§-Unit

Unit 1: Surds, algebra, functions and probability

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

Topic 1: Surds and quadratic functions

Sketch and analyse linear and quadratic functions, finding gradient, intercepts, vertex and discriminant, and solving linear and quadratic equations and inequalities

Linear and quadratic functions. Finds gradient, intercepts and parallel/perpendicular relationships for linear functions; converts between standard, factored and vertex form and uses the discriminant for quadratics.

8 min read →

Simplify expressions involving surds and apply the laws of indices to rational and negative exponents

Surds and exponents. Simplifies surds, rationalises denominators, and applies the seven index laws to rational and negative powers; works the standard QCAA simplification problem.

7 min read →

Topic 2: Binomial expansion and cubic functions

Sketch and analyse polynomial functions of degree 3 and 4, using factored form to read roots and multiplicities, and applying the factor and remainder theorems

Polynomial functions. Sketches cubics and quartics from factored form, applies the factor and remainder theorems, and works the standard QCAA factor-a-cubic problem.

8 min read →

Expand expressions of the form

(a + b)^n

for a positive integer

n

using Pascal's triangle and binomial coefficients, and find specific terms and coefficients

How to expand $(a+b)^n$ using Pascal's triangle and the binomial coefficient $\binom{n}{k}$ , find a required term or coefficient without expanding everything, and use the expansion for approximations. Worked examples with checked numbers and practice questions with model answers.

7 min read →

Topic 3: Functions and relations

Sketch reciprocal functions of the form

y = \dfrac{a}{x - h} + k

, identify asymptotes, domain and range, and find the inverse of a one-to-one function

How to sketch reciprocal functions with their vertical and horizontal asymptotes, state domain and range, test whether a function is one-to-one, and find an inverse function by swapping variables, including domain restrictions. Checked worked examples and practice questions with model answers.

7 min read →

Functions and graphs introduced in Year 11, including linear, quadratic, cubic, polynomial, exponential and logarithmic functions; their key features, intercepts and transformations

QCE Math Methods Unit 1 subject-matter point on functions and graphs. Linear, quadratic, polynomial, exponential and logarithmic functions; identification of intercepts, turning points and asymptotes; the four standard transformations; foundation for Unit 3 / 4 calculus work.

9 min read →

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

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

and the effect of each parameter

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

7 min read →

Topic 5: Probability

Counting techniques (multiplication principle, permutations and combinations), simple probability, conditional probability and the addition and multiplication rules

QCE Math Methods Unit 1 subject-matter point on counting and probability. The multiplication principle, permutations and combinations, set notation, simple and conditional probability, the addition rule, independence, and worked QCAA-style selections.

9 min read →

Apply the rules of probability (addition, multiplication, conditional), permutations and combinations to calculate probabilities of compound events

Probability and counting. States addition, multiplication and conditional probability rules, defines permutations and combinations, and works the standard QCAA card-and-committee problem.

8 min read →

How are arithmetic and geometric sequences analysed?

Define arithmetic and geometric sequences, find the

n

th term and the sum of the first

n

terms, and apply to real-world contexts

Sequences. States the $n$ th-term and sum formulas for arithmetic and geometric sequences, distinguishes the two types, and works the standard QCAA applications to salary growth, depreciation and convergent series.

8 min read →

How are simultaneous equations solved?

Solve systems of simultaneous linear equations in two and three variables, including by substitution, elimination, and matrix methods, and interpret the results graphically

Simultaneous equations. Solves $2 \times 2$ systems by elimination and substitution, identifies parallel-line and identical-line cases, and works a standard QCAA worded problem.

7 min read →

How are arithmetic and geometric sequences and series defined and computed in QCE Math Methods Unit 1?

Arithmetic and geometric sequences and series, including the general term formulas, sum formulas, and applications to growth and decay problems

QCE Math Methods Unit 1 subject-matter point on sequences and series. General term and sum formulas for arithmetic and geometric sequences, and the infinite geometric series formula $S_\infty = a/(1-r)$ for $|r| < 1$ , with applications to compound interest and exponential growth.

8 min read →

How are financial calculations done?

Apply simple interest, compound interest and depreciation models to financial calculations, including future value, present value and effective annual rate

Financial applications. Applies simple interest, compound interest, future and present value, the effective annual rate, and straight-line and declining-balance depreciation, with worked QCAA-style investment and depreciation problems.

8 min read →

What additional algebraic skills does QCE Math Methods Unit 1 introduce?

Index and logarithm laws, factorisation techniques, solving polynomial equations, and the relationship between exponential and logarithmic functions

QCE Math Methods Unit 1 subject-matter point on algebraic manipulation and equation solving. Index laws, logarithm laws, factorisation (common factor, grouping, quadratic, difference of squares, sum/difference of cubes), and solving polynomial / exponential / logarithmic equations.

9 min read →