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Master Path Counting: Efficient Methods for Complex Problems

Year 12 Maths: Master Advanced Concepts & Skills

Introduction to Matrices

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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

Properties of Matrices

Zero matrix

Identity matrix

Properties of matrix addition

Properties of scalar multiplication

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 Cramer"s Rule

Cramer"s Rule with 2 x 2 matrices

Solving linear systems using 2 x 2 inverse matrices

Transformations with Matrices

Transforming vectors with matrices

Transforming shapes with matrices

Finding the transformation matrix

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

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

Functions

One to one functions

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Inverse functions

Operations with functions

Inequalities of combined functions

Inequalities of Combined Functions

Difference quotient: applications of functions

Transformations of Functions

Combining 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

Even and odd functions

Factorising Polynomial

Factoring difference of cubes

Factoring sum of cubes

Factorise by taking out the greatest common factor

Factor by taking out the greatest common factor

Factor by taking out GCF

Factorise by grouping

Factor by grouping

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

Factoring difference of squares: <katex>x^2 - y^2</katex>

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

Factorising trinomials

Factoring trinomials

Factoring Trinomials

Quadratic Functions

Characteristics of quadratic functions

Transformations of quadratic functions

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

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing parabolas for given quadratic functions

Finding the quadratic functions for given parabolas

Applications of quadratic functions

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

Completing the square

Completing the Square

Polynomial Functions

Solving quadratic equations by factorising

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Radicals

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Conversion between entire radicals and mixed radicals

Adding and subtracting radicals

Combining Like Radicals

Adding and Subtracting Radicals

Multiplying radicals

Multiplying Radicals

Operations with radicals

Operations with Radicals

Radical Functions

Transformations of radical functions

Square root of a function

Basic radical functions

Basic Radical Function

Basic Radical Functions

Exponents

Product rule of exponents

Quotient rule of exponents

Power of a product rule

Power of a quotient rule

Power of a power rule

Simplify the power

Negative exponent rule

Combining the exponent rules

Scientific notation

Convert between radicals and rational exponents

Solving for exponents

Rational Functions

What is a rational function?

Point of discontinuity

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Solving equations with algebraic fractions

Solving rational equations

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Exponential Functions

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

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

Exponents: Division 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: Power rule: (a^x)^y = a^(xy)

Exponents: Power rule

Exponents: Negative exponents

Exponents: Zero exponent: a^0 = 1

Exponents: Zero exponent: a^0 = 1

Exponents: Rational exponents

Graphing exponential functions

Graphing transformations of exponential functions

Finding an exponential function given its graph

Logarithmic Functions

Solving logarithmic equations

What is a logarithm?

Logarithms and Exponential Growth

Understanding Logarithms through Exponential Growth

Understanding Logarithms

Converting from logarithmic form to exponential form

Year 12 Maths

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