Precalculus – 5 (hrs)
Sep 25, 2020 20201124 13:01Precalculus – 5 (hrs)
Precalculus – 5 (hrs)
This leads to a return to the study of complex numbers and a study of linear transformations in the complex plane. Students develop an understanding that when complex numbers are considered points in the Cartesian plane, complex number multiplication has the geometric effect of a rotation followed by a dilation in the complex plane.
Students viewed matrices as representing transformations in the plane and developed an understanding of multiplication of a matrix by a vector as a transformation acting on a point in the plane.
Students look at incidence relationships in networks and encode information about them via highdimensional matrices. Questions on counting routes, the results of combining networks, payoffs, and other applications, provide context and use for matrix manipulations: matrix addition and subtraction, matrix product, and multiplication of matrices by scalars.
Students back to the study of complex roots of polynomial functions. Students first briefly review quadratic and cubic functions and then extend familiar polynomial identities to both complex numbers and to general polynomial functions. Students use polynomial identities to find square roots of complex numbers. The binomial theorem and its relationship to Pascal’s triangle are explored using roots of unity. Revisits, unites, and further expands those ideas and introduces new tools for solving geometric and modeling problems through the power of trigonometry. Helps students recall how to use special triangles positioned within the unit circle to determine geometrically the values of sine, cosine, and tangent at special angles.
The multiplication rule for independent events introduced in Algebra II is generalized to a rule that can be used to calculate the probability of the intersection of two events in situations where the two events are not independent. In this topic, students are also introduced to three techniques for counting outcomes—the fundamental counting principle, permutations, and combinations. These techniques are then used to calculate probabilities, and these probabilities are interpreted in context.

Complex Numbers and Transformations
 Complex Numbers as Vectors
 Complex Number Division
 The Geometric Effect of Some Complex Arithmetic
 Distance and Complex Numbers
 Trigonometry and Complex Numbers
 Discovering the Geometric Effect of Complex Multiplication
 Justifying the Geometric Effect of Complex Multiplication
 Representing Reflections with Transformations
 The Geometric Effect of Multiplying by a Reciprocal
 The Power of the Right Notation
 Exploiting the Connection to Trigonometry
 Exploiting the Connection to Cartesian Coordinates
 Modeling Video Game Motion with Matrices
 Matrix Notation Encompasses New Transformations

Vectors and Matrices
 Networks and Matrix Arithmetic
 Coordinates of Points in Space
 Linear Transformations as Matrices
 Linear Transformations Applied to Cubes
 Composition of Linear Transformations
 Matrix Addition Is Commutative
 Matrix Multiplication Is Distributive and Associative
 Using Matrix Operations for Encryption
 Solving Equations Involving Linear Transformations of the Coordinate Plane
 Solving General Systems of Linear Equations
 Vectors in the Coordinate Plane

Rational and Exponential Functions
 Solutions to Polynomial Equations
 Roots of Unity
 The Binomial Theorem
 Curves in the Complex Plane
 Curves from Geometry
 Volume and Cavalieri’s Principle
 The Structure of Rational Expressions
 Rational Functions
 End Behavior of Rational Functions
 Horizontal and Vertical Asymptotes of Graphs of Rational Functions
 Graphing Rational Functions
 Transforming Rational Functions
 Function Composition
 Solving Problems by Function Composition
 Inverse Functions
 Inverses of Logarithmic and Exponential Functions
 An Area Formula for Triangles
 Law of Sines
 Law of Cosines

Trigonometry
 Special Triangles and the Unit Circle
 Properties of Trigonometric Functions
 Addition and Subtraction Formulas
 Tangent Lines and the Tangent Function
 Waves, Sinusoids, and Identities
 Revisiting the Graphs of the Trigonometric Functions
 Inverse Trigonometric Functions
 Modeling with Inverse Trigonometric Functions

Probability and Statistics
 The General Multiplication Rule
 Counting Rules—The Fundamental Counting Principle and Permutations
 Counting Rules―Combinations
 Using Permutations and Combinations to Compute Probabilities
 Discrete Random Variables
 Probability Distribution of a Discrete Random Variable
 Expected Value of a Discrete Random Variable
 Interpreting Expected Value
 Determining Discrete Probability Distributions
 Estimating Probability Distributions Empirically
 Games of Chance and Expected Value
 Using Expected Values to Compare Strategies
 Making Fair Decisions
 Analyzing Decisions and Strategies Using Probability