Pre-Calculus Concepts

          Module 1: Introduction to Functions and Graphical Representations

Section 1: What Are Functions?
- Lesson 1: Understanding the Concept of a Function  
  Module 1, Section 1, Lesson 1: Define functions as rules that assign each input one output.
- Lesson 2: Exploring Domain and Range  
  Module 1, Section 1, Lesson 2: Identify sets of possible inputs (domain) and outputs (range) using simple examples.

Section 2: Function Notation and Terminology
- Lesson 1: Introduction to Function Notation  
  Module 1, Section 2, Lesson 1: Learn to read and write function notation (f(x)) and understand its meaning.
- Lesson 2: Recognizing Input-Output Relationships  
  Module 1, Section 2, Lesson 2: Practice matching inputs with their corresponding outputs through interactive examples.

Section 3: Basics of the Coordinate Plane
- Lesson 1: Understanding the Coordinate System  
  Module 1, Section 3, Lesson 1: Introduce axes, origin, and plotting points on a two-dimensional grid.
- Lesson 2: Plotting Points and Interpreting Graphs  
  Module 1, Section 3, Lesson 2: Practice graphing simple points and connecting them to form basic images.

Section 4: Types of Functions — An Overview
- Lesson 1: Comparing Linear and Nonlinear Functions  
  Module 1, Section 4, Lesson 1: Explore the differences between linear functions and other function types using simple graphs.
- Lesson 2: Identifying Common Function Types  
  Module 1, Section 4, Lesson 2: Recognize functions such as constant, linear, and basic quadratic functions through examples.

Section 5: Key Graph Features
- Lesson 1: Finding Intercepts and Understanding Slope  
  Module 1, Section 5, Lesson 1: Learn how to determine x- and y-intercepts and calculate slope on a graph.
- Lesson 2: Recognizing Symmetry in Graphs  
  Module 1, Section 5, Lesson 2: Identify symmetric features of functions and how they affect the overall graph.

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Module 2: Fundamentals of Linear and Quadratic Functions

Section 1: Exploring Linear Functions
- Lesson 1: Slope-Intercept Form and Its Meaning  
  Module 2, Section 1, Lesson 1: Introduce the slope-intercept formula (y = mx + b) and its components.
- Lesson 2: Graphing Linear Equations  
  Module 2, Section 1, Lesson 2: Practice plotting linear equations and interpreting their slopes and y-intercepts.

Section 2: Introduction to Quadratic Functions
- Lesson 1: Parabolic Shapes and Vertex Form  
  Module 2, Section 2, Lesson 1: Understand the basic structure of quadratic functions and identify the vertex.
- Lesson 2: Factoring Quadratic Equations  
  Module 2, Section 2, Lesson 2: Learn simple factoring techniques to solve quadratic equations.

Section 3: Properties and Characteristics of Linear Functions
- Lesson 1: Understanding Parallel and Perpendicular Lines  
  Module 2, Section 3, Lesson 1: Explore relationships between slopes that indicate parallelism or perpendicularity.
- Lesson 2: Real-Life Applications of Linear Relationships  
  Module 2, Section 3, Lesson 2: Apply linear function concepts to everyday situations and problem scenarios.

Section 4: Characteristics of Quadratic Functions
- Lesson 1: Identifying the Axis of Symmetry  
  Module 2, Section 4, Lesson 1: Discover how and why quadratic graphs have an axis of symmetry.
- Lesson 2: Using the Quadratic Formula  
  Module 2, Section 4, Lesson 2: Learn to apply the quadratic formula to find the roots of a quadratic equation.

Section 5: Transformations of Functions
- Lesson 1: Vertical and Horizontal Shifts  
  Module 2, Section 5, Lesson 1: Explore how functions move on the graph when constants are added or subtracted.
- Lesson 2: Stretching and Compressing Graphs  
  Module 2, Section 5, Lesson 2: Understand the effects of scaling factors on the shape and orientation of graphs.

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Module 3: Exploring Polynomial and Rational Functions

Section 1: Introduction to Polynomial Functions
- Lesson 1: Understanding Degrees and Terms  
  Module 3, Section 1, Lesson 1: Define polynomial functions and differentiate between various degrees and terms.
- Lesson 2: Sketching Simple Polynomial Graphs  
  Module 3, Section 1, Lesson 2: Practice drawing graphs of simple polynomials using key characteristics.

Section 2: Factoring and Finding Zeros
- Lesson 1: Techniques for Factoring Polynomials  
  Module 3, Section 2, Lesson 1: Introduce basic methods for factoring polynomials.
- Lesson 2: Locating Zeros or Roots on a Graph  
  Module 3, Section 2, Lesson 2: Learn how to determine the zeros of a function using factoring and graph interpretation.

Section 3: Understanding Rational Functions
- Lesson 1: Defining Rational Functions  
  Module 3, Section 3, Lesson 1: Learn what makes a function rational and how it is formed from polynomials.
- Lesson 2: Identifying Key Features such as Asymptotes  
  Module 3, Section 3, Lesson 2: Study vertical and horizontal asymptotes in rational functions.

Section 4: Graphing Rational Functions
- Lesson 1: Plotting Rational Functions Using Key Points  
  Module 3, Section 4, Lesson 1: Learn to graph rational functions by identifying characteristics and critical points.
- Lesson 2: Analyzing Behavior Near Asymptotes  
  Module 3, Section 4, Lesson 2: Understand the behavior of functions as they approach their asymptotes.

Section 5: Real-World Applications of Polynomials and Rationals
- Lesson 1: Modeling Situations with Polynomial Functions  
  Module 3, Section 5, Lesson 1: Apply polynomial functions to model real-life scenarios.
- Lesson 2: Solving Word Problems with Rational Functions  
  Module 3, Section 5, Lesson 2: Engage in problem-solving exercises using rational functions and context-based examples.

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Module 4: Function Analysis and Problem-Solving Strategies

Section 1: Operations on Functions
- Lesson 1: Adding and Subtracting Functions  
  Module 4, Section 1, Lesson 1: Learn how to combine functions through addition and subtraction.
- Lesson 2: Multiplying Functions Together  
  Module 4, Section 1, Lesson 2: Explore techniques for multiplying functions and interpreting the results.

Section 2: Composition and Inverses of Functions
- Lesson 1: Understanding the Composition of Functions  
  Module 4, Section 2, Lesson 1: Introduce function composition and how outputs of one function become inputs for another.
- Lesson 2: Finding and Verifying Inverse Functions  
  Module 4, Section 2, Lesson 2: Practice determining inverse functions and confirming their validity.

Section 3: Solving Equations Involving Functions
- Lesson 1: Developing Equations from Graphical Data  
  Module 4, Section 3, Lesson 1: Learn to form equations based on graph reading and interpretation.
- Lesson 2: Analyzing Solutions and Constraints  
  Module 4, Section 3, Lesson 2: Explore methods to check solutions within given restrictions.

Section 4: Introduction to Inequalities with Functions
- Lesson 1: Solving Linear Inequalities  
  Module 4, Section 4, Lesson 1: Review techniques for solving and graphing linear inequalities.
- Lesson 2: Representing Inequality Solutions on a Number Line  
  Module 4, Section 4, Lesson 2: Learn to visually express inequality solutions using number lines.

Section 5: Interactive Problem-Solving in Pre-Calculus
- Lesson 1: Using Graphing Calculators Effectively  
  Module 4, Section 5, Lesson 1: Familiarize with graphing calculators and other interactive tools to explore functions.
- Lesson 2: Applying Concepts Through Real-Life Examples  
  Module 4, Section 5, Lesson 2: Engage in group activities that apply function concepts to practical situations.

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Module 5: Consolidating Pre-Calculus Foundations

Section 1: Review of Core Concepts
- Lesson 1: Recap of Functions, Graphs, and Notation  
  Module 5, Section 1, Lesson 1: Summarize the main ideas of functions, graphing, and notation covered in previous modules.
- Lesson 2: Key Terms and Definitions  
  Module 5, Section 1, Lesson 2: Revisit important terminology to solidify foundational vocabulary.

Section 2: Building Connections Across Topics
- Lesson 1: Linking Algebra and Functions  
  Module 5, Section 2, Lesson 1: Explore how algebraic techniques support the understanding of functions.
- Lesson 2: Integrating Graphs and Equations  
  Module 5, Section 2, Lesson 2: Practice translating between equations and their corresponding graphs.

Section 3: Applied Problem Solving
- Lesson 1: Tackling Word Problems Using Function Models  
  Module 5, Section 3, Lesson 1: Use real-life scenarios to set up and solve equations involving functions.
- Lesson 2: Collaborative Model Construction  
  Module 5, Section 3, Lesson 2: Engage in group projects to create and analyze simple mathematical models.

Section 4: Interactive Explorations of Pre-Calculus Concepts
- Lesson 1: Utilizing Technology for Graphing and Analysis  
  Module 5, Section 4, Lesson 1: Explore interactive tools and software to visualize pre‑calculus ideas.
- Lesson 2: Simulation-Based Learning Activities  
  Module 5, Section 4, Lesson 2: Participate in simulation exercises that reinforce the concepts learned.

Section 5: Final Review and Preparatory Assessment
- Lesson 1: Comprehensive Review of Pre‑Calculus Foundations  
  Module 5, Section 5, Lesson 1: Recap and integrate all topics to ensure a strong understanding for future studies.
- Lesson 2: Q&A Session and Feedback  
  Module 5, Section 5, Lesson 2: Conduct an interactive session to address questions, reinforce key ideas, and provide feedback for improvement.