bythreedu · Math Tracks · Curriculum v1.0 · 2026

Functions,
trig, and the
language of limits.

A ground-up precalculus education — starting with how functions actually behave, moving through every major function family, and building toward a confident entry into Calculus I. Designed for 2 sessions per week with structured assignments and a full resource ecosystem between lessons.

Total Lessons29at your pace
Units9progressive phases
Assignments9+problem sets
DestinationCalculus I
Ready
00Approach
Course Philosophy

Functions First, Formulas Second

Most students hit precalc and try to memorize formulas. That works until it doesn't — usually around trig identities or limits. This course builds understanding instead. Every formula is derived or explained before it's used. You'll know why the unit circle works before you're asked to use it.

Built for Calculus, Not Just the Test

The goal isn't to pass precalc. The goal is to walk into Calculus I with no gaps. That means taking limits seriously, not treating trig as a memorization exercise, and understanding function behavior at a level most precalc courses don't require. Every unit is sequenced with Calc I in mind.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–3
Functions: The Core Idea
3 lessons · Domain and range · Function behavior · Transformations
domain & rangefunction notationtransformations
+
L1
What a Function Is
Domain, range, function notation, and the vertical line test as a diagnostic — grounded in what a function actually means conceptually before touching any algebra.
L2
Function Behavior
Increasing and decreasing intervals, even and odd functions, end behavior — reading a function's personality from its equation before graphing anything.
L3
Transformations
Shifts, reflections, and stretches on any parent function. The transformation toolkit that applies to every function family in the course.
Assignment 1
Take 6 parent functions. Sketch each with and without a given transformation. Write one sentence on what changed and why. No graphing calculator on the first pass.
You read a function's personality from its equation before graphing anything.
02
Unit 2 · Weeks 4–6
Polynomial and Rational Functions
3 lessons · Polynomial graphs · Asymptotes · Sign charts
multiplicityasymptote behaviorsign chart
+
L4
Polynomial Graphs
Degree, leading term, end behavior, and multiplicity of roots — reading the graph of a polynomial from its factored form before plotting a single point.
L5
Rational Functions
Vertical and horizontal asymptotes, holes, and sign analysis. Why holes exist. How to find every feature algebraically before touching a graph.
L6
Polynomial and Rational Inequalities
Sign charts as the method, not guessing from a graph. Solving any polynomial or rational inequality by reading sign changes across critical values.
Assignment 2
For 3 rational functions: find all asymptotes and holes algebraically, sketch the graph without a calculator, then verify. Write one sentence explaining where each asymptote comes from.
You analyze any rational or polynomial function completely before graphing.
03
Unit 3 · Weeks 7–9
Exponential and Logarithmic Functions
3 lessons · Base e · Log rules · Solving equations
base elog rulesexponential equations
+
L7
Exponential Functions
Base e unpacked — where it comes from and why it matters. Growth and decay models. The number e as a limit, not a magic constant someone handed you.
L8
Logarithms
The logarithm defined as the inverse of exponentiation. Log rules derived, not memorized: product, quotient, power, and change of base, each with a proof.
L9
Solving Exponential and Log Equations
When to use ln vs. log. Checking for extraneous solutions — why they appear and how to catch them. Applications to real growth and decay problems.
Assignment 3
Solve 10 equations — 5 exponential, 5 logarithmic. On each, write the first decision you made and why. Flag any extraneous solutions and explain where they came from.
You move freely between exponential and logarithmic form and know when each is the right tool.
04
Unit 4 · Weeks 10–12
Trigonometry: The Unit Circle
3 lessons · Radian measure · Unit circle · Six trig functions
radian measureunit circle derivationsix trig functions
+
L10
Angles and Radian Measure
Why radians exist — the actual reason, not "because calculus needs it." Arc length. Converting between degrees and radians in both directions without a formula sheet.
L11
Building the Unit Circle
Derived from 30-60-90 and 45-45-90 triangles, not memorized. You build it from scratch in this session. By the end, you understand every coordinate and why it's there.
L12
The Six Trig Functions
Sin, cos, tan, and their reciprocals defined on the unit circle. Reading values off the circle. Reference angles. Quadrant signs without memorizing ASTC as a rule.
Assignment 4
Reconstruct the unit circle from scratch three times over two days. On the fourth attempt, use it to evaluate 20 trig expressions without reference. No shortcuts until you've built it yourself.
You own the unit circle. You built it, you didn't memorize it.
05
Unit 5 · Weeks 13–15
Trig Graphs and Identities
3 lessons · Amplitude and period · Trig graphs · Fundamental identities
amplitudeperiod & phase shiftPythagorean identities
+
L13
Graphs of Sin and Cos
Amplitude, period, phase shift, and vertical shift as parameters. Reading a trig equation and sketching the result without plotting points.
L14
Graphs of Tan, Sec, Csc, Cot
Asymptotes, period, and behavior for the remaining four functions. Why tan and cot have different periods. How asymptotes connect back to the unit circle.
L15
Fundamental Identities
Pythagorean, quotient, and co-function identities derived, not listed. Using identities to simplify expressions and verify equations. The difference between simplifying and solving.
Assignment 5
Sketch 5 trig functions from equations — no graphing calculator. Simplify 8 expressions using identities. Each simplification step must be labeled with the identity used.
You read the graph of any trig function from its equation and simplify with identities without a reference sheet.
06
Unit 6 · Weeks 16–18
Triangle Trig and Applications
3 lessons · Right triangle trig · Law of Sines · Law of Cosines
SOH-CAH-TOAambiguous caselaw selection
+
L16
Right Triangle Trig
SOH-CAH-TOA grounded in the unit circle, not presented as a separate rule. Elevation and depression problems. Setting up the diagram before writing any equation.
L17
Law of Sines and the Ambiguous Case
When the Law of Sines applies. The ambiguous case dissected and understood — why two triangles are sometimes possible, and how to find both.
L18
Law of Cosines
When to use Law of Cosines vs. Law of Sines — the decision made explicit. SAS and SSS cases. Applications where no right angle exists.
Assignment 6
Solve 6 triangles — 2 right, 2 using Law of Sines (including one ambiguous case), 2 using Law of Cosines. For each, document your method selection decision before solving.
You solve any triangle and know before you start which tool applies.
07
Unit 7 · Weeks 19–21
Analytic Geometry and Systems
3 lessons · Conic sections · Nonlinear systems · Sequences and series
conic standard formsnonlinear systemssigma notation
+
L19
Conic Sections
Parabola, circle, ellipse, hyperbola — standard form and graphing. Each parameter explained geometrically. Completing the square to convert general form to standard form.
L20
Systems of Nonlinear Equations
Substitution for polynomial and conic intersections. How many solutions to expect based on the geometry — predicting the answer before solving it.
L21
Sequences and Series
Arithmetic and geometric sequences. Sigma notation introduced and read fluently. Finite sums derived, not looked up. The connection between series and later integral concepts.
Assignment 7
Graph all four conic sections from given equations. Write the meaning of each parameter in plain English — not "a controls the width" but what that actually means geometrically.
You see conics as a family and work with sequences without a formula sheet.
08
Unit 8
Discrete Mathematics and Foundations of Calculus
3 lessons · Logic and sets · Combinatorics · Rates of change
logic & proofcombinatoricsaverage rate of changedifference quotient
+
L22
Logic, Sets, and Proof
Propositional logic, set notation, and the basics of mathematical proof — direct, contrapositive, and contradiction. The language of mathematics before calculus demands it.
L23
Combinatorics and the Binomial Theorem
Counting principles, permutations, combinations, and Pascal's triangle. The binomial theorem derived and applied. How counting connects to polynomial expansions.
L24
Rates of Change — The Bridge to Calculus
Average rate of change, the difference quotient, and instantaneous rate of change as a limit. Slope as a dynamic quantity. This is the conceptual entry point to the derivative — without calling it that yet.
Assignment 8
Compute the difference quotient for 4 functions. For each, describe in plain language what happens as h approaches 0. Sketch the secant line becoming a tangent line. You're seeing the derivative before you've named it.
You understand what calculus is actually measuring before you take Calculus I.
09
Unit 9
Introduction to Limits and Capstone
4 lessons · Limit intuition · Limit laws · Continuity · Capstone exam
limit notationone-sided limitscontinuity types
+
L26
Limits Intuitively
Approaching a value from both sides. One-sided limits. The distinction between a limit and a function value — the entire foundation of calculus, explained from scratch.
L27
Limit Laws and Algebraic Evaluation
Limit rules applied rigorously. Indeterminate forms and how to resolve them: factoring, rationalizing. The squeeze theorem introduced with a geometric example.
L28
Continuity
Three conditions for continuity. Types of discontinuity — removable, jump, infinite — and how to identify each from a graph or equation. The Intermediate Value Theorem as intuition before formal statement.
L29
Capstone Planning and Gap Review
Student selects three hardest units for a focused final review. Then a full 90-minute timed mock exam covering all nine units.
Capstone — Mock Exam
A timed 90-minute mock exam covering all 9 units. Every missed problem gets a written correction with reasoning — not just the right answer, but the decision that should have been made at each step.
You step into Calculus I already fluent in functions, trig, and the language of limits.
02Reference
Core Concepts
f(x)
Function Notation
A function maps each input to exactly one output. Domain is the set of valid inputs. Range is the set of resulting outputs. f(x) is not "f times x."
Vertical line test · Domain restrictions · Piecewise
e ≈ 2.718
The Number e
Euler's number arises naturally from compound interest as the compounding interval approaches zero. It is the base of natural growth, not an arbitrary constant.
e = lim(1 + 1/n)^n as n → ∞
π rad
Radian Measure
One radian is the angle subtended by an arc equal in length to the radius. Radians make calculus clean — trig derivatives only simplify in radians.
π = 180° · 2π = full rotation · π/2 = 90°
sin² + cos²
Pythagorean Identity
sin²θ + cos²θ = 1 follows directly from the unit circle. It is the Pythagorean theorem applied to a radius of 1. Every other Pythagorean identity derives from this one.
Divide by cos²: tan²+1=sec² · Divide by sin²: 1+cot²=csc²
lim →
The Limit
The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to, regardless of what happens at x=a. The limit and the function value are different things.
One-sided · Two-sided · Limits at infinity
↕
Asymptotes
Vertical asymptotes occur where a rational function's denominator is zero and numerator is not. Horizontal asymptotes describe end behavior as x grows large.
Vertical · Horizontal · Oblique (slant)
03Capstone
Mock Exam and Final Review

The capstone is a full-length timed mock exam built from every unit in the course. Not a review sheet — a real exam simulation. Every missed problem becomes a lesson: we go through the reasoning together and you write out the corrected approach in your own words. The goal is to walk into Calculus I without a single gap.

○
Gap Identification Session
Before the exam, you select the three units you're least confident in. We do a targeted review of each before the clock starts.
○
90-Minute Timed Exam
Full coverage across all 9 units. Timed and taken independently — same conditions as a real exam.
○
Written Corrections
Every missed problem gets a written correction with reasoning — not just the right answer, but the decision that should have been made at each step.
○
Calculus I Readiness Review
A final session mapping your precalc skills to what Calc I will demand — so you know exactly where you stand on day one.
04Ecosystem
Between-Lesson Resources
►
Curated Reading and Reference
Every unit ships with a reference packet mapped to that week's topic.
  • Annotated worked examples for each lesson
  • Concept summaries with derivations, not just formulas
  • Recommended Khan Academy and Paul's Math Notes sections
  • Recommended book list delivered Week 1
◇
Practice Assignments
Between sessions, a focused problem set targeting one specific skill.
  • One assignment per unit, focused enough to build one skill
  • Submit to shared folder before next session
  • Each session opens with a 10-min review of your work
  • Running feedback notes doc updated every lesson
↗
Async Support
Questions don't wait until next session.
  • Text or DM for quick concept or problem questions
  • Problem feedback via email — 24-hr turnaround
□
Student Hub
A living library organized by unit — everything in one place.
  • Session notes and key takeaways per lesson
  • Formula sheets with derivations attached
  • Unit circle, trig identity, and conic reference sheets
  • Exam correction archive — every capstone feedback saved
05Investment
Pricing Options
per Session
$85
1-hr lesson · online only
Flexible · No commitment
Session length
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing to a plan.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hrs/lesson
Online + In-Person available
Frequency
Session length
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Adjust frequency and session length above — price updates automatically.

Get started
Full Course
$3,775
29 lessons · 1.5 hrs/lesson
Save vs. monthly · Online + In-Person available
Session length
  • All 9 units — full course arc
  • Session notes after every lesson
  • Capstone mock exam + written corrections
  • Calculus I readiness review at completion

The full Precalculus arc — from functions to limits — in one commitment.

Enroll now
In-person sessions are available on Monthly and Semester plans. In-person adds $150/mo to the monthly plan ($800/mo) and $4,625 for the full course. In-person is available within NYC. Per-session bookings are online only.
06Contact
Let's Talk

Not sure which plan is right for you? Start with one session and see how it feels. No pressure, no commitment — just an hour and a clearer sense of where you're headed.

Do I need to be currently enrolled in a precalculus course? +
No. Some students use this track as a standalone course before college. Others are enrolled in precalc and use it to go deeper than the classroom pace allows. Both work. We'll establish where you are in the first session and move from there.
What do I need to know coming in? +
Solid algebra. You should be comfortable solving equations, working with functions at a basic level, and handling fractions and exponents. If you're shaky on any of that, let me know before the first session and we'll run a quick diagnostic.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC. In-person is priced slightly higher to account for travel and setup — see pricing above for details.
Can I switch plans later? +
Yes. You can upgrade from per-hour to Monthly or Semester at any point. Any paid sessions apply toward your first month. If you start Monthly and want to commit to the full semester, we'll prorate what you've already paid.
What if I need to reschedule? +
Life happens. Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.