## PreCalculus B: Functions & Graphical Analysis

### Enrollment Message:

Students will require a computer device with headphones, a microphone, webcam, up-to-date Chrome Web Browser, and access to YouTube. Students will also require a graphing calculator, such as TI-84 Plus, TI-83, or TI-83 Plus. Students will be required to participate in one or more synchronous, live session(s) with the instructor.This course is the second in a two-course sequence that broadens student understanding of functions and mathematical models of real-world phenomena. Designed to prepare a student for Calculus, this course will build on student understanding of algebra, extending their knowledge to new mathematical topics, while deepening their knowledge in others. The course contains units on polynomial, rational, logarithmic and exponential functions, sequences, series, conics and limits, as well as systems and vectors. The majority of the course is asynchronous with a few synchronous elements.
**Prerequisites: **Precalculus A

**Course Objectives:** Upon completion of this course, students will be able to...

- Perform algebraic operations (including compositions) on functions and apply transformations.
- Write an expression for the composition of one given function with another and find the domain, range, and graph of the composite function. Recognize components when a function is composed of two or more elementary functions.
- Identify and describe discontinuities of a function and how these relate to the graph.
- Use the idea of limit to analyze a graph as it approaches an asymptote. Compute limits of simple functions informally.
- Use the inverse relationship between exponential and logarithmic functions to solve equations and problems.
- Graph logarithmic functions. Graph translations and reflections of these functions.
- Compare the large-scale behavior of exponential and logarithmic functions with different bases and recognize that different growth rates are visible in the graphs of the functions.
- Solve exponential and logarithmic equations when possible. For those that cannot be solved analytically, use graphical methods to find approximate solutions.
- Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay.
- Solve quadratic-type equations by substitution.
- Apply quadratic functions and their graphs in the context of motion under gravity and simple optimization problems.
- Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find a quadratic function to model a given data set or situation.
- Solve polynomial equations and inequalities of degree greater than or equal to three. Graph polynomial functions given in factored form using zeros and their multiplicities, testing the sign-on intervals and analyzing the function’s large-scale behavior.
- Recognize and apply fundamental facts about polynomials: the Remainder Theorem, the Factor Theorem, and the Fundamental Theorem of Algebra.
- Solve equations and inequalities involving rational functions. Graph rational functions given in factored form using zeros, identifying asymptotes, analyzing their behavior for large x values, and testing intervals.
- Given vertical and horizontal asymptotes, find an expression for a rational function with these features.
- Recognize and apply the definition and geometric interpretation of difference quotient. Simplify difference quotients and interpret difference quotients as rates of change and slopes of secant lines.
- Perform operations (addition, subtraction, and multiplication by scalars) on vectors in the plane.
- Solve applied problems using vectors.
- Recognize and apply the algebraic and geometric definitions of the dot product of vectors.
- Define matrix addition and multiplication. Add, subtract, and multiply matrices.
- Multiply a vector by a matrix.
- Represent rotations of the plane as matrices and apply to find the equations of rotated conics.
- Define the inverse of a matrix and compute the inverse of two-by-two and three-by-three matrices when they exist.
- Explain the role of determinants in solving systems of linear equations using matrices and compute determinants of two-by-two and three-by-three matrices.
- Write systems of two and three linear equations in matrix form. Solve such systems using Gaussian elimination or inverse matrices.
- Represent and solve systems of inequalities in two variables and apply these methods in linear programming situations to solve problems.
- Recognize, explain, and use sigma and factorial notation.
- Given an arithmetic, geometric, or recursively defined sequence, write an expression for the nth term when possible. Write a particular term of a sequence when given the nth term.
- Explain, and use the formulas for the sums of finite arithmetic and geometric sequences.
- Compute the sums of infinite geometric series. Apply the convergence criterion for geometric series.
- Convert between polar and rectangular coordinates. Graph functions given in polar coordinates.
- Write complex numbers in polar form. Know and use De Moivre’s Theorem.
- Evaluate parametric equations for given values of the parameter.
- Convert between parametric and rectangular forms of equations.
- Graph curves described by parametric equations and find parametric equations for a given graph.
- Explain and apply the locus definitions of parabolas, ellipses, and hyperbolas and recognize these conic sections in applied situations.
- Identify parabolas, ellipses, and hyperbolas from equations, write the equations in standard form, and sketch an appropriate graph of the conic section.
- Derive the equation for a conic section from given geometric information. Identify key characteristics of a conic section from its equation or graph.

**Course Outline:**

Unit 9: Polynomial Functions

Unit 10: Rational Functions

Unit 11: Exponential & Logarithmic Functions

Unit 12: Sequences & Series

Unit 13: Conics

Unit 14: Limits

Unit 15: Systems

Unit 16: Trigonometry & The Polar Plane

**Resources Included:** Online lesson instruction and activities, opportunities to engage with a certified, online instructor and classmates, when appropriate, and online assessments to measure student performance of course objectives and readiness for subsequent academic pursuits.

**Additional Costs:** A graphing calculator, such as TI-84 Plus, TI-83, or TI-83 Plus, is needed.

**Scoring System: **Michigan Virtual does not assign letter grades, grant credit for courses, nor issue diplomas. A final score out of total points earned will be submitted to your school mentor for conversion to their own letter grading system.

**Time Commitment:** Semester sessions are 18-weeks long: Students must be able to spend 1 or more hours per day in the course to be successful. Summer sessions are 10 weeks long: Students must be able to spend a minimum of 2 or more hours per day, or about 90 hours during the summer, for the student to be successful in any course. Trimester sessions are 12-weeks long: Students must be able to spend 1.5 or more hours per day in the course to be successful.

**Technology Requirements: ** Students will require a computer device with headphones, a microphone, webcam, up-to-date Chrome Web Browser, and access to YouTube. Students will also require a graphing calculator, such as TI-84 Plus, TI-83, or TI-83 Plus.

Please review the Michigan Virtual Technology Requirements: https://michiganvirtual.org/about/support/knowledge-base/technical-requirements/

**Instructor Support System: **For technical issues within your course, contact the Customer Care Center by email at CustomerCare@michiganvirtual.org or by phone at (888) 889-2840.

**Instructor Contact Expectations: **Students can use email or the private message system within the Student Learning Portal to access highly qualified teachers when they need instructor assistance. Students will also receive feedback on their work inside the learning management system. The Instructor Info area of their course may describe additional communication options.

**Academic Support Available: **In addition to access to a highly qualified, Michigan certified teacher, students have access to academic videos and outside resources verified by Michigan Virtual. For technical issues within the course, students can contact the Michigan Virtual Customer Care by email at customercare@michiganvirtual.org or by phone at (888) 889-2840.

**Required Assessment: **Online assessments consist of formative and summative assessments represented by computer-graded multiple choice, instructor-graded writing assignments including hands-on projects, model building and other forms of authentic assessments.

**Technical Skills Needed:** No special skills are required beyond being able to operate a computer and use word processing software.

**Additional Information:** A graphing calculator is needed for both Pre-Calculus A and Pre-Calculus B. Students will be required to participate in one or more synchronous, live session(s) with the instructor.