HSC · HSC Year 12

Mathematics Standard 2

Algebra, measurement, financial mathematics and statistics — full HSC papers with step-by-step worked solutions.

20full-length model exams with mark-by-mark answer guides

20detailed note sets — ~120 pages across every topic

64exam-style practice questions with worked solutions

60flashcards for every key term & formula

6official past papers

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Sample revision note

Exponential functions and growth/decay models

The General Form y = ab^x — What Each Parameter Does

Every exponential relationship studied in Mathematics Standard 2 is written in the form:

y = ab^x

Understanding the role of a and b separately is the foundation of this topic.

a (the initial value / y-intercept): When x = 0, b⁰ = 1, so y = a × 1 = a. The value of a is always the y-intercept. In practical contexts it is the starting quantity — the population at time zero, the purchase price of a car, the original investment, and so on. a must be positive in all HSC Standard 2 contexts.
b (the base / multiplying factor): Each time x increases by 1, y is multiplied by b. The base controls both the direction and the rate of change.
- If b > 1 → exponential growth (the multiplier is bigger than 1, so the quantity increases each period).
- If 0 < b < 1 → exponential decay (the multiplier is a fraction, so the quantity decreases each period).
- b = 1 gives a horizontal line (constant), not covered as exponential in this course.
- b ≤ 0 is not applicable in Standard 2 contexts.
x (the independent variable): Usually represents time — years, months, days. It can also represent number of generations, number of doses, etc.
y (the dependent variable): The quantity being modelled — population size, dollar value, bacterial count, etc.

The relationship between b and the percentage rate of change r is important for linking to financial contexts:

Growth: b = 1 + r, where r is the growth rate as a decimal (e.g. 8% growth → b = 1.08).
Decay: b = 1 − r, where r is the decay rate as a decimal (e.g. 15% depreciation → b = 0.85).

Identifying and Distinguishing Growth from Decay

In exam questions you will need to identify whether a given equation, table of values, or graph represents growth or decay. Use the following checklist:

Feature	Exponential Growth (b > 1)	Exponential Decay (0 < b < 1)
Value of b	Greater than 1 (e.g. 1.08, 2, 1.5)	Between 0 and 1 (e.g. 0.85, 0.6, 0.5)
Graph direction	Rises steeply from left to right	Falls steeply, approaches x-axis from left to right
Successive ratio	Each y-value is larger than previous	Each y-value is smaller than previous
y-intercept	a (positive)	a (positive, same formula)
Horizontal asymptote	y = 0 (curve rises away from x-axis)	y = 0 (curve approaches but never touches x-axis)
Practical examples	Population, compound interest, bacterial growth	Depreciation, radioactive decay, cooling

Quick identification from a table: Calculate the ratio of consecutive y-values (y_next ÷ y_current). If this ratio is constant and greater than 1, growth; if constant and less than 1, decay. A constant ratio (as opposed to a constant difference) is the hallmark of an exponential — not a linear — relationship.

Identifying from an equation: Rewrite in y = ab^x form if needed, then inspect b directly.

Sample exam question

A small business sells handmade candles. Its total weekly cost is given by C = 8n + 1200, where n is the number of candles made, and its weekly revenue is R = 20n. The break-even point occurs when cost equals revenue. How many candles must be sold each week to break even?

A. 60 candles
B. 100 candles
C. 150 candles
D. 240 candles

Show the worked answer

Answer: B

Set 8n + 1200 = 20n, so 1200 = 12n, giving n = 100 candles.

All 20 practice exams

Exam 1 — Algebra: Types of Relationships (emphasis); Measurement: Non-right-angled Trigonometry; Rates and Ratios; Financial Mathematics: Investments and Loans; Annuities
Exam 2 — Measurement: Non-right-angled Trigonometry (heaviest weighting - cosine/sine rule, area, bearings, combined modelling); Measurement: Rates and Ratios (heaviest weighting - fuel/energy rates, scale areas, best buy, ratio division); Financial Mathematics: Investments and Loans, and Annuities (reducing-balance recurrence, FV annuity, compound comparison)
Exam 3 — Financial Mathematics: Investments and Loans, and Annuities (42% of marks); Measurement: Non-right-angled Trigonometry, and Rates and Ratios; Statistical Analysis: Bivariate Data Analysis, and The Normal Distribution
Exam 4 — Statistical Analysis: Bivariate Data Analysis (scatterplots, Pearson's r, least-squares regression, interpolation vs extrapolation, causation); Statistical Analysis: The Normal Distribution (empirical 68-95-99.7 rule, z-scores, comparing scores, proportions of a population); Financial Mathematics: Investments and Loans, and Annuities (compound interest, reducing-balance tables, future-value annuities)
Exam 5 — Networks: Critical Path Analysis (forward/backward pass, float, crashing); Networks: Network Concepts (minimum spanning tree, maximum flow / minimum cut); Statistical Analysis (least-squares regression, correlation, normal distribution / empirical rule)
Exam 6 — Algebra: Types of Relationships (linear modelling, simultaneous equations, quadratic and exponential models, variation) — emphasised; Financial Mathematics (compound interest, annuities via FV tables, reducing-balance loans); Statistical Analysis (the normal distribution and z-scores, bivariate data with least-squares regression)
Exam 7 — Measurement: Non-right-angled Trigonometry (emphasis) — area rule, cosine rule, bearings; Measurement: Rates and Ratios (emphasis) — fuel/medication rates, unit conversion, scale, dividing in a ratio; Financial Mathematics — compound vs simple interest, reducing-balance loan tables, future value of an annuity
Exam 8 — Financial Mathematics: Investments and Loans (compound vs simple interest, appreciation, reducing-balance loan tables); Financial Mathematics: Annuities (future value, present value, table and formula methods); Measurement and Statistics (non-right-angled trigonometry, bivariate data, the normal distribution, rates and ratios)
Exam 9 — Statistical Analysis emphasis: least-squares regression line, Pearson's correlation, and the normal distribution (z-scores, empirical 68-95-99.7 rule, standardised comparison); Realistic Standard 2 weighting across all eight Year 12 topics: Algebra, Measurement (non-right trig), Rates & Ratios, Financial Maths (loans & annuities), Statistics, Networks & Critical Path; Multi-part 5+ mark modelling questions (Q20 regression 9 marks, Q25 critical path 9 marks, Q27 normal distribution 9 marks) with graduated difficulty
Exam 10 — Networks emphasis (40% of Section II): Q24 Network Concepts (MST + shortest path), Q25 Critical Path Analysis (forward/backward pass, float, critical path), Q26 maximum flow / minimum cut; Complete Year 12 syllabus coverage: Algebra (Types of Relationships), Measurement (Non-right-angled Trig, Rates & Ratios), Financial Maths (Investments/Loans, Annuities), Statistics (Bivariate, Normal Distribution); At least one multi-part modelling question worth 5+ marks (Q23 quadratic profit model, plus Q25 CPA at 12 marks)
Exam 11 — Algebra: Types of Relationships (emphasis — direct variation, linear simultaneous/break-even, quadratic and exponential modelling); Financial Mathematics: Investments and Loans, and Annuities (compound interest, future-value annuity, reducing-balance loan schedule); Statistical Analysis: The Normal Distribution and Bivariate Data Analysis (z-scores, empirical rule, least-squares regression, correlation)
Exam 12 — Measurement: Non-right-angled Trigonometry (cosine rule, sine rule, area rule, bearings); Measurement: Rates and Ratios (best buy, unit rates, fuel consumption, scale, flow, dividing in a ratio); Financial Mathematics: Investments and Loans, and Annuities
Exam 13 — Financial Mathematics: Investments and Loans, and Annuities (heavily weighted - 33 of 85 marks); Non-right-angled Trigonometry with bearings (sine/cosine rule, area); Statistical Analysis: Bivariate Data Analysis and The Normal Distribution
Exam 14 — Statistical Analysis: Bivariate Data Analysis (scatterplots, Pearson correlation, least-squares regression, interpolation vs extrapolation, correlation vs causation); Statistical Analysis: The Normal Distribution (z-scores, empirical 68-95-99.7 rule, comparing scores across distributions); Financial Mathematics: Investments & Loans and Annuities (future-value annuity, reducing-balance loan tables)
Exam 15 — Networks: Critical Path Analysis (forward/backward pass, float, crashing) — heaviest weighting; Networks: Network Concepts (minimum spanning tree, connected graphs); Financial Mathematics: reducing-balance loans, compound interest and annuities
Exam 16 — Algebra: Types of Relationships (emphasis ~28% of section); Full coverage of all Year 12 Standard 2 modules with realistic weighting; Two multi-part modelling questions worth 5+ marks (Critical Path 9, Algebra 10)
Exam 17 — Measurement: Non-right-angled Trigonometry (cosine/sine rule, area, Heron's, bearings) — heaviest weighting; Measurement: Rates and Ratios (fuel, dosage, scale, proportional sharing, flow rates); Financial Mathematics: Investments and Loans, and Annuities (compound interest, reducing-balance tables, FV of annuity)
Exam 18 — Financial Mathematics: Investments and Loans (reducing-balance tables, compound interest); Financial Mathematics: Annuities (FV factors, superannuation accumulation and drawdown); Measurement: Non-right-angled Trigonometry, Rates and Ratios
Exam 19 — Statistical Analysis: The Normal Distribution (z-scores, empirical 68-95-99.7 rule, expected counts); Statistical Analysis: Bivariate Data Analysis (Pearson's r, least-squares regression, interpolation vs extrapolation); Financial Mathematics (reducing-balance loans, annuities/future value)
Exam 20 — Networks: Critical Path Analysis (forward/backward pass, float, crashing); Networks: Network Concepts (minimum spanning tree, degree, shortest path); Financial Mathematics (reducing-balance loans and annuity tables)

All 20 revision notes

Exponential functions and growth/decay models
Gradient, intercepts and the equation y = mx + b
Quadratic and reciprocal relationships
Solving simultaneous equations graphically and algebraically
Applying the Cosine Rule to find sides and angles
Applying the Sine Rule to find sides and angles
Area formula using two sides and an included angle
Solving problems using bearings and survey techniques
Dividing quantities in a given ratio and scale drawing calculations
Interpreting and calculating rates, and rate conversions
Future Value and Present Value formulas for annuities
Using FV/PV tables and spreadsheets to model annuities
Reducing-balance loans and credit card calculations
Simple interest formula and compound interest formula
Activity networks: EST, LST, float and the critical path
Definitions, Euler's formula and shortest path problems
Constructing scatterplots and interpreting Pearson's correlation coefficient
Least-squares regression line: equation, interpolation and extrapolation
Calculating z-scores and applying the 68-95-99.7 rule
Using z-scores to compare scores across different data sets