Precalculus

A full year of Precalculus: polynomial, rational, exponential and logarithmic functions, trigonometry and identities, polar and parametric curves, vectors and matrices, conics, and induction, with a mid-year capstone and many applied projects.

← Algebra II

180 lessons

One a day for a school year

10 units

Each opens to its lessons

26 applied projects

Plus 10 explainers and a capstone

Standards

Common Core high school math

What's taught, unit by unit

1
Polynomial, rational, exponential & log functionsDays 1–10
Average rate of change, end behavior of polynomials, rational functions as real-world rates, exponential growth and decay, logarithms, exponentials and logs as inverses, and modeling with them.
- 1 · Welcome to Precalculus: The Bridge to Higher Math
- 2 · Polynomial Functions: Average Rate of Change
- 3 · Polynomial Functions: End Behavior Revisited
- 4 · Rational Functions: Modeling Real-World Rates
- 6 · Exponential Functions: Growth, Decay, and Transformations
- 7 · Logarithmic Functions: Solving and Graphing
- 8 · The Inverse Relationship Between Exponentials and Logarithms
- 9 · Modeling with Exponential and Logarithmic Functions
Plus 2 review days, not listed.
2
Trig graphs, polar coordinates & parametric equationsDays 11–20
Trigonometric functions beyond the unit circle, graphing them with transformations, polar coordinates and converting to rectangular, graphing polar curves, and graphing and modeling motion with parametric equations.
- 11 · Introducing Trigonometric Functions Beyond the Unit Circle
- 12 · Graphing Trig Functions with Transformations
- 13 · Introducing Polar Coordinates
- 14 · Converting Between Polar and Rectangular Coordinates
- 16 · Graphing Polar Equations
- 17 · Introducing Parametric Functions
- 18 · Graphing Parametric Functions
- 19 · Modeling Motion with Parametric Functions
Plus 2 review days, not listed.
3
Vectors, matrices & infinite seriesDays 21–30
Vectors and their operations in 3D, determinants and inverse matrices, solving systems with matrices, piecewise and composite functions, and when an infinite geometric series adds up to a finite sum.
- 21 · Introducing Vectors in 3 Dimensions
- 22 · Vector Operations in 3 Dimensions
- 23 · Matrices Extended: Determinants
- 24 · Matrices Extended: Inverse Matrices
- 26 · Using Inverse Matrices to Solve Systems
- 27 · Piecewise and Composite Functions Revisited
- 28 · Introducing Sequences and Series Convergence
- 29 · Mixed Practice: Vectors, Matrices, and Convergence
Plus 2 review days, not listed.
4
Trig identities, the laws of sines & cosines, polar formDays 31–45
Pythagorean, sum and difference, double-angle and half-angle identities, solving trig equations, inverse trig functions, the Laws of Sines and Cosines, triangle area, vectors in polar form, and complex numbers in polar form.
- 31 · Trig Identities: The Pythagorean Identity Extended
- 32 · Sum and Difference Formulas for Sine and Cosine
- 33 · Double-Angle and Half-Angle Formulas
- 34 · Solving Trigonometric Equations
- 36 · Inverse Trigonometric Functions
- 37 · The Law of Sines
- 38 · The Law of Cosines
- 39 · Applications of the Laws of Sines and Cosines
- 41 · Area of a Triangle Using Trig
- 42 · Vectors in Polar Form
- 43 · Complex Numbers in Polar Form
- 44 · Multiplying and Dividing Complex Numbers in Polar Form
Plus 3 review days, not listed.
5
Induction, binomial theorem, conics & function analysisDays 46–65
Mathematical induction, the Binomial Theorem, turning points, continuity, average rate of change over shrinking intervals, nonlinear systems, parabolas, ellipses and hyperbolas, even and odd functions, asymptotes, and recursive sequences.
- 46 · Mathematical Induction: Introduction
- 47 · Mathematical Induction: Completing Simple Proofs
- 48 · The Binomial Theorem
- 49 · Mixed Practice: Polar Forms, Induction, and the Binomial Theorem
- 51 · Analyzing Turning Points and Local Extrema
- 52 · Continuity of Functions
- 53 · Average Rate of Change Over Shrinking Intervals
- 54 · Modeling with Systems of Nonlinear Equations
- 56 · Parabolas: Definition, Focus, and Directrix
- 57 · Ellipses: Definition, Foci, and Standard Form
- 58 · Hyperbolas: Definition, Foci, and Asymptotes
- 59 · Applied Conics: Eccentricity and Real-World Curves
- 61 · Analyzing Function Symmetry: Even and Odd Functions
- 62 · Analyzing Asymptotic Behavior in Depth
- 63 · Piecewise-Defined Function Modeling
- 64 · Introducing Sequences Defined Recursively
Plus 4 review days, not listed.
6
Applied projects, mixed review & halfway checkpointDays 66–90
Applied projects in music, ecology, roller coasters, radio signals, animation, game design and cryptography, explainer projects, mixed review sets, a timed practice assessment, halfway cumulative reviews and a growth reflection.
- 66 · Applied Project: Precalculus in Music Theory
- 67 · Applied Project: Precalculus in Population Ecology
- 68 · Explainer Project: Teaching Even and Odd Function Symmetry
- 69 · Practicing a Comprehensive Mixed Precalculus Review Set 2
- 71 · Applied Project: Precalculus in Roller Coaster Design
- 72 · Applied Project: Precalculus in Radio and Signal Transmission
- 73 · Reviewing Common Errors in Polar/Complex Conversions
- 74 · Practicing a Comprehensive Mixed Precalculus Review Set 3
- 76 · Timed Practice Assessment: A Full-Year Mixed Set
- 77 · Explainer Project: Teaching Mathematical Induction
- 78 · Applied Project: Precalculus in Computer Animation
- 79 · Practicing a Comprehensive Mixed Precalculus Review Set 4
- 81 · Cumulative Review: Rates of Change, Exponentials, and Logs
- 82 · Cumulative Review: Trig Identities and the Laws of Sines/Cosines
- 83 · Cumulative Review: Polar Coordinates, Vectors, and Matrices
- 84 · Timed Practice Assessment: The Halfway Comprehensive
- 85 · Halfway-Year Growth Reflection
- 86 · Applied Project: Precalculus in Game Design Mechanics
- 87 · Applied Project: Precalculus in Cryptography
- 88 · Explainer Project: Teaching Polar Coordinates
- 89 · Practicing a Comprehensive Mixed Precalculus Review Set 5
Plus 4 review days, not listed.
7
Capstone projectDays 91–100
A multi-day project on a real-world field chosen by the student: researching the math behind it, solving applied problems, building visuals, writing it up and presenting. A second capstone presentation comes at the end of the year.
- 91 · Capstone Prep: Choosing a Real-World Precalculus Field
- 92 · Capstone: Researching Foundations
- 93 · Capstone: Solving Applied Problems
- 94 · Capstone: Building Visuals
- 96 · Capstone: Writing Up the Project
- 97 · Capstone: Presentation Prep
- 98 · Capstone: Presentation Delivery
- 99 · Practicing a Comprehensive Mixed Precalculus Review Set 6
Plus 2 review days, not listed.
8
Applied projects, practice & cumulative reviewDays 101–130
Applied projects in meteorology, finance, engineering, sports, medical imaging and video compression, explainers, common-error days, timed practice assessments, a three-part cumulative review and a final applied project.
- 101 · Applied Project: Precalculus in Meteorology
- 102 · Applied Project: Precalculus in Financial Modeling
- 103 · Reviewing Common Errors in Recursive Sequences
- 104 · Practicing a Comprehensive Mixed Precalculus Review Set 7
- 106 · Explainer Project: Teaching Even and Odd Function Symmetry Revisited
- 107 · Applied Project: Precalculus in Structural Engineering
- 108 · Practicing a Comprehensive Mixed Precalculus Review Set 8
- 109 · Timed Practice Assessment 2: A Second Full-Year Mixed Set
- 111 · Applied Project: Precalculus in Sports Analytics
- 112 · Applied Project: Precalculus in Medical Imaging
- 113 · Reviewing Common Errors in Trig Equation Solving
- 114 · Practicing a Comprehensive Mixed Precalculus Review Set 9
- 116 · Applied Project: Precalculus in Video Compression
- 117 · Explainer Project: Teaching the Binomial Theorem
- 118 · Practicing a Comprehensive Mixed Precalculus Review Set 10
- 119 · Timed Practice Assessment: A Full-Year Mixed Set 2
- 121 · Full-Year Precalculus Cumulative Review, Part 1
- 122 · Full-Year Precalculus Cumulative Review, Part 2
- 123 · Full-Year Precalculus Cumulative Review, Part 3
- 124 · Timed Practice Assessment: The Comprehensive Final
- 126 · Explainer Project: Teaching Polar/Rectangular Conversion
- 127 · Full-Year Precalculus Growth Reflection
- 128 · Practicing a Comprehensive Mixed Precalculus Review Set 11
- 129 · Applied Project: Precalculus Bringing It All Together
Plus 6 review days, not listed.
9
More applied projects & timed practiceDays 131–170
Applied projects in robotics, satellites, weather radar, architecture, traffic timing, 3D printing, optics, sound synthesis, GPS and bridge cables, with explainers, common-error days, mixed review sets and timed practice. Little new content.
- 131 · Applied Project: Precalculus in Robotics Sensor Fusion
- 132 · Applied Project: Precalculus in Satellite Communication
- 133 · Reviewing Common Errors in Matrix Determinants and Inverses
- 134 · Practicing a Comprehensive Mixed Precalculus Review Set 12
- 136 · Applied Project: Precalculus in Weather Radar
- 137 · Explainer Project: Teaching Complex Numbers in Polar Form
- 138 · Practicing a Comprehensive Mixed Precalculus Review Set 13
- 139 · Timed Practice Assessment: A Full-Year Mixed Set 3
- 141 · Applied Project: Precalculus in Architecture Revisited
- 142 · Applied Project: Precalculus in Traffic Light Timing
- 143 · Explainer Project: Teaching the Law of Sines and Cosines Applications
- 144 · Practicing a Comprehensive Mixed Precalculus Review Set 14
- 146 · Applied Project: Precalculus in 3D Printing Design
- 147 · Reviewing Common Errors in Parametric Function Modeling
- 148 · Practicing a Comprehensive Mixed Precalculus Review Set 15
- 149 · Timed Practice Assessment: A Full-Year Mixed Set 4
- 151 · Applied Project: Precalculus in Camera Lens Optics
- 152 · Applied Project: Precalculus in Video Game Motion Design
- 153 · Explainer Project: Teaching Matrix Transformations Applications
- 154 · Practicing a Comprehensive Mixed Precalculus Review Set 16
- 156 · Applied Project: Precalculus in Sound Wave Synthesis
- 157 · Reviewing Common Errors in Logarithmic and Exponential Modeling
- 158 · Practicing a Comprehensive Mixed Precalculus Review Set 17
- 159 · Timed Practice Assessment: A Full-Year Mixed Set 5
- 161 · Applied Project: Precalculus in GPS Navigation
- 162 · Applied Project: Precalculus in Population Growth Modeling
- 163 · Explainer Project: Teaching Vector Applications in Physics
- 164 · Practicing a Comprehensive Mixed Precalculus Review Set 18
- 166 · Applied Project: Precalculus in Bridge Cable Design
- 167 · Reviewing Common Errors in Vector and Matrix Operations
- 168 · Practicing a Comprehensive Mixed Precalculus Review Set 19
- 169 · Timed Practice Assessment: A Full-Year Mixed Set 6
Plus 8 review days, not listed.
10
Final review, presentation & what's nextDays 171–180
Three-part final review of functions, trigonometry, vectors, logarithms, conics, matrices, parametric equations and complex numbers, a capstone presentation, reflections, a scavenger hunt and a look ahead to calculus.
- 171 · Final Review: Functions, Trigonometry, and Vectors
- 172 · Final Review: Logarithms, Conics, and Matrices
- 173 · Final Review: Parametric Equations and Complex Numbers
- 174 · Preparing a Capstone Presentation: A Year of Precalculus
- 175 · Delivering Your Capstone Presentation
- 176 · Reflecting on a Year of Precalculus
- 177 · Precalculus Scavenger Hunt: Math All Around You
- 178 · Previewing What’s Next: Calculus and Beyond
- 179 · Final Reflection: My Precalculus Journey
- 180 · Congratulations! You completed Precalculus!

By the end of the year, your child can

Analyze polynomial, rational, exponential and logarithmic functions and use them to model data
Use identities and the Laws of Sines and Cosines to solve trigonometry problems
Convert between polar and rectangular form and graph polar and parametric curves
Add and multiply vectors and matrices and solve systems with inverse matrices
Write simple proofs by induction and expand binomials with the Binomial Theorem
Identify and graph parabolas, ellipses and hyperbolas from their equations

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