Precalculus Course

Lesson 1 • Properties of Real Numbers

Covers commutative, associative, distributive, and identity properties. Students apply these to justify algebraic steps in later chapters.

Lesson 2 • Absolute Value and Inequalities

Defines absolute value geometrically and algebraically, then extends to inequality notation. Prepares students for solving absolute value equations in algebra chapters.

Lesson 3 • Order of Operations and Expressions

Establishes the standard evaluation hierarchy for arithmetic and algebraic expressions. Correct application prevents errors in every subsequent topic.

Lesson 1 • Solving Linear Equations

Applies inverse operations to isolate variables in one-variable linear equations. Provides the procedural backbone for solving all equation types ahead.

Lesson 2 • Factorising Polynomials

Introduces GCF, trinomial, and special-product factorising strategies. Factorising is essential for solving quadratics and simplifying rational expressions.

Lesson 3 • Solving Quadratic Equations

Solves quadratics by factorising, completing the square, and the quadratic formula. Students identify real and complex solutions and interpret discriminant values.

Lesson 4 • Linear and Absolute Value Inequalities

Extends equation-solving techniques to inequalities and absolute value cases. Solutions are expressed in interval notation and graphed on a number line.

Lesson 1 • Relations and Function Definition

Distinguishes relations from functions using mapping diagrams and the vertical line test. Establishes the function concept central to all remaining chapters.

Lesson 2 • Combining and Composing Functions

Defines arithmetic combinations and composition of functions with domain restrictions. Composition is prerequisite knowledge for inverse functions and logarithms.

Lesson 3 • Transformations of Functions

Applies shifts, reflections, stretches, and compressions to parent functions. Understanding transformations enables rapid graphing of complex functions.

Lesson 4 • Inverse Functions

Determines whether a function is one-to-one and finds its inverse algebraically and graphically. Inverse functions underpin exponential-logarithmic relationships ahead.

Lesson 1 • Linear Functions and Their Graphs

Connects slope and intercepts to the graph and equation of a line. Builds graphical intuition applied to all function families in later chapters.

Lesson 2 • Quadratic Functions and Parabolas

Converts between standard, vertex, and intercept forms of quadratic functions. Students graph parabolas and identify vertex, axis of symmetry, and intercepts.

Lesson 3 • Systems of Linear Equations

Solves two- and three-variable systems by substitution, elimination, and matrices. Systems modelling prepares students for optimisation and conic section intersections.

Lesson 1 • Finding Zeros of Polynomials

Uses Rational Zero Theorem, Descartes' Rule, and complex zeros to find all roots. Students connect zeros to factored form and graph x-intercepts.

Lesson 2 • Solving Polynomial and Rational Inequalities

Applies sign-chart analysis to solve polynomial and rational inequalities. Solutions are expressed in interval notation consistent with earlier inequality work.

Lesson 3 • Rational Functions and Asymptotes

Analyses domain, vertical, horizontal, and oblique asymptotes of rational functions. Asymptote behaviour is essential for graphing and limit intuition.

Lesson 4 • Dividing Polynomials

Performs long division and synthetic division to factor and evaluate polynomials. Division is the gateway to the Remainder and Factor Theorems.

Lesson 1 • Exponential and Logarithmic Models

Applies exponential growth, decay, logistic, and logarithmic models to data. Students fit models and interpret parameters in scientific and financial contexts.

Lesson 2 • Solving Exponential and Logarithmic Equations

Solves equations using one-to-one properties, change of base, and logarithm rules. Students check for extraneous solutions arising from domain restrictions.

Lesson 3 • Exponential Functions and Graphs

Defines exponential functions, identifies base restrictions, and graphs transformations. Growth and decay behaviour is connected to real-world contexts immediately.

Lesson 4 • Properties of Logarithms

Applies product, quotient, and power rules to expand and condense logarithmic expressions. These properties are required for solving logarithmic equations.

Lesson 1 • The Unit Circle

Establishes coordinates on the unit circle for standard angles in all four quadrants. Memorising key values enables exact evaluation of all trig functions.

Lesson 2 • The Six Trigonometric Functions

Defines sine, cosine, tangent, and their reciprocals using unit circle coordinates. Students evaluate all six functions and identify undefined values.

Lesson 3 • Graphs of Other Trig Functions

Extends graphing to tangent, cotangent, secant, and cosecant with asymptotes. Students identify all six function graphs and their key features.

Lesson 4 • Graphs of Sine and Cosine

Identifies amplitude, period, phase shift, and vertical shift from equations and graphs. Graphing mastery is required for modelling periodic phenomena.

Lesson 5 • Angles and Radian Measure

Defines degree and radian measures, converts between them, and computes arc length. Radian measure is the standard for all calculus-level trigonometry.

Lesson 1 • Solving Trigonometric Equations

Solves trig equations on restricted and general domains using identities and inverses. Students express general solutions using period-based notation.

Lesson 2 • Conic Sections

Derives and graphs parabolas, ellipses, hyperbolas, and circles from standard equations. Students identify key features and connect conics to their geometric definitions.

Lesson 3 • Laws of Sines and Cosines

Applies the Law of Sines and Law of Cosines to solve oblique triangles. Students determine which law applies and handle the ambiguous SSA case.

Lesson 4 • Trigonometric Identities

Introduces Pythagorean, reciprocal, quotient, and co-function identities for simplification. Mastery of identities is required for proving and solving trig equations.