AP Precalculus

Units

1. 1

   Polynomial and rational functions

   12 lessons

   1. Change in tandem: how inputs and outputs vary togetherDescribe sampled direction, rate trends, and recorded zeros from a contextual table or graph.
   2. Average rate of change and estimating rates at a pointCalculate average rates with units and use nearby averages to estimate a rate at a point.
   3. Rates of change in linear and quadratic functionsUse equal-step changes and average rates to distinguish linear and quadratic patterns and justify concavity.
   4. Polynomial functions and rates of change: extrema and points of inflectionLocate turns and inflections, compare growth rates, recognize symmetry, and apply degree bounds.
   5. Polynomial functions and complex zerosConstruct real polynomials from zeros and a point; account for every zero and predict intercept behavior.
   6. End behavior of polynomial and rational functionsDetermine both end limits and any horizontal asymptote from leading terms.
   7. Rational functions and zeros; solving rational inequalitiesFind rational-function zeros and solve rational inequalities with sign charts, keeping excluded inputs out.
   8. Rational functions: vertical asymptotes and holesClassify excluded inputs, locate holes, and describe behavior on each side of a vertical asymptote.
   9. Equivalent representations of polynomial and rational expressionsRewrite expressions into forms that reveal zeros, intercepts, end behavior, or slant asymptotes.
   10. Transformations of functionsMap key points under transformations and determine the edited function’s domain and range.
   11. Function model selection and assumption articulationSelect a function family from data or graph features, justify the choice, and state assumptions and meaningful inputs.
   12. Function model construction and applicationConstruct models from given features or data, make predictions, and explain their limits with units.
2. 2

   Exponential and logarithmic functions

   14 lessons

   1. Change in arithmetic and geometric sequencesModel fixed-addition and fixed-multiplication sequences, then compare their values at a given year.
   2. Change in linear and exponential functionsUse table values, including unequal input gaps, to choose and write a linear or exponential model.
   3. Exponential functions: features, growth and decayIdentify an exponential function’s initial value and factor, then describe its direction, concavity, and end behavior.
   4. Exponential function manipulationRewrite exponential functions in a·bˣ form and explain the equivalent graph transformations.
   5. Exponential function context and data modelingChoose an exponential model from data, use it to predict, and match its factor to the time unit.
   6. Competing function model validation: residualsCalculate residuals and justify a model choice using error size, residual patterns, and context.
   7. Composition of functionsEvaluate and build compositions, then explain why their order matters.
   8. Inverse functionsFind inverse values and formulas, check them by composition, and restrict a domain when needed.
   9. Logarithmic expressionsEvaluate logarithms, estimate them between powers, and interpret factors on a logarithmic scale.
   10. Logarithmic functions as inverses of exponential functionsFind logarithmic inverses and explain their domains, asymptotes, concavity, and equal-ratio changes.
   11. Logarithmic function manipulationRewrite logarithmic functions in equivalent forms and describe their graph transformations.
   12. Exponential and logarithmic equations and inequalitiesSolve exponential and logarithmic equations and inequalities, check log inputs, and construct inverse functions.
   13. Logarithmic function context and data modelingChoose and fit a logarithmic model, predict an output, and interpret its coefficient.
   14. Semi-log plotsRead a logarithmic count axis, recover an exponential model from its line, and judge whether data support it.
3. 3

   Trigonometric and polar functions

   15 lessons

   1. Periodic phenomenaFind a period, use it to predict later values, and describe how one cycle repeats.
   2. Radians and the unit circleFind exact sine, cosine and tangent values from circle points and special angles, including their signs.
   3. Exact values and signs of sine, cosine and tangentWrite exact sine, cosine and tangent values at multiples of π/6, π/4 and π/3 and at the axis angles, with the right sign, by reference angle and quadrant.
   4. Sine and cosine function graphsGraph sine and cosine from the unit circle; describe their features, direction, concavity, and rates of change.
   5. Sinusoidal functions and their transformationsConstruct and interpret sinusoidal equations from extrema, period, and a marked crossing or crest.
   6. Sinusoidal function context and data modelingBuild a sinusoidal model from context, interpret its behavior, and check a technology fit against data.
   7. The tangent functionSketch tangent graphs and find their periods and vertical asymptotes from a ray’s slope.
   8. Inverse trigonometric functionsEvaluate inverse trig functions, identify their selected angles, reflect their graphs, and distinguish selected angles from all solutions.
   9. Trigonometric equations and inequalitiesSolve trigonometric equations and inequalities, giving every solution or qualifying interval in the stated domain.
   10. The secant, cosecant and cotangent functionsEvaluate, sketch and solve equations involving secant, cosecant and cotangent using sine and cosine.
   11. Pythagorean and sum identitiesRewrite trigonometric expressions to find exact values and solve equations.
   12. Double-angle identities and solving with identitiesRewrite sin 2θ and cos 2θ with θ, choose the useful form of cos 2θ, and solve trigonometric equations by factoring after an identity.
   13. Trigonometry and polar coordinatesConvert among grid, polar, and complex forms, and give equivalent polar labels.
   14. Polar function graphsSketch polar curves from rectangular r–θ graphs, including negative readings, and recognize cardioids, roses, and circles.
   15. Rates of change in polar functionsCalculate average change in signed radius per radian and determine whether a polar point moves toward or away from the pole.
4. 4

   Functions involving parameters, vectors and matrices

   12 lessons

   1. Parametric functionsPlot a parametric path in order, show its direction, and find extreme coordinate positions.
   2. Parametric functions modeling planar motionLocate a moving particle and describe its path, direction, and moments of rest using x(t) and y(t).
   3. Parametric functions and rates of changeCalculate horizontal and vertical average velocities and a path’s secant slope from timed positions or coordinate equations.
   4. Parametrically defined circles and linesWrite and check parametric equations for a line segment or circle, including its starting point and direction.
   5. Implicitly defined functions and conic sectionsRewrite an implicit equation to identify a conic, use focus information, and estimate a rate from nearby points.
   6. Parametrization of implicitly defined functionsParametrize ellipses, hyperbolas, and functions; verify that the coordinates satisfy the original equation.
   7. Vectors: components, magnitude, operations and the dot productFind a vector’s length, direction and unit vector; combine motions and use a dot product to find their angle.
   8. Vector-valued functionsEvaluate position vectors and find displacement, straight-line separation, and average velocity.
   9. Matrices: operations and the determinantMultiply matrices row by column, find 2 × 2 determinant areas, and compare multiplication orders.
   10. The inverse and determinant of a matrixUse a determinant to decide whether a 2 × 2 matrix is invertible, then use its inverse to decode pairs and solve systems.
   11. Linear transformations and matrices as functionsBuild a transformation matrix from unit-arrow images, combine moves in order, and use inverses and determinants to interpret the result.
   12. Matrices modeling contexts: transition matricesUse a transition matrix to predict future, steady, and earlier population states.