Unit 1: Functions and Their Graphs
Function fundamentals
Domain, range, and function notation
Even and odd functions
Increasing, decreasing, and constant intervals
Average rate of change

Parent functions and transformations
Translations, reflections, stretches, compressions
Piecewise and absolute value functions

Combining functions
Arithmetic operations and composition
Inverse functions and one-to-one functions

Modeling with functions
Polynomial functions
End behavior and turning points
Zeros, multiplicity, and graphing
Long and synthetic division
Remainder and Factor Theorems
Rational Root Theorem
Fundamental Theorem of Algebra
Complex zeros and conjugate pairs

Rational functions
Vertical, horizontal, and slant asymptotes
Holes and intercepts
Graphing rational functions

Inequalities
Polynomial and rational inequalities

Partial fractions (in some courses)
Exponential functions
Growth and decay
The number e and continuous compounding

Logarithmic functions
Definition, graphs, and transformations
Properties and change of base

Equations
Solving exponential and logarithmic equations

Modeling
Logistic growth
Regression and real-world applications
Angles and the unit circle
Radian and degree measure
Arc length and angular speed
Reference angles and exact values

Graphs of trig functions
Sine, cosine, tangent
Cosecant, secant, cotangent
Amplitude, period, phase shift, vertical shift

Inverse trigonometric functions
Domains, ranges, and graphs
Evaluating compositions

Applications
Right triangle trigonometry
Harmonic motion and periodic modeling
Fundamental identities
Reciprocal, quotient, Pythagorean
Verifying identities

Sum and difference formulas

Double-angle and half-angle formulas

Product-to-sum and sum-to-product formulas

Solving trigonometric equations
Basic and multiple-angle equations
Equations using identities
Law of Sines
Ambiguous case (SSA)

Law of Cosines

Area of a triangle using trigonometry (including Heron’s formula)

Vectors
Magnitude and direction
Vector operations
Dot product and angle between vectors
Applications (force, velocity)

Complex numbers in trigonometric form
Polar formDe Moivre’s Theorem
Roots of complex numbers
Polar coordinates
Converting between polar and rectangular
Graphing polar equations (circles, cardioids, limaçons, roses)

Parametric equations
Graphing and eliminating the parameter
Modeling motion (projectiles)
Systems of equations and inequalities
Three-variable systems
Nonlinear systems

Matrices
Operations and row reduction
Determinants and inverses
Solving systems with matrices
Matrix transformations

Conic sections
Parabolas, circles, ellipses, hyperbolas
Standard form and graphing
Eccentricity
Rotation of axes (in some courses)
Sequences and series
Arithmetic and geometric sequences
Sigma notationInfinite geometric series
Recursive definitions

Mathematical induction

Binomial Theorem and Pascal’s triangle

Counting and probability
Permutations and combinations
Basic probability
Limits
Limits from graphs and tables
One-sided limits
Limits at infinity
Evaluating limits algebraically

Continuity

Rates of change
Instantaneous rate of change and the tangent line concept

Area under a curve (intuitive introduction)