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Year 12 Maths Topics

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1. Introduction to Relations and Functions

1.1

Relationship between two variables

1.2

Understand relations between x- and y-intercepts

1.3

Domain and range of a function

1.4

Identifying functions

1.5

Function notation

2. Functions

2.1

Function notation

2.2

Operations with functions

2.3

Adding functions

2.4

Subtracting functions

2.5

Multiplying functions

2.6

Dividing functions

2.7

Composite functions

2.8

Inequalities of combined functions

2.9

Inverse functions

2.10

One to one functions

2.11

Difference quotient: applications of functions

3. Transformations of Functions

3.1

Transformations of functions: Horizontal translations

3.2

Transformations of functions: Vertical translations

3.3

Reflection across the y-axis: y = f(-x)

3.4

Reflection across the x-axis: y = -f(x)

3.5

Transformations of functions: Horizontal stretches

3.6

Transformations of functions: Vertical stretches

3.7

Combining transformations of functions

3.8

Even and odd functions

4. Factorising Polynomial

4.1

Factorise by taking out the greatest common factor

4.2

Factorise by grouping

4.3

Factorising difference of squares: x^2 - y^2

4.4

Factorising trinomials

4.5

Factoring difference of cubes

4.6

Factoring sum of cubes

5. Quadratic Functions

5.1

Characteristics of quadratic functions

5.2

Transformations of quadratic functions

5.3

Quadratic function in general form: y = ax^2 + bx + c

5.4

Quadratic function in vertex form: y = a(x-p)^2 + q

5.5

Completing the square

5.6

Converting from general to vertex form by completing the square

5.7

Shortcut: Vertex formula

5.8

Graphing parabolas for given quadratic functions

5.9

Finding the quadratic functions for given parabolas

5.10

Applications of quadratic functions

6. Polynomial Functions

6.1

Solving quadratic equations by factorising

6.2

Solving quadratic equations by completing the square

6.3

Using quadratic formula to solve quadratic equations

6.4

Nature of roots of quadratic equations: The discriminant

6.5

Applications of quadratic equations

6.6

Solving quadratic inequalities

7. Radicals

7.1

Operations with radicals

7.2

Conversion between entire radicals and mixed radicals

7.3

Adding and subtracting radicals

7.4

Multiplying radicals

7.5

Solving radical equations

8. Radical Functions

8.1

Basic radical functions

8.2

Transformations of radical functions

8.3

Square root of a function

9. Exponents

9.1

Product rule of exponents

9.2

Quotient rule of exponents

9.3

Power of a product rule

9.4

Power of a quotient rule

9.5

Power of a power rule

9.6

Negative exponent rule

9.7

Combining the exponent rules

9.8

Scientific notation

9.9

Convert between radicals and rational exponents

9.10

Solving for exponents

10. Rational Functions

10.1

What is a rational function?

10.2

Point of discontinuity

10.3

Vertical asymptote

10.4

Horizontal asymptote

10.5

Slant asymptote

10.6

Graphs of rational functions

10.7

Solving equations with algebraic fractions

10.8

Solving rational inequalities

11. Exponential Functions

11.1

Exponents: Product rule (a^x)(a^y) = a^(x+y)

11.2

Exponents: Division rule (a^x / a^y) = a^(x-y)

11.3

Exponents: Power rule (a^x)^y = a^(x * y)

11.4

Exponents: Negative exponents

11.5

Exponents: Zero exponent: a^0 = 1

11.6

Exponents: Rational exponents

11.7

Graphing exponential functions

11.8

Graphing transformations of exponential functions

11.9

Finding an exponential function given its graph

12. Logarithmic Functions

12.1

What is a logarithm?

12.2

Converting from logarithmic form to exponential form

12.3

Evaluating logarithms without a calculator

12.4

Common logarithms

12.5

Natural log: ln

12.6

Evaluating logarithms using change-of-base formula

12.7

Converting from exponential form to logarithmic form

12.8

Solving exponential equations with logarithms

12.9

Product rule of logarithms

12.10

Quotient rule of logarithms

12.11

Combining product rule and quotient rule in logarithms

12.12

Evaluating logarithms using logarithm rules

12.13

Solving logarithmic equations

12.14

Graphing logarithmic functions

12.15

Finding a logarithmic function given its graph

13. Applications of Exponential and Logarithmic Functions

13.1

Exponential growth and decay by a factor

13.2

Exponential decay: Half-life

13.3

Exponential growth and decay by percentage

13.4

Finance: Compound interest

13.5

Continuous growth and decay

13.6

Logarithmic scale: Richter scale (earthquake)

13.7

Logarithmic scale: pH scale

13.8

Logarithmic scale: dB scale

13.9

Finance: Future value and present value

14. Solving Simultaneous Equations

14.1

Determining number of solutions to linear equations

14.2

Solving simultaneous equations by graphing

14.3

Solving simultaneous equations by elimination

14.4

Solving simultaneous equations by substitution

14.5

Money related questions in linear equations

14.6

Unknown number related questions in linear equations

14.7

Distance and time related questions in linear equations

14.8

Rectangular shape related questions in linear equations

15. Inequalities in Two Variables

15.1

Graphing linear inequalities in two variables

15.2

Graphing simultaneous linear inequalities

15.3

Graphing quadratic inequalities in two variables

15.4

Graphing simultaneous quadratic inequalities

15.5

Applications of inequalities

15.6

What is linear programming?

15.7

Linear programming word problems

16. Introduction to Trigonometry

16.1

Use sine ratio to calculate angles and sides (Sin =

\frac{o}{h}

16.2

Use cosine ratio to calculate angles and sides (Cos =

\frac{a}{h}

16.3

Use tangent ratio to calculate angles and sides (Tan =

\frac{o}{a}

16.4

Combination of SohCahToa questions

16.5

Solving expressions using 45-45-90 special right triangles

16.6

Solving expressions using 30-60-90 special right triangles

16.7

Word problems relating ladder in trigonometry

16.8

Word problems relating guy wire in trigonometry

16.9

Other word problems relating angles in trigonometry

17. Trigonometric Ratios and Angle Measure

17.1

17.2

Coterminal angles

17.3

Reference angles

17.4

Find the exact value of trigonometric ratios

17.5

ASTC rule in trigonometry (All Students Take Calculus)

17.6

17.7

17.8

17.9

Applications of the sine rule and cosine rule

18. Bearings

18.1

Introduction to bearings

18.2

Bearings and direction word problems

18.3

Angle of elevation and depression

19. Graphing Trigonometric Functions

19.1

Sine graph: y = sin x

19.2

Cosine graph: y = cos x

19.3

Tangent graph: y = tan x

19.4

Cotangent graph: y = cot x

19.5

Secant graph: y = sec x

19.6

Cosecant graph: y = csc x

19.7

Graphing transformations of trigonometric functions

19.8

Determining trigonometric functions given their graphs

20. Applications of Trigonometric Functions

20.1

Ferris wheel trig problems

20.2

Tides and water depth trig problems

20.3

Spring (simple harmonic motion) trig problems

21. Trigonometric Identities

21.1

Quotient identities and reciprocal identities

21.2

Pythagorean identities

21.3

Sum and difference identities

21.4

Cofunction identities

21.5

Double-angle identities

22. Conics

22.1

Conics - Parabola

22.2

Conics - Ellipse

22.3

Conics - Circle

22.4

Conics - Hyperbola

23. Imaginary and Complex Numbers

23.1

Introduction to imaginary numbers

23.2

Complex numbers and complex planes

23.3

Adding and subtracting complex numbers

23.4

Complex conjugates

23.5

Multiplying and dividing complex numbers

23.6

Distance and midpoint of complex numbers

23.7

Angle and absolute value of complex numbers

23.8

Polar form of complex numbers

23.9

Operations on complex numbers in polar form

24. Vectors

24.1

Introduction to vectors

24.2

Magnitude of a vector

24.3

Direction angle of a vector

24.4

Scalar multiplication

24.5

Equivalent vectors

24.6

Adding and subtracting vectors in component form

24.7

Operations on vectors in magnitude and direction form

24.8

24.9

Word problems on vectors

25. Sequences and Series

25.1

Arithmetic sequences

25.2

Arithmetic series

25.3

Geometric sequences

25.4

Geometric series

25.5

Infinite geometric series

25.6

25.7

Arithmetic mean vs. Geometric mean

26. Permutations and Combinations

26.1

Fundamental counting principle

26.2

Factorial notation

26.3

Path counting problems

26.4

Permutation vs. Combination

26.5

26.6

26.7

Problems involving both permutations and combinations

26.8

Pascal's triangle

26.9

Binomial theorem

27. Introduction to Probability

27.1

Introduction to probability

27.2

Organizing outcomes

27.3

Probability of independent events

27.4

Comparing experimental and theoretical probability

28. Probability

28.1

Determining probabilities using tree diagrams and tables

28.2

Probability of independent events

28.3

Probability with Venn diagrams

29. Derivatives

29.1

Definition of derivative

29.2

29.3

Gradient and equation of tangent line

29.4

29.5

Derivative of trigonometric functions

29.6

Derivative of exponential functions

29.7

29.8

29.9

Implicit differentiation

29.10

Derivative of inverse trigonometric functions

29.11

Derivative of logarithmic functions

29.12

Higher order derivatives

30. Introduction to Matrices

30.1

Notation of matrices

30.2

Adding and subtracting matrices

30.3

Scalar multiplication

30.4

Matrix multiplication

30.5

The three types of matrix row operations

30.6

Representing a linear system as a matrix

30.7

Solving a linear system with matrices using Gaussian elimination

31. Properties of Matrices

31.1

31.2

Identity matrix

31.3

Properties of matrix addition

31.4

Properties of scalar multiplication

32. Determinants and Inverses of Matrices

32.1

The determinant of a 2 x 2 matrix

32.2

The determinant of a 3 x 3 matrix (General & Shortcut Method)

32.3

The inverse of a 2 x 2 matrix

32.4

The inverse of 3 x 3 matrices with matrix row operations

32.5

The inverse of 3 x 3 matrix with determinants and adjugate

32.6

2 x 2 invertible matrix

32.7

Solving linear systems using Cramer's Rule

32.8

Solving linear systems using 2 x 2 inverse matrices

33. Transformations with Matrices

33.1

Transforming vectors with matrices

33.2

Transforming shapes with matrices

33.3

Finding the transformation matrix

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