Mathematical Methods

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)

examine examples of quadratically related variables (ACMMM006)

recognise features of the graphs of y=x2, y=a(x-b)2+c, and y=ax-bx-c, 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)

examine examples of inverse proportion (ACMMM012)

recognise features of the graphs of y=xn for n∈N,  n=-1 and n=½, including shape, and behaviour as x→∞ and x→-∞ (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=x3, y=a(x-b)3+c and y=kx-ax-bx-c, including shape, intercepts and behaviour as x→∞ and x→-∞ (ACMMM017)

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

recognise features of the graphs of x2+y2=r2 and x-a2+y-b2=r2, including their circular shapes, their centres and their radii (ACMMM020)

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=fx+a and y=f(x+b) (ACMMM025)

examine dilations and the graphs of y=cfx and y=fkx (ACMMM026)

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

understand the unit circle definition of cos⁡θ, sin⁡θ and tan⁡θ and periodicity using degrees (ACMMM029)

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

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

understand the unit circle definition of cos⁡θ, sin⁡θ and tan⁡θ and periodicity using radians (ACMMM034)

recognise the exact values of sin⁡θ, cos⁡θ and tan⁡θ at integer multiples of π6 and π4 (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=asin⁡x and y=acos⁡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 (ACMMM039)

y = tan  ⁡ ( x + c )   and the relationships sin⁡x+π2=cos⁡x and cos⁡x-π2=sin⁡x (ACMMM040)

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)

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  nr and the formula  nr=n!r!n-r! for the number of combinations of r objects taken from a set of n distinct objects (ACMMM045)

expand x+yn for small positive integers n (ACMMM046)

recognise the numbers  nr as binomial coefficients, (as coefficients in the expansion of x+yn) (ACMMM047)

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

use set language and notation for events, including 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)

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

review the probability scale: 0≤P(A)≤1 for each event A, with PA=0 if A is an impossibility and PA=1 if A is a certainty (ACMMM053)

review the rules: PA¯=1-PA and PA∪B=PA+PB-PA∩B (ACMMM054)

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

use the notation PAB and the formula PA∩B=PABP(B) (ACMMM057)

understand the notion of independence of an event A from an event B, as defined by PAB=PA (ACMMM058)

establish and use the formula PA∩B=PAP(B) for independent events A and B, and recognise the symmetry of independence (ACMMM059)

review indices (including fractional indices) and the index laws (ACMMM061)

use radicals and convert to and from fractional indices (ACMMM062)

establish and use the algebraic properties of exponential functions (ACMMM064)

recognise the qualitative features of the graph of y=ax (a>0) including asymptotes, and of its translations (y=ax+b and y=ax+c) (ACMMM065)

identify contexts suitable for modelling by exponential functions and use them to solve practical problems (ACMMM066)

recognise and use the recursive definition of an arithmetic sequence: tn+1=tn+d (ACMMM068)

use the formula tn=t1+n-1d for the general term of an arithmetic sequence and recognise its linear nature (ACMMM069)

use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest (ACMMM070)

recognise and use the recursive definition of a geometric sequence: tn+1=rtn (ACMMM072)

use the formula tn=rn-1t1 for the general term of a geometric sequence and recognise its exponential nature (ACMMM073)

understand the limiting behaviour as n→∞ of the terms tn in a geometric sequence and its dependence on the value of the common ratio r (ACMMM074)

establish and use the formula Sn=t1rn-1r-1 for the sum of the first n terms of a geometric sequence (ACMMM075)

interpret the difference quotient fx+h-f(x)h as the average rate of change of a function f (ACMMM077)

use the Leibniz notation δx and δy for changes or increments in the variables x and y (ACMMM078)

use the notation δyδx for the difference quotient fx+h-f(x)h where y=fx (ACMMM079)

examine the behaviour of the difference quotient fx+h-f(x)h as h→0 as an informal introduction to the concept of a limit (ACMMM081)

define the derivative f'x as limh→0⁡fx+h-f(x)h (ACMMM082)

use the Leibniz notation for the derivative: dydx=limδx→0⁡δyδx and the correspondence dydx=f'x where y=f(x) (ACMMM083)

interpret the derivative as the instantaneous rate of change (ACMMM084)

estimate numerically the value of a derivative, for simple power functions (ACMMM086)

examine examples of variable rates of change of non-linear functions (ACMMM087)

understand the concept of the derivative as a function (ACMMM089)

recognise and use linearity properties of the derivative (ACMMM090)

find instantaneous rates of change (ACMMM092)

find the slope of a tangent and the equation of the tangent (ACMMM093)

construct and interpret position-time graphs, with velocity as the slope of the tangent (ACMMM094)

sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as x→∞ and x→-∞ (ACMMM095)

estimate the limit of ah-1h as h→0 using technology, for various values of a >0 (ACMMM098)

recognise that e is the unique number a for which the above limit is 1 (ACMMM099)

establish and use the formula ddxex=ex (ACMMM100)

establish the formulas ddxsin⁡x=cos⁡x,   and  ddxcos⁡x=-sin⁡x by numerical estimations of the limits and informal proofs based on geometric constructions (ACMMM102)

understand and use the product and quotient rules (ACMMM104)

understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions (ACMMM105)

use the increments formula: δy≅dydx×δx to estimate the change in the dependent variable y resulting from changes in the independent variable x (ACMMM107)

understand the concept of the second derivative as the rate of change of the first derivative function (ACMMM108)

recognise acceleration as the second derivative of position with respect to time (ACMMM109)

understand the concepts of concavity and points of inflection and their relationship with the second derivative (ACMMM110)

understand and use the second derivative test for finding local maxima and minima (ACMMM111)

sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection (ACMMM112)

recognise anti-differentiation as the reverse of differentiation (ACMMM114)

use the notation ∫fxdx for anti-derivatives or indefinite integrals (ACMMM115)

establish and use the formula ∫xndx=1n+1xn+1+c for n≠-1 (ACMMM116)

establish and use the formula ∫exdx=ex+c (ACMMM117)

establish and use the formulas ∫sin⁡xdx=-cos⁡x+c and ∫cos⁡xdx=sin⁡x+c (ACMMM118)

recognise and use linearity of anti-differentiation (ACMMM119)

determine indefinite integrals of the form ∫fax+bdx (ACMMM120)

identify families of curves with the same derivative function (ACMMM121)

determine fx, given f' (x) and an initial condition fa=b (ACMMM122)

examine the area problem, and use sums of the form ∑ifxi δxi to estimate the area under the curve y=f(x) (ACMMM124)

interpret the definite integral ∫abfxdx as area under the curve y=fx if fx>0  (ACMMM125)

recognise the definite integral ∫abfxdx  as a limit of sums of the form ∑ifxi δxi (ACMMM126)

interpret ∫abfxdx as a sum of signed areas (ACMMM127)

understand the concept of the signed area function Fx=∫axftdt (ACMMM129)

understand and use the theorem: F'x=ddx∫axftdt=fx, and illustrate its proof geometrically (ACMMM130)

calculate the area under a curve (ACMMM132)

calculate total change by integrating instantaneous or marginal rate of change (ACMMM133)

calculate the area between curves in simple cases (ACMMM134)

understand the concepts of a discrete random variable and its associated probability function, and their use in modelling data (ACMMM136)

use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable (ACMMM137)

recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes (ACMMM138)

examine simple examples of non-uniform discrete random variables (ACMMM139)

recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases (ACMMM140)

recognise the variance and standard deviation of a discrete random variable as a measures of spread, and evaluate them in simple cases (ACMMM141)

use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)

identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)

recognise the mean p and variance p1-p of the Bernoulli distribution with parameter p (ACMMM145)

understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in n independent Bernoulli trials, with the same probability of success p in each trial (ACMMM147)

identify contexts suitable for modelling by binomial random variables (ACMMM148)

determine and use the probabilities PX=r=nrpr(1-p)n-r associated with the binomial distribution with parameters n and p ; note the mean np and variance  np1-p of a binomial distribution (ACMMM149)

define logarithms as indices: ax=b is equivalent to x=loga⁡b i.e. aloga⁡b=b  (ACMMM151)

establish and use the algebraic properties of logarithms (ACMMM152)

recognise the inverse relationship between logarithms and exponentials: y=ax is equivalent to x=loga⁡y (ACMMM153)

interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake magnitude, octaves in music, pH in chemistry (ACMMM154)

solve equations involving indices using logarithms (ACMMM155)

recognise the qualitative features of the graph of y=loga⁡x (a>1) including asymptotes, and of its translations y=loga⁡x+b and y=loga⁡(x+c) (ACMMM156)

solve simple equations involving logarithmic functions algebraically and graphically (ACMMM157)

define the natural logarithm ln⁡x=loge⁡x (ACMMM159)

recognise and use the inverse relationship of the functions y=ex and y=ln⁡x (ACMMM160)

establish and use the formula ddxln⁡x=1x (ACMMM161)

establish and use the formula ∫1xdx=ln⁡ x +c, for x>0 (ACMMM162)

use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable (ACMMM164)

understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts (ACMMM165)

recognise the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases (ACMMM166)

identify contexts such as naturally occurring variation that are suitable for modelling by normal random variables (ACMMM168)

recognise features of the graph of the probability density function of the normal distribution with mean μ and standard deviation σ and the use of the standard normal distribution (ACMMM169)

understand the concept of a random sample (ACMMM171)

discuss sources of bias in samples, and procedures to ensure randomness (ACMMM172)

understand the concept of the sample proportion p̂ as a random variable whose value varies between samples, and the formulas for the mean p and standard deviation (p(1-p)/n of the sample proportion p̂ (ACMMM174)

examine the approximate normality of the distribution of p̂ for large samples (ACMMM175)

the concept of an interval estimate for a parameter associated with a random variable (ACMMM177)

use the approximate confidence interval p̂-z(p̂(1-p̂)/n,  p̂+z(p̂(1-p̂)/n, as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution (ACMMM178)

define the approximate margin of error E=z(p̂(1-p̂)/n and understand the trade-off between margin of error and level of confidence (ACMMM179)