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1. About this Course

This course, which covers the fundamentals taught in a first-semester calculus course, will provide you with an understanding of functions and their properties.

Many of the questions of the Precalculus CLEP exam will test your knowledge of specific properties of these functions:  linear, quadratic, absolute value, square root, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined. Questions on the exam will present these types of functions symbolically, graphically, verbally, or in tabular form. A solid understanding of these types of functions is at the core of all precalculus courses, and it is a prerequisite for enrolling in calculus and other college-level mathematics courses.

"Precalculus is a transitional course. There is a big difference between life before and after calculus, and I want to prepare you as much as possible for this fascinating subject. The precalculus CLEP exam will require us to study algebra and function theory in great detail. A huge emphasis is also placed on trigonometry. This is because of the central role the trigonometric functions play in calculus, and their role in linking geometry and algebra. We will also spend time applying our mathematical theories to real world problems, through the study of mathematical modeling. Finally, we will review conic sections, a branch of geometry with ancient roots," explains Dr. James Murphy, professor at Johns Hopkins University and instructor of this course.

The Precalculus course –completely self-paced and entirely free– is organized into six topical chapters or modules that contain short video lessons, exercises, readings and other interactive resources. 

The goal of the creator of this course – Modern States Education Alliance, a non-profit organization – is to prepare you to pass the College Board's CLEP examination and obtain college credit for free


2. About the examination

This exam contains approximately 48 questions, in two sections, to be answered in 90 minutes. Although most of the questions on the exam are multiple-choice, there are some questions that require students to enter a numerical answer.

Section 1: 25 questions, 50 minutes. The use of an online graphing calculator (non-CAS), integrated into the exam software, is allowed for this section. Only some of the questions will require the use of the calculator. It will be used to perform calculations (e.g., exponents, roots, trigonometric values, logarithms): graph functions and analyze the graphs; find zeros of functions; find points of intersection of graphs of functions; find minima/maxima of functions; find numerical solutions to equations; generate a table of values for a function.

The graphing calculator, together with brief video tutorials, is available to students as a free download for a 30-day trial period

Section 2: 23 questions, 40 minutes. No calculator is allowed for this section.

Depending on your institution's policy, a passing score on the exam can:

• Show your college math professors that you're ready for a calculus class

• Allow you to opt out of a math proficiency core requirement

• Earn you three college credits

3. Required Knowledge and Skills

Questions on the Precalculus examination require candidates to demonstrate the following abilities:

  • Recalling factual knowledge and/or performing routine mathematical manipulation

  • Solving problems that demonstrate comprehension of mathematical ideas and/or concepts

  • Solving nonroutine problems or problems that require insight, ingenuity, or higher mental processes

4. Course Modules

Following are the main topics and percentages of the exam’s questions, mostly based on the College Board's description of the course:

Module 1: Algebra  

  1.1 Basic Algebra

     1.1.1 Algebraic Operations

     1.1.2 Factoring and Expanding Polynomials

     1.1.3 Introduction to Exponentials

     1.1.4 Logarithms

  1.2 Equations

     1.2.1 Linear Equations and Inequalities

     1.2.2 Quadratic Equations

     1.2.3 Higher Order Polynomials

     1.2.4 Exponential and Logarithmic Equations

     1.2.5 Absolute Value Equations

  1.3 Systems of Equations

     1.3.1 Systems of Equations and Inequalities

     1.3.2 Higher Order Systems


Module 2: Function Theory  

  2.1 Introduction to Functions   

  2.2 Algebra of Functions

  2.3 Domain and Range of a Function

  2.4 Inverse Functions


Module 3: Trigonometry 

  3.1 Introduction to Trigonometry

     3.1.1 Idea of Trigonometry

     3.1.2 Trigonometric Functions

  3.2 Trigonometric Examples   

  3.3 Radians

  3.4 Periodicity and Plotting Trigonometric Functions

  3.5 Major Trigonometric Identities

  3.6 Inverse Trigonometric Functions

  3.7 Domain and Range of Trigonometric Functions

     3.7.1 Domain and Range of Trigonometric Functions

     3.7.2 Domain and Range of Inverse Trigonometric Functions

  3.8 Laws of Sines and Cosines

  3.9 Trigonometric Equations 


Module 4: Function Representations  

  4.1 Plotting Functions with Symmetry and Transformations

     4.1.1 Plotting Functions with Symmetry and Transformations

     4.1.2 Plotting Functions with Asymptotes and Extrema

  4.2 Return to Function Algebra

  4.3 Tabular Representations


Module 5: Modeling with Functions  

  5.1 Linear Growth

  5.2 Exponential Growth   

  5.3 Trigonometric Models


Module 6: Analytic Geometry 

  6.1 Lines   

  6.2 Idea of Conic Sections

  6.3 Circles

  6.4 Ellipses

  6.5 Parabolas

  6.6 Hyperbolas


5. About James Murphy

Dr. James Murphy is an applied mathematician. His research interests include harmonic analysis, machine learning, and the analysis of remote sensing data. Recent work has focused on fast algorithms for unsupervised learning and development of distance metrics for stochastic data models, with applications to hyperspectral imagery.

Dr. Murphy has taught at some of the world’s finest academic institutions, including the University of Chicago, the University of Maryland College Park, Duke University, and most recently at the Johns Hopkins University. His courses range from introductory statistics and precalculus to research topics courses for Ph.D. students. He was honored with the Aziz/Osborn Gold Medal in Teaching Excellence at the University of Maryland in 2013. He also works closely with undergraduates on research projects in both pure and applied mathematics. Several of his students are currently working on Ph.D.s at top universities. Dr. Murphy earned his Ph.D. in mathematics at the University of Maryland College Park in 2015. He earned his B.S. in mathematics from the University of Chicago in 2011, where he was a Student Marshal and Phi Beta Kappa.


6. How CLEP Works

Developed by the College Board, CLEP (College-Level Examination Program®) is the most widely accepted credit-by-examination program.

CLEP’s credits are accepted by 2,900 colleges and universities, according to the College Board. These tests assess college-level knowledge in 33 subject areas.

Modern States Education Alliance is the non-profit organization behind these edX-style courses. Its project is called “Freshman Year for Free” and its mission is to make college more accessible and affordable through free, high-quality online education.

• CLEP® Precalculus: at a Glance

• 'Passing the CLEP and Learning with Modern States' orientation course