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Year 12 Maths Topics

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video
Introduction to Relations and Functions
Relationship between two variables
Understand relations between x- and y-intercepts
Domain and range of a function
Identifying functions
Function notation
video
Functions
Function notation
Operations with functions
Adding functions
Subtracting functions
Multiplying functions
Dividing functions
Composite functions
Inequalities of combined functions
Inverse functions
One to one functions
Difference quotient: applications of functions
video
Transformations of Functions
Transformations of functions: Horizontal translations
Transformations of functions: Vertical translations
Reflection across the y-axis: y = f(-x)
Reflection across the x-axis: y = -f(x)
Transformations of functions: Horizontal stretches
Transformations of functions: Vertical stretches
Combining transformations of functions
Even and odd functions
video
Factorising Polynomial
Factorise by taking out the greatest common factor
Factorise by grouping
Factorising difference of squares: x^2 - y^2
Factorising trinomials
Factoring difference of cubes
Factoring sum of cubes
video
Quadratic Functions
Characteristics of quadratic functions
Transformations of quadratic functions
Quadratic function in general form: y = ax^2 + bx + c
Quadratic function in vertex form: y = a(x-p)^2 + q
Completing the square
Converting from general to vertex form by completing the square
Shortcut: Vertex formula
Graphing parabolas for given quadratic functions
Finding the quadratic functions for given parabolas
Applications of quadratic functions
video
Polynomial Functions
Solving quadratic equations by factorising
Solving quadratic equations by completing the square
Using quadratic formula to solve quadratic equations
Nature of roots of quadratic equations: The discriminant
Applications of quadratic equations
Solving quadratic inequalities
video
Radicals
Operations with radicals
Conversion between entire radicals and mixed radicals
Adding and subtracting radicals
Multiplying radicals
Solving radical equations
video
Radical Functions
Basic radical functions
Transformations of radical functions
Square root of a function
video
Exponents
Product rule of exponents
Quotient rule of exponents
Power of a product rule
Power of a quotient rule
Power of a power rule
Negative exponent rule
Combining the exponent rules
Scientific notation
Convert between radicals and rational exponents
Solving for exponents
video
Rational Functions
What is a rational function?
Point of discontinuity
Vertical asymptote
Horizontal asymptote
Slant asymptote
Graphs of rational functions
Solving equations with algebraic fractions
Solving rational inequalities
video
Exponential Functions
Exponents: Product rule (a^x)(a^y) = a^(x+y)
Exponents: Division rule (a^x / a^y) = a^(x-y)
Exponents: Power rule (a^x)^y = a^(x * y)
Exponents: Negative exponents
Exponents: Zero exponent: a^0 = 1
Exponents: Rational exponents
Graphing exponential functions
Graphing transformations of exponential functions
Finding an exponential function given its graph
video
Logarithmic Functions
What is a logarithm?
Converting from logarithmic form to exponential form
Evaluating logarithms without a calculator
Common logarithms
Natural log: ln
Evaluating logarithms using change-of-base formula
Converting from exponential form to logarithmic form
Solving exponential equations with logarithms
Product rule of logarithms
Quotient rule of logarithms
Combining product rule and quotient rule in logarithms
Evaluating logarithms using logarithm rules
Solving logarithmic equations
Graphing logarithmic functions
Finding a logarithmic function given its graph
video
Applications of Exponential and Logarithmic Functions
Exponential growth and decay by a factor
Exponential decay: Half-life
Exponential growth and decay by percentage
Finance: Compound interest
Continuous growth and decay
Logarithmic scale: Richter scale (earthquake)
Logarithmic scale: pH scale
Logarithmic scale: dB scale
Finance: Future value and present value
video
Solving Simultaneous Equations
Determining number of solutions to linear equations
Solving simultaneous equations by graphing
Solving simultaneous equations by elimination
Solving simultaneous equations by substitution
Money related questions in linear equations
Unknown number related questions in linear equations
Distance and time related questions in linear equations
Rectangular shape related questions in linear equations
video
Inequalities in Two Variables
Graphing linear inequalities in two variables
Graphing simultaneous linear inequalities
Graphing quadratic inequalities in two variables
Graphing simultaneous quadratic inequalities
Applications of inequalities
What is linear programming?
Linear programming word problems
video
Introduction to Trigonometry
Use sine ratio to calculate angles and sides (Sin = oh \frac{o}{h}ho​ )
Use cosine ratio to calculate angles and sides (Cos = ah \frac{a}{h}ha​ )
Use tangent ratio to calculate angles and sides (Tan = oa \frac{o}{a}ao​ )
Combination of SohCahToa questions
Solving expressions using 45-45-90 special right triangles
Solving expressions using 30-60-90 special right triangles
Word problems relating ladder in trigonometry
Word problems relating guy wire in trigonometry
Other word problems relating angles in trigonometry
video
Trigonometric Ratios and Angle Measure
Standard angle
Coterminal angles
Reference angles
Find the exact value of trigonometric ratios
ASTC rule in trigonometry (All Students Take Calculus)
Unit circle
Sine rule
Cosine rule
Applications of the sine rule and cosine rule
video
Bearings
Introduction to bearings
Bearings and direction word problems
Angle of elevation and depression
video
Graphing Trigonometric Functions
Sine graph: y = sin x
Cosine graph: y = cos x
Tangent graph: y = tan x
Cotangent graph: y = cot x
Secant graph: y = sec x
Cosecant graph: y = csc x
Graphing transformations of trigonometric functions
Determining trigonometric functions given their graphs
video
Applications of Trigonometric Functions
Ferris wheel trig problems
Tides and water depth trig problems
Spring (simple harmonic motion) trig problems
video
Trigonometric Identities
Quotient identities and reciprocal identities
Pythagorean identities
Sum and difference identities
Cofunction identities
Double-angle identities
video
Conics
Conics - Parabola
Conics - Ellipse
Conics - Circle
Conics - Hyperbola
video
Imaginary and Complex Numbers
Introduction to imaginary numbers
Complex numbers and complex planes
Adding and subtracting complex numbers
Complex conjugates
Multiplying and dividing complex numbers
Distance and midpoint of complex numbers
Angle and absolute value of complex numbers
Polar form of complex numbers
Operations on complex numbers in polar form
video
Vectors
Introduction to vectors
Magnitude of a vector
Direction angle of a vector
Scalar multiplication
Equivalent vectors
Adding and subtracting vectors in component form
Operations on vectors in magnitude and direction form
Unit Vector
Word problems on vectors
video
Sequences and Series
Arithmetic sequences
Arithmetic series
Geometric sequences
Geometric series
Infinite geometric series
Sigma notation
Arithmetic mean vs. Geometric mean
video
Permutations and Combinations
Fundamental counting principle
Factorial notation
Path counting problems
Permutation vs. Combination
Permutations
Combinations
Problems involving both permutations and combinations
Pascal's triangle
Binomial theorem
video
Introduction to Probability
Introduction to probability
Organizing outcomes
Probability of independent events
Comparing experimental and theoretical probability
video
Probability
Determining probabilities using tree diagrams and tables
Probability of independent events
Probability with Venn diagrams
video
Derivatives
Definition of derivative
Power rule
Gradient and equation of tangent line
Chain rule
Derivative of trigonometric functions
Derivative of exponential functions
Product rule
Quotient rule
Implicit differentiation
Derivative of inverse trigonometric functions
Derivative of logarithmic functions
Higher order derivatives
video
Introduction to Matrices
Notation of matrices
Adding and subtracting matrices
Scalar multiplication
Matrix multiplication
The three types of matrix row operations
Representing a linear system as a matrix
Solving a linear system with matrices using Gaussian elimination
video
Properties of Matrices
Zero matrix
Identity matrix
Properties of matrix addition
Properties of scalar multiplication
video
Determinants and Inverses of Matrices
The determinant of a 2 x 2 matrix
The determinant of a 3 x 3 matrix (General & Shortcut Method)
The inverse of a 2 x 2 matrix
The inverse of 3 x 3 matrices with matrix row operations
The inverse of 3 x 3 matrix with determinants and adjugate
2 x 2 invertible matrix
Solving linear systems using Cramer's Rule
Solving linear systems using 2 x 2 inverse matrices
video
Transformations with Matrices
Transforming vectors with matrices
Transforming shapes with matrices
Finding the transformation matrix

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