Table of Contents
Functions and graphs
1 The adsolute value
2 Lines and functions
3 Functions by formulas and graphs
4 Introduction to the TI-84
5 Basic functions and transformations
6 Operations on functions
7 The inverse of a function
Polynomials and rational functions
8 Dividing polynomials
9 Graphing polynomials
10 Roots of polynomials
11 Rational functions
12 Polynomial and rational inequalities
Exponential and logarithmic functions
13 Exponential and logarithmic functions
14 Properties of exp and log
15 Applications of exp and log
16 Half-life and compound interest
17 Trigonometric functions
18 Addition of angles and multiple angles
19 Inverse trigonometric functions
20 Trigonometric equations
Complex numbers, sequences, and the binomial theorem
21 Complex numbers
22 Vectors in the plane
23 Sequences and series
24 The geometric series
25 The binomial theorem
A Introduction to the TI-84 349
Answers to exercises
Important formulas used in precalculus
About the Book
These are notes for a course in precalculus, as it is taught at New York City College of Technology - CUNY (where it is offered under the course number MAT 1375). Our approach is calculator based. For this, we will use the currently standard TI-84 calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen to use. An introduction to the TI-84 calculator together with the most common applications needed for this course is provided in appendix A. In the future we may expand on this by providing introductions to other calculators or computer algebra systems.
This course in precalculus has the overarching theme of “functions.” This means that many of the often more algebraic topics studied in the previous courses are revisited under this new function theoretic point of view. However, in order to keep this text as self contained as possible we always recall all results that are necessary to follow the core of the course even if we assume that the student has familiarity with the formula or topic at hand. After a first introduction to the abstract notion of a function, we study polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions with the function viewpoint. Throughout, we will always place particular importance to the corresponding graph of the discussed function which will be analyzed with the help of the TI-84 calculator as mentioned above. These are in fact the topics of the first four (of the five) parts of this precalculus course.
In the fifth and last part of the book, we deviate from the above theme and collect more algebraically oriented topics that will be needed in calculus or other advanced mathematics courses or even other science courses. This part includes a discussion of the algebra of complex numbers (in particular complex numbers in polar form), the 2-dimensional real vector space R 2 sequences and series with focus on the arithmetic and geometric series (which are again examples of functions, though this is not emphasized), and finally the generalized binomial theorem.
About the Contributors
Thomas Tradler, Professor, Mathematics, New York City College of Technology.
Holly Carley, Associate Professor, Mathematics, New York City College of Technology.