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Year 13 Maths Topics

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1. Functions

1.1

Function notation

1.2

Operations with functions

1.3

Adding functions

1.4

Subtracting functions

1.5

Multiplying functions

1.6

Dividing functions

1.7

Composite functions

1.8

Inequalities of combined functions

1.9

Inverse functions

1.10

One to one functions

1.11

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. Polynomial Functions

4.1

What is a polynomial function?

4.2

Polynomial long division

4.3

Polynomial synthetic division

4.4

Remainder theorem

4.5

4.6

Rational zero theorem

4.7

Characteristics of polynomial graphs

4.8

Multiplicities of polynomials

4.9

Imaginary zeros of polynomials

4.10

Determining the equation of a polynomial function

4.11

Applications of polynomial functions

4.12

Solving polynomial inequalities

4.13

Fundamental theorem of algebra

4.14

Descartes' rule of signs

5. Rational Functions

5.1

What is a rational function?

5.2

Point of discontinuity

5.3

Vertical asymptote

5.4

Horizontal asymptote

5.5

Slant asymptote

5.6

Graphs of rational functions

5.7

Solving equations with algebraic fractions

5.8

Solving rational inequalities

6. Exponential Functions

6.1

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

6.2

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

6.3

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

6.4

Exponents: Negative exponents

6.5

Exponents: Zero exponent: a^0 = 1

6.6

Exponents: Rational exponents

6.7

Graphing exponential functions

6.8

Graphing transformations of exponential functions

6.9

Finding an exponential function given its graph

7. Logarithmic Functions

7.1

What is a logarithm?

7.2

Converting from logarithmic form to exponential form

7.3

Evaluating logarithms without a calculator

7.4

Common logarithms

7.5

Natural log: ln

7.6

Evaluating logarithms using change-of-base formula

7.7

Converting from exponential form to logarithmic form

7.8

Solving exponential equations with logarithms

7.9

Product rule of logarithms

7.10

Quotient rule of logarithms

7.11

Combining product rule and quotient rule in logarithms

7.12

Evaluating logarithms using logarithm rules

7.13

Solving logarithmic equations

7.14

Graphing logarithmic functions

7.15

Finding a logarithmic function given its graph

8. Applications of Exponential and Logarithmic Functions

8.1

Exponential growth and decay by a factor

8.2

Exponential decay: Half-life

8.3

Exponential growth and decay by percentage

8.4

Finance: Compound interest

8.5

Continuous growth and decay

8.6

Logarithmic scale: Richter scale (earthquake)

8.7

Logarithmic scale: pH scale

8.8

Logarithmic scale: dB scale

8.9

Finance: Future value and present value

9. Quadratic Equations and Inequalities

9.1

Solving quadratic equations by factorising

9.2

Solving quadratic equations by completing the square

9.3

Using quadratic formula to solve quadratic equations

9.4

Nature of roots of quadratic equations: The discriminant

9.5

Applications of quadratic equations

9.6

Solving quadratic inequalities

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. Simultaneous Equations (Advanced)

11.1

Simultaneous linear equations

11.2

Simultaneous linear-quadratic equations

11.3

Simultaneous quadratic-quadratic equations

11.4

Solving 3 variable simultaneous equations by substitution

11.5

Solving 3 variable simultaneous equations by elimination

11.6

Solving 3 variable simultaneous equations with no or infinite solutions

11.7

Word problems relating 3 variable simultaneous equations

12. Introduction to Trigonometry

12.1

Use sine ratio to calculate angles and sides (Sin =

\frac{o}{h}

12.2

Use cosine ratio to calculate angles and sides (Cos =

\frac{a}{h}

12.3

Use tangent ratio to calculate angles and sides (Tan =

\frac{o}{a}

12.4

Combination of SohCahToa questions

12.5

Solving expressions using 45-45-90 special right triangles

12.6

Solving expressions using 30-60-90 special right triangles

12.7

Word problems relating ladder in trigonometry

12.8

Word problems relating guy wire in trigonometry

12.9

Other word problems relating angles in trigonometry

13. Trigonometric Ratios and Angle Measure

13.1

13.2

Coterminal angles

13.3

Reference angles

13.4

Find the exact value of trigonometric ratios

13.5

ASTC rule in trigonometry (All Students Take Calculus)

13.6

13.7

13.8

13.9

Applications of the sine law and cosine law

14. Bearings

14.1

Introduction to bearings

14.2

Bearings and direction word problems

14.3

Angle of elevation and depression

15. Graphing Trigonometric Functions

15.1

Sine graph: y = sin x

15.2

Cosine graph: y = cos x

15.3

Tangent graph: y = tan x

15.4

Cotangent graph: y = cot x

15.5

Secant graph: y = sec x

15.6

Cosecant graph: y = csc x

15.7

Graphing transformations of trigonometric functions

15.8

Determining trigonometric functions given their graphs

16. Trigonometric Identities

16.1

Quotient identities and reciprocal identities

16.2

Pythagorean identities

16.3

Sum and difference identities

16.4

Cofunction identities

16.5

Double-angle identities

17. Conics

17.1

Conics - Parabola

17.2

Conics - Ellipse

17.3

Conics - Circle

17.4

Conics - Hyperbola

18. Imaginary and Complex Numbers

18.1

Introduction to imaginary numbers

18.2

Complex numbers and complex planes

18.3

Adding and subtracting complex numbers

18.4

Complex conjugates

18.5

Multiplying and dividing complex numbers

18.6

Distance and midpoint of complex numbers

18.7

Angle and absolute value of complex numbers

18.8

Polar form of complex numbers

18.9

Operations on complex numbers in polar form

19. Sequences and Series

19.1

Arithmetic sequences

19.2

Arithmetic series

19.3

Geometric sequences

19.4

Geometric series

19.5

Infinite geometric series

19.6

19.7

Arithmetic mean vs. Geometric mean

20. Probability

20.1

Determining probabilities using tree diagrams and tables

20.2

Probability of independent events

20.3

Probability with Venn diagrams

21. Statistics

21.1

Median and mode

21.2

21.3

Range and outliers

21.4

Application of averages

22. Limits

22.1

Finding limits from graphs

22.2

22.3

Finding limits algebraically - direct substitution

22.4

Finding limits algebraically - when direct substitution is not possible

22.5

Infinite limits - vertical asymptotes

22.6

Limits at infinity - horizontal asymptotes

22.7

Intermediate value theorem

22.8

Squeeze theorem

23. Derivatives

23.1

Definition of derivative

23.2

23.3

Gradient and equation of tangent line

23.4

23.5

Derivative of trigonometric functions

23.6

Derivative of exponential functions

23.7

23.8

23.9

Implicit differentiation

23.10

Derivative of inverse trigonometric functions

23.11

Derivative of logarithmic functions

23.12

Higher order derivatives

23Chapters

186Topics

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