# Year 13 Maths Help & Practice

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##### 1Functions

##### 2Transformations of Functions

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

- 2.1Transformations of functions: Horizontal translations
##### 3Quadratic Functions

- 3.1Characteristics of quadratic functions
- 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
- 3.8Graphing parabolas for given quadratic functions
- 3.9Finding the quadratic functions for given parabolas
- 3.10Applications of quadratic functions

- 3.1Characteristics of quadratic functions
##### 4Polynomial Functions

- 4.1What is a polynomial function?
- 4.2Polynomial long division
- 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

- 4.1What is a polynomial function?
##### 5Rational Functions

##### 6Exponential Functions

- 6.1Exponents: Product rule
*(a^x)(a^y) = a^(x+y)* - 6.2Exponents: Division rule (a^x / a^y) = a^(x-y)
- 6.3Exponents: Power rule
*(a^x)^y = a^(x * y)* - 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

- 6.1Exponents: Product rule
##### 7Logarithmic Functions

- 7.1What is a logarithm?
- 7.2Converting from logarithmic form to exponential form
- 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
- 7.8Solving exponential equations with logarithms
- 7.9Product rule of logarithms
- 7.10Quotient rule of logarithms
- 7.11Combining product rule and quotient rule in logarithms
- 7.12Evaluating logarithms using logarithm rules
- 7.13Solving logarithmic equations
- 7.14Graphing logarithmic functions
- 7.15Finding a logarithmic function given its graph

- 7.1What is a logarithm?
##### 8Applications of Exponential and Logarithmic Functions

- 8.1Exponential growth and decay by a factor
- 8.2Exponential decay: Half-life
- 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

- 8.1Exponential growth and decay by a factor
##### 9Quadratic Equations and Inequalities

##### 10Solving Simultaneous Equations

- 10.1Determining number of solutions to linear equations
- 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
- 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

- 10.1Determining number of solutions to linear equations
##### 11Simultaneous Equations (Advanced)

- 11.1Simultaneous linear equations
- 11.2Simultaneous linear-quadratic equations
- 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

- 11.1Simultaneous linear equations
##### 12Introduction to Trigonometry

- 12.1Use sine ratio to calculate angles and sides (Sin = $\frac{o}{h}$ )
- 12.2Use cosine ratio to calculate angles and sides (Cos = $\frac{a}{h}$ )
- 12.3Use tangent ratio to calculate angles and sides (Tan = $\frac{o}{a}$ )
- 12.4Combination of SohCahToa questions
- 12.5Solving expressions using 45-45-90 special right triangles
- 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

- 12.1Use sine ratio to calculate angles and sides (Sin = $\frac{o}{h}$ )
##### 13Trigonometric Ratios and Angle Measure

##### 14Bearings

##### 15Graphing Trigonometric Functions

##### 16Trigonometric Identities

##### 17Conics

##### 18Imaginary and Complex Numbers

- 18.1Introduction to imaginary numbers
- 18.2Complex numbers and complex planes
- 18.3Adding and subtracting complex numbers
- 18.4Complex conjugates
- 18.5Multiplying and dividing complex numbers
- 18.6Distance and midpoint of complex numbers
- 18.7Angle and absolute value of complex numbers
- 18.8Polar form of complex numbers
- 18.9Operations on complex numbers in polar form

- 18.1Introduction to imaginary numbers
##### 19Sequences and Series

##### 20Probability

##### 21Statistics

##### 22Limits

- 22.1Finding limits from graphs
- 22.2Continuity
- 22.3Finding limits algebraically - direct substitution
- 22.4Finding limits algebraically - when direct substitution is not possible
- 22.5Infinite limits - vertical asymptotes
- 22.6Limits at infinity - horizontal asymptotes
- 22.7Intermediate value theorem
- 22.8Squeeze theorem

- 22.1Finding limits from graphs
##### 23Derivatives

- 23.1Definition of derivative
- 23.2Power rule
- 23.3Gradient and equation of tangent line
- 23.4Chain rule
- 23.5Derivative of trigonometric functions
- 23.6Derivative of exponential functions
- 23.7Product rule
- 23.8Quotient rule
- 23.9Implicit differentiation
- 23.10Derivative of inverse trigonometric functions
- 23.11Derivative of logarithmic functions
- 23.12Higher order derivatives

- 23.1Definition of derivative