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Test Information Guide: College-Level Examination ProgramÂŽ 2011-12 Precalculus
Š 2011 The College Board. All rights reserved. College Board, College-Level Examination Program, CLEP, and the acorn logo are registered trademarks of the College Board.
CLEP TEST INFORMATION GUIDE FOR PRECALCULUS
worldwide through computer-based testing programs and also — in forward-deployed areas — through paper-based testing. Approximately one-third of all CLEP candidates are military service members.
History of CLEP Since 1967, the College-Level Examination Program (CLEP®) has provided over six million people with the opportunity to reach their educational goals. CLEP participants have received college credit for knowledge and expertise they have gained through prior course work, independent study or work and life experience.
2010-11 National CLEP Candidates by Age* Under 18 9% 30 years and older 30%
Over the years, the CLEP examinations have evolved to keep pace with changing curricula and pedagogy. Typically, the examinations represent material taught in introductory college-level courses from all areas of the college curriculum. Students may choose from 33 different subject areas in which to demonstrate their mastery of college-level material.
18-22 years 39%
23-29 years 22%
* These data are based on 100% of CLEP test-takers who responded to this survey question during their examinations.
2010-11 National CLEP Candidates by Gender
Today, more than 2,900 colleges and universities recognize and grant credit for CLEP.
41%
Philosophy of CLEP Promoting access to higher education is CLEP’s foundation. CLEP offers students an opportunity to demonstrate and receive validation of their college-level skills and knowledge. Students who achieve an appropriate score on a CLEP exam can enrich their college experience with higher-level courses in their major field of study, expand their horizons by taking a wider array of electives and avoid repetition of material that they already know.
58%
Computer-Based CLEP Testing The computer-based format of CLEP exams allows for a number of key features. These include: • a variety of question formats that ensure effective assessment • real-time score reporting that gives students and colleges the ability to make immediate creditgranting decisions (except College Composition, which requires faculty scoring of essays twice a month) • a uniform recommended credit-granting score of 50 for all exams • “rights-only” scoring, which awards one point per correct answer • pretest questions that are not scored but provide current candidate population data and allow for rapid expansion of question pools
CLEP Participants CLEP’s test-taking population includes people of all ages and walks of life. Traditional 18- to 22-year-old students, adults just entering or returning to school, homeschoolers and international students who need to quantify their knowledge have all been assisted by CLEP in earning their college degrees. Currently, 58 percent of CLEP’s test-takers are women and 52 percent are 23 years of age or older. For over 30 years, the College Board has worked to provide government-funded credit-by-exam opportunities to the military through CLEP. Military service members are fully funded for their CLEP exam fees. Exams are administered at military installations
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CLEP Exam Development
The Committee
Content development for each of the CLEP exams is directed by a test development committee. Each committee is composed of faculty from a wide variety of institutions who are currently teaching the relevant college undergraduate courses. The committee members establish the test specifications based on feedback from a national curriculum survey; recommend credit-granting scores and standards; develop and select test questions; review statistical data and prepare descriptive material for use by faculty (Test Information Guides) and students planning to take the tests (CLEP Official Study Guide).
The College Board appoints standing committees of college faculty for each test title in the CLEP battery. Committee members usually serve a term of up to four years. Each committee works with content specialists at Educational Testing Service to establish test specifications and develop the tests. Listed below are the current committee members and their institutional affiliations.
College faculty also participate in CLEP in other ways: they convene periodically as part of standard-setting panels to determine the recommended level of student competency for the granting of college credit; they are called upon to write exam questions and to review forms and they help to ensure the continuing relevance of the CLEP examinations through the curriculum surveys.
Karen Bolinger, Chair
Clarion University
Donald Campbell
Middle Tennessee State University
Lisa Townsley
University of Georgia
The primary objective of the committee is to produce tests with good content validity. CLEP tests must be rigorous and relevant to the discipline and the appropriate courses. While the consensus of the committee members is that this test has high content validity for a typical introductory Precalculus course or curriculum, the validity of the content for a specific course or curriculum is best determined locally through careful review and comparison of test content, with instructional content covered in a particular course or curriculum.
The Curriculum Survey The first step in the construction of a CLEP exam is a curriculum survey. Its main purpose is to obtain information needed to develop test-content specifications that reflect the current college curriculum and to recognize anticipated changes in the field. The surveys of college faculty are conducted in each subject every three to five years depending on the discipline. Specifically, the survey gathers information on: • the major content and skill areas covered in the equivalent course and the proportion of the course devoted to each area • specific topics taught and the emphasis given to each topic • specific skills students are expected to acquire and the relative emphasis given to them • recent and anticipated changes in course content, skills and topics • the primary textbooks and supplementary learning resources used • titles and lengths of college courses that correspond to the CLEP exam
The Committee Meeting The exam is developed from a pool of questions written by committee members and outside question writers. All questions that will be scored on a CLEP exam have been pretested; those that pass a rigorous statistical analysis for content relevance, difficulty, fairness and correlation with assessment criteria are added to the pool. These questions are compiled by test development specialists according to the test specifications, and are presented to all the committee members for a final review. Before convening at a two- or three-day committee meeting, the members have a chance to review the test specifications and the pool of questions available for possible inclusion in the exam.
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At the meeting, the committee determines whether the questions are appropriate for the test and, if not, whether they need to be reworked and pretested again to ensure that they are accurate and unambiguous. Finally, draft forms of the exam are reviewed to ensure comparable levels of difficulty and content specifications on the various test forms. The committee is also responsible for writing and developing pretest questions. These questions are administered to candidates who take the examination and provide valuable statistical feedback on student performance under operational conditions.
developing, administering and scoring the exams. Effective July 2001, ACE recommended a uniform credit-granting score of 50 across all subjects, with the exception of four-semester language exams, which represents the performance of students who earn a grade of C in the corresponding college course. The American Council on Education, the major coordinating body for all the nation’s higher education institutions, seeks to provide leadership and a unifying voice on key higher education issues and to influence public policy through advocacy, research and program initiatives. For more information, visit the ACE CREDIT website at www.acenet.edu/acecredit.
Once the questions are developed and pretested, tests are assembled in one of two ways. In some cases, test forms are assembled in their entirety. These forms are of comparable difficulty and are therefore interchangeable. More commonly, questions are assembled into smaller, content-specific units called testlets, which can then be combined in different ways to create multiple test forms. This method allows many different forms to be assembled from a pool of questions.
CLEP Credit Granting CLEP uses a common recommended credit-granting score of 50 for all CLEP exams. This common credit-granting score does not mean, however, that the standards for all CLEP exams are the same. When a new or revised version of a test is introduced, the program conducts a standard setting to determine the recommended credit-granting score (“cut score”).
Test Specifications Test content specifications are determined primarily through the curriculum survey, the expertise of the committee and test development specialists, the recommendations of appropriate councils and conferences, textbook reviews and other appropriate sources of information. Content specifications take into account: • the purpose of the test • the intended test-taker population • the titles and descriptions of courses the test is designed to reflect • the specific subject matter and abilities to be tested • the length of the test, types of questions and instructions to be used
A standard-setting panel, consisting of 15–20 faculty members from colleges and universities across the country who are currently teaching the course, is appointed to give its expert judgment on the level of student performance that would be necessary to receive college credit in the course. The panel reviews the test and test specifications and defines the capabilities of the typical A student, as well as those of the typical B, C and D students.* Expected individual student performance is rated by each panelist on each question. The combined average of the ratings is used to determine a recommended number of examination questions that must be answered correctly to mirror classroom performance of typical B and C students in the related course. The panel’s findings are given to members of the test development committee who, with the help of Educational Testing Service and College Board psychometric specialists, make a final determination on which raw scores are equivalent to B and C levels of performance.
Recommendation of the American Council on Education (ACE) The American Council on Education’s College Credit Recommendation Service (ACE CREDIT) has evaluated CLEP processes and procedures for
*Student performance for the language exams (French, German and Spanish) is defined only at the B and C levels.
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Precalculus Description of the Examination
Graphing Calculator
The Precalculus examination assesses student mastery of skills and concepts required for success in a first-semester calculus course. A large portion of the exam is devoted to testing a student’s understanding of functions and their properties. Many of the questions test a student’s knowledge of specific properties of the following types of 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.
A graphing calculator, which is integrated into the exam software, is available to students only during Section 1 of the exam. Students are expected to know how and when to make use of it. The graphing calculator, together with a brief tutorial, is available to students as a free download for a 30-day trial period. Students are expected to become familiar with its functionality prior to taking the exam. For more information about downloading the practice version of the graphing calculator, please visit the Precalculus exam description on the CLEP website, www.collegeboard.org/clep. In order to answer some of the questions in Section 1 of the exam, students may be required to use the online graphing calculator in the following ways:
The examination contains approximately 48 questions, in two sections, to be answered in 90 minutes. Any time candidates spend on tutorials and providing personal information is in addition to the actual testing time.
• Perform calculations (e.g., exponents, roots, trigonometric values, logarithms). • Graph functions and analyze the graphs. • Find zeros of functions.
• Section 1: 25 questions, 50 minutes. The use of an online graphing calculator (non-CAS) is allowed for this section. Only some of the questions will require the use of the calculator.
• Find points of intersection of graphs of functions.
• Section 2: 23 questions, 40 minutes. No calculator is allowed for this section.
• Generate a table of values for a function.
• Find minima/maxima of functions. • Find numerical solutions to equations.
Although most of the questions on the exam are multiple-choice, there are some questions that require students to enter a numerical answer.
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Knowledge and Skills Required
C
Analytic Geometry Ability to demonstrate an understanding of the analytic geometry of lines, circles, parabolas, ellipses and hyperbolas
15%
Trigonometry and its Applications* Ability to demonstrate an understanding of the basic trigonometric functions and their inverses and to apply the basic trigonometric ratios and identities (in right triangles and on the unit circle) Ability to apply trigonometry in various problem-solving contexts
10%
Functions as Models Ability to interpret and construct functions as models and to translate ideas among symbolic, graphical, tabular and verbal representations of functions
• Solving nonroutine problems or problems that require insight, ingenuity or higher mental processes.
15%
Functions: Concept, Properties and Operations Ability to demonstrate an understanding of the concept of a function, the general properties of functions (e.g., domain, range), function notation, and to perform symbolic operations with functions (e.g., evaluation, inverse functions)
S
10%
• Solving problems that demonstrate comprehension of mathematical ideas and/or concepts.
Algebraic Expressions, Equations and Inequalities Ability to perform operations on algebraic expressions Ability to solve equations and inequalities, including linear, quadratic, absolute value, polynomial, rational, radical, exponential, logarithmic and trigonometric Ability to solve systems of equations, including linear and nonlinear
U
Representations of Functions: Symbolic, Graphical and Tabular Ability to recognize and perform operations and transformations on functions presented symbolically, graphically or in tabular form Ability to demonstrate an understanding of basic properties of functions and to recognize elementary functions (linear, quadratic, absolute value, square root, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions) that are presented symbolically, graphically or in tabular form
• Recalling factual knowledge and/or performing routine mathematical manipulation.
20%
L
30%
Questions on the examination require candidates to demonstrate the following abilities.
The subject matter of the Precalculus examination is drawn from the following topics. The percentages next to the topics indicate the approximate percentage of exam questions on that topic.
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*Note that trigonometry permeates most of the major topics and accounts for more than 15 percent of the exam. The actual proportion of exam questions that requires knowledge of either right triangle trigonometry or the properties of the trigonometric functions is approximately 30–40 percent.
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Notes and Reference Information
Sample Test Questions
The following information will be available for reference during the exam.
The following sample questions do not appear on an actual CLEP examination. They are intended to give potential test-takers an indication of the format and difficulty level of the examination and to provide content for practice and review. Knowing the correct answers to all of the sample questions is not a guarantee of satisfactory performance on the exam.
(1) Figures that accompany questions are intended to provide information useful in answering the questions. All figures lie in a plane unless otherwise indicated. The figures are drawn as accurately as possible EXCEPT when it is stated in a specific question that the figure is not drawn to scale. Straight lines and smooth curves may appear slightly jagged on the screen.
Section 1 Directions: A graphing calculator will be available for the questions in this section. Some questions will require you to select from among five choices. For these questions, select the BEST of the choices given. If the exact numerical value of your answer is not one of the choices, select the choice that best approximates this value. Some questions will require you to enter a numerical answer in the box provided.
(2) Unless otherwise specified, all angles are measured in radians, and all numbers used are real numbers. For some questions in this test, you may have to decide whether the calculator should be in radian mode or degree mode. (3) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers [ for which I [ is a real number. The range of f is assumed to be the set of all real numbers I [ where [ is in the domain of f. (4) In this test, ORJ [ denotes the common logarithm of [ (that is, the logarithm to the base 10) and OQ [ denotes the natural logarithm of [ (that is, the logarithm to the base e). (5) The inverse of a trigonometric function f may be indicated using the inverse function notation I 1 or with the prefix “arc� (e.g., VLQ 1 [ DUFVLQ [ ).
p p (6) The range of VLQ 1 [ is Ă‹ Ă› Ă?ĂŒ 2 2 Ă?Ăœ 1 The range of FRV [ is >0 p @
The range of WDQ 1 [ is
D (7) Law of Sines: VLQ $ Law of Cosines: F 2
1. The figure above shows the complete graphs of the functions f and g. Based on the graphs, the equation I [ J [ 0 has how many roots?
p p 2 2
E VLQ %
F VLQ &
(A) (B) (C) (D) (E)
D 2 E2 2 DE FRV &
(8) Sum and Difference Formulas:
VLQ a b
VLQ a FRV b FRV a VLQ b
VLQ a b
VLQ a FRV b FRV a VLQ b
FRV a b
FRV a FRV b VLQ a VLQ b
FRV a b
FRV a FRV b VLQ a VLQ b
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One Two Four Five Seven
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I [
J [
x –2 –1 0 1 2
g( x)
5. The functions f and g are defined above. What are all values of [ for which I [ J [ "
–4 0 2 2 4
(A) (B) (C) (D) (E)
2. The graph of the function f and a table of values for the function g are shown above. What is the value of I J 0
"
K [
3. The domain of the function f is ^ [ 1 Â… [ Â… 5 ` If J [ 2 I [ what is the domain of the function g ?
(B) (C) (D) (E) 4.
(E)
[ 2H [ [
7. The function h is defined above. Which of the following are true about the graph of \ K [ "
^ [ 10 Â… [ Â… 2 ` ^ [ 5 Â… [ Â… 1 ` ^ [ 2 Â… [ Â… 10 ` ^ [ 1 Â… [ Â… 5 ` ^ [ 1 Â… [ Â… 5`
I. The graph has a vertical asymptote at [ II. The graph has a horizontal asymptote at \ 0 III. The graph has a minimum point. (A) (B) (C) (D) (E)
VLQ W FRV W 2 (A) (B) (C) (D)
[ 0 RU [ ! 1 [ 0 RU [ ! 2 0 [ 1 0 [ 2 1 [ 2
6. If π ≤ θ ≤ 2π and cos V ⍽ cos 1, what is the value of V ? Round your answer to the nearest hundredth.
(A) – 4 (B) – 2 (C) 0 (D) 2 (E) 4
(A)
[ [ 1
[
1 1 2 VLQ W 1 VLQ 2W
VLQ W 2 2 VLQ W FRV W FRV W 2
VLQ W 2 FRV W 2
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None I and II only I and III only II and III only I, II, and III
0
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8. An antenna that is 90 feet high is on top of a hill. From a point at the base of the hill, the angles of elevation to the top and bottom of the antenna are 28.5° and 25°, respectively. To the nearest whole number of feet, how high is the hill? (A) (B) (C) (D) (E)
189 ft 213 ft 548 ft 623 ft 697 ft
9. Let g be the function defined by J [ 10 VLQ 20 [ 30 The maximum value of g is attained at which of the following values of x ? (A) (B) (C) (D) (E)
11. The figure above shows the graph of \ 0 5 [ 4 2 5 [3 0 5 [ 2 10 5 [ N where k is a constant. Which of the following could be the value of k ?
p 2 p 10 p 20 p 30 p 40
(A) – 18 (B) – 16 (C) – 9 (D) 9 (E) 16
10. In the xy-plane, the equation of line A is y 6 x 3. What is the measure, in degrees, of the acute angle formed between A and the x-axis? (A) (B) (C) (D) (E)
12. Let f be the function defined by I [ [ The graph of the function g in the xy-plane is obtained by first translating the graph of f horizontally 3 units to the left and then vertically translating this result 2 units up. What is the value of J 2 "
26.6° 60.0° 63.4° 71.6° 80.5°
(A) – 7 (B) – 3 (C) 0 (D) 1 (E) 3
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L I [
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S Î 3 VLQ [ IRU [ 0 � � [ IRU [ • 0
16. What is the range of the function f defined above? (A) All real numbers greater than or equal to – 3 (B) All real numbers greater than or equal to 0 (C) All real numbers greater than or equal to – 3 and less than or equal to 0 (D) All real numbers greater than or equal to – 3 and less than or equal to 3 (E) All real numbers
13. In the figure above, line A passes through the origin and intersects the graph of \ 2 [ at the point D 0 4 What is the slope of line A " (A) (B) (C) (D) (E)
0.200 0.303 0.528 1.322 3.305
14. In the xy-plane, the graph of \ [ 2 E[ F is symmetric about the line [ 3 and passes through the point 5 2 What is the value of c ?
$ W
NH 0 001W where k is a constant.
15. When a certain radioactive element decays, the amount, in milligrams, that remains after t years can be approximated by the function A above. Approximately how many years would it take for an initial amount of 800 milligrams of this element to decay to 400 milligrams?
K W
64 46 FRV
5p W ZKHUH 0 Â… W Â… 10
17. The function h above gives the height above the ground, in feet, of a passenger on a Ferris wheel t minutes after the ride begins. During one revolution of the Ferris wheel, for how many minutes is the passenger at least 100 feet above the ground? Round your answer to the nearest hundredth of a minute.
18. How many different values of x satisfy the equation sin x + 2 sin ( 2 x ) = x ? (A) (B) (C) (D) (E)
(A) 173 (B) 347 (C) 693 (D) 1,386 (E) 2,772
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One Two Three Five Infinitely many
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19. A ball is dropped from an initial height of d feet above the floor and repeatedly bounces off the floor. Each time the ball hits the floor, 3 it rebounds to a maximum height that is of 4 the height from which it previously fell. The function h models the maximum height, in feet, to which the ball rebounds on the nth bounce. Which of the following is an expression for h( n ) ?
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(B) S =
12 x2
(C) S = x 2 +
24 x
(D) S = x 2 +
48 x
2 (E) S = x +
48 x2
n
(C) h( n ) = 3 d n 4 (D) h( n ) =
3 n d4 3
(E) h( n ) = n 4
d
20. In the xy-plane, the vertex of the parabola x = y 2 + 4 y + 1 is the point ( h, k ) . What is the value of k ?
23. Let the functions f and g be defined by 1 f ( x ) = x − and g ( x ) = x . Which of the x following is not in the domain of the composite function ( g f )( x ) ? (A) −1 1 (B) − 2
(A) −13 (B) −5 (C) −2 (D) 2 (E) 5
(C) (D) (E)
21. The measure of a certain angle is 25°. What is the corresponding radian measure of the angle? (A) (B) (C) (D) (E)
5π 36 5π 18 5π 9 18 5π 36 5π
S
(A) S = 5 x 2
n
3 4
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22. A rectangular box with a square base is open at the top and has a volume of 12 cubic feet. Each side of the base has a length of x feet. Which of the following expresses the surface area, S, in square feet, of the outside of the box in terms of x ?
( )d 3 (B) h( n ) = ( d ) 4 (A) h( n ) =
U
x p(x)
1 2 1 2
0 11
1 10
2 11
3 14
24. The table above shows selected values for the function p. If p is a quadratic polynomial, what is the value of p(10 ) ?
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28. The graph in the xy-plane of which of the following equations is a parabola?
(A)
( 2b ) 5
(B)
T 2b 5
(B) [ 2 2 [ 3 \
2b5T
(C) [ 4 [ \ \
(D)
( 2b )5T
(D) [ \ 6 \ (E)
(E)
( 2b )T
(C)
(A) 2 [\ 2 2
5
Section 2 Directions: A calculator will not be available for the questions in this section. Some questions will require you to select from among five choices. For these questions, select the BEST of the choices given. Some questions will require you to enter a numerical answer in the box provided. 26. If [ 5 [ 5
of x ? (A) (B) (C) (D) (E)
1
2
2
[ 2 2
1
1
\2
29. An experiment designed to measure the growth of bacteria began at 2:00 p.m. and ended at 8:00 p.m. on the same day. The number of bacteria is given by the function N, where 1 W 1000 â´˘ 32W 3 and t represents the number of hours that have elapsed since the experiment began. How many more bacteria were there at the end of the experiment than at the beginning of the experiment?
5 what is the value
5“ 5 5 “ 5 “5 “ 10 “ 30
27. If f x 2 x 1 and g x
then f g x
(A) (B) (C) (D) (E)
1
5x x 2 6x 1 6x 2 6 x2 x 1
3 x 1,
30. The equation of the line shown in the graph above is \ D[ E Which of the following is always true for this line? (A) (B) (C) (D) (E)
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DE DE DE D D
0 !0 0 E E
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31. What is the x-intercept of the graph of 1 3 2 \ [ 8" 8
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34. In the xy-plane, the lines with equations 2 [ 2 \ 1 and 4 [ \ 4 intersect at the point with coordinates D E What is the value of b ?
(A) 16 (B) 8 1 (C) 16 (D) 16 (E) 512
35. Which of the following is the graph in the xy-plane of \ 3 VLQ 2 [ p "
32. The function h is given by h x log 2 x 2 2 . For what positive value of x does h x 3 ? (A) 1 (B) 2 (C) 8 (D) 6 (E) 7 33. Which of the following relations define y as a function of x ? I. [ 2 \ 3 2 II.
x y
4
0 1 2 3 4 10 20 30 20 10
III.
(A) (B) (C) (D) (E)
II only III only I and II I and III II and III
36. The function f is given by f x x x 10 . Which of the following defines f x for all x Â… 10 ? (A) (B) (C) (D) (E)
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f x
f x
f x
10 10 10 2 x
f x
f x
10 2 x 10 2 x
P
[ 5 10 15
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I [
a 32 b
37. The table above shows some values for the function f. If f is a linear function, what is the value of D E " (A) (B) (C) (D) (E)
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and OQ E
(B) J [
(C) J [
(D) J [
[3 4 [ [ 4 4 [2
(E) J [
[ 4 4 [2
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2 1 Ăˆ D2 Ă˜ 1 4 what is the value of OQ É Ă™ " ĂŠ EĂš
p 40. If 0 q and 10VLQ q 2 in terms of ] " (A) (B)
38. The figure above shows the graph of a polynomial function g. Which of the following could define J [ "
[3 4 [3 4 [
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39. If a and b are numbers such that OQ D
32 42 48 64 It cannot be determined from the information given.
(A) J [
U
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] 100 ] 2 10 2 ] 100
(C)
100 ] 2 10
(D)
] 2 100 10
(E)
100 ] 2 ]
] what is WDQ q
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A
41. Based on past sales, a convenience store has observed a linear relationship between the number of units of Product X that will be sold to customers each week and the price per unit. The figure above models this linear relationship. Based on the model, how many dollars would the convenience store expect to earn from its sales of Product X in a week when the price per unit is $5 ? (A) (B) (C) (D) (E)
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42. The figure above shows the graph of the function f defined by I [
2 [ 4 1 If I is the inverse function of f , what is the value of I 1 2 " (A) 8 (B) 2
$125 $250 $350 $600 $720
15
(C)
0
(D)
1 8
(E)
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45. In the xy-plane, which of the following is an equation of a vertical asymptote to the graph of y = sec ( 6 x − π ) ?
π 6 π = 4 π = 3 π = 2 =π
(A) x = (B) x (C) x (D) x 43. The Statue of Liberty is 46 meters tall and stands on a pedestal that is 47 meters above the ground. An observer is located d meters from the pedestal and is standing level with the base, as shown in the figure above. Which of the following best expresses the angle q in terms of d ? (A) q (B) q (C) q (D) q (E) q
47G DUFVLQ 46G
93 47 DUFVLQ DUFVLQ
G G 47 46 DUFWDQ DUFWDQ
G G G G DUFWDQ DUFWDQ
93 47 93 47 DUFWDQ DUFWDQ
G G
(E) x
f ( x) =
46. The function f is defined above. Which of the following statements are true?
DUFVLQ
I. The graph of f in the xy-plane has two x-intercepts. II. The graph of f in the xy-plane is the same as the graph of y = x − 3. III. The range of f is the set of all real numbers. (A) (B) (C) (D) (E)
44. The value of log (1, 732 ) is between what two integers? (A) (B) (C) (D) (E)
x2 − 5x + 6 x−2
2 and 3 3 and 4 4 and 5 17 and 18 173 and 174
None II only I and III only II and III only I, II, and III
x− y =1 x2 + y 2 = 5 47. The point ( x, y ) lies in the third quadrant of the xy-plane and satisfies the equations above. What is the value of y ?
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48. For all x ≠ 0, the function f is defined by x f ( x) = . What is the range of f ? x
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50. In the xy-plane, the graph of
(
)(
)
y = x x 2 − 2 x 2 + x + 1 intersects the x-axis in how many different points?
(A) −1 and 1 only (B) All real numbers between −1 and 1, inclusive (C) All real numbers greater than or equal to 0. (D) All real numbers except 0 (E) All real numbers
(A) (B) (C) (D) (E)
49. Let the function f be given by f ( x ) = sin ( x ) . What are all values of x such that f (−x) = f ( x) ? (A) (B) (C) (D) (E)
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0 All integer multiples of π π All integer multiples of 2 All real numbers There are no such values of x.
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One Two Three Four Five
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Study Resources
Answer Key
Most textbooks used in college-level precalculus courses cover the topics in the outline given earlier, but the approaches to certain topics and the emphases given to them may differ. To prepare for the Precalculus exam, it is advisable to study one or more college textbooks, which can be found in most college bookstores. When selecting a textbook, check the table of contents against the knowledge and skills required for this test.
Section 1 1. B 2. B 3. B 4. C 5. D 6. 5.28 7. D 8. C 9. E 10. E 11. C 12. D 13. B 14. 7 15. C 16. A 17. 2.14 18. D 19. A 20. C 21. A 22. D 23. C 24. 91 25. A
Visit www.collegeboard.org/clepprep for additional precalculus resources. You can also find suggestions for exam preparation in Chapter IV of the Official Study Guide. In addition, many college faculty post their course materials on their schools’ websites.
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Section 2 26. D 27. C 28. B 29. 80000 30. A 31. D 32. D 33. E 34. –0.4 35. C 36. A 37. D 38. C 39. 2.8 40. A 41. C 42. C 43. E 44. B 45. B 46. A 47. –2 48. A 49. B 50. C
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Test Measurement Overview Format There are multiple forms of the computer-based test, each containing a predetermined set of scored questions. The examinations are not adaptive. There may be some overlap between different forms of a test: any of the forms may have a few questions, many questions, or no questions in common. Some overlap may be necessary for statistical reasons. In the computer-based test, not all questions contribute to the candidate’s score. Some of the questions presented to the candidate are being pretested for use in future editions of the tests and will not count toward his or her score.
Scoring Information CLEP examinations are scored without a penalty for incorrect guessing. The candidate’s raw score is simply the number of questions answered correctly. However, this raw score is not reported; the raw scores are translated into a scaled score by a process that adjusts for differences in the difficulty of the questions on the various forms of the test.
Scaled Scores The scaled scores are reported on a scale of 20–80. Because the different forms of the tests are not always exactly equal in difficulty, raw-to-scale conversions may in some cases differ from form to form. The easier a form is judged to be, the higher the raw score required to attain a given scaled score. Table 1 indicates the relationship between number correct (raw score) and scaled score across all forms.
The Recommended Credit-Granting Score Table 1 also indicates the recommended credit-granting score, which represents the performance of students earning a grade of C in the corresponding course. The recommended B-level score represents B-level performance in equivalent course work. These scores were established as the result of a Standard Setting Study, the most recent having been conducted in 2005. The recommended credit-granting scores are based upon the judgments of a panel of experts currently teaching equivalent courses at various colleges and universities. These
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experts evaluate each question in order to determine the raw scores that would correspond to B and C levels of performance. Their judgments are then reviewed by a test development committee, which, in consultation with test content and psychometric specialists, makes a final determination. The standard-setting study is described more fully in the earlier section entitled “CLEP Credit Granting” on page 4. Panel members participating in the most recent study were: Edward Anderson
Northern Virginia Community College John Annulis University of Arkansas at Monticello Rajappa Asthagiri Miami University Eisso Atzema University of Maine Jeffrey Baumgartner Hesston College Mark Bollman Albion College Judy Broadwin Baruch College of CUNY Donald Campbell Middle Tennessee State University Blayne Carroll Berry College Keith Chavey University of Wisconsin — River Falls Roger Contreras University of Texas — Brownsville Pam Crawford Jacksonville University Roger Day Illinois State University Joseph Fiedler California State University — Bakersfield Angela Hare Messiah College Ed Harri Whatcom Community College Allen Hibbard Central College Carl Libis University of Rhode Island Connie Meade College of Southern Idaho Daniel Russow Arizona Western College Ronda Sanders University of South Carolina — Columbia To establish the exact correspondences between raw and scaled scores, a scaled score of 50 is assigned to the raw score that corresponds to the recommended credit-granting score for C-level performance. Then a high (but in some cases, possibly less than perfect) raw score will be selected and assigned a scaled score of 80. These two points — 50 and 80 — determine a function that generates a raw-to-scale conversion for the test.
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Table 1: Precalculus Interpretive Score Data American Council on Education (ACE) Recommended Number of Semester Hours of Credit: 3 Course Grade
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80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50* 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20
44 43 42 41 41 40 39 38 37 36 35-36 35 34 33 32 31 31 30 29 28 27 27 26 25 24 23 23 22 21 20 19 18-19 18 17 16 15-16 15 14 13 12 11-12 11 10 9 8 8 7 6 5 4 3 2 1 0 -
*Credit-granting score recommended by ACE. Note: The number-correct scores for each scaled score on different forms may vary depending on form difďŹ culty.
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Validity
Reliability
Validity is a characteristic of a particular use of the test scores of a group of examinees. If the scores are used to make inferences about the examinees’ knowledge of a particular subject, the validity of the scores for that purpose is the extent to which those inferences can be trusted to be accurate.
The reliability of the test scores of a group of examinees is commonly described by two statistics: the reliability coefficient and the standard error of measurement (SEM). The reliability coefficient is the correlation between the scores those examinees get (or would get) on two independent replications of the measurement process. The reliability coefficient is intended to indicate the stability/consistency of the candidates’ test scores, and is often expressed as a number ranging from .00 to 1.00. A value of .00 indicates total lack of stability, while a value of 1.00 indicates perfect stability. The reliability coefficient can be interpreted as the correlation between the scores examinees would earn on two forms of the test that had no questions in common.
One type of evidence for the validity of test scores is called content-related evidence of validity. It is usually based upon the judgments of a set of experts who evaluate the extent to which the content of the test is appropriate for the inferences to be made about the examinees’ knowledge. The committee that developed the CLEP Precalculus examination selected the content of the test to reflect the content of precalculus courses at most colleges, as determined by a curriculum survey. Since colleges differ somewhat in the content of the courses they offer, faculty members should, and are urged to, review the content outline and the sample questions to ensure that the test covers core content appropriate to the courses at their college. Another type of evidence for test-score validity is called criterion-related evidence of validity. It consists of statistical evidence that examinees who score high on the test also do well on other measures of the knowledge or skills the test is being used to measure. Criterion-related evidence for the validity of CLEP scores can be obtained by studies comparing students’ CLEP scores with the grades they received in corresponding classes, or other measures of achievement or ability. CLEP and the College Board conduct these studies, called Admitted Class Evaluation Service or ACES, for individual colleges that meet certain criteria at the college’s request. Please contact CLEP for more information.
Statisticians use an internal-consistency measure to calculate the reliability coefficients for the CLEP exam. This involves looking at the statistical relationships among responses to individual multiple-choice questions to estimate the reliability of the total test score. The formula used is known as Kuder-Richardson 20, or KR-20, which is equivalent to a more general formula called coefficient alpha. The SEM is an index of the extent to which students’ obtained scores tend to vary from their true scores.1 It is expressed in score units of the test. Intervals extending one standard error above and below the true score (see below) for a test-taker will include 68 percent of that test-taker’s obtained scores. Similarly, intervals extending two standard errors above and below the true score will include 95 percent of the test-taker’s obtained scores. The standard error of measurement is inversely related to the reliability coefficient. If the reliability of the test were 1.00 (if it perfectly measured the candidate’s knowledge), the standard error of measurement would be zero. Scores on the CLEP examination in Precalculus are estimated to have a reliability coefficient of 0.88. The standard error of measurement is 3.69 scaled-score points. 1
True score is a hypothetical concept indicating what an individual’s score on a test would be if there were no errors introduced by the measuring process. It is thought of as the hypothetical average of an infinite number of obtained scores for a test-taker with the effect of practice removed.
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