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Year 13 Maths Topics

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1. Functions

1.1

Domain and range of a function

1.2

Function notation

1.3

Operations with functions

1.4

Adding functions

1.5

Subtracting functions

1.6

Multiplying functions

1.7

Dividing functions

1.8

Composite functions

1.9

Inequalities of combined functions

1.10

Inverse functions

1.11

One to one functions

1.12

Difference quotient: applications of functions

2. Transformations of Functions

2.1

Transformations of functions: Horizontal translations

2.2

Transformations of functions: Vertical translations

2.3

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

2.4

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

2.5

Transformations of functions: Horizontal stretches

2.6

Transformations of functions: Vertical stretches

2.7

Combining transformations of functions

2.8

Even and odd functions

3. Quadratic Functions

3.1

Characteristics of quadratic functions

3.2

Transformations of quadratic functions

3.3

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

3.4

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

3.5

Completing the square

3.6

Converting from general to vertex form by completing the square

3.7

Shortcut: Vertex formula

3.8

Graphing parabolas for given quadratic functions

3.9

Finding the quadratic functions for given parabolas

3.10

Applications of quadratic functions

4. Quadratic Equations and Inequalities

4.1

Solving quadratic equations by factorising

4.2

Solving quadratic equations by completing the square

4.3

Using quadratic formula to solve quadratic equations

4.4

Nature of roots of quadratic equations: The discriminant

4.5

Applications of quadratic equations

4.6

Solving quadratic inequalities

5. Polynomial Functions

5.1

What is a polynomial function?

5.2

Polynomial long division

5.3

Polynomial synthetic division

5.4

Remainder theorem

5.5

5.6

Rational zero theorem

5.7

Characteristics of polynomial graphs

5.8

Multiplicities of polynomials

5.9

Imaginary zeros of polynomials

5.10

Determining the equation of a polynomial function

5.11

Applications of polynomial functions

5.12

Solving polynomial inequalities

5.13

Fundamental theorem of algebra

5.14

Descartes' rule of signs

6. Rational Functions

6.1

What is a rational function?

6.2

Point of discontinuity

6.3

Vertical asymptote

6.4

Horizontal asymptote

6.5

Slant asymptote

6.6

Graphs of rational functions

6.7

Solving equations with algebraic fractions

6.8

Solving rational inequalities

7. Exponential Functions

7.1

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

7.2

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

7.3

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

7.4

Exponents: Negative exponents

7.5

Exponents: Zero exponent: a^0 = 1

7.6

Exponents: Rational exponents

7.7

Solving exponential equations using exponent rules

7.8

Graphing exponential functions

7.9

Graphing transformations of exponential functions

7.10

Finding an exponential function given its graph

8. Logarithmic Functions

8.1

What is a logarithm?

8.2

Converting from logarithmic form to exponential form

8.3

Evaluating logarithms without a calculator

8.4

Common logarithms

8.5

Natural log: ln

8.6

Evaluating logarithms using change-of-base formula

8.7

Converting from exponential form to logarithmic form

8.8

Solving exponential equations with logarithms

8.9

Product rule of logarithms

8.10

Quotient rule of logarithms

8.11

Combining product rule and quotient rule in logarithms

8.12

Evaluating logarithms using logarithm rules

8.13

Solving logarithmic equations

8.14

Graphing logarithmic functions

8.15

Finding a logarithmic function given its graph

9. Radical Functions

9.1

Basic radical functions

9.2

Transformations of radical functions

9.3

Square root of a function

9.4

Solving radical equations

10. Solving Simultaneous Equations

10.1

Determining number of solutions to linear equations

10.2

Solving simultaneous equations by graphing

10.3

Solving simultaneous equations by elimination

10.4

Solving simultaneous equations by substitution

10.5

Money related questions in linear equations

10.6

Unknown number related questions in linear equations

10.7

Distance and time related questions in linear equations

10.8

Rectangular shape related questions in linear equations

11. Trigonometry

11.1

11.2

11.3

Applications of the sine rule and cosine rule

12. Bearings

12.1

Introduction to bearings

12.2

Bearings and direction word problems

12.3

Angle of elevation and depression

13. Trigonometric Ratios and Angle Measure

13.1

Angle in standard position

13.2

Coterminal angles

13.3

Reference angle

13.4

Find the exact value of trigonometric ratios

13.5

ASTC rule in trigonometry (All Students Take Calculus)

13.6

13.7

Converting between degrees and radians

13.8

Trigonometric ratios of angles in radians

13.9

Radian measure and arc length

14. Graphing Trigonometric Functions

14.1

Sine graph: y = sin x

14.2

Cosine graph: y = cos x

14.3

Tangent graph: y = tan x

14.4

Cotangent graph: y = cot x

14.5

Secant graph: y = sec x

14.6

Cosecant graph: y = csc x

14.7

Graphing transformations of trigonometric functions

14.8

Determining trigonometric functions given their graphs

15. Trigonometric Identities

15.1

Quotient identities and reciprocal identities

15.2

Pythagorean identities

15.3

Sum and difference identities

15.4

Cofunction identities

15.5

Double-angle identities

16. Parabolas and Circles

16.1

Conics - Parabola

16.2

Conics - Circle

17. Two-dimensional vectors

17.1

Introduction to vectors

17.2

Magnitude of a vector

17.3

Direction angle of a vector

17.4

Scalar multiplication

17.5

Equivalent vectors

17.6

Adding and subtracting vectors in component form

17.7

Operations on vectors in magnitude and direction form

17.8

17.9

Word problems on vectors

18. Sequences and Series

18.1

Arithmetic sequences

18.2

Arithmetic series

18.3

Geometric sequences

18.4

Geometric series

18.5

Infinite geometric series

18.6

18.7

Arithmetic mean vs. Geometric mean

19. Data Interpretation

19.1

Center of a data set: mean, median, mode

19.2

Spread of a data set - standard deviation & variance

19.3

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

19.4

Bivariate, scatter plots and correlation

19.5

Regression analysis

19.6

Equation of the best fit line

20. Discrete Probabilities

20.1

Probability distribution - histogram, mean, variance & standard deviation

20.2

Binomial distribution

20.3

Mean and standard deviation of binomial distribution

20.4

Poisson distribution

20.5

Geometric distribution

20.6

Negative binomial distribution

21. Probability

21.1

Addition rule for "OR"

21.2

Multiplication rule for "AND"

21.3

Conditional probability

21.4

Probability with permutations and combinations

21.5

Law of total probability

21.6

22. Permutations and Combinations

22.1

Fundamental counting principle

22.2

Factorial notation

22.3

Path counting problems

22.4

Permutation vs. Combination

22.5

22.6

22.7

Problems involving both permutations and combinations

22.8

Pascal's triangle

22.9

Binomial theorem

23. Integration Techniques

23.1

23.2

Integration by parts

23.3

Integration using trigonometric identities

23.4

Trigonometric substitution

23.5

Integration of rational functions by partial fractions

23.6

Improper integrals

24. Derivatives

24.1

Definition of derivative

24.2

24.3

Slope and equation of tangent line

24.4

24.5

Derivative of trigonometric functions

24.6

Derivative of exponential functions

24.7

24.8

24.9

Implicit differentiation

24.10

Derivative of inverse trigonometric functions

24.11

Derivative of logarithmic functions

24.12

Higher order derivatives

24.13

Tangent and concavity of parametric equations

25. Integration Applications

25.1

Areas between curves

25.2

Volumes of solids with known cross-sections

25.3

Volumes of solids of revolution - Disc method

25.4

Volumes of solids of revolution - Shell method

25.5

Average value of a function

25.6

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