Precalculus Study Guide
Everything you need to ace precalculus — all 13 chapters organized by topic, with worked examples, practice problems, and explanations for every concept.
13
Chapters
657+
Practice Problems
68
Topic Sections
Foundations
Chapter 1
Fundamentals
Real numbers, algebra review, equations, inequalities, coordinate geometry
Chapter 2
Functions
Definition, domain/range, graphs, transformations, composition, inverse functions
Chapter 3
Polynomial and Rational Functions
Zeros, end behavior, factoring, asymptotes, partial fractions
Chapter 4
Exponential and Logarithmic Functions
Growth/decay, log laws, solving exp/log equations, natural log, applications
Trigonometry
Chapter 5
Trigonometric Functions: Unit Circle
Angles, radian measure, unit circle, sin/cos/tan values, graphs
Chapter 6
Trigonometric Functions: Right Triangle
SOH-CAH-TOA, special angles, Law of Sines, Law of Cosines, Heron's formula
Chapter 7
Analytic Trigonometry
Identities, sum/difference formulas, double-angle, half-angle, inverse trig, equations
Advanced Topics
Chapter 8
Polar Coordinates and Parametric Equations
Polar form, polar graphs, DeMoivre's theorem, complex numbers, parametric curves
Chapter 9
Vectors
2D and 3D vectors, dot product, cross product, applications
Chapter 10
Systems of Equations and Inequalities
Linear systems, Gaussian elimination, matrix operations, nonlinear systems
Chapter 11
Conic Sections
Parabolas, ellipses, hyperbolas, completing the square, applications
Calculus Preview
How to Study Precalculus Effectively
Do problems daily — not marathon sessions
Precalculus requires pattern recognition that builds with repetition. 30 minutes every day beats 4 hours once a week.
Write everything by hand first
Graphs, unit circle values, trig identities — write them out. The physical act of writing cements spatial/visual concepts better than reading.
Don't skip chapters
Each chapter builds on the last. Students who struggle with trig almost always have a gap in Chapter 1-2 (algebra and functions). Go back and fix it.
Master the unit circle
Memorize sin/cos values at 0, π/6, π/4, π/3, π/2 (and symmetry fills the rest). This unlocks Chapters 5-7 completely.
Focus extra time on trig identities
Chapter 7 (analytic trig) trips up more students than any other chapter. Budget double the time you think you need.
Use your private tutor for concepts, not just checking answers
When you don't understand why, ask why. Your private tutor on NailTheTest adapts to how you learn and guides you to understanding rather than giving you answers.
Most Common Precalculus Mistakes
Distributing incorrectly through parentheses
Fix: (a + b)² ≠ a² + b². It's a² + 2ab + b².
Dividing both sides by a variable without considering zero
Fix: If you divide by x, you must assume x ≠ 0 — and check x = 0 separately.
Confusing f(a + b) with f(a) + f(b)
Fix: Functions don't distribute over addition. f(a + b) requires substituting (a + b) into the full expression.
Mixing degrees and radians
Fix: Trig functions in calculus always use radians. π radians = 180°. Never mix units in the same calculation.
Wrong domain of inverse trig functions
Fix: arcsin returns values in [-π/2, π/2]. arccos returns [0, π]. arctan returns (-π/2, π/2). These are not negotiable.
Log of a sum is not sum of logs
Fix: log(a + b) ≠ log(a) + log(b). The product rule is log(ab) = log(a) + log(b).
Common Questions
Is precalculus hard?
For most students, precalculus is harder than high school algebra but easier than calculus. The difficulty spike happens with trig identities and polar coordinates. The key is consistent practice — it's skill-based, not memorization-based.
What comes after precalculus?
Calculus I (differential calculus). Chapter 13 of NailTheTest — Limits — is specifically designed as a bridge to calculus, previewing the fundamental theorem that calculus is built on.
Do I need to memorize the unit circle?
Yes. You need to know the sin, cos, and tan values at the standard angles (0, 30, 45, 60, 90 degrees and their equivalents in radians). Most trig topics become much easier once the unit circle is automatic.
How do I pass my precalculus exam?
Do the homework problems (not just read the examples), retake practice tests until you can do them without looking at notes, and focus extra time on the topics you get wrong rather than reviewing what you already know.
Topic Deep Dives
Functions (Chapter 2)
Domain/range, transformations, composition, inverse functions — the foundation of all precalculus.
Trigonometry (Chapters 5–7)
Unit circle, identities, inverse trig, Law of Sines/Cosines — the longest section of precalc.
Exponential & Log Functions (Ch. 4)
Log laws, solving exp/log equations, growth/decay, compound interest, half-life.
Conic Sections (Ch. 11)
Parabolas, ellipses, hyperbolas — standard forms, graphing, completing the square.
Sequences & Series (Ch. 12)
Arithmetic, geometric, sigma notation, infinite geometric series convergence.
Vectors (Ch. 8)
Component form, magnitude, dot product, angles between vectors, and applications.
Polar Coordinates (Ch. 8)
Polar/rectangular conversion, rose curves, limaçons, cardioids, lemniscates.
Parametric Equations (Ch. 8)
Eliminating the parameter, orientation, projectile motion, and common forms.
Limits — Intro to Calculus (Ch. 13)
Limit definition, four evaluation methods, one-sided limits, continuity.
Polynomial Functions (Ch. 3)
End behavior, rational root theorem, synthetic division, zeros, multiplicity, and graphing.
Systems of Equations (Ch. 9)
Substitution, elimination, Gaussian elimination, matrices, Cramer's rule, and 3-variable systems.
Rational Functions (Ch. 3)
Vertical and horizontal asymptotes, holes, oblique asymptotes, and graphing steps.
Trig Identities (Chs. 6–7)
Pythagorean, reciprocal, double angle, half angle, sum/difference, and verification strategies.
Logarithms & Log Laws (Ch. 4)
Log laws, solving exponential/log equations, change of base, and applications.
Right Triangle Trig (Ch. 5)
SOH CAH TOA, special triangles, Law of Sines & Cosines, area formulas.
Unit Circle (Ch. 5–6)
All 16 standard angles with exact values, ASTC rule, and memory tricks.
Domain and Range (Ch. 2)
Find domain of rational, radical, log, and composite functions. Interval notation guide.
Complex Numbers (Ch. 8)
Imaginary unit i, operations, polar form, De Moivre's theorem.
Inverse Functions (Ch. 2)
Find inverses algebraically, verify with composition, domain restrictions, log/trig pairs.
Function Transformations (Ch. 2)
Vertical/horizontal shifts, reflections, stretches, compressions — y = a·f(b(x−h)) + k.
Absolute Value (Ch. 2)
Equations, inequalities (AND vs OR), V-shaped graph, transformations.
Quadratic Equations (Ch. 3)
4 solving methods: factoring, square root, completing the square, quadratic formula.
Completing the Square (Ch. 3)
Vertex form, solving equations, circle equations, and deriving the quadratic formula.
Piecewise Functions (Ch. 2)
Evaluate, graph, and determine continuity. Real-world applications and step functions.
Geometric Sequences (Ch. 12)
Common ratio, nth term, partial sums, infinite series convergence, repeating decimals.
Arithmetic Sequences (Ch. 12)
Common difference, nth term formula, sum formula, arithmetic means.
Binomial Theorem (Ch. 12)
Pascal's Triangle, binomial coefficients, expanding (a+b)^n, finding specific terms.
Probability (Ch. 12)
Addition/multiplication rules, conditional probability, permutations, combinations, binomial.
Exponential Growth & Decay (Ch. 4)
Compound interest, continuous compounding, half-life, population models, Newton's law of cooling.
Sigma Notation (Ch. 12)
Summation notation, index of summation, closed-form formulas for Σk, Σk², Σk³, and series connections.
Polynomial Long Division (Ch. 3)
Long division, synthetic division, Remainder Theorem, Factor Theorem, Rational Root Theorem.
Matrix Operations (Ch. 9)
Addition, multiplication, determinants (2×2 and 3×3), inverse matrices, and solving linear systems.
Inverse Trig Functions (Ch. 6)
arcsin, arccos, arctan — domains, ranges, common values, compositions, and De Moivre applications.
Complex Plane & Polar Form (Ch. 8)
Rectangular vs polar form, modulus, argument, De Moivre's theorem, and nth roots of complex numbers.
Combinations & Permutations (Ch. 12)
Factorial notation, Fundamental Counting Principle, P(n,r), C(n,r), and multi-step counting problems.
End Behavior of Functions (Ch. 3)
Leading term test for polynomials, rational function asymptotes, exponential/log end behavior, arrow notation.
Solving Trig Equations (Ch. 6)
Isolate, reference angle, quadrants, general solutions, quadratic trig, and identity substitutions.
Function Composition (Ch. 1)
(f∘g)(x), domain of composite functions, decomposing functions, and table-based evaluation.
Law of Sines & Cosines (Ch. 6)
AAS/ASA/SSA via Law of Sines, ambiguous case, SAS/SSS via Law of Cosines, Heron's formula.
Partial Fraction Decomposition (Ch. 11)
Distinct linear, repeated linear, irreducible quadratic factors; cover-up method; integration connection.
Hyperbolic Functions (Advanced)
sinh, cosh, tanh definitions, key identities, Osborn's Rule, derivatives, inverse hyperbolic functions, catenary applications.
Rational Expressions (Ch. 3)
Simplifying, multiplying, dividing, adding/subtracting, complex fractions, solving rational equations, extraneous solutions.
Trig Graphs & Transformations (Ch. 5)
Sine/cosine/tangent graphs, amplitude, period 2π/|B|, phase shift, vertical shift, graphing y = A sin(Bx + C) + D.
Linear Equations & Slope (Ch. 1)
Slope-intercept, point-slope, standard form, parallel/perpendicular lines, linear inequalities, and applications.
Exponential Equations (Ch. 4)
Solving exponential equations by matching bases, using logarithms, exponential growth/decay applications, and compound interest.
Number Theory Fundamentals (Advanced)
Primes, divisibility rules, GCD/LCM, Euclidean algorithm, modular arithmetic, and Diophantine equations.
Solving Inequalities (Ch. 1)
Linear, compound, polynomial (sign chart), rational, and absolute value inequalities with interval notation.
Mathematical Proof Techniques (Advanced)
Direct proof, contradiction, contrapositive, weak and strong induction, well-ordering, and exhaustion methods.
Analytic Geometry (Ch. 1 & 10)
Distance and midpoint formulas, equations of lines, circles in standard/general form, coordinate proofs, locus problems, and section formula.
Statistics & Probability (Ch. 12)
Descriptive stats, probability rules, conditional probability, Bayes' theorem, binomial/geometric distributions, normal distribution, and expected value.
3D Vectors & Space (Ch. 9)
Vectors in three dimensions, dot product, cross product, lines and planes in 3D, distance from point to plane.
Calculus Preview & Readiness
Limits, one-sided limits, continuity, difference quotient, derivatives intro, integrals intro, IVT, and precalculus readiness checklist.
Trigonometric Substitution (Calculus)
Three substitution cases (sinθ, tanθ, secθ), completing the square, back-substitution with reference triangles, and worked examples.
Matrix Determinants & Applications (Ch. 11)
2×2 and 3×3 determinants, cofactor expansion, Cramer's Rule, matrix inverse via adjugate, area formula, and eigenvalues introduction.
Series Convergence Tests (Calculus)
Geometric, p-series, divergence test, integral, comparison, limit comparison, ratio, root, alternating series, and absolute vs. conditional convergence.
Taylor & Maclaurin Series (Calculus)
Power series, radius of convergence, all 5 key Maclaurin series, series manipulation, Taylor's remainder theorem, and applications.
Multivariable Functions (Calculus III)
Functions of two variables, level curves, partial derivatives, gradient, directional derivative, critical points, second derivative test, Lagrange multipliers.
Intro to Differential Equations (Calculus)
Separable ODEs, linear first-order integrating factor, direction fields, equilibrium solutions, exponential growth/decay, Newton's cooling, logistic growth, Euler's method.
Integration Techniques (Calculus 2)
u-substitution, integration by parts (LIATE + tabular), trig integrals, trig substitution, partial fractions, improper integrals, and decision flowchart.
Introduction to Linear Algebra
Vectors, matrices, Gaussian elimination, rank/nullity, linear transformations, vector spaces, null/column space, orthogonality, eigenvalues/eigenvectors.
Complex Analysis (Advanced)
Complex functions, Cauchy-Riemann equations, contour integration, Cauchy's theorem, Laurent series, poles, residues, and residue theorem applications.
Fourier Series (Advanced)
Periodic functions, Fourier coefficients, even/odd functions, Dirichlet conditions, Gibbs phenomenon, Parseval's theorem, and Fourier transform intro.
Numerical Methods (Applied Math)
Bisection, Newton-Raphson, secant method, Lagrange interpolation, cubic splines, Trapezoidal/Simpson's rules, RK4, and error analysis.
Combinatorics (Advanced)
Counting principles, permutations, combinations, stars-and-bars, pigeonhole, inclusion-exclusion, derangements, Catalan numbers, generating functions, graph theory.
Abstract Algebra (Advanced)
Groups, rings, fields, homomorphisms, isomorphism theorems, quotient groups, polynomial rings, ideals, Galois theory, and applications to coding theory.
Topology (Advanced)
Topological spaces, open/closed sets, continuity, homeomorphisms, connectedness, compactness, Heine-Borel, metric spaces, fundamental group, and homotopy.
Real Analysis (Advanced)
Real number axioms, supremum/infimum, Cauchy sequences, series convergence tests, epsilon-delta continuity, Riemann integration, FTC, and uniform convergence.
Advanced Number Theory
GCD, Euclidean algorithm, congruences, CRT, Fermat/Euler theorems, quadratic reciprocity, primitive roots, RSA cryptography, Pell's equation, and prime distribution.
Graph Theory (Advanced)
Vertices, edges, paths, cycles, trees, Euler/Hamiltonian paths, graph coloring, planar graphs, BFS/DFS, Dijkstra, Kruskal, network flow, and CS applications.
Mathematical Optimization
Unconstrained/constrained optimization, Lagrange multipliers, KKT conditions, linear programming, simplex method, convex optimization, gradient descent, and multi-objective optimization.
Probability Theory (Advanced)
Kolmogorov axioms, conditional probability, Bayes theorem, random variables, distributions, expected value, law of large numbers, CLT, Markov chains, and convergence modes.
Stochastic Processes (Advanced)
Markov chains, Poisson processes, Brownian motion, martingales, Ito calculus, SDEs, Black-Scholes, stationary processes, ergodicity, and queuing theory (M/M/1, Little's Law).
Mathematical Logic (Advanced)
Propositional/predicate logic, Gödel's incompleteness theorems, computability, halting problem, ZFC set theory, ordinals, axiom of choice, compactness theorem, and Löwenheim-Skolem.
Information Theory (Advanced)
Shannon entropy, mutual information, channel capacity, source/channel coding theorems, Huffman coding, KL divergence, Fisher information, rate-distortion, and ML applications.
Algebraic Geometry (Advanced)
Affine varieties, Nullstellensatz, coordinate rings, projective space, Riemann-Roch, elliptic curves and group law, divisors, sheaves, Weil conjectures, and elliptic curve cryptography.
Game Theory (Advanced)
Nash equilibrium, dominant strategies, mixed strategies, minimax theorem, extensive form games, backward induction, Shapley value, mechanism design, auction theory, evolutionary game theory, and Arrow impossibility theorem.
Mathematical Cryptography
RSA algorithm, Euler's theorem, Diffie-Hellman, discrete logarithm, elliptic curve cryptography (ECC/ECDH/ECDSA), AES/GF(2^8), hash functions, zero-knowledge proofs, lattice cryptography, and post-quantum standards.
Signal Processing Mathematics
Nyquist-Shannon theorem, aliasing, Fourier series, DFT, FFT, Z-transform, Laplace transform, convolution theorem, FIR/IIR filters, windowing functions, power spectral density, Wiener filter, and wavelet analysis.
Mathematics of Machine Learning
Linear algebra for ML (SVD, PCA, LDA), backpropagation derivation, MLE/MAP/conjugate priors, gradient descent variants, linear/logistic regression, neural networks, SVMs, VC dimension, and PAC learning theory.
Control Theory Mathematics
Transfer functions, poles/zeros, Routh-Hurwitz, root locus, Bode plots, Nyquist criterion, PID tuning (Ziegler-Nichols), state-space, LQR, Kalman filter, lead/lag compensators, and nonlinear control.
Statistical Inference
MLE, MOM, sufficiency, Rao-Blackwell, CRLB, confidence intervals, Neyman-Pearson, GLRT, ANOVA, regression inference, nonparametric tests, bootstrap methods, multiple testing corrections, and Bayesian inference with MCMC.
Discrete Mathematics
Set theory, logic and proofs, number theory, combinatorics, graph theory, recurrence relations, generating functions, Boolean algebra, equivalence relations, and Cantor's cardinality theorems.
Measure Theory
Sigma-algebras, Lebesgue measure, measurable functions, integration theory, convergence theorems, Lp spaces, product measures, Radon-Nikodym theorem, and foundations of probability.
Functional Analysis
Banach spaces, Hilbert spaces, bounded linear operators, spectral theory, Hahn-Banach theorem, open mapping theorem, uniform boundedness, and applications to differential equations.
Stochastic Processes
Markov chains, Poisson processes, Brownian motion, martingales, stochastic differential equations, Ito's lemma, queuing theory, and applications in finance and biology.
Topology
Point-set topology, open and closed sets, compactness, connectedness, separation axioms, metric spaces, fundamental group, covering spaces, homology, and topological data analysis.
Advanced Numerical Methods
Floating-point arithmetic, LU decomposition, iterative solvers, eigenvalue algorithms, interpolation, Gaussian quadrature, Runge-Kutta, finite elements, FFT, and optimization.
Advanced Combinatorics
Generating functions, Ramsey theory, extremal combinatorics, probabilistic method, Burnside and Polya enumeration, combinatorial designs, partition theory, and RSK correspondence.
Algebraic Topology
Homotopy theory, fundamental group, van Kampen's theorem, covering spaces, singular homology, Mayer-Vietoris, CW complexes, cohomology, Brouwer fixed-point, and TDA.
Differential Geometry
Curves and surfaces, first and second fundamental forms, Gaussian curvature, Gauss-Bonnet theorem, geodesics, Riemannian manifolds, Christoffel symbols, differential forms, and Lie groups.
Representation Theory
Group representations, Schur's lemma, character tables, orthogonality relations, irreducible representations, Young tableaux, SU(2) and SU(3), Lie algebra representations, and quantum mechanics applications.
Partial Differential Equations
Heat, wave, and Laplace equations; separation of variables, Fourier methods, method of characteristics, variational methods, Sobolev spaces, and applications to physics and finance.
Category Theory
Categories, functors, natural transformations, Yoneda lemma, adjoint functors, limits and colimits, monads, abelian categories, topos theory, and applications in programming and logic.
Advanced Complex Analysis
Cauchy's theorem, residue theorem, conformal mappings, Riemann mapping theorem, meromorphic functions, Picard's theorems, analytic continuation, Riemann surfaces, and applications.
Set Theory & Foundations
ZFC axioms, ordinal and cardinal numbers, Cantor's theorem, Axiom of Choice, Zorn's lemma, Godel's incompleteness theorems, forcing, and alternative foundations (HoTT, ETCS).
Harmonic Analysis
Fourier series on the circle, Fourier transform, tempered distributions, Hardy-Littlewood maximal function, Hilbert transform, Calderon-Zygmund theory, wavelets, and Pontryagin duality.
Analytic Number Theory
Riemann zeta function, prime number theorem, Dirichlet L-functions, sieve methods, exponential sums, circle method, modular forms, and the Riemann Hypothesis.
Calculus of Variations
Euler-Lagrange equation, brachistochrone, geodesics, minimal surfaces, Lagrangian and Hamiltonian mechanics, Noether's theorem, direct methods, and optimal control applications.
Ergodic Theory
Measure-preserving transformations, Poincare recurrence, ergodicity, Birkhoff's theorem, mixing, Kolmogorov-Sinai entropy, Weyl equidistribution, and Furstenberg's proof of Szemeredi's theorem.
Dynamical Systems
Phase portraits, stability theory, bifurcation theory, chaos and Lyapunov exponents, Lorenz system, discrete maps, Hamiltonian systems, KAM theorem, and applications in ecology and physics.
Algebraic Number Theory
Number fields, ring of integers, ideal theory, class group, Dirichlet unit theorem, quadratic and cyclotomic fields, p-adic numbers, and applications to Fermat Last Theorem and cryptography.
Convex Optimization
Convex sets and functions, Lagrangian duality, KKT conditions, linear and quadratic programming, SOCP, SDP, gradient descent, proximal methods, ADMM, and applications in machine learning and finance.
Abstract Algebra
Groups, rings, fields, Galois theory, Sylow theorems, polynomial rings, modules, classification of finite simple groups, and applications in coding theory and cryptography.
Numerical Stability and ODE Methods
Condition numbers, A-stability, Runge-Kutta methods, linear multistep methods, BDF for stiff systems, LU decomposition, Krylov subspace methods, and error analysis.
Information Theory
Shannon entropy, mutual information, KL divergence, source coding, channel capacity, Shannon-Hartley theorem, error-correcting codes, differential entropy, rate-distortion theory, and Fisher information.
Advanced Topology
Compactness, Tychonoff theorem, separation axioms, Urysohn lemma, Baire category theorem, homotopy theory, fundamental group, van Kampen, singular homology, cohomology, CW complexes, and Brouwer fixed point.
Game Theory
Nash equilibrium, mixed strategies, extensive form games, repeated games, Bayesian games, mechanism design, cooperative game theory, evolutionary game theory, zero-sum games, and applications in economics and biology.
Graph Theory
Trees, connectivity, Eulerian and Hamiltonian graphs, graph coloring, planar graphs, matching theory, network flows, spectral graph theory, random graphs, BFS and DFS algorithms, and applications.
Ocean Navigation Mathematics
Spherical trigonometry, celestial sphere, navigation triangle PZX, sight reduction, HO 229, great circle sailing, Mercator sailing, current triangle, tide harmonics, dead reckoning, and TVMDC.
Advanced Linear Algebra
Inner product spaces, spectral theorem, SVD, matrix norms, Jordan normal form, positive definite matrices, tensor products, exterior algebra, Lie algebras, randomized algorithms, and PCA.
Advanced Differential Equations
First and second-order ODEs, systems, Laplace transforms, power series, Sturm-Liouville, heat and wave equations, Laplace equation, nonlinear ODEs, Lotka-Volterra, and RLC circuits.
Mathematical Logic
Propositional and predicate logic, Godel completeness and incompleteness theorems, Turing machines, decidability, model theory, proof theory, type theory, Curry-Howard, and SAT solvers.
Fourier Analysis (Advanced)
Fourier transform, convolution theorem, DFT and FFT, Parseval identity, heat and wave equations, windowed Fourier transform, wavelets, Riemann-Lebesgue lemma, and signal processing applications.
Probability Theory (Advanced)
Measure-theoretic probability, convergence modes, law of large numbers, central limit theorem, martingales, Brownian motion, Markov chains, stochastic calculus, and concentration inequalities.
Mathematical Physics
Lagrangian and Hamiltonian mechanics, Maxwell equations, quantum mechanics, special functions, Green's functions, tensors, Lie groups, statistical mechanics, differential geometry, and path integrals.
Complex Analysis (Advanced)
Cauchy-Riemann equations, residue theorem, contour integration, conformal mappings, Riemann mapping theorem, analytic continuation, entire functions, Hadamard factorization, and Hardy spaces.
Partial Differential Equations
Heat, wave, and Laplace equations, d'Alembert formula, method of characteristics, Sobolev spaces, weak formulations, finite element method, Navier-Stokes, and conservation laws.
Measure Theory (Advanced)
Caratheodory extension, Lebesgue integral, convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Fubini-Tonelli, Borel regularity, and differentiation of measures.
Trigonometry & the Unit Circle
Unit circle definition, angles in standard position, radians vs degrees, all 16 key angles, special right triangles, reference angles, ASTC sign rule, Pythagorean identity, and graphs of sine and cosine.
Polynomial Functions
End behavior, zeros and multiplicity, polynomial division, Rational Zeros Theorem, Intermediate Value Theorem, graphing strategy, and real-world applications. Covers Stewart Chapter 3 content.
Analytic Trigonometry
Fundamental identities, verifying identities, sum and difference formulas, double-angle and half-angle formulas, product-to-sum formulas, inverse trig, and solving trig equations. Stewart Chapter 7.
Quadratic Functions
Standard form, vertex form, finding the vertex, axis of symmetry, max and min values, completing the square, converting between forms, graphing, and applications in projectile motion and optimization.
Exponential and Logarithmic Functions
Exponential growth and decay, natural base e, logarithm laws, change of base, solving exponential and log equations, half-life, doubling time, compound interest. Stewart Chapter 4.
Systems of Equations
Substitution and elimination methods, linear systems, nonlinear systems, systems of inequalities, Gaussian elimination, matrix methods, and applications in geometry and modeling. Stewart Chapter 10.
Double-Angle and Half-Angle Formulas
Double-angle formulas for sin, cos, and tan; three forms of cos(2A); half-angle formulas; power-reducing formulas; product-to-sum; sum-to-product; and proving identities.
Mathematical Induction
Principle of induction, base case and inductive step, strong induction, proving summation formulas, divisibility statements, inequalities, and well-ordering principle. Stewart Chapter 12.
Rational Functions
Domain and excluded values, vertical and horizontal asymptotes, slant asymptotes, holes, intercepts, graphing strategy, end behavior, and applications. Stewart Chapter 3.
Vectors and the Dot Product
Vector components, magnitude, unit vectors, dot product formula, angle between vectors, orthogonal vectors, vector projection, work as a dot product, and navigation applications. Stewart Chapter 9.
Conic Sections Equations
Parabolas (focus and directrix), ellipses (foci, eccentricity), hyperbolas (asymptotes), circles, completing the square from general form, identifying conics, and real-world applications. Stewart Chapter 11.
Polar Graphs and Limacons
Limacon types (cardioid, inner loop, dimpled, convex), symmetry tests, rose curves, lemniscates, circles and lines in polar form, graphing strategy, and converting to rectangular. Stewart Chapter 8.
Complex Numbers
Imaginary unit i, standard form, complex conjugate, FOIL multiplication, dividing by conjugate, powers of i, Argand diagram, polar form, De Moivre's theorem, nth roots, and Fundamental Theorem of Algebra.
Limits and Continuity
One-sided limits, limit existence, limits at infinity, infinite limits, limit laws, direct substitution, indeterminate forms, Squeeze Theorem, continuity types, IVT, and derivative preview. Stewart Chapter 13.
Sequences and Series
Arithmetic and geometric sequences, partial sums, infinite geometric series convergence, sigma notation, telescoping series, Fibonacci, alternating series, compound interest, and annuities. Stewart Chapter 12.
Function Models and Applications
Four ways to represent a function, average rate of change, difference quotient, linear and quadratic models, exponential growth, power functions, variation, and the 4-stage modeling process. Stewart Chapter 2.
Exponential Growth and Decay Models
Growth and decay formula, doubling time, half-life, Newton cooling, carbon-14 dating, compound and continuous interest, logistic growth, and 6 fully worked application problems. Stewart Chapter 4.6.
Right Triangle Applications
Angle of elevation and depression, bearing and navigation, Law of Sines (including ambiguous case), Law of Cosines, Heron formula, physics force components, and 3D problems. Stewart Chapter 6.
Logarithm Properties
Product, quotient, and power rules; change of base formula; cancellation laws; expanding and condensing expressions; solving log equations; natural log; and graphing. Stewart Chapter 4.3-4.4.
Trigonometric Identities
Reciprocal and quotient identities, all three Pythagorean forms, negative angle and cofunction identities, 7-step verification strategy, 10 worked proofs, and common mistakes. Stewart Chapter 7.
Inverse Functions
One-to-one functions, horizontal line test, 6-step algebraic method, domain restriction, graphical reflection, cancellation equations, inverse trig, and exponential-log inverse pair. Stewart Chapter 2.7.
Binomial Theorem
Pascal triangle, binomial coefficients, expansion formula, finding specific terms, sum identities (2 to the n, alternating sum), probability applications, and induction proof. Stewart Chapter 12.5.
Parametric Equations
Definition, eliminating the parameter, direction of motion, cycloid, line segments, conic sections in parametric form, projectile motion, particle motion, Lissajous figures. Stewart Chapter 8.
Matrices & Systems of Equations
Augmented matrices, elementary row operations, Gaussian elimination, Gauss-Jordan RREF, matrix inverse, Cramer rule, and determinants. Stewart Chapter 10.
Law of Sines & Law of Cosines
Oblique triangles, AAS/ASA/SSA/SAS/SSS cases, ambiguous case, Heron formula, area using sine, navigation bearing problems, and triangulation. Stewart Chapter 6.
Function Transformations
Vertical and horizontal shifts, reflections over x-axis and y-axis, vertical and horizontal stretch and shrink, order of transformations, even and odd functions, and toolkit parent functions. Stewart Chapter 2.
Polar Coordinates
Polar system, plotting (r, theta), multiple representations, rectangular conversion, limacons, rose curves, lemniscate, symmetry tests, and area in polar coordinates. Stewart Chapter 8.
Probability & Counting
Fundamental counting principle, permutations, combinations, factorial, classical probability, addition rule, multiplication rule, conditional probability, expected value, and binomial probability. Stewart Chapter 13.
Solving Trigonometric Equations
Linear trig equations, general solutions, multiple angles, factoring, Pythagorean identity substitutions, squaring both sides, extraneous solutions, and calculator methods. Stewart Chapter 7.
Piecewise Functions
Definition, evaluating by domain, graphing with open and closed circles, continuity conditions, absolute value, greatest integer floor function, step functions, and real-world models. Stewart Chapter 2.
Inequalities
Linear inequalities, compound inequalities, absolute value inequalities, polynomial and rational sign charts, quadratic inequalities, interval notation, and real-world applications. Stewart Chapter 1.
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