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AU Year 12 Maths topics

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1Introduction to Relations and Functions

1.1Relationship between two variables (free lessons)
1.2Understand relations between x- and y-intercepts
1.3Domain and range of a function (free lessons)
1.4Identifying functions
1.5Function notation (free lessons)

2Functions

2.1Function notation (free lessons)
2.2Operations with functions
2.3Adding functions
2.4Subtracting functions
2.5Multiplying functions
2.6Dividing functions
2.7Composite functions (free lessons)
2.8Inequalities of combined functions
2.9Inverse functions (free lessons)
2.10One to one functions
2.11Difference quotient: applications of functions

3Transformations of Functions

3.1Transformations of functions: Horizontal translations
3.2Transformations of functions: Vertical translations
3.3Reflection across the y-axis: y=f(−x)y = f(-x) (free lessons)
3.4Reflection across the x-axis: y=−f(x)y = -f(x)
3.5Transformations of functions: Horizontal stretches
3.6Transformations of functions: Vertical stretches
3.7Combining transformations of functions
3.8Even and odd functions

4Factorising Polynomial

4.1Factorise by taking out the greatest common factor (free lessons)
4.2Factorise by grouping
4.3Factorising difference of squares: x2−y2x^2 - y^2
4.4Factorising trinomials
4.5Factoring difference of cubes
4.6Factoring sum of cubes

5Quadratic Functions

5.1Characteristics of quadratic functions (free lessons)
5.2Transformations of quadratic functions
5.3Quadratic function in general form: y=ax2+bx+cy = ax^2 + bx+c
5.4Quadratic function in vertex form: y = a(x−p)2+qa(x-p)^2 + q
5.5Completing the square (free lessons)
5.6Converting from general to vertex form by completing the square
5.7Shortcut: Vertex formula (free lessons)
5.8Graphing parabolas for given quadratic functions
5.9Finding the quadratic functions for given parabolas (free lessons)
5.10Applications of quadratic functions

6Polynomial Functions

6.1Solving quadratic equations by factorising
6.2Solving quadratic equations by completing the square
6.3Using quadratic formula to solve quadratic equations
6.4Nature of roots of quadratic equations: The discriminant
6.5Applications of quadratic equations
6.6Solving quadratic inequalities

7Radicals

7.1Operations with radicals (free lessons)
7.2Conversion between entire radicals and mixed radicals
7.3Adding and subtracting radicals (free lessons)
7.4Multiplying radicals
7.5Solving radical equations

8Radical Functions

8.1Basic radical functions
8.2Transformations of radical functions
8.3Square root of a function

9Exponents

9.1Product rule of exponents (free lessons)
9.2Quotient rule of exponents
9.3Power of a product rule
9.4Power of a quotient rule
9.5Power of a power rule (free lessons)
9.6Negative exponent rule
9.7Combining the exponent rules
9.8Scientific notation
9.9Convert between radicals and rational exponents (free lessons)
9.10Solving for exponents

10Rational Functions

10.1What is a rational function? (free lessons)
10.2Point of discontinuity
10.3Vertical asymptote
10.4Horizontal asymptote
10.5Slant asymptote
10.6Graphs of rational functions
10.7Solving equations with algebraic fractions
10.8Solving rational inequalities

11Exponential Functions

11.1Exponents: Product rule (ax)(ay)=a(x+y) (a^x)(a^y)=a^{(x+y)} (free lessons)
11.2Exponents: Division rule axay=a(x−y){a^x \over a^y}=a^{(x-y)}
11.3Exponents: Power rule (ax)y=a(x⋅y) (a^x)^y = a^{(x\cdot y)} (free lessons)
11.4Exponents: Negative exponents
11.5Exponents: Zero exponent: a0=1 a^0 = 1
11.6Exponents: Rational exponents
11.7Graphing exponential functions
11.8Graphing transformations of exponential functions
11.9Finding an exponential function given its graph

12Logarithmic Functions

12.1What is a logarithm?
12.2Converting from logarithmic form to exponential form (free lessons)
12.3Evaluating logarithms without a calculator
12.4Common logarithms
12.5Natural log: ln
12.6Evaluating logarithms using change-of-base formula
12.7Converting from exponential form to logarithmic form (free lessons)
12.8Solving exponential equations with logarithms
12.9Product rule of logarithms (free lessons)
12.10Quotient rule of logarithms
12.11Combining product rule and quotient rule in logarithms
12.12Evaluating logarithms using logarithm rules
12.13Solving logarithmic equations (free lessons)
12.14Graphing logarithmic functions
12.15Finding a logarithmic function given its graph

13Applications of Exponential and Logarithmic Functions

13.1Exponential growth and decay by a factor
13.2Exponential decay: Half-life (free lessons)
13.3Exponential growth and decay by percentage
13.4Finance: Compound interest
13.5Continuous growth and decay
13.6Logarithmic scale: Richter scale (earthquake)
13.7Logarithmic scale: pH scale
13.8Logarithmic scale: dB scale
13.9Finance: Future value and present value

14Solving Simultaneous Equations

14.1Determining number of solutions to linear equations (free lessons)
14.2Solving simultaneous equations by graphing
14.3Solving simultaneous equations by elimination
14.4Solving simultaneous equations by substitution
14.5Money related questions in linear equations (free lessons)
14.6Unknown number related questions in linear equations
14.7Distance and time related questions in linear equations
14.8Rectangular shape related questions in linear equations

15Inequalities in Two Variables

15.1Graphing linear inequalities in two variables (free lessons)
15.2Graphing simultaneous linear inequalities (free lessons)
15.3Graphing quadratic inequalities in two variables
15.4Graphing simultaneous quadratic inequalities
15.5Applications of inequalities
15.6What is linear programming?
15.7Linear programming word problems

16Introduction to Trigonometry

16.1Use sine ratio to calculate angles and sides (Sin = oh \frac{o}{h} ) (free lessons)
16.2Use cosine ratio to calculate angles and sides (Cos = ah \frac{a}{h} )
16.3Use tangent ratio to calculate angles and sides (Tan = oa \frac{o}{a} )
16.4Combination of SohCahToa questions (free lessons)
16.5Solving expressions using 45-45-90 special right triangles (free lessons)
16.6Solving expressions using 30-60-90 special right triangles
16.7Word problems relating ladder in trigonometry
16.8Word problems relating guy wire in trigonometry
16.9Other word problems relating angles in trigonometry

17Trigonometric Ratios and Angle Measure

17.1Standard angle
17.2Coterminal angles (free lessons)
17.3Reference angles
17.4Find the exact value of trigonometric ratios
17.5ASTC rule in trigonometry (All Students Take Calculus) (free lessons)
17.6Unit circle (free lessons)
17.7Sine rule
17.8Cosine rule
17.9Applications of the sine rule and cosine rule

18Bearings

18.1Introduction to bearings
18.2Bearings and direction word problems
18.3Angle of elevation and depression

19Graphing Trigonometric Functions

19.1Sine graph: y = sin x
19.2Cosine graph: y = cos x
19.3Tangent graph: y = tan x
19.4Cotangent graph: y = cot x
19.5Secant graph: y = sec x
19.6Cosecant graph: y = csc x
19.7Graphing transformations of trigonometric functions
19.8Determining trigonometric functions given their graphs

20Applications of Trigonometric Functions

20.1Ferris wheel trig problems
20.2Tides and water depth trig problems
20.3Spring (simple harmonic motion) trig problems

21Trigonometric Identities

21.1Quotient identities and reciprocal identities (free lessons)
21.2Pythagorean identities
21.3Sum and difference identities
21.4Cofunction identities
21.5Double-angle identities (free lessons)

22Conics

22.1Conics - Parabola
22.2Conics - Ellipse
22.3Conics - Circle
22.4Conics - Hyperbola

23Imaginary and Complex Numbers

23.1Introduction to imaginary numbers (free lessons)
23.2Complex numbers and complex planes (free lessons)
23.3Adding and subtracting complex numbers
23.4Complex conjugates
23.5Multiplying and dividing complex numbers
23.6Distance and midpoint of complex numbers
23.7Angle and absolute value of complex numbers
23.8Polar form of complex numbers
23.9Operations on complex numbers in polar form

24Vectors

24.1Introduction to vectors
24.2Magnitude of a vector
24.3Direction angle of a vector
24.4Scalar multiplication
24.5Equivalent vectors
24.6Adding and subtracting vectors in component form
24.7Operations on vectors in magnitude and direction form
24.8Unit Vector
24.9Word problems on vectors

25Sequences and Series

25.1Arithmetic sequences (free lessons)
25.2Arithmetic series (free lessons)
25.3Geometric sequences
25.4Geometric series
25.5Infinite geometric series
25.6Sigma notation
25.7Arithmetic mean vs. Geometric mean

26Permutations and Combinations

26.1Fundamental counting principle
26.2Factorial notation
26.3Path counting problems
26.4Permutation vs. Combination
26.5Permutations
26.6Combinations
26.7Problems involving both permutations and combinations
26.8Pascal's triangle
26.9Binomial theorem

27Introduction to Probability

27.1Introduction to probability
27.2Organizing outcomes
27.3Probability of independent events
27.4Comparing experimental and theoretical probability

28Probability

28.1Determining probabilities using tree diagrams and tables (free lessons)
28.2Probability of independent events
28.3Probability with Venn diagrams

29Derivatives

29.1Definition of derivative
29.2Power rule
29.3Gradient and equation of tangent line
29.4Chain rule
29.5Derivative of trigonometric functions
29.6Derivative of exponential functions
29.7Product rule
29.8Quotient rule
29.9Implicit differentiation
29.10Derivative of inverse trigonometric functions
29.11Derivative of logarithmic functions
29.12Higher order derivatives

30Introduction to Matrices

30.1Notation of matrices
30.2Adding and subtracting matrices
30.3Scalar multiplication
30.4Matrix multiplication
30.5The three types of matrix row operations
30.6Representing a linear system as a matrix
30.7Solving a linear system with matrices using Gaussian elimination

31Properties of Matrices

31.1Zero matrix
31.2Identity matrix
31.3Properties of matrix addition
31.4Properties of scalar multiplication

32Determinants and Inverses of Matrices

32.1The determinant of a 2 x 2 matrix
32.2The determinant of a 3 x 3 matrix (General & Shortcut Method)
32.3The inverse of a 2 x 2 matrix
32.4The inverse of 3 x 3 matrices with matrix row operations
32.5The inverse of 3 x 3 matrix with determinants and adjugate
32.62 x 2 invertible matrix
32.7Solving linear systems using Cramer's Rule
32.8Solving linear systems using 2 x 2 inverse matrices

33Transformations with Matrices

33.1Transforming vectors with matrices
33.2Transforming shapes with matrices
33.3Finding the transformation matrix
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