§-Unit

Unit 2: Functions, calculus and probability

12 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How are discrete random variables analysed?

Define a discrete random variable and its probability distribution, and compute expected value (mean) and variance for given distributions

Discrete random variables. Defines a probability distribution, computes expected value $E[X] = \sum x_i p_i$ and variance $\text{Var}(X) = E[X^2] - (E[X])^2$ , and works the VCAA SAC-style fair-die and lottery-EV problems.

7 min read →

How is the binomial distribution used to model repeated trials?

Define and apply the binomial distribution to model the number of successes in

n

independent Bernoulli trials, including computing probabilities, expected value

np

and variance

np(1-p)

Binomial distribution. States $P(X = k) = ^nC_k p^k (1-p)^{n-k}$ , identifies $E[X] = np$ and $\text{Var}(X) = np(1-p)$ , and works the VCAA SAC-style " $10$ coin tosses, exactly $7$ heads" problem.

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How are logarithmic functions used to solve exponential equations?

Define logarithms as the inverse of exponentials, apply the laws of logarithms, sketch logarithmic graphs and solve exponential equations using logs

Logarithms. Defines $\log_b x = y \iff b^y = x$ , lists the three log laws and change of base, sketches $y = \log_b x$ , and works the VCAA SAC-style exponential-equation problem $5^x = 28$ .

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How are exponential and logarithmic functions differentiated?

Differentiate exponential (

e^x

a^x

) and logarithmic (

\ln x

\log_b x

) functions, including composite functions via the chain rule

Derivatives of exponential and logarithmic functions. States $\frac{d}{dx} e^x = e^x$ , $\frac{d}{dx} \ln x = 1/x$ , the chain-rule extensions, and works the VCAA SAC-style continuous-decay derivative problem.

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How are exponential functions analysed?

Sketch and analyse exponential functions of the form

y = a \cdot b^{x - h} + k

, identifying key features (intercepts, asymptote, domain, range) and applying transformations

Exponential functions. Sketches $y = b^x$ for $b > 1$ and $0 < b < 1$ , identifies the y-intercept, horizontal asymptote, domain and range, and works the VCAA SAC-style transformation problem.

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How are circular functions extended to model periodic phenomena?

Sketch and analyse trigonometric functions

y = a\sin(b(x - h)) + k

and

y = a\cos(b(x - h)) + k

, identifying amplitude, period, phase and vertical translation, and solve trig equations over a specified interval

Circular functions. Sketches transformed sine and cosine graphs, identifies amplitude $|a|$ , period $2\pi/|b|$ , phase shift $h$ and vertical translation $k$ , and works the VCAA SAC-style trig equation $\sin(2x) = 1/2$ on a stated interval.

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How are trigonometric functions differentiated?

Differentiate sine, cosine and tangent functions and their compositions via the chain rule

Derivatives of trig functions. States $\frac{d}{dx} \sin x = \cos x$ , $\frac{d}{dx} \cos x = -\sin x$ , $\frac{d}{dx} \tan x = \sec^2 x$ , the chain-rule extensions, and works the VCAA SAC-style oscillation-derivative problem.

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How are antidifferentiation and the integral introduced in VCE Math Methods Unit 2?

Antidifferentiation as the reverse of differentiation, the antiderivative of polynomial functions via the power rule, the constant of integration, and the use of an initial condition to determine a specific antiderivative

VCE Math Methods Unit 2 key-knowledge point on antidifferentiation. The reverse of the power rule, the constant of integration $C$ , and the use of an initial condition to determine $C$ ; foundation for Unit 4 definite integration.

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How are trigonometric functions defined and graphed in VCE Math Methods Unit 2?

Trigonometric functions

y = \sin(x)

y = \cos(x)

and

y = \tan(x)

, the unit circle, exact values at standard angles, transformations of trig graphs, and solving trigonometric equations

VCE Math Methods Unit 2 key-knowledge point on trigonometric functions. The unit circle, exact values at standard angles, the standard graphs of $\sin$ , $\cos$ and $\tan$ with their amplitude, period and asymptotes, transformations, and solving trig equations.

9 min read →

How is differentiation used to analyse and optimise functions?

Use differentiation to analyse the behaviour of functions, including locating and classifying stationary points, finding tangent and normal equations, and solving optimisation problems

Applications of differentiation. Locates stationary points by $f'(x) = 0$ , classifies with the first-derivative sign test or the second-derivative test, writes tangent and normal equations, and works the VCAA SAC-style box-maximisation optimisation problem.

6 min read →

How are inverse and composite functions defined and used in VCE Math Methods Unit 2?

Composite functions

f \circ g

and

g \circ f

, the existence and form of inverse functions

f^{-1}

, the relationship between a function and its inverse (reflection in

y = x

, domain and range swap), and the one-to-one restriction

VCE Math Methods Unit 2 key-knowledge point on inverse and composite functions. Composite function notation $f(g(x))$ , the conditions for $f^{-1}$ to exist (one-to-one), the procedure for finding $f^{-1}$ algebraically, and the graphical relationship (reflection in $y = x$ ).

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How are Bernoulli trials, sample data and simulation introduced in VCE Math Methods Unit 2?

Bernoulli trials and sequences of Bernoulli trials, sample data analysis (mean, median, mode, range), simulation of random processes, and the relationship between theoretical probability and observed relative frequency

VCE Math Methods Unit 2 key-knowledge point on Bernoulli trials, sample data and simulation. Bernoulli trial probabilities, summary statistics of sample data (mean, median, mode, range), and how simulation (physical or computational) approximates theoretical probabilities; foundation for the Unit 3 binomial distribution.

8 min read →