Precalculus

Theory part 15Practice part 16

average rate of change, secant slope, output change divided by nonzero input change, units, signed rate and comparing intervals

Builds on Linear Functions and Constant Rate of Change and Reading Graphs and Function Behavior.

Theory part 17Practice part 18

piecewise definition, branch domain, endpoints, open and closed graph points, evaluation, jumps and overlapping and missing branch conditions

Builds on Sets, Intervals, and Inequalities, Function Notation and Evaluation and Reading Graphs and Function Behavior.

Theory part 19Practice part 20

absolute value as distance, piecewise definition, V-shaped graph, equations, inequalities, inside versus outside an interval, zero and negative comparison values and extraneous candidates

Builds on Sets, Intervals, and Inequalities and Piecewise-Defined Functions.

Theory part 21Practice part 22

even function, odd function, symmetric domain, y-axis symmetry, origin symmetry, algebraic test, neither and zero function is both on a symmetric domain

Builds on Natural and Specified Domains, Function Notation and Evaluation and Reading Graphs and Function Behavior.

Theory part 23Practice part 24

parent function, rigid transformation, vertical shift, horizontal shift and reflection across axes

Builds on Function Notation and Evaluation and Reading Graphs and Function Behavior.

Theory part 25Practice part 26

vertical stretch, horizontal compression, scale factor, dilation and multiplier

Builds on Shifts and Reflections of Graphs.

Theory part 27Practice part 28

sum, difference, product, quotient, domain intersection, nonzero divisor, graph of pointwise operations and restrictions survive cancellation

Builds on Natural and Specified Domains and Function Notation and Evaluation.

Theory part 29Practice part 30

composition, inner function, outer function, ordered evaluation, noncommutativity, composite domain and decomposing a function

Builds on Algebra of Functions.

Theory part 31Practice part 32

one-to-one, injective, distinct inputs give distinct outputs, horizontal line test and strict monotonicity as a sufficient condition

Builds on Range and Codomain and Reading Graphs and Function Behavior.

Theory part 33Practice part 34

inverse mapping, inverse versus reciprocal, domain and range swap, reflection across y=x, solving for the inverse, restriction to an injective branch and composition identities

Builds on Function Composition and One-to-One Functions and the Horizontal Line Test.

Theory part 35Practice part 36

quadratic function, vertex, axis of symmetry, standard and vertex forms, factoring, completing the square, quadratic formula, discriminant, real zeros and modeling constraints

Builds on Linear Functions and Constant Rate of Change and Shifts and Reflections of Graphs.

Theory part 37Practice part 38

distance formula, midpoint formula, circle, center, radius, standard equation, general equation and completing the square

Builds on Quadratic Functions and Vertex Form.

Theory part 39Practice part 40

power function, integer exponent, rational exponent, principal root, even and odd roots, domain, inverse relationship and sqrt(x squared)=absolute value of x

Builds on Natural and Specified Domains and Inverse Functions and Domain Restriction.

Theory part 41Practice part 42

original domain, least common denominator, equivalent transformations, squaring, extraneous solutions and checking candidates in the original equation

Builds on Natural and Specified Domains, Quadratic Functions and Vertex Form and Power Functions and Root Functions.

Theory part 43Practice part 44

degree, even/odd degree, leading coefficient test and limits at infinity intuition

Builds on Stretches and Compressions of Graphs and Power Functions and Root Functions.

Theory part 45Practice part 46

zero, root, x-intercept, multiplicity, sign change for odd multiplicity, no sign change for even multiplicity, local flattening and real versus nonreal roots

Builds on Quadratic Functions and Vertex Form and End Behavior of Polynomials.

Theory part 47Practice part 48

long division, synthetic division, dividend, divisor, quotient and remainder

Builds on Quadratic Functions and Vertex Form.

Theory part 49Practice part 50

Remainder Theorem, Factor Theorem, evaluating polynomials and extracting linear factors

Builds on Polynomial Division.

Theory part 51Practice part 52

rational root candidates, reduced fraction p/q, factors of constant and leading coefficients, testing candidates, synthetic division and theorem gives candidates, not all roots

Builds on Zeros and Multiplicity of Polynomials and The Remainder and Factor Theorems.

Theory part 53Practice part 54

imaginary unit i, i squared=-1, rectangular form a+bi, real and imaginary parts, addition, multiplication, conjugate, division, complex plane, modulus and quadratic roots

Builds on Quadratic Functions and Vertex Form and Power Functions and Root Functions.

Theory part 55Practice part 56

nonconstant complex polynomial, existence of a complex zero, degree counts roots with multiplicity, complex factorization, real coefficients and conjugate pairs and real irreducible quadratic factors

Builds on Rational Zeros Theorem and Factoring and Complex Numbers and the Complex Plane.

Theory part 57Practice part 58

rational function, numerator, denominator, original domain, factors, cancellation, removable hole and excluded value retained in simplified expression

Builds on Natural and Specified Domains and Polynomial Division.

Theory part 59Practice part 60

vertical asymptote, one-sided unbounded behavior, denominator zero after cancellation, removable hole, sign analysis near an excluded value and infinity is not a function value

Builds on Zeros and Multiplicity of Polynomials and Rational Functions and Domain Restrictions.

Theory part 61Practice part 62

horizontal asymptote, dominant terms, degree comparison, separate positive and negative directions and graph may cross an asymptote

Builds on End Behavior of Polynomials and Rational Functions and Domain Restrictions.

Theory part 63Practice part 64

polynomial division, oblique asymptote, degree difference one, remainder over divisor tends to zero and higher-degree polynomial asymptote as extension

Builds on Polynomial Division and Horizontal Asymptotes.

Theory part 65Practice part 66

intercepts, test intervals, approaching asymptotes and crossing horizontal asymptotes

Builds on Vertical Asymptotes and Holes, Horizontal Asymptotes and Slant (Oblique) Asymptotes.

Theory part 67Practice part 68

polynomial inequality, critical values, test points, sign chart and number line

Builds on Sets, Intervals, and Inequalities and Zeros and Multiplicity of Polynomials.

Theory part 69Practice part 70

rational inequality, critical intervals and undefined points vs zero points

Builds on Rational Functions and Domain Restrictions and Polynomial Inequalities.

Theory part 71Practice part 72

exponential function, positive base other than one, growth factor, equal input increments, equal output ratios, growth, decay, domain, range and horizontal asymptote

Builds on Stretches and Compressions of Graphs, Power Functions and Root Functions and Average Rate of Change and Secant Lines.

Theory part 73Practice part 74

number e, compound interest, finite compounding, continuous-compounding idea, numerical convergence, e approximately 2.71828 and approximation versus proof

Builds on Exponential Functions and Growth/Decay.

Theory part 75Practice part 76

logarithm as exponent, positive base other than one, positive argument, inverse relationship, common logarithm, natural logarithm, domain and range

Builds on Inverse Functions and Domain Restriction and Exponential Functions and Growth/Decay.

Theory part 77Practice part 78

log graph, exponential graph, reflections, translations and domains and ranges

Builds on Shifts and Reflections of Graphs, Stretches and Compressions of Graphs and Logarithmic Functions as Inverses.

Theory part 79Practice part 80

logarithm of a product, logarithm of a quotient, positive arguments, common base, expanding, condensing and sums do not split into logarithms

Builds on Logarithmic Functions as Inverses.

Theory part 81Practice part 82

power law with positive argument, change of base, valid logarithm bases, exact expressions and calculator approximations

Builds on Product and Quotient Properties of Logarithms.

Theory part 83Practice part 84

common base, isolating exponential, applying ln and exact form

Builds on The Power Rule and Change of Base for Logarithms.

Theory part 85Practice part 86

exponential form conversion, extraneous solutions and domain validation

Builds on The Power Rule and Change of Base for Logarithms and Solving Rational and Radical Equations.

Theory part 87Practice part 88

growth and decay model, initial value, percent rate and growth factor, discrete versus continuous rate, half-life, doubling time, compound interest, logarithmic scales and interpreting model assumptions

Builds on The Natural Base e, Solving Exponential Equations and Solving Logarithmic Equations.

Theory part 89Practice part 90

variables and parameters, units, model assumptions, linear versus exponential growth, residual, interpolation, extrapolation, exact and approximate values and data limitations

Builds on Average Rate of Change and Secant Lines, Reading Graphs and Function Behavior and Exponential Growth and Decay Models.

Theory part 91Practice part 92

logistic function, carrying capacity, positive parameters, early growth, saturation, symmetry about midpoint and model interpretation without differential equations

Builds on Exponential Growth and Decay Models and Choosing and Interpreting Function Models.

Theory part 93Practice part 94

standard position, degree, radian, converting units and coterminal angle

Builds on Sets, Intervals, and Inequalities.

Theory part 95Practice part 96

arc length, sector area, angular speed and linear speed

Builds on Angles, Radians, and Coterminal Angles.

Theory part 97Practice part 98

right triangle, similar triangles, opposite and adjacent sides, hypotenuse, six trigonometric ratios, angle of elevation, angle of depression and contextual units

Builds on Quadratic Functions and Vertex Form and Angles, Radians, and Coterminal Angles.

Theory part 99Practice part 100

30-60-90, 45-45-90, exact ratios and rationalizing denominators

Builds on Right Triangle Trigonometry.

Theory part 101Practice part 102

general angle, terminal point, radius, reference angle, quadrant signs, exact values, six trigonometric functions and excluded denominators

Builds on Right Triangle Trigonometry and Special Right Triangles.

Theory part 103Practice part 104

unit circle, coordinates cos theta and sin theta, radian arc length, periodicity, signed projections and exact values

Builds on Angles, Radians, and Coterminal Angles and General Angles and Reference Angles.

Theory part 105Practice part 106

sine wave, cosine wave, periodic function, period, amplitude and midline

Builds on The Unit Circle and Reading Graphs and Function Behavior.

Theory part 107Practice part 108

amplitude, midline, period, frequency, horizontal shift, A sin(B(x-C))+D, positive and negative parameters and fitting a sinusoidal model

Builds on Stretches and Compressions of Graphs and Graphs of Sine and Cosine.

Theory part 109Practice part 110

tangent graph, cotangent graph, vertical asymptotes and period pi

Builds on General Angles and Reference Angles and Graphs of Sine and Cosine.

Theory part 111Practice part 112

secant graph, cosecant graph, reciprocal functions and U-shaped branches

Builds on General Angles and Reference Angles and Graphs of Sine and Cosine.

Theory part 113Practice part 114

simple harmonic motion, displacement, equilibrium position, amplitude, period, phase, sinusoidal model and damping as an optional exponential-envelope example

Builds on Amplitude, Period, and Phase Shift and Exponential Growth and Decay Models.

Theory part 115Practice part 116

Pythagorean identities, reciprocal identities, quotient identities, parity identities, common domain and unit-circle derivation

Builds on The Unit Circle, Graphs of Tangent and Cotangent and Graphs of Secant and Cosecant.

Theory part 117Practice part 118

principal branch, arcsine, arccosine, domain [-1,1], restricted sine and cosine, principal output ranges and inverse versus reciprocal

Builds on Inverse Functions and Domain Restriction and Graphs of Sine and Cosine.

Theory part 119Practice part 120

arctangent, real input and principal range, inverse cotangent, inverse secant, inverse cosecant, explicit branch conventions and reciprocal-function domains

Builds on Inverse Functions and Domain Restriction, Graphs of Tangent and Cotangent and Graphs of Secant and Cosecant.

Theory part 121Practice part 122

composition of trigonometric and inverse functions, principal range, exact evaluation, reference triangle, absolute values and branch-dependent simplification

Builds on Inverse Sine and Cosine Functions, Inverse Tangent and Reciprocal Trigonometric Functions and Fundamental Trigonometric Identities.

Theory part 123Practice part 124

identity versus equation, common domain, rewriting one side, factoring, common denominators, conjugate and numerical evidence is not proof

Builds on Fundamental Trigonometric Identities.

Theory part 125Practice part 126

sum formula, difference formula and cofunction proof

Builds on Fundamental Trigonometric Identities.

Theory part 127Practice part 128

double-angle, sin(2x) and cos(2x) three forms

Builds on Sum and Difference Formulas.

Theory part 129Practice part 130

power reduction, half-angle identities, square-root sign chosen by angle and domain conditions

Builds on Double-Angle Formulas.

Theory part 131Practice part 132

product-to-sum identities, sum-to-product identities, derivation from angle addition and oscillation and beat examples

Builds on Sum and Difference Formulas.

Theory part 133Practice part 134

isolating a trigonometric function, principal values, all solutions in an interval, general periodic solutions and excluding undefined inputs

Builds on Inverse Sine and Cosine Functions and Inverse Tangent and Reciprocal Trigonometric Functions.

Theory part 135Practice part 136

factoring trigonometric equations, substitution by a trig expression, multiple angles, full interval for the multiple angle, extraneous candidates and domain checks

Builds on Quadratic Functions and Vertex Form, Verifying Trigonometric Identities, Double-Angle Formulas and Solving Linear Trigonometric Equations.

Theory part 137Practice part 138

oblique triangle, opposite side and angle, law of sines, AAS and ASA, SSA ambiguity and zero, one, or two triangles

Builds on Right Triangle Trigonometry, General Angles and Reference Angles and Inverse Sine and Cosine Functions.

Theory part 139Practice part 140

law of cosines, SAS and SSS, included angle, triangle inequality, area one-half ab sin C, Heron formula and bearings

Builds on Sum and Difference Formulas and The Law of Sines and the Ambiguous Case.

Theory part 141Practice part 142

simultaneous equations, substitution, elimination, intersections, inconsistent systems, dependent systems, unique, no, or infinitely many solutions and units

Builds on Linear Functions and Constant Rate of Change and Quadratic Functions and Vertex Form.

Theory part 143Practice part 144

line and parabola, circle and line, substitution, elimination, intersection points, tangency, checking all equations and extraneous solutions

Builds on Solving Rational and Radical Equations, Linear Systems in Two and Three Variables and Circles, Distance, and Midpoints.

Theory part 145Practice part 146

boundary curve, solid and dashed boundary, test point, half-plane, feasible region, intersection, union and linear and nonlinear inequalities

Builds on Sets, Intervals, and Inequalities, Rational Inequalities, Reading Graphs and Function Behavior and Nonlinear Systems and Curve Intersections.

Theory part 147Practice part 148

proper rational function, polynomial division first, distinct linear factors, repeated linear factors, coefficient matching, solving a linear system and original-domain equality

Builds on Polynomial Division, Rational Functions and Domain Restrictions and Linear Systems in Two and Three Variables.

Theory part 149Practice part 150

irreducible real quadratic factor, linear numerator, repeated powers, coefficient matching, proper rational function and real factorization

Builds on The Fundamental Theorem of Algebra and Partial Fraction Decomposition (Linear Factors).

Theory part 151Practice part 152

vector, magnitude, direction, directed line segment and parallelogram law

Builds on Right Triangle Trigonometry and General Angles and Reference Angles.

Theory part 153Practice part 154

component form, standard unit vectors and scalar multiplication

Builds on Linear Functions and Constant Rate of Change and Geometric Vectors and Vector Addition.

Theory part 155Practice part 156

dot product, angle between nonzero vectors, orthogonality, component formula, geometric formula, zero-vector caveat and parallel vectors

Builds on Algebraic Vectors and Component Form and The Law of Cosines and Triangle Area.

Theory part 157Practice part 158

scalar component, vector projection onto a nonzero vector, orthogonal decomposition, constant-force work, units and sign of work

Builds on The Dot Product and Orthogonality.

Theory part 159Practice part 160

matrix, dimensions, scalar multiplication and matrix addition

Builds on Linear Systems in Two and Three Variables.

Theory part 161Practice part 162

compatible dimensions, row-column dot product, composition of linear maps, noncommutativity, identity matrix and zero matrix

Builds on The Dot Product and Orthogonality and Introduction to Matrices and Matrix Addition.

Theory part 163Practice part 164

augmented matrix, elementary row operations, row echelon form, reduced row echelon form, pivots, free variables, back-substitution and inconsistent row

Builds on Introduction to Matrices and Matrix Addition and Linear Systems in Two and Three Variables.

Theory part 165Practice part 166

2 by 2 and 3 by 3 determinants, signed area and volume scaling, orientation, expansion by minors, zero determinant, singularity and Cramer rule as an optional method

Builds on Matrix Multiplication and Gaussian Elimination and Solution Structure.

Theory part 167Practice part 168

inverse matrix, identity, square and nonsingular conditions, 2 by 2 inverse formula, matrix equation, solving a system and inverse versus entrywise reciprocal

Builds on Determinants and Their Geometric Meaning.

Theory part 169Practice part 170

pole, polar axis, radius, angle, negative radius, nonunique coordinates, origin and plotting points

Builds on Angles, Radians, and Coterminal Angles and The Unit Circle.

Theory part 171Practice part 172

x=r cos theta, y=r sin theta, r squared=x squared+y squared, atan2 and quadrant choice, point conversion, equation conversion and domain and tracing checks

Builds on Quadratic Functions and Vertex Form and Introduction to Polar Coordinates.

Theory part 173Practice part 174

modulus, argument, polar representation, cis notation, nonunique arguments, principal argument convention, zero has no unique argument and rectangular conversion

Builds on Complex Numbers and the Complex Plane and Converting Between Polar and Rectangular Coordinates.

Theory part 175Practice part 176

complex multiplication, complex division by a nonzero number, modulus multiplication and division, argument addition and subtraction and rotation and scaling

Builds on Sum and Difference Formulas and Polar Form of Complex Numbers.

Theory part 177Practice part 178

De Moivre theorem for integer powers, nth roots of nonzero complex numbers, n distinct roots, roots of unity, regular polygons and angular spacing

Builds on Products and Quotients in Polar Form.

Theory part 179Practice part 180

symmetry tests, limacon, cardioid and maximum r

Builds on Amplitude, Period, and Phase Shift and Converting Between Polar and Rectangular Coordinates.

Theory part 181Practice part 182

rose curves, integer angular frequency, odd/even petal counts under stated form, lemniscates, negative radius, valid real branches and tracing interval

Builds on Power-Reducing and Half-Angle Formulas and Polar Graphs of Limacons and Cardioids.

Theory part 183Practice part 184

conic section, parabola, focus, directrix, vertex and latus rectum

Builds on Circles, Distance, and Midpoints.

Theory part 185Practice part 186

ellipse as constant sum of distances, foci, semimajor and semiminor axes, center, c squared=a squared-b squared, eccentricity and circle as special case

Builds on Circles, Distance, and Midpoints.

Theory part 187Practice part 188

hyperbola as constant absolute difference of distances, foci, transverse and conjugate axes, vertices, asymptotes, c squared=a squared+b squared and eccentricity

Builds on Circles, Distance, and Midpoints and Slant (Oblique) Asymptotes.

Theory part 189Practice part 190

translated conics, completing the square and shifting vertices

Builds on Parabolas as Geometric Loci, Ellipses as Geometric Loci and Hyperbolas as Geometric Loci.

Theory part 191Practice part 192

rotated coordinates, mixed xy term, angle convention, classification by discriminant with degeneracy caveats and geometric rotation

Builds on Sum and Difference Formulas and Shifted Conics and Completing the Square.

Theory part 193Practice part 194

focus at the pole, eccentricity e and directrix distance d

Builds on Converting Between Polar and Rectangular Coordinates, Parabolas as Geometric Loci, Ellipses as Geometric Loci and Hyperbolas as Geometric Loci.

Theory part 195Practice part 196

parameter, paired coordinate functions, parameter interval, orientation, tracing speed and same curve with different parameterizations

Builds on Function Notation and Evaluation and The Unit Circle.

Theory part 197Practice part 198

elimination by substitution or identity, rectangular equation, inherited restrictions, orientation information lost and extraneous branches

Builds on Fundamental Trigonometric Identities and Parametric Equations and Oriented Paths.

Theory part 199Practice part 200

3D space, octants, distance formula in 3D and spheres

Builds on Circles, Distance, and Midpoints and Algebraic Vectors and Component Form.

Theory part 201Practice part 202

sequence as a function of an integer index, indexing convention, explicit term, finite and infinite sequences, factorial and arithmetic on terms

Builds on Function Notation and Evaluation and Power Functions and Root Functions.

Theory part 203Practice part 204

initial value, recursive rule, Fibonacci recurrence, generating terms, explicit versus recursive description and recursion alone does not guarantee convergence

Builds on Introduction to Sequences and Factorials.

Theory part 205Practice part 206

summation index, summand, lower and upper bounds, dummy index, finite sum, partial sum, sum rules, reindexing and sequence versus series

Builds on Introduction to Sequences and Factorials.

Theory part 207Practice part 208

common difference, explicit nth term, arithmetic partial sums, endpoint averaging, pairing argument and indexing

Builds on Linear Functions and Constant Rate of Change and Sigma Notation and Partial Sums.

Theory part 209Practice part 210

common ratio, explicit nth term, finite geometric sum, r=1 exception, derivation by subtraction and applications

Builds on Exponential Functions and Growth/Decay and Sigma Notation and Partial Sums.

Theory part 211Practice part 212

partial sums, convergence, divergence, absolute value of ratio less than one, infinite geometric sum, r=1 and r=-1 and repeating decimals

Builds on Geometric Sequences and Series.

Theory part 213Practice part 214

statement indexed by integers, base case, inductive hypothesis, inductive step, starting index and proof versus checking examples

Builds on Introduction to Sequences and Factorials and Sigma Notation and Partial Sums.

Theory part 215Practice part 216

nonnegative integer exponent, binomial theorem, Pascal triangle, binomial coefficient, factorial formula, combinations and finite expansion versus infinite binomial series

Builds on Introduction to Sequences and Factorials and Sigma Notation and Partial Sums.

Theory part 217Practice part 218

sequence limit intuition, convergence, divergence, oscillation, bounded does not imply convergent, geometric examples and numerical evidence versus proof

Builds on Recursive Sequences and the Fibonacci Sequence and Infinite Geometric Series and Convergence.

Theory part 219Practice part 220

nearby inputs, function value versus limiting value, left and right approach, numerical tables, graph window, nonexistence and evidence versus proof

Builds on Piecewise-Defined Functions, Vertical Asymptotes and Holes, Horizontal Asymptotes and Sequence Behavior and Limiting Values.

Theory part 221Practice part 222

difference quotient, h nonzero, original domain, algebraic simplification, shrinking secants, instantaneous-rate limit when it exists and corners and failure of a single tangent slope

Builds on Average Rate of Change and Secant Lines and Approaching a Limit Numerically and Graphically.

Theory part 223Practice part 224

nonnegative function, finite partition, left and right rectangles, lower and upper estimates under stated monotonicity, refinement, area intuition and finite sum versus limiting area

Builds on Sigma Notation and Partial Sums and Approaching a Limit Numerically and Graphically.