§-Unit

Unit 1: Functions, relations and graphs

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

How are inverse and composite functions defined?

Define inverse and composite functions, identify when an inverse function exists (one-to-one), find inverse functions algebraically, and graph inverse and composite functions

Inverse and composite functions. Defines when an inverse exists, finds $f^{-1}$ algebraically by swapping $x$ and $y$ and solving, sketches the inverse as the reflection of $y = f(x)$ in the line $y = x$ , and works the VCAA SAC-style domain-restriction problem.

7 min read →

How are cubic and quartic polynomials analysed?

Sketch cubic and quartic polynomials, identifying intercepts, end behaviour, turning points and points of inflection, and using factored form to read roots and multiplicities

Cubic and quartic polynomials. Sketches $y = a(x - p)(x - q)(x - r)$ and quartic equivalents, reads end behaviour from the leading term, identifies turning points and inflection points, and interprets root multiplicities.

5 min read →

How are linear functions analysed?

Sketch and analyse linear functions of the form

y = mx + c

, including finding gradient,

x

- and

y

-intercepts, equations of parallel and perpendicular lines, and solving linear equations and inequalities

Linear functions. Sketches $y = mx + c$ , finds gradient and intercepts, derives equations of parallel and perpendicular lines, and works the VCAA SAC-style line-through-two-points and perpendicular-bisector problems.

6 min read →

How are quadratic functions analysed?

Sketch and analyse quadratic functions in standard, factored and turning-point form, including finding vertex, axis of symmetry, intercepts and using the discriminant to classify roots

Quadratic functions. Sketches $y = ax^2 + bx + c$ , converts between forms, finds the vertex from $x = -b/(2a)$ , applies the discriminant $b^2 - 4ac$ , and works the VCAA SAC-style turning-point and roots problem.

7 min read →

What algebraic skills does VCE Math Methods Unit 1 introduce, including index laws, logarithm laws, and the solution of equations?

Algebraic manipulation of polynomial, exponential and logarithmic expressions, including index laws, logarithm laws, factorisation, and the solution of linear, quadratic, polynomial, exponential and logarithmic equations

VCE Math Methods Unit 1 key-knowledge point on algebra. Index and logarithm laws, factorisation techniques (common factor, grouping, quadratic factorisation, sum and difference of cubes), and methods for solving linear, quadratic, polynomial, exponential and logarithmic equations.

8 min read →

How is the concept of a rate of change introduced in VCE Math Methods Unit 1, leading to the derivative?

Average rates of change between two points, the gradient of a chord, the gradient at a point as a limit, and the derivative of polynomial functions using the power rule

VCE Math Methods Unit 1 key-knowledge point introducing calculus. The average rate of change as the gradient of a chord, the instantaneous rate of change as a limit, and the power rule for derivatives of polynomial functions; foundation for the full Unit 3 differentiation toolkit.

8 min read →

How are probabilities computed using counting and combinations?

Apply the rules of probability (addition, multiplication, conditional), the counting principles (permutations and combinations), and use these to find probabilities in compound experiments

Probability and counting. States addition, multiplication and conditional probability rules, defines permutations ( $^nP_r$ ) and combinations ( $^nC_r$ ), and works the VCAA SAC-style card-and-committee problems.

7 min read →

What functions and relations are introduced in VCE Math Methods Unit 1, and how are they graphed and transformed?

Linear, quadratic, cubic and quartic polynomial functions, basic exponential functions

y = a^x

, logarithmic functions

y = \log_a(x)

, and the standard transformations (dilation, reflection, translation)

VCE Math Methods Unit 1 key-knowledge point on functions and graphs. Linear, quadratic, polynomial, exponential and logarithmic functions, their key features (axes intercepts, turning points, asymptotes), and the four standard transformations that prepare for Unit 3 graphical work.

8 min read →

How are polynomial factors found?

Apply the factor theorem and the remainder theorem to factorise polynomials and to solve polynomial equations

Factor and remainder theorems. States both theorems, demonstrates polynomial long division, and works the VCAA SAC-style problem of factoring a cubic by finding a rational root and dividing.

5 min read →

How are surds and rational exponents manipulated?

Simplify and operate on surd expressions and apply the laws of indices to rational and negative exponents

Surds and rational exponents. Simplifies surds using $\sqrt{ab} = \sqrt a \sqrt b$ , rationalises denominators, applies the index laws to fractional and negative powers, and works the VCAA SAC-style simplification problem.

6 min read →

What probability and counting principles does VCE Math Methods Unit 1 introduce?

Counting principles (multiplication principle, permutations and combinations), set notation, simple probability, conditional probability and the addition / multiplication rules

VCE Math Methods Unit 1 key-knowledge point on probability and counting. The multiplication principle, permutations and combinations, set notation, simple probability, conditional probability $P(A|B)$ , and the addition and multiplication rules; foundation for Unit 3 discrete random variables and Unit 4 sampling.

8 min read →

How do transformations affect the graph of a function?

Apply translations, dilations and reflections to the graph of a function

y = f(x)

, including the form

y = a f(b(x - h)) + k

and the effect of each parameter on the graph

Transformations. Maps the four parameters of $y = af(b(x - h)) + k$ to vertical dilation/reflection, horizontal dilation/reflection, horizontal translation and vertical translation, and works the VCAA SAC-style sequence-of-transformations problem.

5 min read →