Unit 1
Function Foundations & Algebraic Readiness
- Function notation and representations
- Domain, range and restrictions
- Transformations and symmetry
- Composition and inverse functions
- Equation and inequality review
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School Mathematics · Grades 6–12
Precalculus
A calculus-readiness pathway combining advanced algebra, function analysis, trigonometry, exponential and logarithmic models, sequences, polar/parametric representations, and introductory limits.
Course overview
This roadmap is intentionally calculus-facing. It combines the algebra and trigonometry foundation students need for calculus and mirrors the breadth of UW–Madison MATH 114, while adding secondary-school modeling and function-analysis practice.
Math Aficionado uses this structure as a tutoring roadmap rather than a fixed school syllabus. Lessons can be reordered around the student's current unit, teacher materials, placement level, assignments, and assessment goals.
Units & topics
Unit 2
Polynomial & Power Functions
- Polynomial structure and zeros
- End behavior and multiplicity
- Factoring and division
- Power functions and rational exponents
- Polynomial modeling
Unit 3
Rational Functions
- Rational expressions and restrictions
- Asymptotes and discontinuities
- Graphing rational functions
- Rational equations and inequalities
- Modeling with rational functions
Unit 4
Exponential & Logarithmic Functions
- Exponential growth and decay
- Logarithm definitions and properties
- Change of base
- Exponential/logarithmic equations
- Applications including finance and growth
Unit 5
Trigonometric Foundations
- Angles and radian measure
- Unit circle
- Exact values
- Right-triangle relationships
- Arc length and sector applications
Unit 6
Trigonometric Functions & Graphs
- Sine, cosine and tangent functions
- Amplitude, period and phase shift
- Inverse trigonometric functions
- Reciprocal functions
- Periodic modeling
Unit 7
Trigonometric Identities & Equations
- Pythagorean identities
- Sum and difference formulas
- Double-angle identities
- Simplifying and proving identities
- Solving trigonometric equations
Unit 8
Applications of Trigonometry
- Law of Sines and Cosines
- Triangle area and surveying
- Vectors and components
- Harmonic and periodic models
- Real-world problem solving
Unit 9
Systems, Matrices & Conic Connections
- Nonlinear systems
- Matrix-based system methods
- Conic sections and coordinate equations
- Parametric descriptions
- Choosing algebraic versus graphical methods
Unit 10
Sequences & Series
- Arithmetic and geometric sequences
- Recursive versus explicit forms
- Finite sums
- Infinite geometric series
- Applications and growth patterns
Unit 11
Polar & Parametric Mathematics
- Polar coordinates and graphs
- Polar equations and symmetry
- Parametric curves
- Motion and elimination of parameters
- Connections among representations
Unit 12
Limits & Calculus Readiness
- Average versus instantaneous rate of change
- Informal limits and continuity
- Local linear behavior
- Derivative intuition from graphs
- Calculus vocabulary and readiness for MATH 221-style work
How the roadmap is aligned
Wisconsin high-school Functions/Algebra/Trigonometry standards; UW–Madison MATH 114: Precalculus, which combines algebra and trigonometry and explicitly prepares students for calculus with trigonometry.
This is an independent tutoring roadmap, not an official UW–Madison or Wisconsin Department of Public Instruction course. Wisconsin standards are used as the K–12 framework; current UW–Madison course descriptions are used where the pathway overlaps with college-level precalculus and calculus preparation.
Wisconsin Standards for Mathematics (2021) · UW–Madison Mathematics Course Guide
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