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Sixth Year 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. Radical Functions

5.1

Basic radical functions

5.2

Transformations of radical functions

5.3

Square root of a function

5.4

Solving radical equations

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 rational equations

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

Graphing exponential functions

7.8

Graphing transformations of exponential functions

7.9

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. Applications of Exponential and Logarithmic Functions

9.1

Exponential growth and decay by a factor

9.2

Exponential decay: Half-life

9.3

Exponential growth and decay by percentage

9.4

Finance: Compound interest

9.5

Continuous growth and decay

9.6

Logarithmic scale: Richter scale (earthquake)

9.7

Logarithmic scale: pH scale

9.8

Logarithmic scale: dB scale

9.9

Finance: Future value and present value

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 (Advance)

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. Inequalities in two variables

12.1

Graphing linear inequalities in two variables

12.2

Graphing simultaneous linear inequalities

12.3

Graphing quadratic inequalities in two variables

12.4

Graphing simultaneous quadratic inequalities

12.5

Applications of inequalities

12.6

What is linear programming?

12.7

Linear programming word problems

13. Trigonometry

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

13.10

13.11

13.12

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. Applications of Trigonometric Functions

16.1

Ferris wheel trig problems

16.2

Tides and water depth trig problems

16.3

Spring (simple harmonic motion) trig problems

17. Trigonometric Identities

17.1

Quotient identities and reciprocal identities

17.2

Pythagorean identities

17.3

Sum and difference identities

17.4

Cofunction identities

17.5

Double-angle identities

18. Conics

18.1

Conics - Parabola

18.2

Conics - Ellipse

18.3

Conics - Circle

18.4

Conics - Hyperbola

19. Imaginary and Complex Numbers

19.1

Introduction to imaginary numbers

19.2

Complex numbers and complex planes

19.3

Adding and subtracting complex numbers

19.4

Complex conjugates

19.5

Multiplying and dividing complex numbers

19.6

Distance and midpoint of complex numbers

19.7

Angle and absolute value of complex numbers

19.8

Polar form of complex numbers

19.9

Operations on complex numbers in polar form

20. Sequences and Series

20.1

Arithmetic sequences

20.2

Arithmetic series

20.3

Geometric sequences

20.4

Geometric series

20.5

Infinite geometric series

20.6

20.7

Arithmetic mean vs. Geometric mean

21. Probability

21.1

Determining probabilities using tree diagrams and tables

21.2

Probability of independent events

21.3

Probability with Venn diagrams

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. Statistics

23.1

Reading and drawing bar graphs

23.2

Reading and drawing histograms

23.3

Reading and drawing line graphs

23.4

Box-and-whisker plots and scatter plots

23.5

Stem-and-leaf plots

23.6

Reading and drawing Venn diagrams

23.7

Frequency tables and dot plots

24. Limits

24.1

Finding limits from graphs

24.2

24.3

24.4

Finding limits algebraically - direct substitution

24.5

Finding limits algebraically - when direct substitution is not possible

24.6

Infinite limits - vertical asymptotes

24.7

Limits at infinity - horizontal asymptotes

24.8

Intermediate value theorem

24.9

Squeeze theorem

25. Derivatives

25.1

Definition of derivative

25.2

25.3

Gradient and equation of tangent line

25.4

25.5

Derivative of trigonometric functions

25.6

Derivative of exponential functions

25.7

25.8

25.9

Implicit differentiation

25.10

Derivative of inverse trigonometric functions

25.11

Derivative of logarithmic functions

25.12

Higher order derivatives

26. Derivative Applications

26.1

Position velocity acceleration

26.2

Critical number & maximum and minimum values

26.3

l'Hospital's rule

26.4

Curve sketching

26.5

26.6

26.7

Mean value theorem

26.8

Linear approximation

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