Foundations for Calculus

Theory part 5Practice part 6

function, input, output, domain, function notation, rule, table and unique output

Builds on Reading Formulas and Substituting Values and Number Lines, Intervals, and Allowed Values.

Theory part 7Practice part 8

coordinate axes, ordered pair, horizontal coordinate, vertical coordinate, graph, scale, input and output

Builds on Functions as Input-Output Relationships and Number Lines, Intervals, and Allowed Values.

Theory part 9Practice part 10

straight line, slope, rise, run, intercept, slope-intercept form, horizontal line and vertical line

Builds on Reading Graphs and Plotting Points and Reading Formulas and Substituting Values.

Theory part 11Practice part 12

changing quantity, initial value, change in input, change in output, average rate of change, units of rate and secant slope

Builds on Drawing Lines from Slope and Intercepts and Functions as Input-Output Relationships.

Theory part 13Practice part 14

absolute value, distance, nonnegative, number line, graph of absolute value, vertex and symmetry

Builds on Number Lines, Intervals, and Allowed Values, Reading Graphs and Plotting Points and Drawing Lines from Slope and Intercepts.

Theory part 15Practice part 16

term, factor, distributive property, expansion, factorization, square of a sum, difference of squares and equivalent expressions

Builds on Reading Formulas and Substituting Values.

Theory part 17Practice part 18

factor, cancellation, denominator, excluded value, equivalent expression, original domain and zero numerator

Builds on Expanding and Factoring for a Purpose and Number Lines, Intervals, and Allowed Values.

Theory part 19Practice part 20

common denominator, equivalent fraction, numerator, complex fraction, division by a fraction and restriction

Builds on Cancellation and Excluded Values and Expanding and Factoring for a Purpose.

Theory part 21Practice part 22

base, exponent, negative exponent, principal square root, real root, rational exponent, parentheses and absolute value

Builds on Reading Formulas and Substituting Values and Absolute Value as Distance and a Graph.

Theory part 23Practice part 24

equation, solution, equivalent step, candidate solution, restriction, squaring and checking in the original equation

Builds on Cancellation and Excluded Values and Powers, Roots, and Signs.

Theory part 25Practice part 26

inequality, solution set, order reversal, strict inequality, weak inequality, compound inequality and interval notation

Builds on Number Lines, Intervals, and Allowed Values and Solving Equations and Checking Solutions.

Theory part 27Practice part 28

conjugate, difference of squares, radical, rationalizing, equivalent expression and domain restriction

Builds on Powers, Roots, and Signs, Cancellation and Excluded Values and Expanding and Factoring for a Purpose.

Theory part 29Practice part 30

completing the square, square term, constant adjustment, leading coefficient, equivalent form and minimum value

Builds on Expanding and Factoring for a Purpose and Powers, Roots, and Signs.

Theory part 31Practice part 32

quadratic equation, coefficient, quadratic formula, discriminant, real solution, repeated root and plus or minus

Builds on Completing the Square and Solving Equations and Checking Solutions.

Theory part 33Practice part 34

subject of a formula, parameter, inverse operation, nonzero condition, units and admissible value

Builds on Solving Equations and Checking Solutions and Solving Inequalities on a Number Line.

Theory part 35Practice part 36

domain, real-valued function, natural domain, stated domain, radicand, denominator restriction and intersection of conditions

Builds on Functions as Input-Output Relationships, Solving Inequalities on a Number Line, Powers, Roots, and Signs and Cancellation and Excluded Values.

Theory part 37Practice part 38

range, attained output, domain restriction, minimum, maximum, endpoint inclusion and output interval

Builds on Finding the Domain, Reading Graphs and Plotting Points and Absolute Value as Distance and a Graph.

Theory part 39Practice part 40

quadratic function, parabola, vertex, axis of symmetry, intercept, opening direction, range and equivalent forms

Builds on Completing the Square, The Quadratic Formula and Real Solutions, Finding the Range and Reading Graphs and Plotting Points.

Theory part 41Practice part 42

integer-power function, cubic function, even power, odd power, symmetry, increasing function, domain and range

Builds on Powers, Roots, and Signs, Finding the Range and Reading Graphs and Plotting Points.

Theory part 43Practice part 44

principal square-root function, radicand, domain, range, endpoint, increasing, decreasing and coordinate pair

Builds on Powers, Roots, and Signs, Finding the Domain, Finding the Range and Reading Graphs and Plotting Points.

Theory part 45Practice part 46

real cube root, cube-root function, negative input, domain, range, origin symmetry, coordinate pair and increasing

Builds on Cubic and Other Integer-Power Functions, Powers, Roots, and Signs, Finding the Domain and Finding the Range.

Theory part 47Practice part 48

reciprocal function, inverse proportion, nonzero input, positive and negative branches, origin symmetry, vertical asymptote, horizontal asymptote, domain and range

Builds on Reading Graphs and Plotting Points, Cancellation and Excluded Values, Finding the Domain and Finding the Range.

Theory part 49Practice part 50

reciprocal-square function, negative integer power, positive output, even symmetry, two branches, domain, range, vertical asymptote, horizontal asymptote and inverse-square change

Builds on The Reciprocal Function, Powers, Roots, and Signs and Reading Graphs and Plotting Points.

Theory part 51Practice part 52

piecewise function, input condition, branch, rule selection, open endpoint, closed endpoint, switching input, matching branch values and jump

Builds on Functions as Input-Output Relationships, Reading Graphs and Plotting Points, Drawing Lines from Slope and Intercepts, Finding the Domain and Absolute Value as Distance and a Graph.

Theory part 53Practice part 54

polynomial, zero, factor, multiplicity, sign chart, crossing the axis, touching the axis, leading term and end behavior

Builds on Expanding and Factoring for a Purpose, Reading Graphs and Plotting Points, The Quadratic Function and Its Graph, Cubic and Other Integer-Power Functions and Solving Equations and Checking Solutions.

Theory part 55Practice part 56

sign chart, critical input, zero of a factor, denominator exclusion, positive and negative interval, product inequality, rational inequality and endpoint test

Builds on Solving Inequalities on a Number Line, Polynomial Zeros, Signs, and Graph Shape and Cancellation and Excluded Values.

Theory part 57Practice part 58

rational function, denominator restriction, reciprocal graph, hole, vertical asymptote, horizontal asymptote, separate branches and intercept

Builds on The Reciprocal Function, The Reciprocal-Square Function, Cancellation and Excluded Values, Sign Charts for Products and Quotients and Polynomial Zeros, Signs, and Graph Shape.

Theory part 59Practice part 60

parent graph, horizontal shift, vertical shift, reflection in the x-axis, reflection in the y-axis, image of a point and new domain

Builds on The Quadratic Function and Its Graph, The Square-Root Function, The Cube-Root Function, The Reciprocal Function and Finding the Domain.

Theory part 61Practice part 62

vertical scaling, horizontal scaling, stretch, compression, scale factor, input scaling, output scaling and point mapping

Builds on Shifting and Reflecting Graphs, Powers, Roots, and Signs, The Square-Root Function and The Quadratic Function and Its Graph.

Theory part 63Practice part 64

sum of functions, difference of functions, product of functions, quotient of functions, pointwise operation, common domain, denominator function and domain restriction

Builds on Functions as Input-Output Relationships, Finding the Domain, Finding the Range, Cancellation and Excluded Values and The Square-Root Function.

Theory part 65Practice part 66

function composition, inner function, outer function, composite function, composite domain, order of operations and output used as input

Builds on Functions as Input-Output Relationships, Finding the Domain, Adding, Multiplying, and Dividing Functions, Powers, Roots, and Signs and Making a Variable the Subject.

Theory part 67Practice part 68

inverse function, undoing a function, one-to-one, horizontal line test, restricted domain, swapped coordinates, reflection across y=x and inverse versus reciprocal

Builds on Composing Functions, Finding the Domain, Finding the Range, The Quadratic Function and Its Graph, The Square-Root Function and Making a Variable the Subject.

Theory part 69Practice part 70

curve intersection, simultaneous equations, common point, same input, same output, vertical difference, which curve is higher, extraneous intersection and interval splitting

Builds on Reading Graphs and Plotting Points, The Quadratic Function and Its Graph, Solving Equations and Checking Solutions, The Quadratic Formula and Real Solutions, The Square-Root Function and Sign Charts for Products and Quotients.

Theory part 71Practice part 72

circle, center, radius, circle equation, Pythagorean relationship, upper semicircle, lower semicircle, branch, principal square root, domain and range

Builds on Powers, Roots, and Signs, Solving Equations and Checking Solutions, Finding the Domain, Finding the Range, The Square-Root Function and Making a Variable the Subject.

Theory part 73Practice part 74

exponential function, base, growth factor, decay factor, fixed-factor change, positive output, y-intercept, horizontal asymptote and contextual domain

Builds on Powers, Roots, and Signs, Reading Graphs and Plotting Points, Finding the Domain and Finding the Range.

Theory part 75Practice part 76

natural exponential, number e, positive base, fixed-factor growth, exponential decay, exact value, approximate value, horizontal asymptote and scaled exponential

Builds on Exponential Functions with Different Bases, Powers, Roots, and Signs and Stretching and Compressing Graphs.

Theory part 77Practice part 78

logarithm, base, argument, exponent as output, inverse relationship, logarithmic graph, positive argument, vertical asymptote and increasing and decreasing logarithms

Builds on Exponential Functions with Different Bases, Inverse Functions and Domain Restrictions, Finding the Domain and Finding the Range.

Theory part 79Practice part 80

natural logarithm, base e, inverse exponential, positive input, zero at one, negative logarithm values, shifted logarithm and vertical asymptote

Builds on Logarithms as Inverse Functions, The Natural Exponential Function, Shifting and Reflecting Graphs, Finding the Domain and Finding the Range.

Theory part 81Practice part 82

product of powers, quotient of powers, power of a power, logarithm product law, logarithm quotient law, logarithm power law, change of base and positive arguments

Builds on Logarithms as Inverse Functions, The Natural Logarithm Function, Expanding and Factoring for a Purpose, Cancellation and Excluded Values and Powers, Roots, and Signs.

Theory part 83Practice part 84

ln absolute x, absolute value inside a logarithm, nonzero domain, positive argument, two branches, even symmetry, piecewise description and logarithmic identity

Builds on The Natural Logarithm Function, Using Exponent and Logarithm Laws, Absolute Value as Distance and a Graph, Reading and Drawing Piecewise Functions and Shifting and Reflecting Graphs.

Theory part 85Practice part 86

exponential equation, logarithmic equation, common base, taking logarithms, one-to-one exponential, candidate solution, argument restriction and checking the original equation

Builds on Using Exponent and Logarithm Laws, The Natural Logarithm Function, Solving Equations and Checking Solutions and Solving Inequalities on a Number Line.

Theory part 87Practice part 88

angle, radian, degree, radius, arc length, full turn, signed rotation, counterclockwise, clockwise, coterminal angles and angle conversion

Builds on Reading Graphs and Plotting Points, Powers, Roots, and Signs and Circles, Semicircles, and Choosing a Branch.

Theory part 89Practice part 90

unit circle, coordinates, sine coordinate, cosine coordinate, reference angle, quadrant signs, exact value, special right triangle, coterminal angles and circle equation

Builds on Radians and Angle Measurement, Circles, Semicircles, and Choosing a Branch, Powers, Roots, and Signs and Reading Graphs and Plotting Points.

Theory part 91Practice part 92

sine, unit-circle height, radian input, sine graph, period, odd symmetry, zeros, amplitude and range

Builds on Unit-Circle Coordinates and Exact Values and Stretching and Compressing Graphs.

Theory part 93Practice part 94

cosine, unit-circle horizontal coordinate, cosine graph, even symmetry, period, zeros, amplitude and midline

Builds on Unit-Circle Coordinates and Exact Values, The Sine Function and Stretching and Compressing Graphs.

Theory part 95Practice part 96

tangent, sine-to-cosine ratio, undefined input, tangent graph, period, zero, vertical asymptote and principal branch

Builds on The Sine Function, The Cosine Function and The Reciprocal Function.

Theory part 97Practice part 98

cotangent, cosine-to-sine ratio, cotangent graph, undefined input, period, zero, vertical asymptote and reciprocal identity

Builds on The Tangent Function, The Sine Function and The Cosine Function.

Theory part 99Practice part 100

secant, reciprocal cosine, secant graph, excluded input, range, vertical asymptote, even symmetry and period

Builds on The Cosine Function, The Reciprocal Function and Stretching and Compressing Graphs.

Theory part 101Practice part 102

cosecant, reciprocal sine, cosecant graph, excluded input, range, vertical asymptote, odd symmetry and period

Builds on The Sine Function and The Secant Function.

Theory part 103Practice part 104

trigonometric identity, Pythagorean identity, quotient identity, reciprocal identity, common domain, replacement and cancellation

Builds on Unit-Circle Coordinates and Exact Values, The Cotangent Function, The Secant Function, The Cosecant Function and Common Denominators and Complex Fractions.

Theory part 105Practice part 106

trigonometric equation, reference angle, unit-circle intersections, periodic solutions, specified interval, scaled input, integer parameter and checking solutions

Builds on The Sine Function, The Cosine Function, The Tangent Function and The Quadratic Formula and Real Solutions.

Theory part 107Practice part 108

arcsine, principal angle, restricted sine branch, inverse function, arcsine graph, domain, range and included endpoints

Builds on Inverse Functions and Domain Restrictions, The Sine Function and Solving Basic Trigonometric Equations.

Theory part 109Practice part 110

arccosine, principal angle, restricted cosine branch, inverse graph, decreasing function, domain, range and endpoint values

Builds on Inverse Functions and Domain Restrictions, The Cosine Function and The Arcsine Function.

Theory part 111Practice part 112

arctangent, principal angle, restricted tangent branch, inverse graph, all-real domain, open output interval, horizontal asymptote and odd symmetry

Builds on Inverse Functions and Domain Restrictions, The Tangent Function and The Arcsine Function.

Theory part 113Practice part 114

nested function, inner value, principal angle, reference triangle, coordinate ratio, Pythagorean relation, sign choice and common domain

Builds on The Arcsine Function, The Arccosine Function, The Arctangent Function, Unit-Circle Coordinates and Exact Values and Composing Functions.

Theory part 115Practice part 116

double-angle identity, power reduction, product-to-sum, squared trigonometric function, angle grouping, purposeful rewrite and equivalent expression

Builds on Trigonometric Identities from the Unit Circle, The Sine Function, The Cosine Function and Expanding and Factoring for a Purpose.

Theory part 117Practice part 118

length, area, volume, radius, diameter, perpendicular height, annulus, cylinder, units and unit conversion

Builds on Reading Formulas and Substituting Values, Powers, Roots, and Signs and Circles, Semicircles, and Choosing a Branch.

Theory part 119Practice part 120

similar triangles, corresponding sides, scale factor, equal angles, proportion, cone, height, radius and shadow

Builds on Geometry Formulas and Units, Making a Variable the Subject and Drawing Lines from Slope and Intercepts.

Theory part 121Practice part 122

variable, parameter, constant, constraint, model, independent quantity, dependent quantity, physical domain, units and piecewise rule

Builds on Geometry Formulas and Units, Making a Variable the Subject, Finding the Domain and Reading and Drawing Piecewise Functions.

Theory part 123Practice part 124

input change, output change, difference quotient, f(a+h), substitution, secant slope, nonzero increment and cancellation

Builds on Changing Quantities and Average Rate, Composing Functions, Expanding and Factoring for a Purpose, Common Denominators and Complex Fractions and Cancellation and Excluded Values.

Theory part 125Practice part 126

polynomial division, dividend, divisor, quotient, remainder, degree, proper rational expression, polynomial identity and excluded value

Builds on Expanding and Factoring for a Purpose, Cancellation and Excluded Values, Cubic and Other Integer-Power Functions and Rational Functions, Holes, and Asymptotes.

Theory part 127Practice part 128

partial fraction, proper rational expression, linear factor, irreducible quadratic, unknown coefficient, coefficient matching, common denominator and domain restriction

Builds on Polynomial Division and Useful Rewrites, Common Denominators and Complex Fractions and Solving Equations and Checking Solutions.

Theory part 129Practice part 130

index, indexed term, starting index, subscript, term formula, shifted index, alternating sign and consecutive-term ratio

Builds on Reading Formulas and Substituting Values, Powers, Roots, and Signs and Common Denominators and Complex Fractions.

Theory part 131Practice part 132

factorial, zero factorial, factor cancellation, binomial coefficient, choosing positions, finite binomial expansion and integer conditions

Builds on Reading Indexed Terms and Shifting the Index, Cancellation and Excluded Values and Expanding and Factoring for a Purpose.

Theory part 133Practice part 134

summation notation, index, lower bound, upper bound, summand, finite sum, dummy index, reindexing and telescoping cancellation

Builds on Reading Indexed Terms and Shifting the Index, Common Denominators and Complex Fractions and Cancellation and Excluded Values.