WhizzNotes

Mathematics Extension 1

Further functions, polynomials, 3D and inverse trigonometry, counting and the binomial theorem, proof by induction, vectors, further calculus and the binomial distribution for Year 11 and 12. Every chapter starts with the background you need, so you can learn it on your own.

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Pick your year on any chapter.

Unit 1 Further Work with Functions

  1. 1Graphing Related FunctionsBuild new graphs from old ones. Sketch 1/f(x), |f(x)|, f(|x|) and the sum or difference of two functions.
  2. 2Inverse FunctionsUndo a function. One-to-one functions, finding f⁻¹(x), reflecting in y = x, and restricting the domain.
  3. 3Parametric EquationsDescribe a curve with a third variable, the parameter. Convert lines, parabolas and circles between parametric and Cartesian form.
  4. 4InequalitiesSolve cubic inequalities, inequalities with x in the denominator and absolute value inequalities, with algebra and with graphs.

Unit 2 Polynomials

  1. 5Polynomials, Division and the Factor TheoremThe language of polynomials, long division, and the remainder and factor theorems for finding roots fast.
  2. 6Graphs of PolynomialsSketch polynomials from their factors. End behaviour, multiple roots that touch or flatten, and reading equations from graphs.
  3. 7Sums and Products of RootsFind the sum and product of the roots straight from the coefficients, for quadratics, cubics and quartics.

Unit 3 Trigonometric Functions

  1. 8Reciprocal Trigonometric FunctionsMeet sec, cosec and cot. Their values, their graphs and asymptotes, and the identities that link them.
  2. 9Trigonometry in Three DimensionsSolve problems in 3D by finding the right triangles inside boxes, pyramids and hills, with angles of elevation and bearings.
  3. 10Compound Angles and Double AnglesExpand sin(A + B) and cos(A + B), use double angle formulas, write a sin x + b cos x as one wave, and solve harder trig equations.
  4. 11Inverse Trigonometric FunctionsThe functions sin⁻¹ x, cos⁻¹ x and tan⁻¹ x. Their domains, ranges, graphs, exact values and key properties.

Unit 4 Combinatorics

  1. 12Counting and PermutationsThe multiplication principle, factorials, arrangements with and without restrictions, repeated letters and circles.
  2. 13Combinations and the Pigeonhole PrincipleChoose groups where order does not matter, handle restrictions on committees, and use the pigeonhole principle.
  3. 14The Binomial TheoremExpand (a + b)ⁿ quickly with Pascal's triangle and combinations, find any term or coefficient, and spot patterns.

Unit 5 Proof

  1. 15Proof by Mathematical InductionProve a statement is true for every whole number with two steps, the base case and the inductive step. Sums and divisibility.

Unit 6 Vectors

  1. 16Introduction to VectorsQuantities with size and direction. Vector notation, components, magnitude, adding, subtracting, scaling and unit vectors.
  2. 17The Scalar Product and ProjectionsMultiply two vectors to get a number. Find angles, test for right angles, project one vector onto another and prove geometry results.
  3. 18Motion in Vector FormDescribe moving objects with position vectors. Find velocity and acceleration with calculus, and handle boats in currents and planes in crosswinds.
  4. 19Vectors and Projectile MotionModel a thrown ball with vectors. Split the launch velocity into parts, find the time of flight, greatest height and range, and the path's equation.

Unit 7 Further Calculus

  1. 20Calculus with Inverse Trig FunctionsDifferentiate sin⁻¹ x, cos⁻¹ x and tan⁻¹ x, and use them to integrate 1/√(a² − x²) and 1/(a² + x²).
  2. 21Integration by SubstitutionTurn a hard integral into an easy one by swapping x for a new variable u. Change the limits, then integrate.
  3. 22Integrating with Trig IdentitiesIntegrate sin² x, cos² x, sin x cos x and tan² x by first rewriting them with the double angle formulas.
  4. 23Volumes of Solids of RevolutionSpin a curve around an axis to make a solid, then find its volume by adding up thin circular slices with an integral.
  5. 24Related Rates of ChangeWhen one quantity changes, others linked to it change too. Use the chain rule to find how fast, for ripples, balloons, tanks and ladders.
  6. 25Newton's Law of Cooling and Further GrowthThings that grow or decay towards a level that is not zero. Model cooling coffee and warming drinks with N = P + Ae^(kt).
  7. 26Differential Equations and Slope FieldsEquations that involve a derivative. Read slope fields, solve by integrating and by separating variables, and model growth with limits.

Unit 8 Statistical Analysis

  1. 27The Binomial DistributionCount successes in a fixed number of yes-or-no trials. Bernoulli trials, the binomial formula, and the mean and variance np and np(1 - p).
  2. 28The Sampling Distribution of the MeanAverages of samples behave better than single values. The mean and standard error of a sample mean, the central limit theorem, and sample proportions.

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