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NZ Year 13 Maths Topics

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1Functions

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)
1.8Inequalities of combined functions
1.9Inverse functions (free lessons)
1.10One to one functions
1.11Difference quotient: applications of functions

2Transformations of Functions

2.1Transformations of functions: Horizontal translations
2.2Transformations of functions: Vertical translations
2.3Reflection across the y-axis: y=f(−x)y = f(-x) (free lessons)
2.4Reflection across the x-axis: y=−f(x)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

3Quadratic Functions

3.1Characteristics of quadratic functions (free lessons)
3.2Transformations of quadratic functions
3.3Quadratic function in general form: y=ax2+bx+cy = ax^2 + bx+c
3.4Quadratic function in vertex form: y=a(x−p)2+qy = a(x-p)^2 + q
3.5Completing the square (free lessons)
3.6Converting from general to vertex form by completing the square
3.7Shortcut: Vertex formula (free lessons)
3.8Graphing parabolas for given quadratic functions
3.9Finding the quadratic functions for given parabolas (free lessons)
3.10Applications of quadratic functions

4Polynomial 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
4.14Descartes’ rule of signs

5Rational 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 equations with algebraic fractions
5.8Solving rational inequalities

6Exponential Functions

6.1Exponents: Product rule (ax)(ay)=a(x+y) (a^x)(a^y)=a^{(x+y)} (free lessons)
6.2Exponents: Division rule axay=a(x−y){a^x \over a^y}=a^{(x-y)}
6.3Exponents: Power rule (ax)y=a(x⋅y) (a^x)^y = a^{(x\cdot y)} (free lessons)
6.4Exponents: Negative exponents
6.5Exponents: Zero exponent: a0=1 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

7Logarithmic 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

8Applications 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

9Quadratic 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

10Solving 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

11Simultaneous 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 with no or infinite solutions
11.7Word problems relating 3 variable simultaneous equations

12Introduction to Trigonometry

12.1Use sine ratio to calculate angles and side (Sin = oh \frac{o}{h} ) (free lessons)
12.2Use cosine ratio to calculate angles and side (Cos = ah \frac{a}{h} )
12.3Use tangent ratio to calculate angles and side (Tan = oa \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

13Trigonometric 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

14Graphing 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

15Trigonometric 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)

16Conics

16.1Conics - Parabola
16.2Conics - Ellipse
16.3Conics - Circle
16.4Conics - Hyperbola

17Imaginary and Complex Numbers

17.1Introduction to imaginary numbers (free lessons)
17.2Complex numbers and complex planes (free lessons)
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

18Sequences 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

19Probability

19.1Determining probabilities using tree diagrams and tables (free lessons)
19.2Probability of independent events
19.3Probability with Venn diagrams

20Statistics

20.1Median and mode
20.2Mean
20.3Range and outliers
20.4Application of averages

21Limits

21.1Finding limits from graphs
21.2Continuity
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

22Derivatives

22.1Definition of derivative
22.2Power rule
22.3Gradient and equation of tangent line
22.4Chain rule
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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