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:

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

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

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

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

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 $(4,-8)$