Algebra and Functions

Equations, graphs and models, and when an algebraic step is allowed

About this course

Equations, inequalities and systems, then functions and their graphs — quadratics, polynomials, rational, power, exponential and logarithmic functions — ending with sequences and defensible models. No calculus is required.

What you will be able to do

  • Justify which algebraic steps preserve the solution set of an equation or inequality
  • Solve linear systems and interpret them as models
  • Transform, compose and invert functions, restricting domains where needed
  • Analyse quadratic, polynomial and rational functions through their roots, factors and missing inputs
  • Use exponentials, logarithms and recurrences to model change

What this course covers

  • Sequences: explicit formulas vs. recurrences
  • Uniqueness via induction structure
  • Arithmetic finite sum via reversed pairing
  • Geometric finite sum via shifted cancellation
  • Retained-stock accumulation model
  • Fixed point and gap analysis for the stock model
  • Synthesis: terms, states, and sums as complementary views
  • Solution sets, domains, and equation equivalence
  • Reversible operations and solution-set preservation
  • Graphical interpretation of equation solutions
  • Division by an expression containing the unknown
  • Squaring equations and extraneous candidates
  • One-way versus two-way equivalence arrows
  • The four-part solution certificate
  • Completing the square as an identity
  • Geometric (area) justification of the procedure
  • Three forms of a quadratic and the vertex
  • Bounds, attainment, and domain membership for extrema
  • Maximum area with a fixed perimeter constraint
  • Solving a quadratic equation via square form
  • Completion certificate and summary of habits
  • Cancellation preserves domain restrictions
  • Functions vs. expressions: domain as part of identity
  • Classifying excluded inputs: holes vs. vertical asymptotes
  • Polynomial division and end-behavior comparison lines
  • Rational inequality sign charts
  • Multiplying inequalities by expressions of unknown sign
  • Complete function certificate for a rational function
  • Function notation and evaluation
  • Domain, range, and the function condition
  • Evaluating vs. solving: one output per input does not mean one input per output
  • Graphs, tables, and the domain-determines-picture principle
  • Piecewise functions and boundary ownership
  • Vertical line test and relations vs. functions
  • Natural domain restrictions from operations
  • Graph as a set of input-output pairs
  • Output (vertical) transformations: shift and scale
  • Horizontal (input) shifts via the address equation
  • Input scaling and horizontal reflection
  • Order of operations and non-commutativity of transformations
  • Combined transformation rule
  • Practical habits and verification strategy
  • Quantity dictionary and equation translation
  • Model verification by reconstruction
  • Equality as agreement between two changing quantities
  • Parameters and classification of solution counts
  • Domain conditions and admissible solutions
  • The model certificate
  • Four modelling habits as closing synthesis
  • Classifying roots of a parametric quadratic family
  • Root count vs. root sign vs. admissibility
  • Deriving the quadratic formula via the discriminant
  • Repeated roots and the touching-axis behaviour
  • Interpreting quadratic roots within a problem domain
  • Handling a zero leading coefficient as a separate case
  • Sign-chart classification for a two-coefficient parameter
  • Model structure: variables, domain, and assumptions
  • Process diagram: tracing price through calculations
  • Surplus as a quadratic: expanding and completing the square
  • Surplus interval via completed square and factoring
  • Feasible set: intersecting all constraints simultaneously
  • Objective-dependent optimisation within the feasible set
  • Sensitivity analysis: fixed cost variation and attendance rule challenge
  • Principal roots versus equation solution sets
  • Square root composition and absolute value
  • Rational exponents as sequences of operations
  • Reducing before rooting: order-of-reduction matters
  • Domain of expressions involving roots and reciprocals
  • Power laws for rational exponents: positive-base conditions
  • Solving radical equations: candidates and extraneous solutions
  • Constant difference vs. constant ratio
  • Exponential formula: initial amount, base, and exponent
  • Equal input intervals and exponential graphs
  • Exponential decay and reciprocal reversal
  • Defending a model: assumptions, domain, and limits
  • Worked synthesis: time-unit conversion and percentage reversal
  • Four habits for exponential modelling
  • Polynomials: definitions, roots, and representations
  • Factors, roots, and multiplicity
  • Polynomial long division and the division identity
  • The remainder theorem and factor theorem
  • Multiplicity parity: crossing versus touching the axis
  • Completeness of real root lists and irreducible quadratic factors
  • Reconstructing a polynomial from remainder conditions
  • Inequalities and number line representation
  • Order preservation and reversal under operations
  • Boundary location vs. boundary inclusion in linear inequalities
  • Intersection and union of intervals
  • Absolute value as distance
  • Four-part certificate for inequality solutions
  • Combining absolute value with a domain restriction
  • Function notation and substitution review
  • Composition of functions and order-dependence
  • Domain propagation through composition
  • Inverse functions: definition and construction
  • Inverse notation vs. reciprocal
  • Restricting domains to create inverse functions for squaring
  • Inverse of a shifted quadratic: capstone example
  • Logarithm as an inverse exponential question
  • Inverse graph as coordinate reflection
  • Base and domain restrictions for real logarithms
  • Product, quotient, and power laws derived from exponential laws
  • Solving logarithmic equations with domain checking
  • Logarithmic scale as coordinates for dimensionless ratios
  • Change of base formula
  • Simultaneous truth and the system solution set
  • Geometric interpretation: lines and their intersection
  • Substitution method and its reversibility
  • Elimination method and equivalence of the system
  • Classifying systems: one, infinitely many, or no solutions
  • Domain restrictions in applied models
  • Solution certificate and complete answer reporting

Syllabus

  1. Equations, Inequalities and Systems

    1. L1 · Which Algebraic Steps Keep the Same Solutions
    2. L2 · Linear Equations as Models
    3. L3 · Inequalities and Sets of Allowed Values
    4. L4 · What a System of Equations Asks

  2. Functions and Their Graphs

    1. L5 · Functions: Rules, Inputs and Outputs
    2. L6 · How Graph Transformations Work
    3. L7 · Composition and the Need for Inverse Restrictions

  3. Quadratics and Polynomials

    1. L8 · Quadratics Through Completing the Square
    2. L9 · How Parameters Change Quadratic Roots
    3. L10 · What Polynomial Factors Reveal

  4. Rational, Power and Exponential Functions

    1. L11 · Rational Functions and Their Missing Inputs
    2. L12 · Powers, Roots and Real Domains
    3. L13 · Exponential Change and Repeated Factors
    4. L14 · Logarithms as Inverse Questions

  5. Sequences and Modelling

    1. L15 · Sequences, Recurrences and Accumulation
    2. L16 · A Model That Can Be Defended