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 $y=a{x}^2+bx+c$.

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 $x\to \mathrm{\infty }$ and $x\to -\mathrm{\infty }$
• 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 $x\to \mathrm{\infty }$ and $x\to -\mathrm{\infty }$
• 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:

$y=x$ $y=x-2$ $y=-4x$

$y=x+5$ $y=2x-4$ $y=-x+6$

$y=6x-2$ $y=4x+8$ $y=-2x+5$

$y=8x-4$ $y=3x-9$ $y=-2x+12$

Graph the following linear equations:

$y=x+1$ $y=x+3$ $y=2x$

$y=0.5x$ $y=x-2$ $y=-x-3$

$y=3x-4$ $y=-0.5x+3$ $y=-3x+5$

$y=-3x-2$ $y=5x+1$ $y=1.5x+3$

Lines and Linear Relationships

Midpoints

Say we have two points $(x_1,y_1)$ and $(x_2,y_2)$. The midpoint between these points is given by:

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Example Find the midpoint between the points $(2,4)$ and $(7,-1)$ Find the $x$ coordinate: $x_m=\frac{x_1+x_2}{2}$ $\therefore x_m=\frac{2+7}{2}$ $\therefore x_m=4\frac{1}{2}$ Find the $y$ coordinate: $y_m=\frac{y_1+y_2}{2}$ $\therefore y_m=\frac{4+(-1)}{2}$ $\therefore y_m=1\frac{1}{2}$ The coordinates of the midpoint are: $(3\frac{1}{2},1\frac{1}{2})$

Find the coordinates of the midpoint between the following points:

$(2,6)$ and $(3,3)$ $(0,5)$ and $(2,1)$ $(3,2)$ and $(1,0)$

$(4,4)$ and $(2,-1)$ $(6,-4)$ and $(7,-2)$ $(-5,6)$ and $(4,-8)$

$(6,1)$ and $(-4,-3)$ $(8,-9)$ and $(11,-4)$ $(-15,2)$ and $(17,-22)$

Linear Functions

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

$y=x+3$ $y=x-2$ $y=2x+4$

$y=0.5x+5$ $y=-x+2$ $y=-0.5x+6$

$y=4-2x$ $y=-0.3x-3$ $y=-6x-8$

Hyperbolic functions

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

$y=\frac{2}{x}+1$ $y=\frac{1}{x}+3$ $y=\frac{2}{x}-2$

$y=\frac{-4}{x}+5$ $y=-\frac{7}{x}-3$ $y=-\frac{3}{x}+2$

$y=\frac{6}{x}-6$ $y=-\frac{1.5}{x}+9$ $y=\frac{4.7}{x}+1.4$

Solutions

1.1

$(2,6)$ and $(3,3)$ $(0,5)$ and $(2,1)$ $(3,2)$ and $(1,0)$

$(4,4)$ and $(2,-1)$ $(6,-4)$ and $(7,-2)$ $(-5,6)$ and $(4,-8)$