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Family Table Math

Functions (Grade 11)

47 of 47 topics written so far. Greyed-out topics are coming soon.

Unit 1: Characteristics of Functions

  1. Relations and FunctionsDraft — What makes a relation a function — ordered pairs, tables, mapping diagrams, graphs, and the vertical line test.
  2. Function NotationDraft — Writing f(x), evaluating functions, and solving f(x) = a value — from equations, tables, and graphs.
  3. Domain and RangeDraft — Finding the set of possible inputs and outputs of a function from graphs, equations, and real-world situations.
  4. Parent FunctionsDraft — The four base graphs — y = x, y = x², y = √x, and y = 1/x — with their shapes, key points, domains, and ranges.
  5. Inverse FunctionsDraft — Undoing a function — finding, graphing, and checking inverses.
  6. Translations of FunctionsDraft — Shifting graphs up, down, left, and right — how c and d move y = f(x − d) + c.
  7. Stretches, Compressions, and ReflectionsDraft — How a and k stretch, compress, and flip graphs in y = af(x) and y = f(kx).
  8. Combining TransformationsDraft — Graphing y = af(k(x − d)) + c by applying stretches, reflections, and translations in the right order.

Unit 2: Equivalent Algebraic Expressions

  1. Adding, Subtracting, and Multiplying PolynomialsDraft — Combining like terms, subtracting carefully, and expanding products of polynomials — including the special products.
  2. Simplifying RadicalsDraft — Writing square roots in simplest form, and adding, subtracting, and multiplying radical expressions.
  3. Simplifying Rational ExpressionsDraft — Factoring and cancelling to simplify rational expressions, and stating the restrictions on the variable.
  4. Multiplying and Dividing Rational ExpressionsDraft — Factor, cancel, and multiply — and for division, multiply by the reciprocal — while keeping track of every restriction.
  5. Adding and Subtracting Rational ExpressionsDraft — Finding the lowest common denominator, combining numerators carefully, and simplifying the result.

Unit 3: Quadratic Functions

  1. Zeros of Quadratics and the DiscriminantDraft — Finding the zeros of a quadratic function by factoring or the quadratic formula, and using the discriminant to count them.
  2. Completing the SquareDraft — Rewriting a quadratic from standard form into vertex form, so you can read off its vertex and maximum or minimum.
  3. Maximum and Minimum of a QuadraticDraft — Finding the vertex of a quadratic by completing the square or averaging the zeros, and solving real-world optimization problems.
  4. Families of Quadratic FunctionsDraft — Quadratics that share the same zeros, and finding the one member that passes through a given point.
  5. Linear–Quadratic SystemsDraft — Finding where a line meets a parabola, and using the discriminant to tell whether they cross twice, touch once, or miss.

Unit 4: Exponential Functions

  1. Rational ExponentsDraft — What fractional exponents like 8^(2/3) mean, how to evaluate and simplify them, and how to rewrite powers in a different base.
  2. Exponential FunctionsDraft — Graphs and key properties of y = a^x, and how to tell exponential, linear, and quadratic patterns apart.
  3. Transformations of Exponential FunctionsDraft — Graphing y = a·b^(k(x − d)) + c from the parent y = b^x, and writing equations from graphs or properties.
  4. Exponential Growth and DecayDraft — Modelling populations, depreciation, half-life, and doubling with exponential functions, and solving real-world problems.

Unit 5: Sequences and Series

  1. Sequences and RecursionDraft — Sequences as discrete functions — general terms, recursion formulas, and how to move between them.
  2. Pascal's TriangleDraft — How Pascal's triangle is built, the patterns hiding in it, and how it counts paths and combinations.
  3. Binomial ExpansionDraft — Expanding powers like (x + y)^n and (2x − 3)^4 quickly using the rows of Pascal's triangle.
  4. Arithmetic SequencesDraft — Sequences that add the same amount each time — the common difference, the general term t_n = a + (n − 1)d, and problems.
  5. Geometric SequencesDraft — Sequences that multiply by the same amount each time — the common ratio, the general term t_n = ar^(n − 1), and problems.
  6. Arithmetic SeriesDraft — Adding the terms of an arithmetic sequence quickly with S_n = n/2 (2a + (n − 1)d), and why the formula works.
  7. Geometric SeriesDraft — Adding the terms of a geometric sequence with S_n = a(r^n − 1)/(r − 1), and why the formula works.

Unit 6: Financial Applications

  1. Simple InterestDraft — Calculating interest with I = Prt, finding the total amount, and seeing why simple interest grows like an arithmetic sequence.
  2. Compound InterestDraft — Interest on interest — A = P(1 + i)^n, compounding periods, present value, and finding the rate or time.
  3. Annuities: Future ValueDraft — Regular deposits that earn compound interest — the future value formula, why it's a geometric series, and finding the payment.
  4. Annuities: Present ValueDraft — How much a series of future payments is worth today — loans, regular withdrawals, and the present value formula.

Unit 7: Trigonometric Ratios

  1. Special AnglesDraft — Exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and 90°, from two special right triangles.
  2. Trig Ratios of Any AngleDraft — Angles in standard position from 0° to 360°, the CAST rule, reference angles, and finding exact values.
  3. Finding Angles from 0° to 360°Draft — Given a trig ratio, finding both angles between 0° and 360° that have it, using reference angles and CAST.
  4. Reciprocal Trig RatiosDraft — Cosecant, secant, and cotangent — the reciprocals of sine, cosine, and tangent — and how to evaluate them.
  5. Proving Trig IdentitiesDraft — The Pythagorean, quotient, and reciprocal identities, and how to prove simple identities step by step.
  6. The Sine LawDraft — Solving any triangle when you know two angles and a side, using a/sin A = b/sin B = c/sin C.
  7. The Cosine LawDraft — Solving triangles from two sides and the angle between them, or from three sides, using c² = a² + b² − 2ab cos C.
  8. The Ambiguous CaseDraft — When two sides and a non-included angle (SSA) give no triangle, one triangle, or two — and how to find them all.
  9. Trig Problems in Three DimensionsDraft — Solving 3-D problems — towers, cliffs, boxes, and pyramids — by splitting them into 2-D right and oblique triangles.

Unit 8: Sinusoidal Functions

  1. Periodic FunctionsDraft — Functions that repeat — cycles, period, amplitude, and the axis of the curve — and how to use them to predict.
  2. Graphs of Sine and CosineDraft — The graphs of y = sin x and y = cos x in degrees, where they come from, and their key properties.
  3. Transformations of Sinusoidal FunctionsDraft — How a, k, d, and c change amplitude, period, phase shift, and axis in y = a sin(k(x − d)) + c, in degrees.
  4. Sinusoidal Equations from GraphsDraft — Writing the equation of a sinusoidal function from its graph, a table, or a description of its properties.
  5. Sinusoidal ModellingDraft — Building and using sinusoidal models for Ferris wheels, tides, temperatures, and daylight, in degrees.