Grade 12 · printable course

Precalculus: Functions, Trigonometry, and Mathematical Modeling

Grade 12 · 571 pages · 81 lessons · PDF, print at home

This course extends the function concept built in Algebra II, polynomial, rational, exponential, and logarithmic, into a unified study of functions as models, adds trigonometric functions defined on the unit circle rather than just right triangles, and closes with the algebraic and geometric machinery (vectors, matrices, polar and parametric representations, conics, and the limit concept) that a calculus or applied statistics course assumes on day one. Students build fluency in reasoning flexibly among graphical, numerical, symbolic, and verbal representations of a relationship, and in choosing a function family to fit a context rather than being told which family to use. The course is proof-light but justification-heavy: students are repeatedly asked to argue why a model or identity must hold, not merely to compute. It is designed for students who have completed Algebra II and Geometry and are headed toward Calculus, Statistics, or a STEM-adjacent postsecondary path.

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Every unit and lesson in this book, in order. The same lessons are taught online at lettersanddevotion.com, so a family can move between paper and screen.

Precalculus: Functions, Trigonometry, and Mathematical Modeling

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Unit 1: Functions as a Family of Structures9 lessons
  1. 1.The four structural tests (constant difference, constant ratio, finite roots, domain/asymptote restriction) used to distinguish linear, exponential, polynomial, and rational families
  2. 2.Family classification of tabular data using structural tests under realistic (non-clean) conditions
  3. 3.The distinguishing structural assumption between two function families whose graphs look alike over a restricted domain
  4. 4.Correspondence between a real-world process description and a function family's structural assumption
  5. 5.End behavior of a function family as a testable real-world prediction, checked against domain constraints
  6. 6.The reverse-order undoing structure of an inverse function, expressed verbally prior to symbolic representation
  7. 7.Symbolic construction and numeric verification of an inverse function for an untaught multi-step process
  8. 8.Family selection and falsifiability justification for real, previously unseen data sets
  9. 9.Comparative rate-of-change behavior across linear, polynomial, and exponential families
Unit 2: Polynomial and Rational Functions: Structure and Behavior10 lessons
  1. 10.The relationship between factor multiplicity (odd vs even) and graph behavior at a zero
  2. 11.The rule linking a polynomial's degree parity and leading coefficient sign to its end behavior
  3. 12.Synthetic division and the remainder theorem as tools for testing and extracting factors of a polynomial
  4. 13.The complete zero-finding and multiplicity-based sketching process for a higher-degree polynomial function
  5. 14.Why a vertical asymptote arises from division by a factor approaching zero
  6. 15.The cancel-before-classify procedure for distinguishing holes from vertical asymptotes in a rational function
  7. 16.The degree-comparison rule for horizontal asymptotes of rational functions, grounded in end-behavior reasoning
  8. 17.The complete set of structural predictors (multiplicity, leading term, degree comparison, cancellation) that jointly determine a polynomial or rational function's graph
  9. 18.The inverse construction problem: building a rational function's algebraic form from specified zero, asymptote, and hole behavior
  10. 19.Unit 1's function-family identification vocabulary, applied to novel equations mixed with this unit's content
Unit 3: Exponential and Logarithmic Functions: Growth, Decay, and Inversion11 lessons
  1. 20.Constant ratio vs. constant rate of change in tabular data
  2. 21.The mechanism connecting percent growth/decay to the exponential function form b^x
  3. 22.Exponent-rule transformation of exponential growth expressions to reveal an equivalent rate
  4. 23.The logarithm as the inverse function of an exponential, and why it requires a new function class
  5. 24.The definition of logarithm and exponential/logarithmic form conversion
  6. 25.The derivation of log properties from exponent rules
  7. 26.Solving exponential and logarithmic equations algebraically
  8. 27.Multi-term exponential/logarithmic equation solving requiring strategic property selection
  9. 28.E as the limit of (1+1/n)^n and its role in continuous-compounding models
  10. 29.Model selection among exponential, logarithmic, and polynomial families based on structural data features
  11. 30.Validity of proposed logarithm properties, tested against the definition of log as inverse exponent
Unit 4: The Unit Circle and Trigonometric Functions11 lessons
  1. 31.Radian measure as arc length on the unit circle
  2. 32.Exact trigonometric values at special unit circle angles
  3. 33.The equivalence and extension of right-triangle trig to unit-circle trig
  4. 34.Reference angles and quadrant sign patterns on the unit circle
  5. 35.The sine and cosine graphs as traces of the unit circle's y- and x-coordinates over rotation
  6. 36.Amplitude, period, phase shift, and midline transformations of sine and cosine
  7. 37.The tangent function's period and asymptotes derived from sin θ / cos θ
  8. 38.Inverse trigonometric functions defined via domain restriction
  9. 39.Periodic real-world phenomena modeled by sine or cosine functions
  10. 40.The structural fit between a candidate function family and a novel periodic or non-periodic dataset
  11. 41.Unit-circle symmetry relationships for supplementary and related angles
Unit 5: Trigonometric Identities and Equations10 lessons
  1. 42.The Pythagorean identity sin^2θ + cos^2θ = 1 and its two rearranged forms
  2. 43.Algebraic simplification of trig expressions using identity substitution
  3. 44.The structural difference between proving an identity and solving a conditional equation
  4. 45.Verification of trigonometric identities as a chain of justified equivalent transformations
  5. 46.The double-angle identities for sine and cosine as a special case of the sum formulas
  6. 47.Solving trigonometric equations over a restricted domain
  7. 48.General solutions to trigonometric equations over all real numbers
  8. 49.Identity substitution as a strategy for converting mixed trig equations into single-function polynomial equations
  9. 50.The logical status of an equivalence claim between two trig expressions (identity vs. false-in-general claim)
  10. 51.The half-angle identities as an algebraic consequence of the double-angle cosine identity
Unit 6: Vectors and Matrices: Representing Multi-Part Quantities11 lessons
  1. 52.Vectors as quantities with magnitude and direction
  2. 53.Vector component resolution using unit-circle trigonometric ratios
  3. 54.Vector addition and the non-additivity of magnitude
  4. 55.Resultant vector models of force or velocity
  5. 56.The boundary between vector-appropriate and vector-inappropriate quantities
  6. 57.Matrices as organized rectangular data
  7. 58.Matrix addition, subtraction, and scalar multiplication
  8. 59.Matrix multiplication and non-commutativity
  9. 60.The determinant and its relationship to matrix invertibility
  10. 61.Solving linear systems via matrices
  11. 62.Translating novel multi-constraint problems into matrix equations
Unit 7: Polar Coordinates and Parametric Equations8 lessons
  1. 63.Conversion formulas between rectangular and polar coordinates
  2. 64.Non-uniqueness of polar coordinate representation
  3. 65.Structural link between polar equation form and curve shape family
  4. 66.Parametric equations as jointly determined by a shared parameter
  5. 67.Method selection for parameter elimination based on equation structure
  6. 68.Parametrizing linear motion using vector direction and speed
  7. 69.Comparative efficiency of coordinate/parametric representations for a given curve
  8. 70.Conversion of whole equations (not just points) between rectangular and polar form
Unit 8: Conics and an Introduction to Limits11 lessons
  1. 71.The circle as the set of points at fixed distance (radius) from a center, and its standard-form equation
  2. 72.Conic sections as cross-sections of a double cone
  3. 73.The parabola as the locus of points equidistant from a focus and a directrix
  4. 74.Ellipses and hyperbolas as loci defined by constant sum or constant difference of distances from two foci
  5. 75.Eccentricity as the unifying parameter across all four conic types
  6. 76.Conic equations applied to an unfamiliar physical modeling context
  7. 77.Limits as the value a function's output is forced toward, evaluated numerically and graphically
  8. 78.The distinction between limit-at-a-point and function-value-at-a-point
  9. 79.Existence and classification of discontinuities using limit reasoning
  10. 80.Instantaneous rate of change as the limit of average rates of change over shrinking intervals
  11. 81.Selecting between conic and limit/rate representations for an unfamiliar quantitative scenario

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