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NZ Year 13 Maths Topics

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1.Functions

1.1Function notation (free lessons)

1.2Operations with functions

1.3Adding functions

1.4Subtracting functions

1.5Multiplying functions

1.6Dividing functions

1.7Composite functions (free lessons)

1.8Inverse functions

1.9One to one functions

1.10Difference quotient: applications of functions

2.Transformations of Functions

2.1Transformations of functions: Horizontal translations

2.2Transformations of functions: Vertical translations

2.3Reflection across the y-axis:

y = f(-x)

(free lessons)

2.4Reflection across the x-axis:

y = -f(x)

2.5Transformations of functions: Horizontal stretches

2.6Transformations of functions: Vertical stretches

2.7Combining transformations of functions

2.8Even and odd functions

3.Quadratic Functions

3.1Characteristics of quadratic functions (free lessons)

3.2Transformations of quadratic functions

3.3Quadratic function in general form:

y = ax^2 + bx+c

3.4Quadratic function in vertex form:

y = a(x-p)^2 + q

3.5Completing the square

3.6Converting from general to vertex form by completing the square

3.7Shortcut: Vertex formula (free lessons)

3.8Finding the quadratic functions for given parabolas (free lessons)

3.9Applications of quadratic functions

4.Polynomial Functions

4.1What is a polynomial function? (free lessons)

4.2Polynomial long division (free lessons)

4.3Polynomial synthetic division

4.4Remainder theorem

4.5Factor theorem

4.6Rational zero theorem

4.7Characteristics of polynomial graphs

4.8Multiplicities of polynomials

4.9Imaginary zeros of polynomials

4.10Determining the equation of a polynomial function

4.11Applications of polynomial functions

4.12Solving polynomial inequalities

4.13Fundamental theorem of algebra

5.Rational Functions

5.1What is a rational function? (free lessons)

5.2Point of discontinuity

5.3Vertical asymptote

5.4Horizontal asymptote

5.5Slant asymptote

5.6Graphs of rational functions

5.7Solving rational equations

5.8Solving rational inequalities

6.Exponential Functions

6.1Exponents: Product rule

(a^x)(a^y)=a^{(x+y)}

(free lessons)

6.2Exponents: Division rule

{a^x \over a^y}=a^{(x-y)}

6.3Exponents: Power rule

(a^x)^y = a^{(x\cdot y)}

(free lessons)

6.4Exponents: Negative exponents

6.5Exponents: Zero exponent:

a^0 = 1

6.6Exponents: Rational exponents

6.7Graphing exponential functions

6.8Graphing transformations of exponential functions

6.9Finding an exponential function given its graph

7.Logarithmic Functions

7.1What is a logarithm?

7.2Converting from logarithmic form to exponential form (free lessons)

7.3Evaluating logarithms without a calculator

7.4Common logarithms

7.5Natural log: ln

7.6Evaluating logarithms using change-of-base formula

7.7Converting from exponential form to logarithmic form (free lessons)

7.8Solving exponential equations with logarithms

7.9Product rule of logarithms (free lessons)

7.10Quotient rule of logarithms

7.11Combining product rule and quotient rule in logarithms

7.12Evaluating logarithms using logarithm rules

7.13Solving logarithmic equations (free lessons)

7.14Graphing logarithmic functions

7.15Finding a logarithmic function given its graph

8.Applications of Exponential and Logarithmic Functions

8.1Exponential growth and decay by a factor

8.2Exponential decay: Half-life (free lessons)

8.3Exponential growth and decay by percentage

8.4Finance: Compound interest

8.5Continuous growth and decay

8.6Logarithmic scale: Richter scale (earthquake)

8.7Logarithmic scale: pH scale

8.8Logarithmic scale: dB scale

8.9Finance: Future value and present value

9.Quadratic Equations and Inequalities

9.1Solving quadratic equations by factorising

9.2Solving quadratic equations by completing the square

9.3Using quadratic formula to solve quadratic equations

9.4Nature of roots of quadratic equations: The discriminant

9.5Applications of quadratic equations

9.6Solving quadratic inequalities

10.Solving Simultaneous Equations

10.1Determining number of solutions to linear equations (free lessons)

10.2Solving simultaneous equations by graphing

10.3Solving simultaneous equations by elimination

10.4Solving simultaneous equations by substitution

10.5Money related questions in linear equations (free lessons)

10.6Unknown number related questions in linear equations

10.7Distance and time related questions in linear equations

10.8Rectangular shape related questions in linear equations

11.Simultaneous Equations (Advanced)

11.1Simultaneous linear equations

11.2Simultaneous linear-quadratic equations (free lessons)

11.3Simultaneous quadratic-quadratic equations

11.4Solving 3 variable simultaneous equations by substitution

11.5Solving 3 variable simultaneous equations by elimination

11.6Solving 3 variable simultaneous equations (no solution, infinite solutions)

11.7Word problems relating 3 variable simultaneous equations

12.Introduction to Trigonometry

12.1Use sine ratio to calculate angles and side (Sin =

\frac{o}{h}

) (free lessons)

12.2Use cosine ratio to calculate angles and side (Cos =

\frac{a}{h}

12.3Use tangent ratio to calculate angles and side (Tan =

\frac{o}{a}

12.4Combination of SohCahToa questions (free lessons)

12.5Solving expressions using 45-45-90 special right triangles (free lessons)

12.6Solving expressions using 30-60-90 special right triangles

12.7Word problems relating ladder in trigonometry

12.8Word problems relating guy wire in trigonometry

12.9Other word problems relating angles in trigonometry

13.Trigonometric Ratios and Angle Measure

13.1Standard angle

13.2Coterminal angles (free lessons)

13.3Reference angles

13.4Find the exact value of trigonometric ratios

13.5ASTC rule in trigonometry (All Students Take Calculus) (free lessons)

13.6Unit circle (free lessons)

13.7Law of sines

13.8Law of cosines

13.9Applications of the sine law and cosine law

14.Graphing Trigonometric Functions

14.1Sine graph: y = sin x

14.2Cosine graph: y = cos x

14.3Tangent graph: y = tan x

14.4Cotangent graph: y = cot x

14.5Secant graph: y = sec x

14.6Cosecant graph: y = csc x

14.7Graphing transformations of trigonometric functions

14.8Determining trigonometric functions given their graphs

15.Trigonometric Identities

15.1Quotient identities and reciprocal identities (free lessons)

15.2Pythagorean identities

15.3Sum and difference identities

15.4Cofunction identities

15.5Double-angle identities (free lessons)

16.Conics

16.1Conics - Parabola

16.2Conics - Ellipse

16.3Conics - Circle

16.4Conics - Hyperbola

17.Imaginary and Complex Numbers

17.1Introduction to imaginary numbers

17.2Complex numbers and complex planes

17.3Adding and subtracting complex numbers

17.4Complex conjugates

17.5Multiplying and dividing complex numbers

17.6Distance and midpoint of complex numbers

17.7Angle and absolute value of complex numbers

17.8Polar form of complex numbers

17.9Operations on complex numbers in polar form

18.Sequences and Series

18.1Arithmetic sequences (free lessons)

18.2Arithmetic series (free lessons)

18.3Geometric sequences

18.4Geometric series

18.5Infinite geometric series

18.6Sigma notation

18.7Arithmetic mean vs. Geometric mean

19.Probability

19.1Determining probabilities using tree diagrams and tables (free lessons)

19.2Probability of independent events

19.3Probability with Venn diagrams

20.Statistics

20.1Median and mode

20.3Range and outliers

20.4Application of averages

21.Limits

21.1Finding limits from graphs

21.3Finding limits algebraically - direct substitution

21.4Finding limits algebraically - when direct substitution is not possible

21.5Infinite limits - vertical asymptotes

21.6Limits at infinity - horizontal asymptotes

21.7Intermediate value theorem

21.8Squeeze theorem

22.Derivatives

22.1Definition of derivative

22.3Gradient and equation of tangent line

22.5Derivative of trigonometric functions

22.6Derivative of exponential functions

22.7Product rule

22.8Quotient rule

22.9Implicit differentiation

22.10Derivative of inverse trigonometric functions

22.11Derivative of logarithmic functions

22.12Higher order derivatives

1.Functions

1.1Function notation (free lessons)

1.2Operations with functions

1.3Adding functions

1.4Subtracting functions

1.5Multiplying functions

1.6Dividing functions

1.7Composite functions (free lessons)

1.8Inverse functions

1.9One to one functions

1.10Difference quotient: applications of functions

2.Transformations of Functions

2.1Transformations of functions: Horizontal translations

2.2Transformations of functions: Vertical translations

2.3Reflection across the y-axis:

y = f(-x)

(free lessons)

2.4Reflection across the x-axis:

y = -f(x)

2.5Transformations of functions: Horizontal stretches

2.6Transformations of functions: Vertical stretches

2.7Combining transformations of functions

2.8Even and odd functions

3.Quadratic Functions

3.1Characteristics of quadratic functions (free lessons)

3.2Transformations of quadratic functions

3.3Quadratic function in general form:

y = ax^2 + bx+c

3.4Quadratic function in vertex form:

y = a(x-p)^2 + q

3.5Completing the square

3.6Converting from general to vertex form by completing the square

3.7Shortcut: Vertex formula (free lessons)

3.8Finding the quadratic functions for given parabolas (free lessons)

3.9Applications of quadratic functions

4.Polynomial Functions

4.1What is a polynomial function? (free lessons)

4.2Polynomial long division (free lessons)

4.3Polynomial synthetic division

4.4Remainder theorem

4.5Factor theorem

4.6Rational zero theorem

4.7Characteristics of polynomial graphs

4.8Multiplicities of polynomials

4.9Imaginary zeros of polynomials

4.10Determining the equation of a polynomial function

4.11Applications of polynomial functions

4.12Solving polynomial inequalities

4.13Fundamental theorem of algebra

5.Rational Functions

5.1What is a rational function? (free lessons)

5.2Point of discontinuity

5.3Vertical asymptote

5.4Horizontal asymptote

5.5Slant asymptote

5.6Graphs of rational functions

5.7Solving rational equations

5.8Solving rational inequalities

6.Exponential Functions

6.1Exponents: Product rule

(a^x)(a^y)=a^{(x+y)}

(free lessons)

6.2Exponents: Division rule

{a^x \over a^y}=a^{(x-y)}

6.3Exponents: Power rule

(a^x)^y = a^{(x\cdot y)}

(free lessons)

6.4Exponents: Negative exponents

6.5Exponents: Zero exponent:

a^0 = 1

6.6Exponents: Rational exponents

6.7Graphing exponential functions

6.8Graphing transformations of exponential functions

6.9Finding an exponential function given its graph

7.Logarithmic Functions

7.1What is a logarithm?

7.2Converting from logarithmic form to exponential form (free lessons)

7.3Evaluating logarithms without a calculator

7.4Common logarithms

7.5Natural log: ln

7.6Evaluating logarithms using change-of-base formula

7.7Converting from exponential form to logarithmic form (free lessons)

7.8Solving exponential equations with logarithms

7.9Product rule of logarithms (free lessons)

7.10Quotient rule of logarithms

7.11Combining product rule and quotient rule in logarithms

7.12Evaluating logarithms using logarithm rules

7.13Solving logarithmic equations (free lessons)

7.14Graphing logarithmic functions

7.15Finding a logarithmic function given its graph

8.Applications of Exponential and Logarithmic Functions

8.1Exponential growth and decay by a factor

8.2Exponential decay: Half-life (free lessons)

8.3Exponential growth and decay by percentage

8.4Finance: Compound interest

8.5Continuous growth and decay

8.6Logarithmic scale: Richter scale (earthquake)

8.7Logarithmic scale: pH scale

8.8Logarithmic scale: dB scale

8.9Finance: Future value and present value

9.Quadratic Equations and Inequalities

9.1Solving quadratic equations by factorising

9.2Solving quadratic equations by completing the square

9.3Using quadratic formula to solve quadratic equations

9.4Nature of roots of quadratic equations: The discriminant

9.5Applications of quadratic equations

9.6Solving quadratic inequalities

10.Solving Simultaneous Equations

10.1Determining number of solutions to linear equations (free lessons)

10.2Solving simultaneous equations by graphing

10.3Solving simultaneous equations by elimination

10.4Solving simultaneous equations by substitution

10.5Money related questions in linear equations (free lessons)

10.6Unknown number related questions in linear equations

10.7Distance and time related questions in linear equations

10.8Rectangular shape related questions in linear equations

11.Simultaneous Equations (Advanced)

11.1Simultaneous linear equations

11.2Simultaneous linear-quadratic equations (free lessons)

11.3Simultaneous quadratic-quadratic equations

11.4Solving 3 variable simultaneous equations by substitution

11.5Solving 3 variable simultaneous equations by elimination

11.6Solving 3 variable simultaneous equations (no solution, infinite solutions)

11.7Word problems relating 3 variable simultaneous equations

12.Introduction to Trigonometry

12.1Use sine ratio to calculate angles and side (Sin =

\frac{o}{h}

) (free lessons)

12.2Use cosine ratio to calculate angles and side (Cos =

\frac{a}{h}

12.3Use tangent ratio to calculate angles and side (Tan =

\frac{o}{a}

12.4Combination of SohCahToa questions (free lessons)

12.5Solving expressions using 45-45-90 special right triangles (free lessons)

12.6Solving expressions using 30-60-90 special right triangles

12.7Word problems relating ladder in trigonometry

12.8Word problems relating guy wire in trigonometry

12.9Other word problems relating angles in trigonometry

13.Trigonometric Ratios and Angle Measure

13.1Standard angle

13.2Coterminal angles (free lessons)

13.3Reference angles

13.4Find the exact value of trigonometric ratios

13.5ASTC rule in trigonometry (All Students Take Calculus) (free lessons)

13.6Unit circle (free lessons)

13.7Law of sines

13.8Law of cosines

13.9Applications of the sine law and cosine law

14.Graphing Trigonometric Functions

14.1Sine graph: y = sin x

14.2Cosine graph: y = cos x

14.3Tangent graph: y = tan x

14.4Cotangent graph: y = cot x

14.5Secant graph: y = sec x

14.6Cosecant graph: y = csc x

14.7Graphing transformations of trigonometric functions

14.8Determining trigonometric functions given their graphs

15.Trigonometric Identities

15.1Quotient identities and reciprocal identities (free lessons)

15.2Pythagorean identities

15.3Sum and difference identities

15.4Cofunction identities

15.5Double-angle identities (free lessons)

16.Conics

16.1Conics - Parabola

16.2Conics - Ellipse

16.3Conics - Circle

16.4Conics - Hyperbola

17.Imaginary and Complex Numbers

17.1Introduction to imaginary numbers

17.2Complex numbers and complex planes

17.3Adding and subtracting complex numbers

17.4Complex conjugates

17.5Multiplying and dividing complex numbers

17.6Distance and midpoint of complex numbers

17.7Angle and absolute value of complex numbers

17.8Polar form of complex numbers

17.9Operations on complex numbers in polar form

18.Sequences and Series

18.1Arithmetic sequences (free lessons)

18.2Arithmetic series (free lessons)

18.3Geometric sequences

18.4Geometric series

18.5Infinite geometric series

18.6Sigma notation

18.7Arithmetic mean vs. Geometric mean

19.Probability

19.1Determining probabilities using tree diagrams and tables (free lessons)

19.2Probability of independent events

19.3Probability with Venn diagrams

20.Statistics

20.1Median and mode

20.3Range and outliers

20.4Application of averages

21.Limits

21.1Finding limits from graphs

21.3Finding limits algebraically - direct substitution

21.4Finding limits algebraically - when direct substitution is not possible

21.5Infinite limits - vertical asymptotes

21.6Limits at infinity - horizontal asymptotes

21.7Intermediate value theorem

21.8Squeeze theorem

22.Derivatives

22.1Definition of derivative

22.3Gradient and equation of tangent line

22.5Derivative of trigonometric functions

22.6Derivative of exponential functions

22.7Product rule

22.8Quotient rule

22.9Implicit differentiation

22.10Derivative of inverse trigonometric functions

22.11Derivative of logarithmic functions

22.12Higher order derivatives

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