SAT® Subject Test: Mathematics Level TENTH EDITION SAT® is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product Table of Contents SAT® Subject Test: Mathematics Level Cover Title Page Part One: The Basics Chapter 1: Getting Ready for SAT Subject Test: Mathematics Understand the SAT Subject Tests Content of the SAT Subject Test: Mathematics Finding Your Level Chapter 2: SAT Subject Test Mastery Use the Structure of the Test to Your Advantage Approaching SAT Subject Test Questions Work Strategically Stress Management The Final Countdown Chapter 3: The Calculator Have the Right Calculator Use Your Calculator Strategically Part Two: Mathematics Level Review Chapter 4: Algebra How to Use This Chapter Algebra Diagnostic Test Algebra Diagnostic Test: Answer Key Find Your Study Plan Test Topics Algebra Follow-Up Test Algebra Follow-Up Test: Answer Key Algebra Follow-Up Test: Answers and Explanations Chapter 5: Plane Geometry How to Use This Chapter Diagnostic Test Diagnostic Test: Answer Key Find Your Study Plan Test Topics Plane Geometry Follow-Up Test Plane Geometry Follow-Up Test: Answer Key Plane Geometry Follow-Up Test: Answers and Explanations Chapter 6: Solid Geometry How to Use This Chapter Solid Geometry Diagnostic Test Solid Geometry Diagnostic Test: Answer Key Find Your Study Plan Test Topics Solid Geometry Follow-Up Test Solid Geometry Follow-Up Test: Answer Key Solid Geometry Follow-Up Test: Answers and Explanations Chapter 7: Coordinate Geometry How to Use This Chapter Coordinate Geometry Diagnostic Test Coordinate Geometry Diagnostic Test: Answer Key Find Your Study Plan Test Topics Coordinate Geometry Follow-Up Test Coordinate Geometry Follow-Up Test: Answer Key Coordinate Geometry Follow-Up Test: Answers and Explanations Chapter 8: Trigonometry How to Use This Chapter Trigonometry Diagnostic Test Trigonometry Diagnostic Test: Answer Key Find Your Study Plan Test Topics Trigonometry Follow-Up Test Trigonometry Follow-Up Test: Answer Key Trigonometry Follow-Up Test: Answers and Explanations Chapter 9: Functions How to Use This Chapter Functions Diagnostic Test Functions Diagnostic Test: Answer Key Find Your Study Plan Test Topics Functions Follow-Up Test Functions Follow-Up Test: Answer Key Functions Follow-Up Test: Answers and Explanations Chapter 10: Miscellaneous Topics How to Use This Chapter Miscellaneous Topics Diagnostic Test Miscellaneous Topics Diagnostic Test: Answer Key Find Your Study Plan Test Topics Miscellaneous Topics Follow-Up Test Miscellaneous Topics Follow-Up Test: Answer Key Miscellaneous Topics Follow-Up Test: Answers and Explanations Part Three: Practice Tests Practice Test How to Take the Practice Tests How to Calculate Your Score Answer Grid Practice Test Practice Test 1: Answer Key Practice Test 1: Answers and Explanations Practice Test How to Calculate Your Score Answer Grid Practice Test Practice Test 2: Answer Key Practice Test 2: Answers and Explanations Practice Test How to Calculate Your Score Answer Grid Practice Test Practice Test 3: Answer Key Practice Test 3: Answers and Explanations Practice Test How to Calculate Your Score Answer Grid Practice Test Practice Test 4: Answer Key Practice Test 4: Answers and Explanations 100 Essential Math Concepts 100 Essential Math Concepts About This Book Copyright Information Part One THE BASICS Chapter Getting Ready for SAT Subject Test: Mathematics Understand the SAT subject tests Content of the SAT Subject Test: Mathematics Finding your level Level of difficulty and scoring You’re serious about going to the college of your choice You wouldn’t have opened this book otherwise You’ve made a wise choice, because this book can help you to achieve your goal It’ll show you how to score your best on the SAT Subject Test: Mathematics But before turning to the math content, let’s look at the SAT subject tests generally Understand the SAT Subject Tests The following background information about the SAT subject test is important to keep in mind as you get ready to prep for the SAT Subject Test: Mathematics WHAT ARE THE SAT SUBJECT TESTS? Known until 1994 as the College Board Achievement Tests and until 2004 as the SAT IIs, the SAT Subject Tests focus on specific disciplines: English, U.S History, World History, Mathematics, Physics, Chemistry, Biology, and many foreign languages Each test lasts one hour and consists entirely of multiple-choice questions On any one test date, you can take one, two, or three subject tests 10 FUNCTIONS 69 Function Notation and Evaluation Standard function notation is written f(x) and read “f of x.” To evaluate the function f(x) = 2x + for f(4), replace x with and simplify: f(4) = 2(4) + = 11 70 Direct and Inverse Variation In direct variation, y = kx, where k is a nonzero constant In direct variation, the variable y changes directly as x does If a unit of Currency A is worth units of Currency B, then A = 2B If the number of units of B were to double, the number of units of A would double, and so on for halving, tripling, etc In inverse variation, xy = k, where x and y are variables and k is a constant A famous inverse relationship is rate × time = distance, where distance is constant Imagine having to cover a distance of 24 miles If you were to travel at 12 miles per hour, you’d need hours But if you were to halve your rate, you would have to double your time This is just another way of saying that rate and time vary inversely 71 Domain and Range of a Function The domain of a function is the set of values for which the function is defined For example, the domain of is all values of x except and −1, because for those values the denominator has a value of and the fraction is therefore undefined The range of a function is the set of outputs or results of the function For example, the range of f(x) = x2 is all numbers greater than or equal to zero, because x2 cannot be negative 486 COORDINATE GEOMETRY 72 Finding the Distance Between Two Points To find the distance between points, use the Pythagorean theorem or special right triangles The difference between the x’s is one leg and the difference between the y’s is the other In the figure above, PQ is the hypotenuse of a 3-4-5 triangle, so PQ = You can also use the distance formula: To find the distance between R(3,6) and S(5,−2): 73 Using Two Points to Find the Slope The slope of the line that contains the points A(2,3) and B(0,−1) is 74 Using an Equation to Find the Slope To find the slope of a line from an equation, put the equation into the slope-intercept form: 487 y = mx + b The slope is m To find the slope of the equation 3x + 2y = 4, rearrange it: The slope is 75 Using an Equation to Find an Intercept To find the y-intercept, you can either put the equation into y = mx + b (slope-intercept) form—in which case b is the y-intercept—or you can just plug x = into the equation and solve for y To find the x-intercept, plug y = into the equation and solve for x 76 Finding the Midpoint The midpoint of two points on a line segment is the average of the x-coordinates of the endpoints and the average of the y-coordinates of the endpoints If the endpoints are (x1,y1) and (x2,y2), the midpoint is The midpoint of (3,5) and (9,1) is 488 or (6, 3) LINES AND ANGLES 77 Intersecting Lines When two lines intersect, adjacent angles are supplementary, and vertical angles are equal In the figure above, the angles marked a° and b° are adjacent and supplementary, so a + b = 180 Furthermore, the angles marked a° and 60° are vertical and equal, so a = 60 78 Parallel Lines and Transversals A transversal across parallel lines forms four equal acute angles and four equal obtuse angles, unless the transversal meets the lines at a right angle; then all eight angles are right angles In the figure above, line is parallel to line Angles a, c, e, and g are obtuse, so they are all equal Angles b, d, f, and h are acute, so they are all equal Furthermore, any of the acute angles is supplementary to any of the obtuse angles Angles a and h are supplementary, as are b and e, c and f, and so on 489 TRIANGLES—GENERAL 79 Interior and Exterior Angles of a Triangle The three angles of any triangle add up to 180 degrees In the figure above, x + 50 + 100 = 180, so x = 30 An exterior angle of a triangle is equal to the sum of the remote interior angles In the figure above, the exterior angle labeled x° is equal to the sum of the remote angles: x = 50 + 100 = 150 The three exterior angles of a triangle add up to 360 degrees In the figure above, a + b + c = 360 80 Similar Triangles Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional 490 The triangles above are similar because they have the same angles The side of length corresponds to the side of length 4, and the side of length corresponds to the side of length s 81 Area of a Triangle The height is the perpendicular distance between the side that’s chosen as the base and the opposite vertex In the triangle above, is the height when is chosen as the base 82 Triangle Inequality Theorem The length of one side of a triangle must be greater than the difference between and less than the sum of the lengths of the other two sides For example, if it is given that the length of one side is and the length of another side is 7, then you know that the length of the third side must be greater than − = and less than + = 10 83 Isosceles and Equilateral Triangles An isosceles triangle is a triangle that has two equal sides Not only are two sides equal, but the angles opposite the equal sides, called base angles, are also equal Equilateral triangles are triangles in which all three sides are equal Since all the sides are equal, all the angles are also equal All three angles in an equilateral triangle measure 60 degrees, regardless of the lengths of the sides 491 RIGHT TRIANGLES 84 Pythagorean Theorem For all right triangles: (leg1)2 + (leg2)2 = (hypotenuse)2 If one leg is and the other leg is 3, then: 85 The 3-4-5 Triangle If a right triangle’s leg-to-leg ratio is 3:4, or if the leg-to-hypotenuse ratio is 3:5 or 4:5, it’s a 3-4-5 triangle, and you don’t need to use the Pythagorean theorem to find the third side Just figure out what multiple of 3-4-5 it is In the right triangle shown, one leg is 30, and the hypotenuse is 50 This is 10 times 3-4-5 The other leg is 40 86 The 5-12-13 Triangle If a right triangle’s leg-to-leg ratio is 5:12, or if the leg-to-hypotenuse ratio is 5:13 or 12:13, then it’s a 5-12-13 triangle, and you don’t need to use the Pythagorean theorem to find the third side Just figure out what multiple of 5-12-13 it is 492 Here one leg is 36, and the hypotenuse is 39 This is times 5-12-13 The other leg is 15 87 The 30-60-90 Triangle The sides of a 30-60-90 triangle are in a ratio of theorem You don’t need the Pythagorean If the hypotenuse is 6, then the shorter leg is half that, or 3, and then the longer leg is equal to the short leg times or 88 The 45-45-90 Triangle The sides of a 45-45-90 triangle are in a ratio of If one leg has a length of 3, then the other leg also has a length of 3, and the hypotenuse is equal to a leg times or 493 OTHER POLYGONS 89 Characteristics of a Rectangle A rectangle is a four-sided figure with four right angles Opposite sides are equal Diagonals are equal Quadrilateral ABCD above is shown to have three right angles The fourth angle therefore also measures 90 degrees, and ABCD is a rectangle The perimeter of a rectangle is equal to the sum of the lengths of the four sides, which is equivalent to 2(length + width) Area of Rectangle = length × width The area of a 7-by-3 rectangle is × = 21 90 Characteristics of a Parallelogram A parallelogram has two pairs of parallel sides Opposite sides are equal Opposite angles are equal Consecutive angles add up to 180 degrees In the figure above, s is the length of the side opposite the 3, so s = Area of Parallelogram = base × height In parallelogram KLMN above, is the height when LM or KN is used as the base 494 Base × height = × = 24 91 Characteristics of a Square A square is a rectangle with four equal sides If PQRS is a square, all sides are the same length as QR The perimeter of a square is equal to four times the length of one side Area of Square = (side)2 The square above, with sides of length 5, has an area of 52 = 25 92 Interior Angles of a Polygon The sum of the measures of the interior angles of a polygon = (n − 2) × 180, where n is the number of sides Sum of the Angles = (n − 2) × 180 The eight angles of an octagon, for example, add up to (8 − 2) × 180 = 1,080 495 CIRCLES 93 Circumference of a Circle Circumference = 2πr In the circle above, the radius has a length of 3, so the circumference is 2π(3) = 6π 94 Length of an Arc An arc is a piece of the circumference If n is the degree measure of the arc’s central angle, then the formula is In the figure above, the radius has a length of 5, and the measure of the central angle is 72 degrees The arc length is of the circumference: 95 Area of a Circle Area of a Circle = πr2 496 The area of the circle is π(4)2 = 16π 96 Area of a Sector A sector is a piece of the area of a circle If n is the degree measure of the sector’s central angle, then the formula is In the figure above, the radius has a length of 6, and the measure of the sector’s central angle is 30 degrees The sector has of the area of the circle: 97 Tangency When a line is tangent to a circle, the radius of the circle is perpendicular to the line at the point of contact 497 SOLIDS 98 Surface Area of a Rectangular Solid The surface of a rectangular solid consists of three pairs of identical faces To find the surface area, find the area of each face and add them up If the length is l, the width is w, and the height is h, the formula is Surface Area = 2lw + 2wh + 2lh The surface area of the box above is: (2 × × 3) + (2 × × 4) + (2 × × 4) = 42 + 24 + 56 = 122 99 Volume of a Rectangular Solid Volume of a Rectangular Solid = lwh The volume of a 4-by-5-by-6 box is × × = 120 A cube is a rectangular solid with length, width, and height all equal If s is the length of an edge of a cube, the volume formula is Volume of a Cube = s3 498 The volume of this cube is 23 = 100 Volume of a Cylinder Volume of a Cylinder = πr2h In the cylinder above, r = 2, h = 5, so Volume = π(22)(5) = 20π 499 SAT® is a trademark registered and/or owned by the College Board, which was not involved in the production of, and does not endorse, this product This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold with the understanding that the publisher is not engaged in rendering legal, accounting, or other professional service If legal advice or other expert assistance is required, the services of a competent professional should be sought © 2017 by Kaplan, Inc Published by Kaplan Publishing, a division of Kaplan, Inc 750 Third Avenue New York, NY 10017 All rights reserved under International and Pan-American Copyright Conventions By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this eBook on screen No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher 500 ... Contents SAT? ? Subject Test: Mathematics Level Cover Title Page Part One: The Basics Chapter 1: Getting Ready for SAT Subject Test: Mathematics Understand the SAT Subject Tests Content of the SAT Subject. .. BASICS Chapter Getting Ready for SAT Subject Test: Mathematics Understand the SAT subject tests Content of the SAT Subject Test: Mathematics Finding your level Level of difficulty and scoring... THE SAT SUBJECT TESTS ARE HIGHLY PREDICTABLE Because the format and directions of the SAT subject tests remain unchanged from test to test, you can learn the tests’ setups in advance On test