Australian Curriculum Mathematics/Mathematical Methods/Functions and Graphs

Australian Curriculum Content^[1]

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 ${\displaystyle y=mx+c}$, 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 ${\displaystyle y=x^{2}}$, ${\displaystyle y=a(x-b)^{2}+c}$, and ${\displaystyle y=a(x-b)(x-c)}$, 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 ${\displaystyle y=ax^{2}+bx+c}$.

Inverse proportion:

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

Powers and polynomials:

• recognise features of the graphs of ${\displaystyle y=x^{n}}$ for ${\displaystyle n\in \mathbb {N} }$, ${\displaystyle n=-1}$ and ${\displaystyle n={\frac {1}{2}}}$, including shape, and behaviour as ${\displaystyle x\to \infty }$ and ${\displaystyle x\to -\infty }$
• identify the coefficients and the degree of a polynomial
• expand quadratic and cubic polynomials from factors
• recognise features of the graphs of ${\displaystyle y=x^{3}}$, ${\displaystyle y=a(x-b)^{3}+c}$ and ${\displaystyle y=k(x-a)(x-b)(x-c)}$, including shape, intercepts and behaviour as ${\displaystyle x\to \infty }$ and ${\displaystyle x\to -\infty }$
• 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 ${\displaystyle x^{2}+y^{2}=r^{2}}$ and ${\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}$, including their circular shapes, their centres and their radii
• recognise features of the graph of ${\displaystyle y^{2}=x}$ 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 ${\displaystyle y=f(x)+a}$ and ${\displaystyle y=f(x+b)}$
• examine dilations and the graphs of ${\displaystyle y=cf(x)}$ and ${\displaystyle y=f(kx)}$
• recognise the distinction between functions and relations, and the vertical line test.

Identify the slope and intercepts of the following equations:

Graph the following linear equations:

Say we have two points ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$. The midpoint between these points is given by:

${\displaystyle (x_{m},y_{m})=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}})}$

Find the coordinates of the midpoint between the following points:

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

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