Honors Pre-Calculus with Python Programming
Prerequisites:
Students should successfully complete Algebra 2 prior to enrolling in this course.
Description:
The course in Honors Pre-calculus provides an in-depth study of the algebraic and graphical characteristics of functions. Students also investigate and explore conic sections, sequences, and series. Students use graphing calculators, explore real world applications of the topics, and complete a semester-long project-based portfolio of their choice in order to achieve a more complete understanding of the mathematical concepts. It also examines trigonometric functions, identities, and applications. Students also investigate and explore polar coordinates, vectors, and parametric equations. The final unit introduces students to the fundamentals of calculus. Mathematical reasoning and problem solving skills are stressed as students prepare for future high school or college coursework in calculus. Students use graphing calculators, explore real world applications of the topics, and complete a semester-long project-based portfolio of their choice in order to achieve a more complete understanding of the mathematical concepts.
Unit 1 Functions:
Welcome to Precalculus
In this unit, you will review many of the prerequisite skills and concepts you will apply in your study of precalculus. The unit begins with an explanation of the name of the course and tips for getting organized for success in the course. This is followed by four lessons to help you recall topics you have learned previously so you are prepared to expand your understanding of them. These include working with the set of complex numbers, solving equations, and analyzing the graphs of functions.
Function Algebra
In this unit, you will explore all aspects of functions, including domain and range, from equations. You will compute the sum, difference, product, and quotient of functions and interpret the results. In addition, you will explore the composition of functions. You will determine whether a function has an inverse. Throughout the unit, you will discover applications for the concepts you are learning.
Graph Behavior
In this unit, you will explore aspects of functions from graphs. You will identify and interpret significant points on the function’s graph. In addition, you will identify and classify points of discontinuity on a graph. Finally, you will explore the graphs of different families of functions, and transformations of the parent functions.
Polynomial and Rational Functions
In this unit, you will explore aspects of polynomial and rational equations with real coefficients. You will evaluate computations with complex numbers, polynomials, and rational expressions. In addition, you will prove that the sets of complex numbers and rational expressions are closed under the four arithmetic operations. You will compare and contrast the families of functions you have studied using various representations. Finally, you will learn about and apply the Intermediate and Extreme Value Theorems.
Exponential, Logarithmic, Piecewise Functions
In this unit, you will compare and contrast various aspects of exponential and logarithmic functions. You will translate among the various representations of these families of functions. You will also graph piecewise-defined functions and analyze them for points of discontinuity. Finally, you will identify and interpret functions for which there are no elementary algorithms for finding their zeroes and employ alternative strategies to find the zeroes.
Unit 2: Series, Sequence, Quantitized Function and Conic Functions
Conic Sections
In this unit, you will explore concepts in analytic geometry. You will write and graph the standard form of an equation for circles, ellipses, hyperbolas, and parabolas. You will identify conic sections and apply your knowledge to solve real-world problems.
Sequences and Series
In this unit, you will be introduced to sequences and series. You will identify and generate arithmetic and geometric sequences. You will use sigma and factorial notation to write the terms and sum of a sequence. In addition, you will analyze a series to determine if it is convergent or divergent and calculate the sum of convergent series. Finally, you will model real-world situations using sequences and series.Unit 3:
Unit 3: Trigonometry
Introduction to Trigonometry
In this unit, you will explore the unit circle. You will review angle and radian measure concepts including radian and degree measure. You will use the unit circle and right triangles to define and evaluate trigonometric functions of real numbers and solve real-world problems. Finally, you will calculate the values for trigonometric ratios using a right triangle and the unit circle.
Trigonometric Functions
In this unit, you will expand your knowledge of trigonometric functions. You will learn how to graph trigonometric functions and describe their behavior in terms of periodicity, amplitude, zeroes, asymptotes, and symmetries. In addition, you will learn how to translate the graphs of the trigonometric functions and interpret vertical and phase shift. You will apply your knowledge to determine, graph, and use inverse trigonometric functions. Finally, you will solve real world problems using trigonometric functions.
Trigonometric Identities and Applications
In this unit, you will continue your study of trigonometric functions and identify fundamental identities. You will learn how to use these identities to verify and develop other trigonometric identities and formulas. In addition, you will prove the Law of Sines and Law of Cosines and use them to solve real-world problems. Finally, you will apply your knowledge of trigonometric functions to solve trigonometric equations.
Unit 4: Analytical Geometry
Polar Coordinates and Functions
In this unit, you will be introduced to polar coordinates and functions. You will plot points in the polar coordinate system and convert between rectangular and polar forms. You will also convert equations between rectangular and polar forms and graph polar equations of conics. In addition, you will use rectangular and polar representations of complex numbers and prove and apply DeMoivre’s Theorem.
Vectors
In this unit, you will explore vectors both algebraically and geometrically. You will be introduced to vector notation, including terms such as magnitude, direction, and resultant. You will learn how to find the components of a vector and use vector addition and scalar multiplication to solve problems. You will also learn how to use vector addition to model translations in the plane and use vectors to solve real-world problems.
Unit 5: Approximation and Limit Theory
Parametric Functions
In this unit, you will be introduced to parametric equations as a way to represent the path of a moving object. You will learn how to convert between Cartesian and parametric equations. In addition, you will graph plane curves described by parametric equations and find parametric equations for a given graph.
Looking Ahead to Calculus
In this unit, you will be introduced to some fundamental concepts in calculus. You will calculate limits using tables, graphs, properties of limits, and algebraic methods. Then, you will explore derivatives of functions and find average and instantaneous rates of change and velocity. Finally, you will apply your knowledge of limits and derivatives to solve real-world problems.