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Family Table Math

Advanced Functions (Grade 12)

45 of 45 topics written so far. Greyed-out topics are coming soon.

Unit 1: Polynomial Functions

  1. Polynomial FunctionsDraft — What makes a function a polynomial, its degree and leading coefficient, and how polynomial graphs differ from exponential and sinusoidal graphs.
  2. Features of Polynomial GraphsDraft — Finite differences, end behaviour, the number of x-intercepts and turning points, and the domain and range of polynomial functions.
  3. Polynomials in Factored FormDraft — Reading zeros and their multiplicity from factored form, and sketching a polynomial from its intercepts, end behaviour, and sign intervals.
  4. Transformations of Cubic and Quartic FunctionsDraft — Graphing y = a f(k(x − d)) + c when f(x) = x³ or x⁴, using mapping notation, and finding equations from key features.
  5. Families of Polynomial FunctionsDraft — Writing the family of polynomials with given zeros, finding the member through a given point, and building equations from conditions or finite differences.
  6. Even and Odd FunctionsDraft — Testing whether a function is even, odd, or neither using f(−x), the symmetry of their graphs, and what happens when you add or multiply them.

Unit 2: Polynomial Equations and Inequalities

  1. Dividing PolynomialsDraft — Long division and synthetic division of polynomials, including missing terms, and writing the result as P(x) = D(x)Q(x) + R.
  2. The Remainder TheoremDraft — Finding the remainder of a polynomial division without dividing, and using remainders to find unknown coefficients.
  3. The Factor TheoremDraft — Using P(a) = 0 to find factors, testing possible integer and rational zeros, and factoring cubics and quartics.
  4. Factoring PolynomialsDraft — A strategy for factoring polynomials up to degree 4 — common factors, special products, trinomials, grouping, sums and differences of cubes, and the factor theorem.
  5. Solving Polynomial EquationsDraft — Solving polynomial equations up to degree 4 by factoring, connecting real roots to x-intercepts, and solving problems such as box volumes.
  6. Polynomial InequalitiesDraft — Solving linear and factorable polynomial inequalities using graphs, intervals and test points, and showing the solution on a number line.

Unit 3: Rational Functions

  1. Reciprocal FunctionsDraft — Graphing y = 1/f(x) from the graph of a linear or quadratic function f, using asymptotes, invariant points, signs, and turning points.
  2. Graphs of Rational FunctionsDraft — Sketching y = (ax + b)/(cx + d) from its asymptotes and intercepts, spotting holes, and a first look at oblique asymptotes.
  3. Solving Rational EquationsDraft — Solving equations with variables in the denominator by multiplying by the LCD, rejecting extraneous roots, and using them for work-rate, speed, and concentration problems.
  4. Rational InequalitiesDraft — Solving inequalities with variables in the denominator, using sign charts built from zeros and asymptotes, and checking with graphs.

Unit 4: Trigonometric Functions

  1. Radian MeasureDraft — What a radian is, converting between degrees and radians, arc length a = rθ, and angular velocity.
  2. Trig Ratios in RadiansDraft — Primary and reciprocal trig ratios of angles in radians, with a calculator and exactly for the special angles, using the CAST rule and related acute angles.
  3. Sinusoidal Functions in RadiansDraft — The graphs of y = sin x and y = cos x in radians, and y = a sin(k(x − d)) + c — amplitude, period 2π/k, phase shift, sketching, and writing equations.
  4. The Tangent FunctionDraft — Graphing y = tan x in radians — asymptotes, period π, zeros, domain and range — and its link to sin x / cos x.
  5. Reciprocal Trig FunctionsDraft — Graphing y = csc x, y = sec x, and y = cot x in radians from sin, cos, and tan — domain, range, period, asymptotes, and notation.
  6. Sinusoidal Applications in RadiansDraft — Modelling Ferris wheels, tides, springs, daylight, and predator–prey cycles with sinusoidal functions in radians, and answering questions with the model and its graph.

Unit 5: Trigonometric Identities and Equations

  1. Equivalent Trig ExpressionsDraft — Cofunction, even–odd, and related-angle identities in radians, from right triangles, the unit circle, and transformations, and how to check equivalence with a graph.
  2. Compound Angle FormulasDraft — The sine, cosine, and tangent of a sum or difference of two angles, where the formulas come from, and how to use them for exact values and simplifying.
  3. Double Angle FormulasDraft — Formulas for sin 2x, cos 2x (in three forms), and tan 2x, derived from the compound angle formulas, with exact values and finding sin 2x from one ratio and a quadrant.
  4. Advanced Trig Identity ProofsDraft — What makes an equation an identity, how a counterexample disproves one, and how to prove identities in L.S./R.S. form using the Pythagorean, quotient, reciprocal, compound angle, and double angle identities.
  5. Solving Linear Trig EquationsDraft — Solving equations like 2 sin x + 1 = 0 for 0 ≤ x ≤ 2π in radians, with exact answers from special angles, decimal answers from related angles, and equations with kx.
  6. Solving Quadratic Trig EquationsDraft — Solving trig equations that are quadratic, like 2 cos²x − cos x − 1 = 0, by factoring, common factoring, and using identities first, for 0 ≤ x ≤ 2π in radians.

Unit 6: Exponential and Logarithmic Functions

  1. Introduction to LogarithmsDraft — A logarithm is an exponent — evaluating logs, switching between exponential and logarithmic form, and estimating logs between whole numbers.
  2. Logarithmic FunctionsDraft — Graphing y = log_b x as the inverse of y = b^x — its asymptote, domain, range, intercept, and how the base changes the graph.
  3. Transformations of Logarithmic FunctionsDraft — Graphing y = a log(k(x − d)) + c from y = log x with mapping notation, and tracking the vertical asymptote, domain, and intercepts.
  4. Laws of LogarithmsDraft — The product, quotient, and power laws, the change of base formula, and using them to simplify and evaluate expressions.

Unit 7: Exponential and Logarithmic Equations

  1. Solving Exponential EquationsDraft — Solving equations with the unknown in the exponent — by finding a common base or by taking logarithms — including factoring and quadratic-type equations.
  2. Solving Logarithmic EquationsDraft — Solving equations with logarithms by rewriting in exponential form, combining logs with the laws, and rejecting extraneous roots.
  3. Logarithmic ScalesDraft — pH, earthquake magnitude, and decibels — why each step on a logarithmic scale means multiplying by 10, and how to compare and solve with them.
  4. Exponential and Logarithmic ApplicationsDraft — Using logarithms to find the time in growth and decay problems — doubling time, half-life, compound interest, populations, and cooling — and reading answers from graphs.

Unit 8: Rates of Change

  1. Average Rate of ChangeDraft — Measuring how fast a quantity changes over an interval, from tables, graphs, and equations, and connecting it to the slope of a secant line.
  2. Instantaneous Rate of ChangeDraft — Estimating the rate of change at a single moment as the slope of a tangent, using secants over shrinking intervals and centred intervals.

Unit 9: Combining Functions

  1. Adding and Subtracting FunctionsDraft — Building new functions with f + g and f − g — algebraically, from tables, and by adding y-values on a graph — plus domains, superposition of waves, and even/odd sums.
  2. Multiplying and Dividing FunctionsDraft — Products and quotients of functions — finding fg and f/g, their domains, holes and zeros, damped oscillations, and which products are even or odd.
  3. Composition of FunctionsDraft — Putting one function inside another — finding f(g(x)) from tables, graphs, and equations, its domain and range, decomposing functions, and real-world chains.
  4. Composition, Inverses, and TransformationsDraft — Why f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, checking inverses by composition, and seeing transformations as compositions with a linear function.
  5. Comparing Function TypesDraft — Polynomial, rational, exponential, logarithmic, and sinusoidal functions side by side — key features, equation forms, and how to identify a function type from a table or graph.
  6. Solving Equations and Inequalities GraphicallyDraft — Solving equations like 2^x = x + 3 and cos x = x, and inequalities like 2x² < 2^x, by graphing and by narrowing in on zeros to two decimal places.
  7. Function ModellingDraft — Building linear, quadratic, exponential, sinusoidal, and polynomial models from data, checking how well they fit, and using them to predict and interpret real situations.