Basic Algebraic Functions Course

Lesson 1 • Properties of Algebraic Operations

Covers commutative, associative, and distributive laws with algebraic proof. Provides the structural rules that justify every manipulation used later.

Lesson 2 • Numbers, Sets, and the Real Line

Classifies real number subsets and maps them on a number line. Establishes the numeric vocabulary needed for all subsequent function definitions.

Lesson 3 • Introduction to Mathematical Reasoning

Develops logical argumentation and pattern recognition as analytical tools. Prepares students to verify function properties through structured reasoning.

Lesson 4 • Variables, Expressions, and Equations

Introduces symbolic representation of quantities and relationships. Connects arithmetic operations to algebraic manipulation techniques.

Lesson 1 • Function Notation and Evaluation

Introduces f(x) notation and the process of substituting values into functions. Builds fluency in reading and writing functional expressions.

Lesson 2 • Graphical Representation of Functions

Translates algebraic functions into accurate graphs and interprets graph features. Links visual behavior to algebraic properties established earlier.

Lesson 3 • Domain and Range

Defines domain as valid inputs and range as resulting outputs for any function. Students determine both sets algebraically and graphically.

Lesson 4 • Relations Versus Functions

Distinguishes general relations from functions using the definition of unique output. Anchors the concept of function as a precise mathematical object.

Lesson 1 • Forms of Linear Equations

Presents slope-intercept, point-slope, and standard forms of linear equations. Students convert between forms and select the most useful form for each task.

Lesson 2 • Linear Modeling and Problem Solving

Applies linear functions to model real-world scenarios involving constant change. Students build, interpret, and evaluate linear models from contextual data.

Lesson 3 • Slope and Rate of Change

Defines slope as a constant rate of change and computes it from two points. Connects the numeric value of slope to the steepness and direction of a line.

Lesson 4 • Graphing Linear Functions

Develops efficient techniques for graphing lines from equations and data. Reinforces the visual interpretation of slope and intercepts on the coordinate plane.

Lesson 1 • Solving Quadratic Equations

Presents factoring, completing the square, and the quadratic formula as solution methods. Students select the most efficient method based on equation structure.

Lesson 2 • Quadratic Modeling and Optimization

Uses quadratic functions to model projectile motion, area, and revenue problems. Students find maximum or minimum values to solve optimization tasks.

Lesson 3 • Standard and Vertex Forms

Introduces f(x) = ax² + bx + c and the vertex form f(x) = a(x−h)² + k. Students convert between forms and extract key features from each.

Lesson 4 • Graphing Parabolas

Builds accurate parabola sketches using vertex, intercepts, and symmetry. Connects the sign of the leading coefficient to the direction of opening.

Lesson 1 • Factoring Polynomials Completely

Develops systematic factoring strategies including GCF, grouping, and special patterns. Complete factoring is the gateway to finding zeros and graphing.

Lesson 2 • Polynomial Division and Remainders

Applies long division and synthetic division to divide polynomials and find remainders. Remainder Theorem links division results to function evaluation.

Lesson 3 • Polynomial Structure and Terminology

Defines degree, leading coefficient, and standard form for polynomials. Establishes vocabulary and classification needed for all polynomial analysis.

Lesson 4 • Graphing Polynomial Functions

Combines end behavior, zeros, and multiplicity to produce accurate polynomial graphs. Students interpret turning points and crossing versus touching behavior.

Lesson 1 • Asymptotes and End Behavior

Derives vertical, horizontal, and oblique asymptotes from polynomial degrees. Students use asymptotes to describe long-run and local function behavior.

Lesson 2 • Rational Function Basics

Defines rational functions as ratios of polynomials and identifies their domains. Introduces the concept of undefined points and their graphical consequences.

Lesson 3 • Solving Rational Equations and Inequalities

Develops algebraic methods for solving rational equations and checking for extraneous solutions. Extends to rational inequalities using sign analysis.

Lesson 4 • Radical Functions and Their Domains

Introduces square root and cube root functions with attention to domain restrictions. Students graph radical functions and identify key points and transformations.

Lesson 1 • Exponential and Logarithmic Modeling

Applies exponential and logarithmic functions to growth, decay, and scaling problems. Students build and interpret models from real-world data.

Lesson 2 • Logarithmic Functions as Inverses

Establishes logarithms as inverses of exponential functions and converts between forms. Students evaluate and graph logarithmic functions accurately.

Lesson 3 • Solving Exponential and Logarithmic Equations

Applies logarithm properties and one-to-one principles to solve equations. Students handle both exact and approximate solutions systematically.

Lesson 4 • Properties of Logarithms

Presents product, quotient, and power rules for logarithms with proofs. Students expand and condense logarithmic expressions to simplify equations.

Lesson 1 • Composition of Functions

Introduces function composition as applying one function to the output of another. Students compute, simplify, and determine the domain of composite functions.

Lesson 2 • Comprehensive Transformation Toolkit

Unifies all transformation types—shifts, reflections, and stretches—into a single framework. Students apply transformations to any function family using the general form.

Lesson 3 • Even, Odd, and Periodic Functions

Classifies functions by symmetry properties and introduces periodicity as a function trait. Students test functions algebraically and interpret symmetry graphically.

Lesson 4 • Arithmetic Operations on Functions

Defines sum, difference, product, and quotient of two functions with domain analysis. Students compute and simplify combined functions algebraically.