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Pre-Calculus

Pre-Calculus Course

Course Features

Course Duration: Full Year
Category: , ,
Prerequisite: Trigonometry
Assessment: Pre-Test, Lesson Practice, Unit Exams, Mid-Term Exam, Final Exam
Language: English

Course Details

Course Overview

Acellus Pre-Calculus serves as a critical bridge between Algebra II and Calculus, equipping students with advanced mathematical tools and conceptual understanding necessary for success in higher mathematics. The course delves deeply into the study of functions and their properties, polynomial and rational expressions, exponential and logarithmic modeling, comprehensive trigonometry, and additional topics such as vectors, matrices, conic sections, and probability. Students strengthen their algebraic manipulation skills, graphical analysis abilities, and problem-solving strategies while exploring real-world applications across various contexts.

Throughout the curriculum, students encounter previews of calculus concepts integrated within each unit, fostering an early appreciation for rates of change, accumulation, and limits. By mastering transformations of functions, analytic trigonometry, systems of equations, and data modeling techniques, learners develop the rigorous foundation required for STEM fields. Emphasis is placed on both procedural fluency and conceptual understanding, preparing students to approach calculus with confidence and mathematical maturity.

Sample Lesson - Introduction to Pre-Calculus

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Scope and Sequence

Unit 1: Functions In Unit 1, students will master the foundational concepts of functions, including notation, domain and range, continuity, increasing/decreasing behavior, odd/even properties, and asymptotes. They will explore the toolkit of parent functions, piecewise definitions, operations on functions (addition, subtraction, multiplication, division, composition), inverses, and various transformations such as translations, dilations, and reflections. The unit also introduces arithmetic and geometric sequences as building blocks for mathematical modeling.

Unit 2: Polynomials and Solving Equations Students in Unit 2 will develop expertise with polynomial functions of various degrees, with particular focus on linear and quadratic functions. They will learn multiple methods for solving equations including factoring, the quadratic formula, and completing the square, as well as advanced techniques for higher-degree polynomials such as long and synthetic division, the Remainder and Factor Theorems, and the Rational Zero Theorem. Power functions and direct variation are also examined.

Unit 3: Complex Numbers & Rational Functions This unit introduces operations with complex numbers, graphing in the complex plane, and applications involving absolute value, distance, and midpoint. Students will apply the Fundamental Theorem of Algebra to analyze zeros of polynomials and construct polynomials from given zeros. For rational functions, topics include identifying intercepts and all types of asymptotes, performing arithmetic operations and partial fraction decomposition, solving rational equations and inequalities, and expressing solutions in interval notation.

Unit 4: Exponential & Logarithmic Functions Unit 4 focuses on exponential and logarithmic functions and their real-world applications. Students will graph and transform these functions, study the natural base e and logistic models, and apply logarithm properties to simplify and solve equations. Modeling with sequences and series (arithmetic, geometric, and logistic) is emphasized, along with practical applications including compound and continuous interest, population growth, and decay.

Unit 5: Trigonometric Functions In Unit 5, students build foundational trigonometric understanding through angle measurement in degrees and radians, arc length and sector area calculations, and right triangle trigonometry. They master the unit circle for evaluating trigonometric values, graph all six trigonometric functions and their transformations, explore harmonic motion applications, and learn to evaluate and apply inverse trigonometric functions.

Unit 6: Analytic Trigonometry Students will deepen their trigonometric knowledge by working with fundamental identities, sum and difference formulas, double- and half-angle formulas, and techniques for proving identities and solving trigonometric equations. Triangle applications include the Law of Sines and Law of Cosines, area formulas (including Heron's), and calculations for regular polygons, culminating in integrated problem-solving and regression modeling.

Unit 7: Vectors, Polar, and Complex Roots Unit 7 covers vector operations including components, addition, and dot products with geometric applications. Students explore parametric equations and convert between rectangular, polar, and complex representations. Key topics include graphing polar equations and applying DeMoivre's Theorem to powers and roots of complex numbers in polar form.

Unit 8: Multivariate Systems and Matrices This unit examines systems of linear equations in two and three variables using substitution, elimination, and matrix methods (row reduction, inverses). Students study matrix arithmetic, geometric transformations using matrices, systems of inequalities, linear programming optimization, and related concepts such as distance, midpoints, and incidence matrices.

Unit 9: Conic Sections Students investigate the four conic sections—parabolas, circles, ellipses, and hyperbolas—learning to derive equations from graphs and vice versa, identify key features (vertices, foci, asymptotes), and graph these curves. Techniques include completing the square and classifying general second-degree equations, with discussion of reflective properties of the conics.

Unit 10: Probability and Combinatorics In Unit 10, students explore counting principles, permutations, combinations, Pascal's Triangle, and the Binomial Theorem. The unit also covers probability concepts through discrete and continuous distributions, including binomial and normal distributions, while incorporating three-dimensional graphing and vector applications in space.

Unit 11: Introduction to the Basics of Calculus The concluding unit provides an accessible introduction to calculus concepts. Students explore limits, instantaneous rates of change (derivative intuition), approximation of areas under curves, relationships between position and velocity functions, and the Fundamental Theorem of Calculus.

Following this unit students are presented with the Final Review and Exam.
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