Mathematical Methods (Version 8.4)

determine the coordinates of the midpoint of two points (ACMMM001)

examine examples of direct proportion and linearly related variables (ACMMM002)

recognise features of the graph of \(y=mx+c\), including its linear nature, its intercepts and its slope or gradient (ACMMM003)

find the equation of a straight line given sufficient information; parallel and perpendicular lines (ACMMM004)

solve linear equations. (ACMMM005)

examine examples of quadratically related variables (ACMMM006)

recognise features of the graphs of \(y=x^2\), \(y=a{(x-b)}^2+c\), and \(y=a\left(x-b\right)\left(x-c\right)\) including their parabolic nature, turning points, axes of symmetry and intercepts (ACMMM007)

solve quadratic equations using the quadratic formula and by completing the square (ACMMM008)

find the equation of a quadratic given sufficient information (ACMMM009)

find turning points and zeros of quadratics and understand the role of the discriminant (ACMMM010)

recognise features of the graph of the general quadratic \(y=ax^2+bx+c\) (ACMMM011)

examine examples of inverse proportion (ACMMM012)

recognise features of the graphs of \(y=\frac1x\) and \(y=\frac a{x-b}\), including their hyperbolic shapes, and their asymptotes. (ACMMM013)

recognise features of the graphs of \(y=x^n\) for \(n\in\boldsymbol N,\) \(n=-1\) and \(n=½\), including shape, and behaviour as \(x\rightarrow\infty\) and \(x\rightarrow-\infty\) (ACMMM014)

identify the coefficients and the degree of a polynomial (ACMMM015)

expand quadratic and cubic polynomials from factors (ACMMM016)

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 \(x\rightarrow\infty\) and \(x\rightarrow-\infty\) (ACMMM017)

factorise cubic polynomials in cases where a linear factor is easily obtained (ACMMM018)

solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained. (ACMMM019)

recognise features of the graphs of \(x^2+y^2=r^2\) and \(\left(x-a\right)^2+\left(y-b\right)^2=r^2\), including their circular shapes, their centres and their radii (ACMMM020)

recognise features of the graph of \(y^2=x\) including its parabolic shape and its axis of symmetry. (ACMMM021)

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 (ACMMM022)

use function notation, domain and range, independent and dependent variables (ACMMM023)

understand the concept of the graph of a function (ACMMM024)

examine translations and the graphs of \(y=f\left(x\right)+a\) and \(y=f(x+b)\) (ACMMM025)

examine dilations and the graphs of \(y=cf\left(x\right)\) and \(y=f\left(kx\right)\) (ACMMM026)

recognise the distinction between functions and relations, and the vertical line test. (ACMMM027)

review sine, cosine and tangent as ratios of side lengths in right-angled triangles (ACMMM028)

understand the unit circle definition of \(\cos\theta,\;\sin\theta\) and \(\tan\theta\) and periodicity using degrees (ACMMM029)

examine the relationship between the angle of inclination of a line and the gradient of that line (ACMMM030)

establish and use the sine and cosine rules and the formula \(Area=\frac12bc\sin A\) for the area of a triangle. (ACMMM031)

define and use radian measure and understand its relationship with degree measure (ACMMM032)

calculate lengths of arcs and areas of sectors in circles. (ACMMM033)

understand the unit circle definition of \(\cos\theta,\;\sin\theta\) and \(\tan\theta\) and periodicity using radians (ACMMM034)

recognise the exact values of \(\cos\theta,\;\sin\theta\) and \(\tan\theta\) at integer multiples of \(\frac\pi6\) and \(\frac\pi4\) (ACMMM035)

recognise the graphs of \(y=\sin x,\;y=\cos x,\) and \(y=\tan x\) on extended domains (ACMMM036)

examine amplitude changes and the graphs of \(y=a\sin x\) and \(y=a\cos x\) (ACMMM037)

examine period changes and the graphs of \(y=\sin bx,\;\), \(y=\cos bx\), and \(y=\tan bx\) (ACMMM038)

examine phase changes and the graphs of \(y=\sin{(x+c)}\), \(y=\cos{(x+c)}\) and \(y=\tan{(x+c)}\) and the relationships \(\sin\left(x+\frac\pi2\right)=\cos x\) and \(\cos\left(x-\frac\pi2\right)=\sin x\) (ACMMM039)

prove and apply the angle sum and difference identities (ACMMM041)

identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems (ACMMM042)

solve equations involving trigonometric functions using technology, and algebraically in simple cases. (ACMMM043)

understand the notion of a combination as an unordered set of \(r\) objects taken from a set of \(n\) distinct objects (ACMMM044)

use the notation \(\begin{pmatrix}n\\r\end{pmatrix}\) and the formula \(\begin{pmatrix}n\\r\end{pmatrix}=\frac{n!}{r!\left(n-r\right)!}\) for the number of combinations of \(r\) objects taken from a set of \(n\) distinct objects (ACMMM045)

expand \(\left(x+y\right)^n\) for small positive integers \(n\) (ACMMM046)

recognise the numbers \(\begin{pmatrix}n\\r\end{pmatrix}\) as binomial coefficients, (as coefficients in the expansion of \(\left(x+y\right)^n)\) (ACMMM047)

use Pascal’s triangle and its properties. (ACMMM048)

review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049)

use set language and notation for events, including \(\overline A\) (or \(A'\)) for the complement of an event \(A,\) \(A?B\) for the intersection of events \(A\) and \(B\), and \(A?B\) for the union, and recognise mutually exclusive events (ACMMM050)

use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051)

review probability as a measure of ‘the likelihood of occurrence’ of an event (ACMMM052)

review the probability scale: \(0\leq P(A)\leq1\) for each event \(A,\) with \(P\left(A\right)=0\) if \(A\) is an impossibility and \(P\left(A\right)=1\) if \(A\) is a certaint (ACMMM053)

review the rules: \(P\left(\overline A\right)=1-P\left(A\right)\) and \(P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\) (ACMMM054)

use relative frequencies obtained from data as point estimates of probabilities. (ACMMM055)

understand the notion of a conditional probability and recognise and use language that indicates conditionality (ACMMM056)

use the notation \(P(A\vert B)\) and the formula \(P(A\vert B)=P(A\cap B) / P(B)\) (ACMMM057)

understand the notion of independence of an event \(A\) from an event \(B\), as defined by \(P(A\vert B)=P(A)\) (ACMMM058)

establish and use the formula \(P(A\cap B)=P(A)P(B)\) for independent events \(A\) and \(B\), and recognise the symmetry of independence (ACMMM059)

use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events. (ACMMM060)