DEKALB COLLEGE DIVISION OF MATHEMATICS COMMON COURSE OUTLINE COURSE ABBREVIATION MATH 130 CREDIT HOURS 5 COURSE TITLE Precalculus PREREQUISITES MATH 118, with a C or better, or satisfactory placement score CATALOG DESCRIPTION This course provides a foundation for the study of calculus. Topics covered include functions and graphs (polynomial, rational, trigonometric, exponential, logarithmic), inverse functions, theory of equations, systems of equations, determinants and matrices, analytic geometry, combinations, the Binomial Theorem, induction, and summation notation. EXPECTED EDUCATIONAL RESULTS As a result of completing this course, the student will be able to: 1. Demonstrate proficiency in problem-solving skills including applications of logarithmic and exponential functions 2. Unify and extend basic concepts from Math 117 and Math 118 including the following a. Graph power functions with integer powers, root functions, exponential and logarithmic functions, the absolute value function, the greatest integer function, trigonometric functions (sine, cosine, and tangent), and inverse trigonometric functions (arcsine, arccosine, and arctangent) b. Graph variations of functions using shifting, reflection, stretching, domain, range, and intercepts c. Graph split-domain functions d. Describe the behavior of functions using domain, range, intervals of increase and decrease, intervals where the graph of the function is above the x-axis or below the x-axis, even or odd definitions, and one-to-one properties e. Solve problems using exponential and logarithmic functions 3. Write and graph the inverse function for a given function 4. Graph polynomial functions using synthetic division, Remainder Theorem, Factor Theorem, Rational Root Theorem, Descartes' Rule of Signs, Intermediate Value Theorem for Polynomials, and zeros 5. Graph rational functions identifying vertical asymptotes, horizontal asymptotes, oblique asymptotes, intercepts, and possible asymptote intercepts 6. Evaluate determinants and perform the matrix operations of addition, subtraction, scalar multiplication, multiplication, and inversion 7. Solve systems of linear equations using elementary row operations, Cramer's Rule, and the inverse matrix method 8. Identify and graph the conics and half-conics 9. Compute a finite sum using summation properties 10. Prove conjectures using mathematical induction 11. Write the terms of a binomial raised to a power using the Binomial Theorem GENERAL EDUCATION OUTCOMES I. This course addresses the general education outcome relating to communication by providing additional support as follows: A. Students develop their listening skills through lecture and through group problem solving. B. Students develop their reading comprehension skills by reading the text and by reading the instructions for text exercises, problems on tests, or on projects. Reading mathematics text requires recognizing symbolic notation as well as analyzing problems written in prose. C. Students develop their writing skills through the use of problems which require written explanations of concepts. II. This course addresses the general education outcome of demonstrating effective individual and group problem-solving and critical-thinking skills as follows: A. Students must apply mathematical concepts previously mastered to new problems and situations. B. In applications, students must analyze problems and describe problems with either pictures, diagrams, or graphs, then determine the appropriate strategy for solving the problem. III. This course addresses the general education outcome of using mathematical concepts to interpret, understand, and communicate quantitative data as follows: A. Students must demonstrate proficiency in problem-solving skills including applications of polynomial, rational, logarithmic, and exponential functions. B. Students must write functions to describe real-world situations and interpret information from both the function rule and the graph of the function. C. Students must solve systems of linear and nonlinear equations which often arise in modeling numerical relationships. COURSE CONTENT 1. Functions and Graphs (polynomial, rational, trigonometric, exponential, logarithmic) 2. Inverse Functions 3. Theory of Equations 4. Systems of Equations 5. Determinants and Matrices 6. Analytic Geometry 7. Combinations and the Binomial Theorem 8. Induction 9. Summation Notation ENTRY LEVEL COMPETENCIES Upon entering this course the student should be able to do the following: 1. Analyze problems using critical-thinking skills 2. Use algebraic symbols and notation to make meaningful statements 3. Analyze applications for which linear and quadratic equations are mathematical models 4. Solve the following types of equations: linear, quadratic, absolute value involving linear expressions, fractional resulting in linear and quadratic equations, quadratic-in-form, literal (both linear and quadratic), polynomial of degree higher than two by factoring, logarithmic, exponential, radical, and equations involving trigonometric or inverse trigonometric functions 5. Solve the following types of inequalities and write the solution in interval notation and set-builder notation as well as graph the solution on a number line: polynomial with degree greater than or equal to two, fractional, and absolute value involving linear expressions 6. Solve a system of two equation in two unknowns 7. Identify and graph the following types of equations in two variables: linear, absolute value, exponential, logarithmic, split-domain, square root, polynomial in factored form, rational, trigonometric, and equations in two unknowns whose graphs are parabolas or circles 8. Graph equations whose graphs are reflections, translations, and expansions or compressions of a previously defined equation in two variables 9. Identify where a function is increasing, decreasing, or constant from its graph 10. Sketch and identify the graph of a one-to-one function and its inverse 11. Identify a function from a rule, a graph, and a set of ordered pairs 12. Compose two functions 13. State the domain and range of a function from a rule and from a graph 14. Find and graph inverse functions for linear, exponential, logarithmic, and trigonometric functions 15. Define exponential and logarithmic functions, emphasizing the relationship between them 16. Use the properties of logarithms to rewrite expressions 17. State the unit circle definitions of the sine and cosine functions 18. State and apply the definitions of the tangent, cotangent, secant, and cosecant functions 19. State and apply the reciprocal, quotient, Pythagorean, cofunction, and even-odd identities for trigonometric functions 20. State and apply the definitions of arcsine, arccosine, and arctangent ASSESSMENT OF EXPECTED EDUCATIONAL RESULTS I. COURSE GRADE The course grade will be determined by the individual instructor using a variety of evaluation methods such as tests, quizzes, projects, homework, and writing assignments. A comprehensive final examination is required which must count at least one-fourth and no more than one-third of the course grade. The final examination will include items that require the student to demonstrate ability in problem solving and critical thinking as evidenced by detailed, worked-out solutions. II. DEPARTMENTAL ASSESSMENT The Math 130 assessment will be conducted every five years in the spring. An appropriate assessment instrument will be determined by the Math 130 course committee. III. USE OF ASSESSMENT FINDINGS The fall quarter following assessment the Math 130 course committee will review the collected assessment materials for Math 130 and will produce a report based upon their review. The assessment report should detail a plan for any needed curriculum changes and include a time frame for implementing them. The assessment report will be submitted to the division dean who will send copies of the report to the faculty and to the Director of Institutional Effectiveness. The division dean will be responsible for providing yearly reports on the implementation of the recommended changes. EFFECTIVE DATE: September 1994 APPROVED DATE: January 1994