Course Overview

This course, Pre-Calculus Algebra/Trigonometry, covers foundational and advanced topics in algebra and trigonometry, preparing students for calculus and further mathematical studies. The curriculum aligns with standard precalculus chapters, including functions, equations, graphs, trigonometric concepts, analytic geometry, systems, sequences, probability, and more.

Course Competencies and Learning Outcomes

Piecewise Defined Functions

Piecewise functions are defined by different expressions for different intervals of the domain. Understanding and graphing these functions is essential for modeling real-world scenarios with abrupt changes.

• Definition: A function composed of multiple sub-functions, each applying to a certain interval.

• Graphing: Plot each segment according to its domain.

• Example:

Exponential and Logarithmic Functions

These functions model growth, decay, and many natural phenomena. Their properties and graphs are fundamental in precalculus.

• Exponential Function: where and

• Logarithmic Function: , the inverse of the exponential function

• Properties: Domain, range, transformations, and inverse relationships

• Solving Equations: Use properties of exponents and logarithms

• Example: Solve ;

• Applications: Exponential growth/decay, compound interest

Polynomial Functions and the Fundamental Theorem of Algebra

Polynomials are central to algebra, with their roots and behavior forming the basis for many mathematical analyses.

• Definition:

• Fundamental Theorem of Algebra: Every non-zero polynomial of degree has $n$ complex roots.

• Graphing: Analyze zeros, multiplicity, and end behavior

• Example:

Rational Functions

Rational functions are quotients of polynomials, with unique behaviors near points of discontinuity.

• Definition: where

• Domain: All real numbers except where

• Graphing: Identify asymptotes and discontinuities

• Example:

Equations and Inequalities

Solving equations and inequalities, both linear and nonlinear, is a fundamental skill in precalculus.

• Linear Equations:

• Nonlinear Equations: Involve powers, roots, or products of variables

• Inequalities: Solve and graph solution sets

• Example: Solve ;

Conic Sections

Conic sections arise from the intersection of a plane and a cone, resulting in parabolas, ellipses, and hyperbolas.

• Types: Parabola, ellipse, hyperbola, circle

• Standard Forms: Parabola: ; Ellipse:

• Applications: Satellite orbits, optics

Matrices and Determinants

Matrices are arrays of numbers used to solve systems of equations and perform linear transformations.

• Definition: Rectangular array of numbers

• Operations: Addition, multiplication, finding determinants

• Solving Systems: Use matrix methods (e.g., Gaussian elimination)

• Example:

Sequences and Series

Sequences are ordered lists of numbers; series are their sums. Arithmetic and geometric sequences are especially important.

• Arithmetic Sequence:

• Geometric Sequence:

• Sum of Series: Arithmetic: ; Geometric:

Mathematical Induction

Induction is a proof technique used to establish the truth of an infinite sequence of statements.

• Principle: Prove base case, then show if true for , true for

• Example: Prove

Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of binomials.

• Formula:

• Finding Terms: Use binomial coefficients

Trigonometric Functions and Their Graphs

Trigonometric functions describe relationships in triangles and periodic phenomena.

• Definitions: Ratios in right triangles, unit circle, arc length

• Domain and Range: Varies by function (e.g., sine: domain , range )

• Graphing: Plot basic and transformed functions

• Example:

Inverse Trigonometric Functions

Inverse trigonometric functions allow for finding angles given ratios.

• Definition: , ,

• Domain and Range: Restricted to ensure functions are invertible

Trigonometric Identities

Identities are equations involving trigonometric functions that hold for all values in their domains.

• Fundamental Identities:

• Sum/Difference Formulas:

• Double Angle:

Solving Trigonometric Equations

Solving equations involving trigonometric functions is essential for applications in geometry and physics.

• General Solutions: Find all solutions, often using identities

• Interval Solutions: Restrict to specified intervals

Solving Triangles

Solving triangles involves finding unknown sides or angles using trigonometric relationships.

• Right Triangles: Use basic trigonometric ratios

• Oblique Triangles: Use Law of Sines and Law of Cosines

• Law of Sines:

• Law of Cosines:

Complex Numbers in Trigonometric Form

Complex numbers can be represented in trigonometric form, facilitating multiplication, division, and finding roots.

• Standard Form:

• Trigonometric Form:

• DeMoivre’s Theorem:

Vectors

Vectors are quantities with magnitude and direction, used in physics and engineering.

• Graphing: Represent vectors in the plane

• Operations: Addition, subtraction, scalar multiplication

• Component Form:

Parametric Equations

Parametric equations express curves using parameters, allowing for more flexible representations.

• Definition:

• Conversion: Between parametric and rectangular forms

Polar Coordinates

Polar coordinates describe points in the plane using radius and angle, useful for circular and spiral patterns.

• Conversion: ,

• Graphing: Plot polar equations

Applications of Trigonometry

Trigonometry is applied in calculating arc lengths, areas, and solving real-world problems involving triangles and vectors.

• Arc Length:

• Area of Sector:

• Applications: Navigation, engineering, physics

Course Materials

• Required Textbook: Sullivan, Michael. Algebra and Trigonometry, 12th Edition, Pearson.

• Optional Calculator: TI-36X Pro Scientific Calculator

Grading and Assessment

• Participation: 15%

• Homework: 15% (lowest 8 dropped)

• Quizzes: 25% (lowest 2 dropped)

• Tests: 45% (lowest dropped)

• Minimum Passing Requirement: Average score on 10 proctored exams must be 60% or higher

Attendance Policy

• Roll taken every class; three absences lower final grade by 2%

• Three tardiness count as one absence

• More than two absences may result in withdrawal

Support Services

• Tutoring labs, virtual tutoring, advisement, technical support

• ACCESS Disability Services for students with documented disabilities

Summary Table: Key Precalculus Topics