Pre-Calculus (Grade 10–12 Bridge to AP Calculus) Pre-Calculus (Grade 10–12 Bridge to AP Calculus)
📌 Course at a glance
· 16 units · key topics drilled across the official outline
· Next exam: 2026-05-12 · 2026-05-12
📝 Full chapter-by-chapter lecture walkthroughs are being written. This page is the course outline — every unit and topic that will be covered, with key concepts you can already start to recognise. Use it as a study map; the chapter notes drop in over the coming weeks.
What you can already do today (free)
- 📺 Watch curated YouTube videos on every topic (links from the chapter pages as they release)
- 🎯 Take 1 free diagnostic quiz to see your starting level
- 📊 Browse the full course outline below to plan your study path
With All-Access (free 7-day trial):
- 🧠 Memory Curve spaced review with email reminders
- 📝 Unlimited MCQ / FRQ practice
- 📑 Mock exams and progress analytics
Course outline
Unit 1 · Algebra & Function Foundations Algebra & Function Foundations
Exam weight: Gr 10 · Foundational
1.1 — Real Numbers, Exponents & Radicals Real Numbers, Exponents & Radicals
· Integer, rational and irrational numbers; the real number line and interval notation · Laws of exponents, including zero, negative and rational (fractional) exponents · Simplifying radicals and rationalizing denominators
1.2 — Factoring & Algebraic Expressions Factoring & Algebraic Expressions
· Expanding and factoring: common factor, grouping, trinomials, difference/sum of squares and cubes · Simplifying rational expressions and combining fractions · Polynomial and rational expression arithmetic
1.3 — Linear & Quadratic Equations and Inequalities Linear & Quadratic Equations and Inequalities
· Solving linear equations/inequalities and absolute-value equations · Quadratics by factoring, completing the square, and the quadratic formula; the discriminant · Quadratic and absolute-value inequalities; sign analysis on a number line
1.4 — Functions: Notation, Domain & Range Functions: Notation, Domain & Range
· Function definition and the vertical line test; function notation f(x) · Finding domain and range from formulas and graphs · Piecewise-defined functions and evaluating them
1.5 — Graphs & Transformations of Functions Graphs & Transformations of Functions
· Parent functions (line, parabola, cubic, root, reciprocal, absolute value) and their shapes · Shifts, reflections, vertical/horizontal stretches and compressions · Even and odd functions; symmetry
1.6 — Composition & Inverse Functions Composition & Inverse Functions
· Composing functions (f ∘ g)(x) and decomposing a composite · One-to-one functions and the horizontal line test · Finding inverse functions and the reflection over y = x
Unit 2 · Polynomial & Rational Functions Polynomial & Rational Functions
Exam weight: Gr 10–11 · Core
2.1 — Polynomial Functions & End Behavior Polynomial Functions & End Behavior
· Degree, leading coefficient and standard form · End behavior from degree parity and leading-coefficient sign · Turning points, multiplicity of zeros, and graph shape
2.2 — Polynomial Division; Remainder & Factor Theorems Polynomial Division; Remainder & Factor Theorems
· Long division and synthetic division of polynomials · Remainder Theorem: f(c) equals the remainder on dividing by (x − c) · Factor Theorem: (x − c) is a factor iff f(c) = 0
2.3 — Zeros of Polynomials & the Fundamental Theorem of Algebra Zeros of Polynomials & the Fundamental Theorem of Algebra
· Rational Root Theorem and finding all real zeros · Complex zeros occur in conjugate pairs · Fundamental Theorem of Algebra: degree n has n roots (with multiplicity)
2.4 — Rational Functions & Asymptotes Rational Functions & Asymptotes
· Domain, holes, and vertical asymptotes from the denominator · Horizontal and slant asymptotes from degree comparison · Sketching rational graphs with intercepts and asymptotes
2.5 — Polynomial & Rational Inequalities Polynomial & Rational Inequalities
· Sign charts using zeros and undefined points · Solving polynomial inequalities · Solving rational inequalities and writing interval solutions
Unit 3 · Exponential & Logarithmic Functions Exponential & Logarithmic Functions
Exam weight: Gr 11 · Core
3.1 — Exponential Functions and e Exponential Functions and e
· Exponential growth/decay graphs and the role of the base · The natural base e and continuous growth · Transformations of exponential functions
3.2 — Logarithmic Functions & Their Graphs Logarithmic Functions & Their Graphs
· Logarithm as the inverse of an exponential; log/exponential form · Common (base 10) and natural (base e) logarithms · Domain, range and graph of logarithmic functions
3.3 — Properties of Logarithms Properties of Logarithms
· Product, quotient and power rules · Change-of-base formula · Expanding and condensing logarithmic expressions
3.4 — Exponential & Logarithmic Equations Exponential & Logarithmic Equations
· Solving exponential equations by taking logs · Solving logarithmic equations and checking domain · Extraneous solutions
3.5 — Modeling with Exponential & Log Functions Modeling with Exponential & Log Functions
· Compound and continuous interest · Population growth, half-life and Newton's law of cooling · Logarithmic scales (pH, decibels, Richter)
Unit 4 · Trigonometric Functions Trigonometric Functions
Exam weight: Gr 11 · Core
4.1 — Angles & Radian Measure Angles & Radian Measure
· Degrees and radians; converting between them · Coterminal and reference angles; standard position · Arc length and sector area
4.2 — The Unit Circle The Unit Circle
· Defining sine, cosine and tangent on the unit circle · Exact values for special angles (30°, 45°, 60° and multiples) · Signs of trig functions by quadrant
4.3 — Right-Triangle Trigonometry Right-Triangle Trigonometry
· SOH-CAH-TOA and solving right triangles · Angles of elevation and depression · The reciprocal ratios (csc, sec, cot)
4.4 — Graphs of Sine & Cosine Graphs of Sine & Cosine
· Amplitude, period, phase shift and vertical shift · Graphing y = A sin(Bx − C) + D · Modeling periodic phenomena
4.5 — Graphs of Other Trig Functions Graphs of Other Trig Functions
· Tangent and cotangent graphs and their asymptotes · Secant and cosecant graphs · Period and domain restrictions
4.6 — Inverse Trigonometric Functions Inverse Trigonometric Functions
· Restricting domains to define arcsin, arccos, arctan · Evaluating inverse trig expressions · Compositions like sin(arccos x)
Unit 5 · Analytic Trigonometry Analytic Trigonometry
Exam weight: Gr 11–12 · Core
5.1 — Fundamental Trig Identities Fundamental Trig Identities
· Reciprocal, quotient and Pythagorean identities · Even/odd and cofunction identities · Simplifying and verifying identities
5.2 — Sum & Difference Formulas Sum & Difference Formulas
· sin(A ± B), cos(A ± B), tan(A ± B) · Finding exact values of non-special angles · Verifying identities with sum/difference formulas
5.3 — Double- & Half-Angle Formulas Double- & Half-Angle Formulas
· Double-angle formulas for sin, cos, tan · Half-angle formulas and sign selection · Power-reduction formulas
5.4 — Solving Trigonometric Equations Solving Trigonometric Equations
· Solving on a restricted interval and over all reals · Using identities to reduce to a single function · General solutions with period notation
5.5 — Law of Sines & Law of Cosines Law of Sines & Law of Cosines
· Law of Sines and the ambiguous (SSA) case · Law of Cosines for SAS and SSS · Triangle area, including Heron's formula
Unit 6 · Vectors, Polar & Parametric (AP Calculus Bridge) Vectors, Polar & Parametric
Exam weight: AP Bridge · beyond BC PreCalc
6.1 — Vectors in the Plane Vectors in the Plane
· Component and magnitude-direction form · Vector addition, subtraction and scalar multiples · Unit vectors and resolving into components
6.2 — The Dot Product The Dot Product
· Dot product formula and the angle between vectors · Orthogonality test · Projection of one vector onto another
6.3 — Polar Coordinates & Polar Graphs Polar Coordinates & Polar Graphs
· Converting between polar and rectangular coordinates · Graphing circles, roses, cardioids and limaçons · Polar equations of common curves
6.4 — Complex Numbers in Polar Form & De Moivre's Theorem Complex Numbers in Polar Form & De Moivre's Theorem
· Modulus and argument; trigonometric form · Multiplying/dividing in polar form · De Moivre's Theorem and nth roots
6.5 — Parametric Equations Parametric Equations
· Graphing parametric curves and direction of motion · Eliminating the parameter · Projectile and motion applications
Unit 7 · Analytic Geometry: Conic Sections Analytic Geometry: Conic Sections
Exam weight: Gr 12 · Advanced
7.1 — Parabolas Parabolas
· Focus-directrix definition · Standard equations and orientation · Vertex, focus and directrix from the equation
7.2 — Ellipses Ellipses
· Foci, vertices, major/minor axes · Standard equation and the a, b, c relationship · Eccentricity
7.3 — Hyperbolas Hyperbolas
· Foci, vertices and the transverse axis · Asymptotes of a hyperbola · Standard equation and orientation
7.4 — Conics in Polar Form & Eccentricity Conics in Polar Form & Eccentricity
· Unified focus-directrix definition by eccentricity · Polar equation of a conic · Identifying a conic from its eccentricity
Unit 8 · Sequences, Series & the Binomial Theorem Sequences, Series & the Binomial Theorem
Exam weight: Gr 12 · Advanced
8.1 — Sequences & Summation Notation Sequences & Summation Notation
· Explicit and recursive definitions · Sigma (summation) notation · Terms, partial sums and patterns
8.2 — Arithmetic Sequences & Series Arithmetic Sequences & Series
· Common difference and the nth term · Sum of a finite arithmetic series · Applications
8.3 — Geometric Sequences & Series Geometric Sequences & Series
· Common ratio and the nth term · Finite geometric sums · Infinite geometric series and convergence
8.4 — The Binomial Theorem The Binomial Theorem
· Pascal's triangle and binomial coefficients · Expanding (a + b)^n · Finding a specific term
8.5 — Mathematical Induction Mathematical Induction
· Base case and inductive step · Proving summation formulas · Proving divisibility and inequalities
Unit 9 · Limits & Rates of Change (AP Calculus Bridge) Bridge to Calculus: Limits & Rates of Change
Exam weight: AP Bridge · Calculus 12 preview
9.1 — The Idea of a Limit (Graphical & Numerical) The Idea of a Limit (Graphical & Numerical)
· Limits from tables and graphs · One-sided limits · When a limit fails to exist
9.2 — Computing Limits Algebraically Computing Limits Algebraically
· Limit laws and direct substitution · Factoring and rationalizing to resolve · Limits at infinity
9.3 — Continuity Continuity
· The three-part definition of continuity at a point · Types of discontinuity · Intermediate Value Theorem (introduction)
9.4 — Average & Instantaneous Rate of Change Average & Instantaneous Rate of Change
· Average rate of change as the slope of a secant line · The difference quotient · Instantaneous rate as a limit (preview of the derivative)
9.5 — Area & the Accumulation Idea Area & the Accumulation Idea
· Estimating area under a curve with rectangles · Left, right and midpoint sums · Accumulation as a preview of the integral
Unit 10 · Grade 10 Foundations: Numbers, Exponents & Polynomials
Exam weight: Gr 10 · Foundational
10.1 — Powers with Integer Exponents & Exponent Laws Powers with Integer Exponents & Exponent Laws
· Positive and negative integer exponents · Product, quotient, and power laws · Evaluation with order of operations, numerical and variable bases
10.2 — Prime Factorization, GCF & LCM Prime Factorization, GCF & LCM
· Prime factorization using powers and factor trees · Greatest common factor (GCF) · Least common multiple (LCM)
10.3 — Multiplying Polynomials Multiplying Polynomials
· Distributive property between two polynomials · FOIL and area models for binomial products · Special products
10.4 — Factoring Polynomials (GCF, Trinomials, Difference of Squares) Factoring Polynomials (GCF, Trinomials, Difference of Squares)
· Greatest common factor of a polynomial · Trinomials x^2 + bx + c · Difference of squares
Unit 11 · Grade 10 Foundations: Relations, Lines, Trig & Money
Exam weight: Gr 10 · Foundational
11.1 — Relations, Functions, Domain & Range Relations, Functions, Domain & Range
· Meaning of a function; identifying functions from relations · Function notation · Domain and range from graphs and contexts
11.2 — Linear Functions: Slope & Equations of Lines Linear Functions: Slope & Equations of Lines
· Slope: positive, negative, zero, undefined · Slope-intercept, point-slope, and general forms · Parallel and perpendicular lines
11.3 — Arithmetic Sequences Arithmetic Sequences
· Common difference, first term, general term · Increasing and decreasing linear patterns · Connection to linear relations and arithmetic series
11.4 — Systems of Linear Equations Systems of Linear Equations
· Solving graphically · Solving by substitution and elimination · Applications in context
11.5 — Primary Trigonometric Ratios (Right Triangles) Primary Trigonometric Ratios (Right Triangles)
· Sine, cosine, and tangent ratios · Finding missing sides and angles · Direct and indirect measurement problems
11.6 — Financial Literacy: Gross & Net Pay Financial Literacy: Gross & Net Pay
· Types of income · Income tax and deductions · Computing net pay from gross pay
Unit 12 · Function Transformations (Pre-Calculus 12)
Exam weight: Gr 12 · Core
12.1 — Vertical & Horizontal Translations Vertical & Horizontal Translations
· y = f(x) + k shifts vertically · y = f(x - h) shifts horizontally (opposite sign) · Effect on domain, range, and key points
12.2 — Reflections Reflections
· y = -f(x) reflects across the x-axis · y = f(-x) reflects across the y-axis · Reflecting key points and intercepts
12.3 — Stretches & Compressions Stretches & Compressions
· y = a f(x): vertical stretch/compression by |a| · y = f(bx): horizontal stretch/compression by 1/|b| · Combined effect of a and b on the graph
12.4 — Combining Transformations Combining Transformations
· Order of operations for multiple transformations · Writing y = a f(b(x - h)) + k · Mapping points through the full transformation
12.5 — Inverse of a Function Inverse of a Function
· Inverse as a reflection across y = x · Finding the inverse by swapping x and y · Domain restrictions so an inverse is a function
Unit 13 · Radical, Absolute Value & Reciprocal Functions (Pre-Calculus 12)
Exam weight: Gr 12 · Core
13.1 — Radical Functions & Their Graphs Radical Functions & Their Graphs
· Graph of y = sqrt(x) and transformations · Domain and range of radical functions · y = sqrt(f(x)) from y = f(x)
13.2 — Solving Radical Equations Solving Radical Equations
· Solving algebraically and graphically · Restrictions on the variable · Checking for extraneous roots
13.3 — Absolute Value Functions Absolute Value Functions
· Graph of y = |x| and transformations · Writing absolute value as a piecewise function · Solving absolute value equations
13.4 — Reciprocal Functions Reciprocal Functions
· Graph of y = (x); vertical asymptotes where f(x)=0 · Invariant points where f(x) = ±1 · Behaviour near asymptotes
Unit 14 · Function Operations & Composition (Pre-Calculus 12)
Exam weight: Gr 12 · Core
14.1 — Sum, Difference, Product & Quotient of Functions Sum, Difference, Product & Quotient of Functions
· (f ± g)(x), (fg)(x), ()(x) · Domain of the combined function · Quotient restrictions where g(x)=0
14.2 — Composite Functions Composite Functions
· (f o g)(x) = f(g(x)) · Evaluating and forming compositions · Decomposing a composite function
14.3 — Restrictions on Composite Functions Restrictions on Composite Functions
· Domain of a composite function · Inner-function range feeding the outer function · Excluded values
Unit 15 · Counting: Permutations, Combinations & Binomial Theorem (Pre-Calculus 12)
Exam weight: Gr 12 · Core
15.1 — The Fundamental Counting Principle The Fundamental Counting Principle
· Multiplying choices for independent stages · Counting with and without restrictions · Tree diagrams
15.2 — Permutations Permutations
· Ordered arrangements; nPr = n!/(n-r)! · Permutations with repetition and restrictions · Factorial notation
15.3 — Combinations Combinations
· Unordered selections; nCr = n!/(r!(n-r)!) · When order does not matter · Relationship nCr = nC(n-r)
15.4 — The Binomial Theorem via Combinations The Binomial Theorem via Combinations
· Binomial coefficients as nCk · Expanding (a+b)^n with combinations · Finding a general term
Unit 16 · Financial Literacy (Pre-Calculus 11)
Exam weight: Gr 11 · Core
16.1 — Simple & Compound Interest Simple & Compound Interest
· Simple interest I = Prt · Compound interest A = P(1 + i)^n · Comparing simple and compound growth
16.2 — Annuities: Investments & Loans Annuities: Investments & Loans
· Future value of an annuity (regular payments) · Present value and loan payments · Effect of compounding frequency
16.3 — Buying vs Leasing, Owning vs Renting Buying vs Leasing, Owning vs Renting
· Total cost of buying vs leasing a vehicle · Owning vs renting a home · Comparing financial options