OhioCoreAcademyOhio K–12 lessons

Grade 12

Grade 12 · Mathematics

Choose a lesson below. Each includes an explanation, a model, worked examples and practice with answers.

190 lessons available160 of 160 statements targeted

Algebra 1 · published lessons

74 lessons and 21 practice sets, reviewed and published. Each lesson names the Ohio standard it teaches.

  1. 01Reading an Expression: Terms, Factors, CoefficientsA.SSE.1a · explanation and practice
  2. 02Seeing a Chunk as One ThingA.SSE.1b · explanation and practice
  3. 03Equivalent Expressions and the Distributive PropertyA.SSE.2 · explanation and practice
  4. 04Adding, Subtracting and Multiplying PolynomialsA.APR.1a · explanation and practice
  5. 05Solving in Several StepsA.REI.3 · explanation and practice
  6. 06Variables on Both Sides — One, None, or Every SolutionA.REI.3 · explanation and practice
  7. 07Justifying Each StepA.REI.1 · explanation and practice
  8. 08Rearranging Formulas and Letter CoefficientsA.CED.4a · explanation and practice
  9. 09From a Situation to an EquationA.CED.1a · explanation and practice
  10. 10Units as a Planning ToolN.Q.1 · explanation and practice
  11. 11Choosing the Quantity to MeasureN.Q.2 · explanation and practice
  12. 12Scale and Origin in a GraphA.CED.2a · explanation and practice
  13. 13Solving and Graphing One-Variable InequalitiesA.REI.3 · explanation and practice
  14. 14The Flip Rule — Why a Negative Reverses the SignA.REI.3 · explanation and practice
  15. 15Compound Inequalities: And, and OrA.REI.3 · explanation and practice
  16. 16Constraints, and Solutions the Situation Will Not AllowA.CED.3 · explanation and practice
  17. 17Dot Plots, Histograms and Box PlotsS.ID.1 · explanation and practice
  18. 18Center: Mean, Median, and Which One to TrustS.ID.2 · explanation and practice
  19. 19Spread: Range, IQR, MAD and Standard DeviationS.ID.2 · explanation and practice
  20. 20Shape, Outliers, and Reporting an Answer HonestlyS.ID.3 · explanation and practice
  21. 21What Makes a Relation a FunctionF.IF.1 · explanation and practice
  22. 22Function Notation and EvaluatingF.IF.2 · explanation and practice
  23. 23Reading f(x) in Context — What f(3) = 20 MeansF.IF.2 · explanation and practice
  24. 24Domain and Range, and the Inputs a Situation AllowsF.IF.5b · explanation and practice
  25. 25Rate of Change and SlopeF.IF.4b · explanation and practice
  26. 26Slope-Intercept Form and Graphing a LineF.IF.7a · explanation and practice
  27. 27Point-Slope and Standard FormA.CED.2a · explanation and practice
  28. 28Key Features from a GraphF.IF.4b · explanation and practice
  29. 29Why a Graph Is the Set of All SolutionsA.REI.10 · explanation and practice
  30. 30Writing a Linear Function from Four Kinds of ClueF.LE.2 · explanation and practice
  31. 31What the Numbers Mean in the SituationF.LE.5 · explanation and practice
  32. 32Equal Differences or Equal FactorsF.LE.1 · explanation and practice
  33. 33Comparing Two Functions, and Working BackwardsF.IF.9b · explanation and practice
  34. 34Windows, Graphs and Key FeaturesF.IF.7a · explanation and practice
  35. 35Intersections, Tables and Lines of FitA.REI.11 · explanation and practice
  36. 36Two-Way Frequency TablesS.ID.5 · explanation and practice
  37. 37Scatter Plots and AssociationS.ID.6 · explanation and practice
  38. 38Fitting a Line, and Using It to PredictS.ID.6c · explanation and practice
  39. 39Reading the Model: Slope, Intercept and CorrelationS.ID.7 · explanation and practice
  40. 40Solving a System by GraphingA.REI.6a · explanation and practice
  41. 41SubstitutionA.REI.6a · explanation and practice
  42. 42Elimination — and Why Adding a Multiple Is LegalA.REI.5 · explanation and practice
  43. 43No Solution, or Infinitely ManyA.REI.6a · explanation and practice
  44. 44Where f(x) Meets g(x)A.REI.11 · explanation and practice
  45. 45Half-Planes and Systems of InequalitiesA.REI.12 · explanation and practice
  46. 46A Sequence Is a FunctionF.IF.3 · explanation and practice
  47. 47Arithmetic SequencesF.BF.2 · explanation and practice
  48. 48Geometric Sequences — and the Doorway to Chapter 14F.BF.2 · explanation and practice
  49. 49What Makes a Function ExponentialF.LE.1c · explanation and practice
  50. 50Graphing: Intercepts, End Behavior, and the Line It Never ReachesF.IF.7e · explanation and practice
  51. 51Percent Rates, and Reading Them Out of the ExpressionF.IF.8b · explanation and practice
  52. 52Rewriting the Base, and Why Exponential Always WinsA.SSE.3c · explanation and practice
  53. 53Greatest Common Factor FirstA.SSE.2 · explanation and practice
  54. 54Factoring x^2 + bx + cA.SSE.2 · explanation and practice
  55. 55When the Leading Coefficient Is Not 1A.SSE.2 · explanation and practice
  56. 56Special Forms, and Expressions Quadratic in FormA.SSE.2 · explanation and practice
  57. 57Completing the Square to Reveal a Maximum or MinimumA.SSE.3b · explanation and practice
  58. 58Square Roots and the Zero-Product PropertyA.REI.4b · explanation and practice
  59. 59Solving by FactoringA.REI.4b · explanation and practice
  60. 60Completing the Square to SolveA.REI.4a · explanation and practice
  61. 61The Quadratic Formula and the DiscriminantA.REI.4b · explanation and practice
  62. 62Choosing a Method, Modeling, and a Line Meeting a CurveA.CED.1b · explanation and practice
  63. 63Parabolas: Vertex, Axis of Symmetry, InterceptsF.IF.7b · explanation and practice
  64. 64The Three Forms and What Each One RevealsF.IF.8a · explanation and practice
  65. 65Maximum and Minimum in ContextF.IF.4b · explanation and practice
  66. 66Transformations: f(x) + k, k·f(x), f(x + k)F.BF.3a · explanation and practice
  67. 67Comparing Quadratic Functions Across RepresentationsF.IF.9b · explanation and practice
  68. 68Reading a Modeling ProblemN.Q.1 · explanation and practice
  69. 69Choosing a Model — Linear, Exponential, or QuadraticF.LE.1 · explanation and practice
  70. 70Justifying a Step, and Writing a Reason That Earns CreditA.REI.1 · explanation and practice
  71. 71Critiquing Reasoning and Error AnalysisA.REI.1 · explanation and practice
  72. 72Number, Quantities, Equations and ExpressionsN.Q.1 · explanation and practice
  73. 73FunctionsA.REI.5 · explanation and practice
  74. 74StatisticsS.ID.1 · explanation and practice

Practice sets

  1. 01Algebra 1 · A.CED.1a practice7 questions · scored on the student site
  2. 02Algebra 1 · A.REI.1 practice7 questions · scored on the student site
  3. 03Algebra 1 · A.REI.11 practice6 questions · scored on the student site
  4. 04Algebra 1 · A.REI.3 practice11 questions · scored on the student site
  5. 05Algebra 1 · A.REI.4b practice4 questions · scored on the student site
  6. 06Algebra 1 · A.SSE.1b practice4 questions · scored on the student site
  7. 07Algebra 1 · A.SSE.2 practice5 questions · scored on the student site
  8. 08Algebra 1 · F.BF.2 practice8 questions · scored on the student site
  9. 09Algebra 1 · F.IF.2 practice6 questions · scored on the student site
  10. 10Algebra 1 · F.IF.4b practice6 questions · scored on the student site
  11. 11Algebra 1 · F.LE.1 practice5 questions · scored on the student site
  12. 12Algebra 1 · S.ID.2 practice6 questions · scored on the student site
  13. 13Algebra 1 · S.ID.5 practice4 questions · scored on the student site
  14. 14Algebra 1 · S.ID.8 practice5 questions · scored on the student site
  15. 15Algebra 1 · Creating Equations (A.CED) mixed practice6 questions · scored on the student site
  16. 16Algebra 1 · Reasoning with Equations and Inequalities (A.REI) mixed practice8 questions · scored on the student site
  17. 17Algebra 1 · Seeing Structure In Expressions (A.SSE) mixed practice7 questions · scored on the student site
  18. 18Algebra 1 · Interpreting Functions (F.IF) mixed practice10 questions · scored on the student site
  19. 19Algebra 1 · Building Functions & Linear, Quadratic, and Exponential Models (F.BF, F.LE) mixed practice9 questions · scored on the student site
  20. 20Algebra 1 · Quantities (N.Q) mixed practice7 questions · scored on the student site
  21. 21Algebra 1 · Interpreting Categorical and Quantitative Data (S.ID) mixed practice8 questions · scored on the student site

Mathematics · 2017 framework

Lessons

  1. 01Prove that any two circles are similar using a translation followed by a dilationExplanation · illustration · worked examples · practice and answers
  2. 02Identify and use the relationships among central, inscribed and circumscribed angles, arcs, chords, radii and tangentsExplanation · illustration · worked examples · practice and answers
  3. 03Construct the inscribed and circumscribed circles of a triangle and prove that opposite angles of an inscribed quadrilateral are supplementaryExplanation · illustration · worked examples · practice and answers
  4. 04Construct a tangent line from a point outside a circle to the circle and prove the construction worksExplanation · illustration · worked examples · practice and answers
  5. 05Find arc lengths and sector areas, define radian measure as a ratio, and derive the formulas that convert between degrees and radiansExplanation · illustration · worked examples · practice and answers
  6. 06Derive the equation of a circle from the Pythagorean Theorem and complete the square to find a center and radiusExplanation · illustration · worked examples · practice and answers
  7. 07Derive the equation of a parabola from its focus and directrixExplanation · illustration · worked examples · practice and answers
  8. 08Derive the equations of ellipses and hyperbolas from their foci using the constant sum or difference of distancesExplanation · illustration · worked examples · practice and answers
  9. 09Justify the slope criteria for parallel and perpendicular lines and use coordinates to prove theorems about triangles, quadrilaterals and circlesExplanation · illustration · worked examples · practice and answers
  10. 10Partition a directed line segment in a given ratio and compute perimeters and areas of polygons from coordinatesExplanation · illustration · worked examples · practice and answers
  11. 11Know the number i, write complex numbers in the form a + bi, and simplify powers of iExplanation · illustration · worked examples · practice and answers
  12. 12Add, subtract and multiply complex numbers using i² = −1 and the commutative, associative and distributive propertiesExplanation · illustration · worked examples · practice and answers
  13. 13Find the conjugate of a complex number and use conjugates to find magnitudes and quotientsExplanation · illustration · worked examples · practice and answers
  14. 14Represent complex numbers on the complex plane in rectangular and polar form and explain why the two forms name the same numberExplanation · illustration · worked examples · practice and answers
  15. 15Represent addition, subtraction, multiplication and conjugation geometrically, and calculate distance and midpoint in the complex planeExplanation · illustration · worked examples · practice and answers
  16. 16Solve quadratic equations with complex solutions, extend polynomial identities to the complex numbers, and show the Fundamental Theorem of Algebra holds for quadraticsExplanation · illustration · worked examples · practice and answers
  17. 17Give precise definitions of segment, ray, angle, circle, perpendicular lines and parallel lines from the undefined notions of point, line, distance and arc lengthExplanation · illustration · worked examples · practice and answers
  18. 18Describe transformations as functions of the plane and compare those that preserve distance and angle with those that do notExplanation · illustration · worked examples · practice and answers
  19. 19Identify the rotations and reflections that carry a figure onto itself and use symmetry to analyze properties of shapesExplanation · illustration · worked examples · practice and answers
  20. 20Define rotations, reflections and translations precisely in terms of angles, circles, perpendicular lines, parallel lines and line segmentsExplanation · illustration · worked examples · practice and answers
  21. 21Draw the image of a figure under a given rigid motion and specify a sequence of transformations carrying one figure onto anotherExplanation · illustration · worked examples · practice and answers
  22. 22Use geometric descriptions of rigid motions to predict their effect on a figure and use the rigid-motion definition of congruence to decide whether two figures are congruentExplanation · illustration · worked examples · practice and answers
  23. 23Show that two triangles are congruent if and only if corresponding sides and angles are congruent, and explain how ASA, SAS and SSS follow from the rigid-motion definitionExplanation · illustration · worked examples · practice and answers
  24. 24Prove and apply theorems about lines and angles: vertical angles, angle pairs formed by a transversal crossing parallel lines, and the perpendicular bisector as the set of points equidistant from the endpointsExplanation · illustration · worked examples · practice and answers
  25. 25Prove and apply theorems about triangles: the angle sum, base angles of isosceles triangles, the midsegment, and the concurrence of the mediansExplanation · illustration · worked examples · practice and answers
  26. 26Prove and apply theorems about parallelograms: opposite sides and angles, diagonals that bisect each other and its converse, and rectangles as parallelograms with congruent diagonalsExplanation · illustration · worked examples · practice and answers
  27. 27Make formal geometric constructions with compass and straightedge and other tools, and inscribe an equilateral triangle, a square and a regular hexagon in a circleExplanation · illustration · worked examples · practice and answers
  28. 28Classify two-dimensional figures in a hierarchy based on their properties and use the hierarchy to decide which properties a figure inheritsExplanation · illustration · worked examples · practice and answers
  29. 29Solve simple rational and radical equations in one variable and explain how extraneous solutions ariseExplanation · illustration · worked examples · practice and answers
  30. 30Complete the square to transform any quadratic equation into (x − p)² = q, and derive the quadratic formulaExplanation · illustration · worked examples · practice and answers
  31. 31Choose a solving method from the initial form of a quadratic equation, and write complex solutions as a ± biExplanation · illustration · worked examples · practice and answers
  32. 32Solve systems of linear equations in two variables algebraically and graphically, and in three variables algebraicallyExplanation · illustration · worked examples · practice and answers
  33. 33Solve a system of a linear equation and a quadratic equation in two variables algebraically and graphicallyExplanation · illustration · worked examples · practice and answers
  34. 34Represent a system of linear equations as a single matrix equation AX = B in a vector variableExplanation · illustration · worked examples · practice and answers
  35. 35Find the inverse of a matrix if it exists and use it to solve systems of linear equationsExplanation · illustration · worked examples · practice and answers
  36. 36Interpret intercepts, intervals, extrema, symmetry, end behavior and periodicity of a function that models a situation, and sketch a graph from a verbal descriptionExplanation · illustration · worked examples · practice and answers
  37. 37Relate the domain of a function to its graph and to the situation it describes, and choose a function type from the behavior of dataExplanation · illustration · worked examples · practice and answers
  38. 38Calculate and interpret the average rate of change of a function over an interval, and estimate it from a graphExplanation · illustration · worked examples · practice and answers
  39. 39Graph linear, quadratic and simple exponential functions from their formulas and indicate intercepts, maxima, minima and end behaviorExplanation · illustration · worked examples · practice and answers
  40. 40Graph square root, cube root and piecewise-defined functions, including step and absolute value functions, and indicate their key featuresExplanation · illustration · worked examples · practice and answers
  41. 41Graph polynomial, exponential, logarithmic, rational and trigonometric functions and indicate zeros, asymptotes, intercepts, end behavior, period, midline and amplitudeExplanation · illustration · worked examples · practice and answers
  42. 42Rewrite quadratic and exponential functions in equivalent forms that reveal zeros, extreme values, symmetry and percent rate of changeExplanation · illustration · worked examples · practice and answers
  43. 43Compare properties of two functions when each is represented differently: algebraically, graphically, in a table or in wordsExplanation · illustration · worked examples · practice and answers
  44. 44Write a function for a relationship between two quantities as an explicit expression, a recursive process or steps for calculation, then build new functions by combining and composing standard typesExplanation · illustration · worked examples · practice and answers
  45. 45Identify the effect on a graph of replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k), find k from graphs, and recognize even and odd functionsExplanation · illustration · worked examples · practice and answers
  46. 46Find inverse functions: determine an input from a known output, read an inverse from a table or graph, find an inverse algebraically, verify it by composition, and restrict a domain to make a function invertibleExplanation · illustration · worked examples · practice and answers
  47. 47Use the inverse relationship between exponents and logarithms to solve problems, and express the solution of ab^(ct) = d as a logarithm with base 2, 10 or e evaluated using technologyExplanation · illustration · worked examples · practice and answers
  48. 48Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadraticallyExplanation · illustration · worked examples · practice and answers
  49. 49Give informal arguments for the circumference and area of a circle and the volumes of a cylinder, a pyramid and a cone using dissection, Cavalieri's principle and informal limitsExplanation · illustration · worked examples · practice and answers
  50. 50Give an informal argument using Cavalieri's principle for the volume of a sphere and of other solid figuresExplanation · illustration · worked examples · practice and answers
  51. 51Use volume formulas for cylinders, pyramids, cones and spheres to solve problems with unitsExplanation · illustration · worked examples · practice and answers
  52. 52Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify the three-dimensional objects generated by rotating two-dimensional figuresExplanation · illustration · worked examples · practice and answers
  53. 53Understand when changes to lengths or angles produce similar or non-similar figures, and apply the scaling of lengths, areas and volumes by k, k² and k³Explanation · illustration · worked examples · practice and answers
  54. 54Describe objects with geometric shapes and their measures, and apply density based on area and volume in modeling situationsExplanation · illustration · worked examples · practice and answers
  55. 55Apply geometric methods to solve design problems: satisfy physical constraints, minimize cost and work with typographic grid systems based on ratiosExplanation · illustration · worked examples · practice and answers
  56. 56Fit linear, quadratic and exponential functions to data on a scatter plot and use the fitted function to solve problems in contextExplanation · illustration · worked examples · practice and answers
  57. 57Compute residuals, draw a residual plot and use it to assess informally how well a function fits the dataExplanation · illustration · worked examples · practice and answers
  58. 58Represent constraints from more complicated situations with equations, inequalities and systems that mix linear, quadratic and exponential relationships, and decide which solutions are viableExplanation · illustration · worked examples · practice and answers
  59. 59Show that linear functions grow by equal differences and exponential functions by equal factors over equal intervals, and recognize constant-rate and constant-percent-rate situationsExplanation · illustration · worked examples · practice and answers
  60. 60Define rational exponents from the properties of integer exponents and rewrite radical expressionsExplanation · illustration · worked examples · practice and answers
  61. 61Explain why sums and products of rationals are rational, and when a sum or product involving an irrational number is guaranteed to be irrationalExplanation · illustration · worked examples · practice and answers
  62. 62Choose a level of accuracy appropriate to the limitations on measurement when reporting quantitiesExplanation · illustration · worked examples · practice and answers
  63. 63Interpret the parts of an expression in context and view complicated expressions as built from single entitiesExplanation · illustration · worked examples · practice and answers
  64. 64Factor and complete the square to reveal the zeros and the maximum or minimum of a quadraticExplanation · illustration · worked examples · practice and answers
  65. 65Transform exponential expressions with the properties of exponents to reveal growth and decay ratesExplanation · illustration · worked examples · practice and answers
  66. 66Derive the finite geometric series formula and use it for payments and totalsExplanation · illustration · worked examples · practice and answers
  67. 67Add, subtract and multiply polynomials and explain why the results are always polynomialsExplanation · illustration · worked examples · practice and answers
  68. 68Use the Remainder Theorem to find remainders and decide whether x − a is a factorExplanation · illustration · worked examples · practice and answers
  69. 69Find the zeros of a factorable polynomial and use them to sketch a rough graphExplanation · illustration · worked examples · practice and answers
  70. 70Prove polynomial identities and use them to describe numerical relationships such as Pythagorean triplesExplanation · illustration · worked examples · practice and answers
  71. 71Expand (x + y)ⁿ with the Binomial Theorem and Pascal's triangleExplanation · illustration · worked examples · practice and answers
  72. 72Rewrite rational expressions as quotient plus remainder, and add, subtract, multiply and divide themExplanation · illustration · worked examples · practice and answers
  73. 73Create and solve equations and inequalities in one variable from linear, quadratic, rational and exponential situationsExplanation · illustration · worked examples · practice and answers
  74. 74Create equations in two or more variables for a relationship and graph them on labeled, scaled axesExplanation · illustration · worked examples · practice and answers
  75. 75Rearrange formulas to highlight a quantity of interest using the reasoning of equation solvingExplanation · illustration · worked examples · practice and answers
  76. 76Describe events as subsets of a sample space using or, and, notExplanation · illustration · worked examples · practice and answers
  77. 77Find conditional probability as a fraction of the given event and interpret itExplanation · illustration · worked examples · practice and answers
  78. 78Decide whether two events are independent and explain conditional probability in everyday situationsExplanation · illustration · worked examples · practice and answers
  79. 79Construct two-way frequency tables and use them to estimate conditional probabilities and test independenceExplanation · illustration · worked examples · practice and answers
  80. 80Apply the Addition Rule and the general Multiplication Rule and interpret the resultsExplanation · illustration · worked examples · practice and answers
  81. 81Use permutations and combinations to compute probabilities of compound eventsExplanation · illustration · worked examples · practice and answers
  82. 82Define a random variable, graph its probability distribution and interpret expected value as its meanExplanation · illustration · worked examples · practice and answers
  83. 83Develop theoretical and empirical probability distributions and find their expected valuesExplanation · illustration · worked examples · practice and answers
  84. 84Weigh decisions by expected value: payoffs of games of chance and comparison of strategiesExplanation · illustration · worked examples · practice and answers
  85. 85Use probability to make fair decisions and to analyze strategies in product testing, medical screening and gamesExplanation · illustration · worked examples · practice and answers
  86. 86Verify experimentally that a dilation fixes lines through its center, takes other lines to parallel lines, and scales every segment by the scale factorExplanation · illustration · worked examples · practice and answers
  87. 87Decide whether two figures are similar using similarity transformations, and explain why similar triangles have equal angles and proportional sidesExplanation · illustration · worked examples · practice and answers
  88. 88Establish the AA criterion for similar triangles from the properties of similarity transformations, and use itExplanation · illustration · worked examples · practice and answers
  89. 89Prove that a line parallel to one side of a triangle divides the other two proportionally, prove the converse, and prove the Pythagorean Theorem using similar trianglesExplanation · illustration · worked examples · practice and answers
  90. 90Use congruence and similarity criteria for triangles to solve problems and to justify relationships in figures that decompose into trianglesExplanation · illustration · worked examples · practice and answers
  91. 91Define sine, cosine and tangent as side ratios that depend only on the acute angle, and explain why the sine of an angle equals the cosine of its complementExplanation · illustration · worked examples · practice and answers
  92. 92Solve right triangles in applied problems using trigonometric ratios and the Pythagorean TheoremExplanation · illustration · worked examples · practice and answers
  93. 93Derive the area formula A = ½ab·sin(C) by drawing an auxiliary perpendicular from a vertex, and use itExplanation · illustration · worked examples · practice and answers
  94. 94Explain proofs of the Laws of Sines and Cosines, and apply them to find unknown measurements in right and non-right trianglesExplanation · illustration · worked examples · practice and answers
  95. 95Fit a normal distribution to a data set and estimate population percentages from areas under the curveExplanation · illustration · worked examples · practice and answers
  96. 96Compute and interpret the correlation coefficient of a linear fit, and distinguish correlation from causationExplanation · illustration · worked examples · practice and answers
  97. 97Understand statistics as inference from a random sample, and decide by simulation whether a model is consistent with dataExplanation · illustration · worked examples · practice and answers
  98. 98Distinguish sample surveys, experiments and observational studies, and explain the role of randomization in eachExplanation · illustration · worked examples · practice and answers
  99. 99Estimate a population proportion or mean from a sample survey and develop a margin of error by simulationExplanation · illustration · worked examples · practice and answers
  100. 100Compare two treatments in a randomized experiment and use simulation to decide whether the difference is statistically significantExplanation · illustration · worked examples · practice and answers
  101. 101Evaluate reports based on dataExplanation · illustration · worked examples · practice and answers
  102. 102Understand radian measure as arc length on the unit circle and convert between radians and degreesExplanation · illustration · worked examples · practice and answers
  103. 103Explain how the unit circle extends sine, cosine and tangent to all real numbersExplanation · illustration · worked examples · practice and answers
  104. 104Use special right triangles to find exact values of sine, cosine and tangent at π/6, π/4 and π/3Explanation · illustration · worked examples · practice and answers
  105. 105Use the unit circle to explain the values at π − x, π + x and 2π − x, odd and even symmetry, and periodicityExplanation · illustration · worked examples · practice and answers
  106. 106Choose sine and cosine functions to model periodic phenomena with a given amplitude, period, frequency and midlineExplanation · illustration · worked examples · practice and answers
  107. 107Restrict trigonometric functions so their inverses exist, and use inverse functions to solve trigonometric equations from modeling contextsExplanation · illustration · worked examples · practice and answers
  108. 108Prove the Pythagorean identity sin²θ + cos²θ = 1 and use it with the quadrant to find the other trigonometric valuesExplanation · illustration · worked examples · practice and answers
  109. 109Prove the addition and subtraction formulas for sine, cosine and tangent, and use them to find exact values and solve problemsExplanation · illustration · worked examples · practice and answers
  110. 110Represent vector quantities by directed line segments and find their components from initial and terminal pointsExplanation · illustration · worked examples · practice and answers
  111. 111Solve problems involving velocity, displacement and force by representing each quantity as a vectorExplanation · illustration · worked examples · practice and answers
  112. 112Add vectors end-to-end, by the parallelogram rule and component-wise, and subtract by adding the additive inverseExplanation · illustration · worked examples · practice and answers
  113. 113Multiply a vector by a scalar graphically and component-wise, and find the magnitude and direction of the resultExplanation · illustration · worked examples · practice and answers
  114. 114Use matrices to represent data and networks, and add, subtract, scalar-multiply and multiply matrices of appropriate dimensionsExplanation · illustration · worked examples · practice and answers
  115. 115Understand that matrix multiplication is not commutative but is associative and distributive, and that the zero and identity matrices and the determinant govern inversesExplanation · illustration · worked examples · practice and answers
  116. 116Multiply a vector by a matrix, treat 2 × 2 matrices as transformations of the plane, and read the determinant as an area factorExplanation · illustration · worked examples · practice and answers