Content

 Lines and linear relationships :
 * determine the coordinates of the midpoint of two points
 * examine examples of direct proportion and linearly related variables
 * recognise features of the graph of $$y=mx+c$$, including its linear nature, its intercepts and its slope or gradient
 * find the equation of a straight line given sufficient information; parallel and perpendicular lines
 * solve linear equations.

 Review of quadratic relationships :
 * examine examples of quadratically related variables
 * recognise features of the graphs of $$y=x^2$$, $$y=a(x-b)^2+c$$, and $$y=a(x-b)(x-c)$$, including their parabolic nature, turning points, axes of symmetry and intercepts
 * solve quadratic equations using the quadratic formula and by completing the square
 * find the equation of a quadratic given sufficient information
 * find turning points and zeros of quadratics and understand the role of the discriminant
 * recognise features of the graph of the general quadratic $$y=ax^2+bx+c$$.

 Inverse proportion :
 * examine examples of inverse proportion
 * recognise features of the graphs of $$y=\frac{1}{x}$$ and $$y=\frac{a}{x-b}$$, including their hyperbolic shapes, and their asymptotes.

 Powers and polynomials :
 * recognise features of the graphs of $$y=x^n$$ for $$n\in\mathbb{N}$$, $$n=-1$$ and $$n=\frac{1}{2}$$, including shape, and behaviour as $$x\to\infty$$ and $$x\to-\infty$$
 * identify the coefficients and the degree of a polynomial
 * expand quadratic and cubic polynomials from factors
 * recognise features of the graphs of $$y=x^3$$, $$y=a(x-b)^{3}+c$$ and $$y=k(x-a)(x-b)(x-c)$$, including shape, intercepts and behaviour as $$x\to\infty$$ and $$x\to-\infty$$
 * factorise cubic polynomials in cases where a linear factor is easily obtained
 * solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained.

 Graphs of relations :
 * recognise features of the graphs of $$x^2+y^2=r^2$$ and $$(x-a)^2+(y-b)^2=r^2$$, including their circular shapes, their centres and their radii
 * recognise features of the graph of $$y^2=x$$ including its parabolic shape and its axis of symmetry.

 Functions :
 * understand the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another
 * use function notation, domain and range, independent and dependent variables
 * understand the concept of the graph of a function
 * examine translations and the graphs of $$y=f(x)+a$$ and $$y=f(x+b)$$
 * examine dilations and the graphs of $$y=cf(x)$$ and $$y=f(kx)$$
 * recognise the distinction between functions and relations, and the vertical line test.

Linear Equations
Identify the slope and intercepts of the following equations:

Graph the following linear equations:

Midpoints
Say we have two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$. The midpoint between these points is given by:
 * $$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

Find the coordinates of the midpoint between the following points:

Linear Functions
Sketch the graph of the following linear functions showing all key features:

Hyperbolic functions
Sketch the graphs of the following hyperbolic functions, including all key details including asymptotes: