DepED Grade 11/12 Pre-Calculus Curriculum Guide

Administrative and Program Framework

• Educational Department: Department of Education (DepED)
• National Movement / Branding: Bagong Pilipinas
• Subject Name: Pre-Calculus
• Academic Track: Academic Elective
• Grade Level Target: Grade 11/12
• Prerequisite Coursework: None
• Time Allotment: 80 hours for a term

Course Description and Foundational Goals

• Primary Purpose: Equips learners with essential mathematical knowledge and skills in preparation for higher-level Calculus courses while fostering critical and analytical thinking, alongside advanced problem-solving skills.
• Core Mathematical Domains:
  • Exploration of graph transformations such as translations and reflections.
  • Detailed analysis of functions including polynomial, rational, exponential, and logarithmic functions.
  • Study of the analytic geometry of lines and circles.
• Key Learning Experiences:
  • Analyzing and solving complex mathematical problems.
  • Justifying solutions through formal logical reasoning.
  • Constructing mathematical proofs.
  • Communicating mathematical ideas effectively using appropriate graphical, algebraic, and tabular representations.
• Technological Integration: Translating real-life situations into mathematical models using appropriate digital tools and technology.
• 21st Century Skills & Career Application: Learners apply mathematical knowledge and understanding to prepare for future academic pursuits and professional careers in specialized fields, including:
  • Engineering
  • Computer Science
  • Economics
  • Statistics
  • Finance
  • Physics
  • Other STEM and quantitative sciences

Content Domain: Measurement and Geometry (MG)

Topic 1: Lines and Circles

• Learning Competency 1: Determine the equations of lines and circles from given geometric conditions, such as:
  • Two given points
  • A given point and a slope
  • Conditions of parallelism or perpendicularity
  • Center and radius of a circle
  • Endpoints of a circle's diameter
• Learning Competency 2: Represent real-life situations using equations of lines and circles.
• Learning Competency 3: Analyze geometric relationships between lines and circles using their respective algebraic equations.
• Learning Competency 4: Graph the solution sets of systems of equations involving lines and circles.
• Learning Competency 5: Solve systems of equations involving lines and circles algebraically and graphically.

Topic 2: Graph Transformations

• Learning Competency 6: Apply graph transformations, specifically translations and reflections, to the graphs of equations.
• Learning Competency 7: Analyze the relationships between equations and their graphs by identifying underlying structural patterns that affect:
  • Orientation
  • Symmetry
  • Shape
  • General graphical behavior
• Learning Competency 8: Solve contextualized and abstract problems involving graphs of various types of equations and their transformations.

Content Domain: Number and Algebra (NA)

Topic 3: Polynomial Functions

• Learning Competency 9: Identify polynomial functions of third degree or higher.
• Learning Competency 10: Apply the Remainder Theorem and Factor Theorem to determine the zeros of polynomial functions, where appropriate.
• Learning Competency 11: Solve polynomial equations and polynomial inequalities.
• Learning Competency 12: Graph and describe the characteristic features of polynomial functions both with and without the assistance of digital graphing tools.
• Learning Competency 13: Solve real-world and mathematical problems involving polynomial equations and functions.

Topic 4: Rational Functions

• Learning Competency 14: Differentiate among rational expressions, rational equations, and rational functions.
• Learning Competency 15: Graph rational functions and describe their key structural features:
  • $x$-intercepts and $y$-intercepts
  • Zeros
  • Vertical, horizontal, and slant asymptotes
  • Domain and range
• Learning Competency 16: Solve rational equations and real-life problems involving rational functions.

Topic 5: Exponential and Logarithmic Functions

• Learning Competency 17: Define the inverse of a function and enumerate its fundamental algebraic properties.
• Learning Competency 18: Identify the inverse of simple one-to-one algebraic functions, including:
  • Linear functions
  • Square root of linear functions
  • Cubic functions of the form $y = x^3$
  • Quotient of two linear functions
• Learning Competency 19: Sketch the graph of an inverse function given the graph of the original function.
• Learning Competency 20: Define and illustrate exponential functions.
• Learning Competency 21: Formulate the inverse of an exponential function using logarithms.
• Learning Competency 22: Sketch graphs of exponential and logarithmic functions, explicitly identifying their domains and ranges.
• Learning Competency 23: Formally prove the algebraic properties of logarithms.
• Learning Competency 24: Solve exponential and logarithmic equations.
• Learning Competency 25: Solve real-life problems using exponential and logarithmic functions and equations.

Comprehensive Performance Standards

Analytic Geometry, Systems, and Graph Transformations

• Real-Life Modeling: Learners investigate and model real-life situations using graphs of equations, polynomial functions, and rational functions.
• Geometric Derivation and Analysis: Learners determine equations of lines and circles directly from specified geometric conditions, model real-life situations using these equations, and analyze geometric relationships using both algebraic and graphical techniques.
• Transformation Mechanics: Learners apply basic graph transformations (translations and reflections) to show and explain how changes in variables affect movement, orientation, and overall graph behavior.
• System Solutions: Learners graph solution sets of systems and accurately solve line-circle systems.

Modeling Framework, Function Analysis, and Polynomial Methods

• Model Selection & Constraints: Learners select appropriate function types for given contexts, define variables clearly, and state all relevant assumptions and constraints.
• Multi-Modular Representation: Learners represent functions using equations, tables, and graphs with appropriate technology.
• Feature Analysis: Learners evaluate key graphical and algebraic features, including domain, range, intercepts, asymptotes, and points of intersection.
• Algebraic Resolution: Learners solve polynomial equations using suitable algebraic methods, including:
  • Remainder Theorem
  • Factor Theorem
  • Synthetic division
  • Long division
  • Contextual interpretation of algebraic solutions
• Validation: Learners evaluate the reasonableness of results by checking against physical or situational constraints, unit consistency, and graphical behavior, as well as by comparing alternate problem-solving approaches.

Advanced Function Modeling and Practical Applications

• In-Depth Situational Analysis: Learners conduct comprehensive analyses of real-life situations involving rational, exponential, and logarithmic functions across diverse fields, including:
  • Resource optimization
  • Electrical resistance
  • Cooling rates (Newton's Law of Cooling)
  • Loan interests and financial growth
  • Population growth models
• Inverse Function Mastery: Learners construct, interpret, and analyze the inverse of one-to-one functions and exponential functions using logarithms and vice-versa, accurately sketching their graphs while specifying correct domains and ranges.
• Logarithmic Parameter Applications: Learners prove and apply the properties of logarithms to solve equations and interpret key physical parameters in real-world contexts, such as:
  • Growth and decay rates
  • Half-life calculations
  • Doubling time
• Technological & Abstract Connections: Using digital tools, learners investigate parameter shifts to determine their exact effects on graph behavior, linking abstract mathematical concepts directly to real-world applications in:
  • Growth and decay dynamics
  • Signal processing
  • Structural and system design
  • Navigation
  • Optimization processes

Communication and Rigor Standards

• Mathematical Communication: Learners present findings using well-labeled graphs, precise mathematical notation, and clear textual explanations that justify all chosen methods and conclusions.
• Academic & Career Readiness: Learners demonstrate both solid theoretical understanding and practical problem-solving competency, establishing complete readiness for advanced studies in Calculus and real-world technical challenges.