The purpose of this book is to help you prepare for the SAT Level 2 Mathematics Subject Test. This book can be used as a self-study guide or as a textbook in a test preparation course. It is a self-contained resource for those who want to achieve their best possible score.

Because the SAT Subject Tests cover specific content, they should be taken as soon as possible after completing the necessary course(s). This means that you should register for the Level 2 Mathematics Subject Test in June after you complete a precalculus course.

You can register for SAT Subject Tests at the College Board’s web site,; by calling (866) 756-7346, if you previously registered for an SAT Reasoning Test or Subject Test; or by completing registration forms in the SAT Registration Booklet, which can be obtained in your high school guidance office. You may register for up to three Subject Tests at each sitting.

Important Reminder

Be sure to check the official College Board web site for the most accurate information about how to register for the test and what documentation to bring on test day.

Colleges use SAT Subject Tests to help them make both admission and placement decisions. Because the Subject Tests are not tied to specific curricula, grading procedures, or instructional methods, they provide uniform measures of achievement in various subject areas. This way, colleges can use Subject Test results to compare the achievement of students who come from varying backgrounds and schools.

You can consult college catalogs and web sites to determine which, if any, SAT Subject Tests are required as part of an admissions package. Many “competitive” colleges require the Level 1 Mathematics Test.

If you intend to apply for admission to a college program in mathematics, science, or engineering, you may be required to take the Level 2 Mathematics Subject Test. If you have been generally successful in high school mathematics courses and want to showcase your achievement, you may want to take the Level 2 Subject Test and send your scores to colleges you are interested in even if it isn’t required.


A Diagnostic Test in Part 1 follows this introduction. This test will help you quickly identify your weaknesses and gaps in your knowledge of the topics. You should take it under test conditions (in one quiet hour). Use the Answer Key immediately following the test to check your answers, read the explanations for the problems you did not get right, and complete the self-evaluation chart that follows the explanations. These explanations include a code for calculator use, the correct answer choice, and the location of the relevant topic in the Part 2 “Review of Major Topics.” For your convenience, a self-evaluation chart is also keyed to these locations.

The majority of those taking the Level 2 Mathematics Subject Test are accustomed to using graphing calculators. Where appropriate, explanations of problem solutions are based on their use. Secondary explanations that rely on algebraic techniques may also be given.

Part 3 contains six model tests. The breakdown of test items by topic approximately reflects the nominal distribution established by the College Board. The percentage of questions for which calculators are required or useful on the model tests is also approximately the same as that specified by the College Board. The model tests are self-contained. Each has an answer sheet and a complete set of directions. Each test is followed by an answer key, explanations such as those found in the Diagnostic Test, and a self-evaluation chart.

This e-Book contains hyperlinks to help you navigate through content, bring you to helpful resources, and click between test questions and their answer explanations.


The SAT Mathematics Level 2 Subject Test is one hour in length and consists of 50 multiple-choice questions, each with five answer choices. The test is aimed at students who have had two years of algebra, one year of geometry, and one year of trigonometry and elementary functions. According to the College Board, test items are distributed over topics as follows:

•     Numbers and Operation: 5–7 questions 
Operations, ratio and proportion, complex numbers, counting, elementary number theory, matrices, sequences, series, and vectors

•     Algebra and Functions: 24–26 questions 
Work with equations, inequalities, and expressions; know properties of the following classes of functions: linear, polynomial, rational, exponential, logarithmic, trigonometric and inverse trigonometric, periodic, piecewise, recursive, and parametric

•     Coordinate Geometry: 5–7 questions 
Symmetry, transformations, conic sections, polar coordinates

•     Three-dimensional Geometry: 2–3 questions 
Volume and surface area of solids (prisms, cylinders, pyramids, cones, and spheres); coordinates in 3 dimensions

•     Trigonometry: 6–8 questions 
Radian measure; laws of sines and law of cosines; Pythagorean theorem, cofunction, and double-angle identities

•     Data Analysis, Statistics, and Probability: 3–5 questions 
Measures of central tendency and spread; graphs and plots; least squares regression (linear, quadratic, and exponential); probability


As noted earlier, most taking the Level 2 Mathematics Subject Test will use a graphing calculator. In addition to performing the calculations of a scientific calculator, graphing calculators can be used to analyze graphs and to find zeros, points of intersection of graphs, and maxima and minima of functions. Graphing calculators can also be used to find numerical solutions to equations, generate tables of function values, evaluate statistics, and find regression equations. The authors assume that readers of this book plan to use a graphing calculator when taking the Level 2 test.


To make them as specific and succinct as possible, calculator instructions in the answer explanations are based on the TI-83 and TI-84 families of calculators.

You should always read a question carefully and decide on a strategy to answer it before deciding whether a calculator is necessary. A calculator is useful or necessary on only 55–65 percent of the questions. You may find, for example, that you need a calculator only to evaluate some expression that must be determined based solely on your knowledge about how to solve the problem.

Most graphing calculators are user friendly. They follow order of operations, and expressions can be entered using several levels of parentheses. There is never a need to round and write down the result of an intermediate calculation and then rekey that value as part of another calculation. Premature rounding can result in choosing a wrong answer if numerical answer choices are close in value.

On the other hand, graphing calculators can be troublesome or even misleading. For example, if you have difficulty finding a useful window for a graph, perhaps there is a better way to solve a problem. Piecewise functions, functions with restricted domains, and functions having asymptotes provide other examples where the usefulness of a graphing calculator may be limited.

Calculators have popularized a multiple-choice problem-solving technique called back-solving, where answer choices are entered into the problem to see which works. In problems where decimal answer choices are rounded, none of the choices may work satisfactorily. Be careful not to overuse this technique.

The College Board has established rules governing the use of calculators on the Mathematics Subject Tests:

•    You may bring extra batteries or a backup calculator to the test. If you wish, you may bring both scientific and graphing calculators.

•    Test centers are not expected to provide calculators, and test takers may not share calculators.

•    Notify the test supervisor to have your score cancelled if your calculator malfunctions during the test and you do not have a backup.

•    Certain types of devices that have computational power are not permitted: cell phones, pocket organizers, powerbooks and portable handheld computers, and electronic writing pads. Calculators that require an electrical outlet, make noise or “talk,” or use paper tapes are also prohibited.

•    You do not have to clear a graphing calculator memory before or after taking the test. However, any attempt to take notes in your calculator about a test and remove it from the room will be grounds for dismissal and cancellation of scores.


Leave your cell phone at home, in your locker, or in your car!


There are 50 questions on the Math Level 2 Subject Test. Your raw score is the number of correct answers minus one-fourth of the number of incorrect answers, rounded to the nearest whole number. For example, if you get 30 correct answers, 15 incorrect answers, and leave 5 blank, your raw score would be , rounded to the nearest whole number.

Raw scores are transformed into scaled scores between 200 and 800. The formula for this transformation changes slightly from year to year to reflect varying test difficulty. In recent years, a raw score of 44 was high enough to transform to a scaled score of 800. Each point less in the raw score resulted in approximately 10 points less in the scaled score. For a raw score of 44 or more, the approximate scaled score is 800. For raw scores of 44 or less, the following formula can be used to get an approximate scaled score on the Diagnostic Test and each model test:

= 800 – 10(44 – R), where S is the approximate scaled score and R is the rounded raw score.

The self-evaluation page for the Diagnostic Test and each model test includes spaces for you to calculate your raw score and scaled score.


•    Budget your time. Although most testing centers have wall clocks, you would be wise to have a watch on your desk. Since there are 50 items on a one-hour test, you have a little over a minute per item. Typically, test items are easier near the beginning of a test, and they get progressively more difficult. Don’t linger over difficult questions. Work the problems you are confident of first, and then return later to the ones that are difficult for you.

•    Guess intelligently. As noted above, you are likely to get a higher score if you can confidently eliminate two or more answer choices, and a lower score if you can’t eliminate any.

•    Read the questions carefully. Answer the question asked, not the one you may have expected. For example, you may have to solve an equation to answer the question, but the solution itself may not be the answer.

•    Mark answers clearly and accurately. Since you may skip questions that are difficult, be sure to mark the correct number on your answer sheet. If you change an answer, erase cleanly and leave no stray marks. Mark only one answer; an item will be graded as incorrect if more than one answer choice is marked.

•    Change an answer only if you have a good reason for doing so. It is usually not a good idea to change an answer on the basis of a hunch or whim.

•    As you read a problem, think about possible computational shortcuts to obtain the correct answer choice. Even though calculators simplify the computational process, you may save time by identifying a pattern that leads to a shortcut.

•    Substitute numbers to determine the nature of a relationship. If a problem contains only variable quantities, it is sometimes helpful to substitute numbers to understand the relationships implied in the problem.

•    Think carefully about whether to use a calculator. The College Board’s guideline is that a calculator is useful or necessary in about 60% of the problems on the Level 2 Test. An appropriate percentage for you may differ from this, depending on your experience with calculators. Even if you learned the material in a highly calculator-active environment, you may discover that a problem can be done more efficiently without a calculator than with one.

•    Check the answer choices. If the answer choices are in decimal form, the problem is likely to require the use of a calculator.


Your first step is to take the Diagnostic Test. This should be taken under test conditions: timed, quiet, without interruption. Correct the test and identify areas of weakness using the cross-references to the Part 2 review. Use the review to strengthen your understanding of the concepts involved.

Ideally, you would start preparing for the test two to three months in advance. Each week, you would be able to take one sample test, following the same procedure as for the Diagnostic Test. Depending on how well you do, it might take you anywhere between 15 minutes and an hour to complete the work after you take the test. Obviously, if you have less time to prepare, you would have to intensify your efforts to complete the six sample tests, or do fewer of them.

The best way to use Part 2 of this book is as reference material. You should look through this material quickly before you take the sample tests, just to get an idea of the range of topics covered and the level of detail. However, these parts of the book are more effectively used after you’ve taken and corrected a sample test.