College Precalculus

Units

1. 1

   Functions

   3 lessons

   1. Transformations of functions: shifts, stretches and reflections, in orderMap key points to graph combined transformations in the correct order.
   2. Composition and decomposition of functionsEvaluate function chains, split functions into simpler rules, and check which inputs survive every step.
   3. Inverse functions: existence, finding, verifying and restricting domainsDetermine when an inverse function exists, find it with its domain, and verify it by composition.
2. 2

   Polynomial and rational functions

   5 lessons

   1. Polynomial functions: end behavior, turning points and the Intermediate Value TheoremPredict polynomial ends and turn bounds, read local extrema, and use a sign change to guarantee a zero.
   2. Finding real and complex zeros: Rational Root Theorem and divisionFind every zero of a polynomial using rational candidates, synthetic division, and a quadratic.
   3. Graphing rational functions: asymptotes, holes and interceptsSketch a rational graph from its holes, asymptotes, intercepts, and behavior near a vertical asymptote.
   4. Polynomial and rational inequalities with sign chartsSolve polynomial and rational inequalities by marking boundaries, testing signs, and choosing endpoints correctly.
   5. Partial fractionsSplit a proper rational expression into simpler fractions for distinct, repeated and irreducible factors, dividing first when needed, and use the split to sum a telescoping series.
3. 3

   Exponential and logarithmic functions

   4 lessons

   1. Graphs of exponential and logarithmic functions with transformationsSketch transformed exponential and logarithmic graphs using their asymptotes, domains, and key points.
   2. Exponential and logarithmic equations, including quadratic formSolve unlike-base, quadratic-form, and logarithmic equations exactly; check which solutions are valid.
   3. Exponential, logarithmic and logistic models from dataChoose a growth model from data, check its predictions and residuals, and interpret its limits.
   4. Semi-log plotsRead a logarithmic count axis, recover an exponential model from its line, and judge whether data support it.
4. 4

   Trigonometric functions

   6 lessons

   1. Angles, radians, arc length and angular speedConvert between degrees and radians, find coterminal angles, and use s = rθ, A = (1/2)r²θ and v = rω with the angle in radians.
   2. Exact values on the unit circle using special triangles and symmetryFind exact unit-circle points, use symmetry, and calculate all six trigonometric values.
   3. Symmetry and periodicity of the trigonometric functionsUse unit-circle reflections and turns to evaluate and simplify trigonometric expressions.
   4. Sinusoidal functions: transformations and modelingBuild a repeating-height model from data and read its amplitude, period, midline, and shift.
   5. Sinusoidal function context and data modelingBuild a sinusoidal model from context, interpret its behavior, and check a technology fit against data.
   6. Graphs of tangent, cotangent, secant and cosecantSketch all four graphs using denominator zeros, key points, periods and branch directions.
5. 5

   Inverse functions, identities and equations

   5 lessons

   1. Inverse trigonometric functions and restricted domainsChoose an inverse-trig angle from its allowed range and evaluate compositions using a reference triangle.
   2. Solving trigonometric equations in contextFind every solution in a stated interval and interpret solutions as tide times.
   3. Pythagorean identities and verifying identitiesVerify trig identities by transforming one side into the other and stating where the expressions are defined.
   4. Sum, difference, double-angle and half-angle formulasFind exact trigonometric values, use double- and half-angle formulas, and solve a double-angle equation.
   5. Half-angle, power-reducing and product-to-sum identitiesRewrite sin²θ and cos²θ with cos 2θ, find half-angle values with the right sign, and turn products of sines and cosines into sums.
6. 6

   Laws of sines and cosines

   3 lessons

   1. Area of a triangle: A = (1/2)ab sin CDerive and use triangle area from two sides and the angle between them, including an obtuse angle.
   2. The Law of Sines and the ambiguous caseUse the Law of Sines to find missing sides and solve every triangle possible from SSA measurements.
   3. The Law of Cosines and solving any triangleFind a missing side or angle with cosines, then choose when sines can finish the triangle.
7. 7

   Polar coordinates and complex numbers

   3 lessons

   1. Polar coordinates and polar graphsPlot polar points and curves, convert coordinates, and recognize circles, cardioids, and roses.
   2. Complex numbers in polar form: multiplication as rotation and scalingWrite complex numbers in polar form and use length and angle to compute products, powers, and quotients.
   3. De Moivre's Theorem and nth roots of complex numbersSolve zⁿ = w for every complex root and explain where the roots lie.
8. 8

   Vectors

   5 lessons

   1. Vectors as quantities with magnitude and directionRepresent vectors, find their components, magnitude and direction, and convert speed and bearing into components.
   2. Adding, subtracting and scaling vectorsAdd, subtract, and scale vectors; find a resultant’s length and direction.
   3. Modeling with vectors: velocity, force and navigationFind a flight heading that corrects for wind and cable tensions that balance a sign.
   4. The dot product: angle between vectors and projectionsCompute a dot product from components, use it to find the angle between vectors and test perpendicularity, and find projections and work.
   5. Vectors in three dimensions and the cross productCompute with vectors in space, find the cross product and check it is perpendicular to both factors, and use its length for areas.
9. 9

   Matrices and systems

   5 lessons

   1. Matrix arithmetic: adding, scaling and multiplyingAdd and scale matrices, calculate products, check dimensions, and explain why factor order matters.
   2. Determinants and inverse matricesCalculate a 2 × 2 determinant, find an inverse when one exists, and decode a column vector.
   3. Solving systems with matricesWrite AX = B, solve with an inverse when possible, and classify systems with no inverse.
   4. Matrices as transformations of the planeUse 2 × 2 matrices to transform points, combine moves, and find area scale.
   5. Matrices modeling contexts: transition matricesUse a transition matrix to predict future, steady, and earlier population states.
10. 10

    Conic sections

    4 lessons

    1. Circles and parabolas from distance definitionsBuild equations from distance definitions and identify a circle’s centre and radius or a parabola’s focus and directrix.
    2. Ellipses from fociUse two foci and a fixed distance sum to write an ellipse’s equation, locate its vertices, and find its eccentricity.
    3. Hyperbolas from fociUse a fixed difference of distances to identify, write and sketch hyperbolas.
    4. Nonlinear systems: lines, circles and conicsSolve a system with a line and a conic by substitution and a system of two conics by elimination, find every intersection point, and decide how many solutions to expect.
11. 11

    Parametric equations

    2 lessons

    1. Parametric equations: graphing, direction and eliminating the parameterGraph a parametric path with arrows, eliminate t, and keep only the points reached.
    2. Modeling motion with parametric equationsModel a projectile and find its height, landing position, maximum height, and average path slope.
12. 12

    Sequences and series

    5 lessons

    1. Sequences as functions: recursive and explicit definitionsGraph sequences as discrete functions, write recursive and explicit rules, and describe geometric long-run behavior.
    2. Sigma notation and finite arithmetic and geometric seriesEvaluate finite sums in sigma notation and use a geometric sum to find a loan payment.
    3. Infinite geometric series and convergenceDecide whether an infinite geometric series has a finite sum, and find that sum for decimals and rebound distances.
    4. The binomial theorem and Pascal's triangleExpand (a + b)ⁿ with Pascal's triangle or the theorem, find one term or coefficient without expanding, and use a few terms to estimate a power.
    5. Mathematical inductionProve a statement about every whole number with a base case and an inductive step, for sum formulas and divisibility.
13. 13

    Introduction to limits

    3 lessons

    1. Limits from graphs and tables, and one-sided limitsEstimate limits from graphs and two-sided tables, and decide when a two-sided limit exists.
    2. Evaluating limits algebraically and continuityEvaluate limits from formulas and decide whether a function is continuous at a point.
    3. Limits at infinity and infinite limitsFind end and one-sided limits, identify horizontal and vertical asymptotes, and justify signs.