Master function composition, domains/ranges, and finding inverse functions.
- Function composition (f∘g)(x)
- Domains/ranges under composition
- One-to-one functions & horizontal line test
- Inverse functions: definition, notation, verification
- Restricting domains to create inverses
- Solving equations with inverses
Deep dive into trig functions, identities, equations, and their graphs.
- Angle measure (degrees ↔ radians) & arc length
- Unit circle values & reference angles
- Trig functions as coordinates on unit circle
- Graphs of sine, cosine, tangent
- Trig identities (Pythagorean, reciprocal, quotient)
- Sum & difference identities
- Inverse trig functions
- Law of sines & law of cosines
Work with a+bi algebraically and geometrically, including polar form.
- Imaginary unit i and powers of i
- Adding, subtracting, multiplying, dividing
- Complex conjugates & modulus
- Graphing in the complex plane
- Polar form r(cosθ+isinθ)
- De Moivre's theorem & complex roots
Analyze, simplify, and graph rational expressions and functions.
- Domains, intercepts, and holes
- Vertical, horizontal, and slant asymptotes
- End behavior and limits
- Graphing rational functions
- Simplifying rational expressions
- Partial fraction decomposition
Circles, ellipses, parabolas, and hyperbolas from algebraic and geometric viewpoints.
- General vs. standard forms
- Parabolas: vertex, focus/directrix form
- Ellipses: center, axes, foci, eccentricity
- Hyperbolas: axes, asymptotes, foci
- Circles: center-radius form
- Graphing and extracting features
Algebraic & geometric operations, dot product, and applications.
- Magnitude & direction; unit vectors
- Component form vs. magnitude–direction form
- Adding, subtracting, and scaling vectors
- Dot product & angle between vectors
- Word problems: forces, velocity, navigation
Matrix arithmetic and using matrices to solve systems & model transformations.
- Matrix addition, subtraction, scalar multiplication
- Matrix multiplication & properties
- Determinants & inverses (2×2 focus)
- Solving linear systems with matrices
- Transformations of vectors with matrices
Counting methods and probability structures, including expected value.
- Basic probability rules
- Permutations & combinations (nPr, nCr)
- Binomial coefficients & Pascal's triangle
- Probability distributions
- Expected value & decision making
Arithmetic & geometric sequences/series, sigma notation, and binomial theorem.
- Sequences (explicit vs. recursive)
- Arithmetic series: partial sums
- Geometric series: finite & infinite sums
- Sigma (Σ) notation manipulation
- Binomial theorem & combinatorics connection
Bridge to Calculus: introduce the language and intuition of limits.
- Limit notation & evaluating basic limits
- One-sided & infinite limits
- Continuity at a point and on intervals
- Piecewise functions & discontinuities
- Using limits to describe end behavior