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Year 13 Maths Topics

Functions

Domain and range of a function

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

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

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

Quadratic Equations and Inequalities

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

Polynomial Functions

What is a polynomial function?

Polynomial long division

Polynomial synthetic division

Remainder theorem

Rational zero theorem

Characteristics of polynomial graphs

Multiplicities of polynomials

Imaginary zeros of polynomials

Determining the equation of a polynomial function

Applications of polynomial functions

Solving polynomial inequalities

Fundamental theorem of algebra

Descartes' rule of signs

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

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

Solving exponential equations using exponent rules

Graphing exponential functions

Graphing transformations of exponential functions

Finding an exponential function given its graph

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

Radical Functions

Basic radical functions

Transformations of radical functions

Square root of a function

Solving radical equations

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

Trigonometry

Applications of the sine rule and cosine rule

Bearings

Introduction to bearings

Bearings and direction word problems

Angle of elevation and depression

Trigonometric Ratios and Angle Measure

Angle in standard position

Coterminal angles

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

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

Trigonometric Identities

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Cofunction identities

Double-angle identities

Parabolas and Circles

Conics - Parabola

Conics - Circle

Two-dimensional 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

Word problems on vectors

Sequences and Series

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

Infinite geometric series

Arithmetic mean vs. Geometric mean

Data Interpretation

Center of a data set: mean, median, mode

Spread of a data set - standard deviation & variance

Measures of relative standing - z-score, quartiles, percentiles

Bivariate, scatter plots and correlation

Regression analysis

Equation of the best fit line

Discrete Probabilities

Probability distribution - histogram, mean, variance & standard deviation

Binomial distribution

Mean and standard deviation of binomial distribution

Poisson distribution

Geometric distribution

Negative binomial distribution

Probability

Addition rule for "OR"

Multiplication rule for "AND"

Conditional probability

Probability with permutations and combinations

Law of total probability

Permutations and Combinations

Fundamental counting principle

Factorial notation

Path counting problems

Permutation vs. Combination

Problems involving both permutations and combinations

Pascal's triangle

Binomial theorem

Integration Techniques

Integration by parts

Integration using trigonometric identities

Trigonometric substitution

Integration of rational functions by partial fractions

Improper integrals

Derivatives

Definition of derivative

Slope and equation of tangent line

Derivative of trigonometric functions

Derivative of exponential functions

Implicit differentiation

Derivative of inverse trigonometric functions

Derivative of logarithmic functions

Higher order derivatives

Tangent and concavity of parametric equations

Integration Applications

Areas between curves

Volumes of solids with known cross-sections

Volumes of solids of revolution - Disc method

Volumes of solids of revolution - Shell method

Average value of a function

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