To the Student This Study Guide and Intervention Workbook gives you additional examples and problems for the concept exercises in each lesson. The exercises are designed to aid your study of mathematics by reinforcing important mathematical skills needed to succeed in the everyday world. The materials are organized by chapter and lesson, with one Study Guide and Intervention worksheet for every lesson in IMPACT Mathematics, Course 3. Always keep your workbook handy. Along with your textbook, daily homework, and class notes, the completed Study Guide and Intervention Workbook can help you in reviewing for quizzes and tests. To the Teacher These worksheets are the same ones found in the Chapter Resource Masters for IMPACT Mathematics, Course 3. The answers to these worksheets are available at the end of each Chapter Resource Masters Booklet. Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 ISBN: 978-0-07-891165-1 MHID: 0-07-891165-6 Study Guide and Intervention Workbook, IMPACT Mathematics, Course 3 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 009 14 13 12 11 10 09 08 Table of Contents Lesson/Title Page 1-1 Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . 1 1-2 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1-3 Write Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3 2-1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2-2 Angle Relationships . . . . . . . . . . . . . . . . . . . . . . . 5 2-3 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3-1 Understand Percents . . . . . . . . . . . . . . . . . . . . . . 7 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3-2 Work with Percents . . . . . . . . . . . . . . . . . . . . . . . 8 4-1 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4-2 Exponential Relationships . . . . . . . . . . . . . . . . . . . 10 4-3 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5-1 Rearrange Algebraic Expressions . . . . . . . . . . . . . . . 12 5-2 Monomials, Binomials, and Trinomials . . . . . . . . . . . . . . . . . . . . . . . . . 13 5-3 Special Products . . . . . . . . . . . . . . . . . . . . . . . . 14 6-1 Symmetry and Reflection . . . . . . . . . . . . . . . . . . . 15 6-2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6-3 Translations, Dilations, and Combined Transformations . . . . . . . . . . . . . . . . . . 17 7-1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7-2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7-3 Solve Systems of Equations . . . . . . . . . . . . . . . . . . 20 8-1 Use Graphs and Tables to Solve Equations . . . . . . . . . . . . . . . . . . . . . . . . 21 8-2 Quadratic Relationships . . . . . . . . . . . . . . . . . . . . 22 8-3 Families of Quadratrics . . . . . . . . . . . . . . . . . . . . 23 8-4 Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . 24 8-5 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . 25 9-1 Backtracking . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9-2 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 9-3 Completing the Square . . . . . . . . . . . . . . . . . . . . 28 9-4 The Quadratic Formula . . . . . . . . . . . . . . . . . . . . 29 10-1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . 31 11-1 Counting Strategies . . . . . . . . . . . . . . . . . . . . . 32 11-2 Modeling with Data . . . . . . . . . . . . . . . . . . . . . 33 12-1 Work with Algebraic Fractions . . . . . . . . . . . . . . . . 34 12-2 Add and Subtract Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 iv Chapter 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 10-2 Name Date Lesson 1.1 Study Guide and Intervention Direct Variation Relationships that have straight-line graphs are called linear relationships. In a linear relationship, the change in y and the change in x form a ratio that is constant. A special type of linear equation is called a direct variation. The graph of a direct variation always passes through the origin and can be expressed as y = kx. The k is called the constant of variation. When x doubles in value, y doubles in value, too. Example 1 A photo printer prints a 3-inch-by-5-inch picture in 3 minutes. a. Complete the table to show the number of minutes required to print p photos. Two photos take 6 minutes, and 3 photos take 9 minutes. No. of Minutes p m 0 0 1 3 2 ? 3 ? c. Write an equation that shows the number of minutes m required to print p photos. The linear relationship between minutes and number of photos is proportional, so we can use the direct variation equation y = kx, or m = kp. The constant of variation is the number of minutes to make 1 photo. So k = 3, and the equation is m = 3p. Exercises For Exercises 1–2, refer to the graph at the right. 1. Determine whether a proportional linear relationship exists between the two quantities. 2. Write an equation that relates the number of shares and the total cost. Cost of Shares 120 Total Cost ($) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. As the number of photos printed increases by 1, does the total number of minutes always increase by a constant number? Explain. Yes, each photo adds 3 more minutes. No. of Photos 100 80 60 40 20 0 1 2 3 4 5 6 Number of Shares IMPACT Mathematics, Course 3 1 Name Date Lesson 1.2 Study Guide and Intervention Slope The slope of a line is the ratio of the rise, or vertical change, to the run, or horizontal change. Find the slope of the line in the graph. Example 1 Choose two points on the line. The vertical change from pointA to point B is 4 units while the horizontal change is 2 units. rise slope = _ run Definition of slope =_ The rise is 4, and the run is 2. =2 The slope of the line is 2. 4 2 run: 2 y 7 6 B 5 4 4 rise: 3 2 A 1 O 1 2 3 4 5 6 7x The points in the table lie on a line. Find the slope of the line. Example 2 +3 +3 +3 x -2 1 4 7 y 5 1 -3 -7 -4 _4 _4 slope = _ run = - or rise 3 3 -4 4 The slope of the line is -_. 3 Find the slope of each line. Exercises 1. 2. y x O 3. y y x O x O The points given in each table lie on a line. Find the slope of the line. 4. 2 x 3 5 7 9 y -1 2 5 8 Chapter 1 5. x -5 0 5 10 y 4 3 2 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -4 Name Date Lesson 1.3 Study Guide and Intervention Write Equations Linear equations are often written in the form y = mx + b. This is called the slope-intercept form. When an equation is written in this form, m is the slope and b is the y-intercept. Example 1 State the slope and y-intercept of the graph of y = x - 3. y=x-3 Write the original equation. y = 1x + (-3) Write the equation in the form y = mx + b. ↑ ↑ y = mx + b m = 1, b = -3 The slope of the graph is 1, and the y-intercept is -3. You can use the slope-intercept form of an equation to graph the equation. Example 2 Graph y = 2x + 1 using the slope and y-intercept. Step 1 Find the slope, 2, and y-intercept, 1. Graph the y-intercept at (0, 1). 2 Write the slope 2 as _ . Use it to 1 locate a second point on the line. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 2 Step 3 2 m=_ 1 Step 4 y 4 3 2 up 2 1 -4 -3 -2 ← change in y: up 2 units ← change in x: right 1 unit right 1 y = 2x = 1 1 2 3 4x O -2 -3 -4 Draw a line through the two points. Exercises State the slope and y-intercept of the graph of each equation. 1. y = x + 1 3. y = _ x-1 2 1 2. y = 2x - 4 Graph each equation using the slope and y-intercept. 4. y = 2x + 2 1 y y y O 6. y = _ x+2 2 5. y = x - 1 x O x O x IMPACT Mathematics, Course 3 3 Name Date Lesson 2.1 Study Guide and Intervention Lines As you learned in the last chapter, linear equations are often written in the form y = mx + b. This is called the slope-intercept form. When an equation is written in this form, m is the slope and b is the y-intercept. Example 1 State the slope and y-intercept of the graph of y = 2x - 5. y = 2x - 5 Write the original equation. y = 2x + (-5) Write the equation in the form y = mx + b. ↑ ↑ y = mx + b m = 2, b = -5 The slope of the graph is 2, and the y-intercept is -5. You can use the slope-intercept form of an equation to graph the equation. Example 2 a. State whether the equation is linear. If it is linear, identify the values of m and b. If an equation is nonlinear, explain how you know. 3x + y = 6 Step 1 b. y = x(x + 3) - 7 Step 1 y = x2 + 3x - 7 Step 2 The equation is not linear because x is squared. Exercises State the slope and y-intercept of the graph of each equation. 1. y = 3x - 2 2. y = x + 4 3. y = _ x-1 3 1 State whether the equation is linear. If it is linear, identify the values of m and b. If an equation is not linear, explain how you know. 4. 2y = _ - 1 x 3 4 Chapter 2 5. y = 6x + 4(x - 5) 6. x(x + 1) = y - 2(x + 6) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 2 Put in slope-intercept form. -3x + 3x + y = -3x + 6 y = -3x + 6 The equation is linear. m= -3, b = 6 Name Date Lesson 2.2 Study Guide and Intervention Angle Relationships The relationship between pairs of angles can be used to find missing measures. Example 1 Find the value of x in the figure at the right. The two angles are supplementary, so their sum is 180°. x + 35 = 180 Write an equation. x - 35 + 35 = 180 - 35 Subtract 35 from each side. x = 145 Simplify. Examples x˚ 35˚ Use the figure at the right. 2. Find m∠3 if m∠7 = 70°. ∠3 and ∠7 are corresponding angles. Since corresponding angles are congruent, their measures are the same. m∠3 = m∠7, so m∠3 = 70°. 5 6 8 3 1 3. Find m∠4 if m∠5 = 120°. 4 2 ∠4 and ∠5 are alternate interior angles. Since alternate interior angles are congruent, their measures are the same. m∠4 = m∠5, so m∠4 = 120°. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 7 Exercises Find the value of x in each figure. 1. 2. x˚ 58˚ 3. 56˚ 67˚ x˚ x˚ For Exercises 4 –8, use the figure at the right. 4. Find m∠5 if m∠3 = 110°. 5. Find m∠2 if m∠6 = 75°. 6. Find m∠1 if m∠7 = 94°. 7. Find m∠8 if m∠4 = 68°. 4 8 5 7 1 3 2 6 8. Find m∠5 if m∠6 = 71°. IMPACT Mathematics, Course 3 5 Name Date Lesson 2.3 Study Guide and Intervention Constructions Example 1 Construct the perpendicular bisector of the line segment AB. " # Step 1: Draw a circle, or an arc of a circle, about point A with radius more than half the distance between A and B. Step 2: Draw a congruent circle, or arc, about B. " # " # Step 3: Draw a line through the two points of intersection of the circles. This is the perpendicular bisector of segment AB. Example 2 Bisect angle A. " Step 1: Draw an arc through the angle with the compass point on A. where the first arc intersects the sides of the angle. The two new arcs must have the same radius. # " $ Step 3: Draw a line from A through the point where the last two arcs intersect. This is the bisector of angle A. # " $ Exercises 1. Construct the perpendicular bisector of segment XY. 2. Construct the bisector of angle W. : 8 9 6 Chapter 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 2: Draw arcs from points B and C, Name Date Lesson 3.1 Study Guide and Intervention Understand Percents You can use the percent proportion to find the percent. part percent a n _ =_ or _ = _ 100 100 b base You can also use the percent proportion to find a missing part or base. Example 1 12 is what percent of 60? n 12 n _a = _ _=_ 100 b 60 Replace a with 12 and b with 60. 100 12 · 100 = 60 · n Find the cross products. 1,200 = 60n Multiply. 1,200 60n _ =_ 60 60 Divide each side by 60. 20 = n Example 2 12 is 20% of 60. What number is 40% of 55? n a 40 _a = _ _=_ b 100 55 100 a · 100 = 55 · 40 a = 22 Replace n with 40 and b with 55. Find the cross products. Use similar steps to solve for a. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. So, 22 is 40% of 55. Exercises Write a percent proportion to solve each problem. Then solve. Round to the nearest tenth if necessary. 1. 3 is what percent of 10? 2. What number is 15% of 40? 3. 24 is 75% of what number? 4. 86 is what percent of 200? 5. What number is 65% of 120? 6. 52 is 13% of what number? 7. 35 is what percent of 56? 8. What number is 12.5% of 88? 9. 161 is 92% of what number? 11. What number is 31.5% of 200? 10. 45 is what percent of 66? 12. 81 is 54% of what number? IMPACT Mathematics, Course 3 7 Name Date Lesson 3.2 Study Guide and Intervention Work with Percents To find the percent of increase or decrease, first find the amount of the increase or decrease. Then find the ratio of that amount to the original amount, and express it as a percent. Two months ago, the bicycle shop sold 50 bicycles. Last month, 55 bicycles were sold. Find the percent of change. State whether the percent of change is an increase or a decrease. Example Example Step 1 Subtract to find the amount of change. 55 - 50 = 5 Step 2 Write a ratio that compares the amount of change to the original number of bicycles. Express the ratio as a percent. amount of change original amount 5 =_ 50 percent of change = __ The amount of change is 5. The original amount is 50. Divide. Write as a percent. The percent of change is 10%. Since the new amount is greater than the original, it is a percent of increase. Exercises Find each percent of change. Round to the nearest tenth of a percent if necessary. State whether the percent of change is an increase or a decrease. 8 1. Original: 4 New: 5 2. Original: 10 New: 13 3. Original: 15 New: 12 4. Original: 30 New: 18 5. Original: 60 New: 63 6. Original: 160 New: 136 7. Original: 77 New: 105 8. Original: 96 New: 59 Chapter 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. = 0.1 or 10% Definition of percent of change Name Date Lesson 4.1 Study Guide and Intervention Exponents Expressions containing repeated factors can be written using exponents. Example 1 Write p . p . p . q . q using exponents. Since p is used as a factor 3 times and q is used as a factor 2 times, p · p · p · q · q = p3 · q2. Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is 1 divided by the number to the nth power. Example 2 Evaluate 62. 62 = 6 · 6 Definition of negative exponents 1 =_ 125 Simplify. Simplify. Simplify 22 . 23. 22 · 23 = 2 · 2 · 2 · 2 · 2 = 25 = 32 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Evaluate 5–3. 1 5-3 = _3 5 Definition of exponents = 36 Example 4 Example 3 Definition of exponents Simplify. Exercises Write each expression using exponents. 1. 8 · 8 · 8 · 8 · 8 2. 4 · 4 · 4 · 4 Evaluate each expression. 3. 53 4. 23 · 32 5. 25 · 23 6. 3-4 Write each expression using a positive exponent. 7. 6–4 8. (–7)–8 9. b–6 10. n–1 Simplify. 11. a2 · a4 12. b–7 · b9 13. (2a)(3a3) 14. (–5x2)(4x3) IMPACT Mathematics, Course 3 9 Name Date Lesson 4.2 Study Guide and Intervention Exponential Relationships Quantities that are repeatedly multiplied by a number greater than 1 are said to grow exponentially, or to show exponential increase, or exponential growth. Fish Marcus has three fish in an aquarium. Each week the number of fish in the aquarium doubles. The chart shows the number of fish Marcus has after each week. Example 1 Number of weeks 0 1 2 3 4 Number of fish 3 6 12 24 48 By what number do you multiply to get from one number of fish to the next? 3 · 2 = 6; 6 · 2 = 12; 12 · 2 = 24; 24 · 2 = 48 You multiply by 2 each time. Quantities that are repeatedly multiplied by a positive number less than 1 are said to decrease exponentially, or to show exponential decrease, or exponential decay. The equation y = 5x represents exponential growth. The growth factor is 5. Example 2 Exercises 1. For a science experiment, Tanya put one amoeba in a dish. Each day she counted and recorded in a table the number of amoebas in the dish. Day 0 1 2 3 Bacteria 1 6 36 216 a. How does the number of amoebas change each day? b. Which expression describes the number of amoebas, 6x, 6x, or x6? c. How many amoebas will be in the dish on Day 6? 2. Suppose you have 100 milligrams of medicine in your blood stream. 1 Every hour, _ of the medicine is eliminated. 4 a. How many milligrams of medicine remain after 1 hour? After n hours? Explain your reasoning. b. Is this an exponential growth or exponential decay situation? 10 Chapter 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The equation y = 10 · 0.5x represents exponential decay. The decay factor is 0.5. Name Date Lesson 4.3 Study Guide and Intervention Radicals The Square root of ... Examples Find each square root. 1. √ 1 Since 1 · 1 = 1, √ 1 = 1. 2. - √ 16 Since 4 · 4 = 16, - √ 16 = -4. 3. √ 0.25 Since 0.5 · 0.5 = 0.25, √ 0.25 = 0.5. 4. 25 5 5 25 5 25 Since _ · _ = _, _ = _. 6 6 36 36 6 36 √_ Example 5 √ Simplify √ 180 . Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 180 = √ 2·2·3·3·5 √ Simplify √ 196 . Example 6 Prime factorization √ 196 = √ 2·2·7·7 2 · √3 2 · √ = √2 5 of 180 Product Property of Square Roots = √ 22 · √ 72 = 2 · 3 · √ 5 Simplify. = 14. = 6 √ 5 Simplify. Example 7 =2·7 6 Find √ 64 . Example 8 3 Find √ 27 . Since 2 · 2 · 2 · 2 · 2 · 2 6 = 64, √ 64 = 2. 3 Since 3 · 3 · 3 = 27, √ 27 = 3. Exercises 1. √ 4 2. √ 9 3. - √ 49 0.01 5. √ 6. - √ 0.64 7. 9 _ 16 √ 4. - √ 25 8. - 1 √_ 25 Simplify. (28) 9. √ 10. (12) √ 11. √ (18) 12. – √ (50) (9x4) 13. √ 14. (100x3y) √ 4 2 15. √(24a b) 16. (18m3) √ IMPACT Mathematics, Course 3 11 Name Date Lesson 5.1 Study Guide and Intervention Rearrange Algebraic Expressions Terms that contain the same variables raised to the same powers are called like terms. Example 1 Simplify 6x - 5 - 2x + 7. 6x - 5 - 2x + 7 = 6x + (-5) + (-2x) + 7 Definition of subtraction = 6x + (-2x) + (-5) + 7 Commutative Property = [6 + (-2)]x + (-5) + 7 Distributive Property = 4x + 2 Example 2 Simplify Simplify 4(x + 1) - 5(2x - 3). 4(x + 1) - 5(2x - 3) = 4x + 4 - 10x + 15 Distributive = 4x - 10x + 4 + 15 Property Commutative Property Distributive Property = -6x + 19 Simplify Exercises Simplify each expression. 1. 9m + 3m 2. 5x - x 3. 8y + 2y + 3y 4. 2 + 6a + 4a 5. m + 4m + 2m + 5 6. 3c + 4d - c + 2 7. 5h - 3g + 2g - h 8. 3w + 4u - 6 9. 4r - 5s + 5s - 2r 10. 4 + m - 3m 11. 13a + 7a + 2a 12. 3y + 1 + 5 + 4y 13. 8d - 4 - d + 5 14. 10 - 4s + 2s - 3 15. 2y + 7 - (3y - 5) + y 12 Chapter 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. = (4 - 10)x + 4 + 15 Name Date Lesson 5.2 Study Guide and Intervention Monomials, Binomials, and Trinomials Multiply 3x(2x + 6). Example 1 3x(2x + 6) = 3x(2x) + 3x(6) = 6x2 + 18x Distributive property Add exponents when base is the same. To multiply two binomials, apply the Distributive Property twice. A useful way to keep track of terms in the product is to use the FOIL method, as illustrated in Example 3. Find (x + 3)(x -4) using the horizontal Example 2 Example 3 method. Find (x - 2)(x + 5) using the FOIL method. (x - 2)(x + 5) (x + 3)(x - 4) First = x(x - 4) + 3(x - 4) Inner Last = (x)(x) + (x)(5) + (-2)(x) + (-2)(5) = x2 + 5x + (-2x) - 10 = x2 - 4x + 3x - 12 = x2 - x - 12 = x2 + 3x - 10 Divide _ . 2 2x + 6 Example 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Outer 2(x + 3) 2x + 6 _ =_ 2 2 =x+3 Distributive property Simplify Exercises Multiply or divide. 1. 2a5 · 6a 2. -3t3 · 2t8 3. 4x2(-5x6) 4. (2w)(3w) 5. a(2a + 3) 6. 5x(4x – 5) 8x + 4 7. _ 4 4x2 - 10x 8. _ 2x 10x + 25x2 9. _ 5x 10. (x + 3)(x + 4) 11. (x – 5)(x + 2) 12. (x – 6)(x – 4) IMPACT Mathematics, Course 3 13 Name Date Lesson 5.3 Study Guide and Intervention Special Products Perfect Square Trinomials Some pairs of binomials have products that follow specific patterns. One such pattern is called the square of a sum. Another is called the square of a difference. (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 (a - b)2 = (a - b)(a - b) = a2 - 2ab + b2 Square of a sum Square of a difference Example 1 Find (3x + 4)(3x + 4). Example 2 Find (2z - 9)(2z - 9). Use the square-of-a-sum pattern with a = 3x and b = 4. Use the square-of-a-difference pattern with a = 2z and b = 9. (3x + 4)(3x + 4) (2z - 9)(2z - 9) = (3x)2 + 2(3x)(4) + (4)2 = (2z)2 - 2(2z)(9) + (9)2 = 9x2 + 24x + 16 = 4z2 - 36z + 81 The product is 9x2 + 24x + 16. The product is 4z2 - 36z + 81. There is also a pattern for the product of the sum and the difference of the same two terms. The product is called the difference of squares. Example 3 + b)(a - b) = a2 - b2 Find (5x + 3y)(5x - 3y). (a + b)(a - b) = a2 - b2 Product of a sum and a difference (5x + 3y)(5x - 3y) = (5x)2 - (3y)2 = 25x2 - 9y2 a = 5x and b = 3y Simplify. The product is 25x2 - 9y2. Exercises Find each product. 1. (x - 6)2 2. (3p + 4)2 3. (4x - 5)2 4. (2x - 1)2 5. (2h + 3)2 6. (m + 5)2 7. (x - 4)(x + 4) 8. (p + 2)( p - 2) 9. (4x - 5)(4x + 5) 14 Chapter 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. (a Difference of squares Name Date Lesson 6.1 Study Guide and Intervention Symmetry and Reflection A figure has line symmetry if it can be folded over a line so that one half of the figure matches the other half. This fold line is called the line of symmetry. Some figures have more than one line of symmetry. Example 1 Determine whether the figure has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none. This figure has three lines of symmetry. Example 2 Draw the image of quadrilateral ABCD after a reflection over the given line. Step 1 Count the number of units between A each vertex and the line of reflection. B 1C D 1 Step 2 To find the corresponding point for Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3 3 A vertex A, move along the line through A perpendicular to the line of reflection until you are 3 units from the line on the opposite side. Draw a point and label it A´. Repeat for each vertex. B D 1 1 3 A Step 3 Connect the new vertices to form quadrilateral A´B´C´D´. C 3 B D C D' C' A' B' Exercises For Exercise 1, draw all lines of symmetry or write none. For Exercises 2–3, draw the image after a reflection over the given line. 1. 2. 3. # $ ( " ' * ) % IMPACT Mathematics, Course 3 15 Name Date Lesson 6.2 Study Guide and Intervention Rotation A figure has rotational symmetry if it can be rotated or turned less than 360° about its center so that the figure looks exactly as it does in its original position. The degree measure of the angle through which the figure is rotated is called the angle of rotation. Example Determine whether the figure has rotational symmetry. Write yes or no. If yes, name its angles of rotation. Yes, this figure has rotational symmetry. It matches itself after being rotated 180°. Exercises For Exercises 1–9, determine whether the figure has rotational symmetry. Write yes or no. If yes, name its angles of rotation. 2. 3. 4. 5. 6. 7. 8. 9. 16 Chapter 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. Name Date Lesson 6.3 Study Guide and Intervention Translations, Dilations, and Combined Transformations When a figure is translated, every point is moved the same distance in the same direction. The translated figure is congruent to the original figure and has the same orientation. Example 1 Translate the figure using the given vector. Step 1 Move each vertex the length and direction of the vector. Step 2 Connect the new vertices to form the translated figure. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A dilation creates a figure that is similar but not necessarily congruent to the original figure. The corresponding sides are proportional. This ratio is called the scale factor. Example 2 Dilate the smaller figure by a scale factor of 2. y Step 1 Multiply the coordinates of the vertices by 2: (6, 0), (2, 8), (8, 6), and (12, 10). Step 2 Graph and connect the new points. 0 x Exercises 1. Translate the figure using the given vector. IMPACT Mathematics, Course 3 17 Name Date Lesson 7.1 Study Guide and Intervention Equations Work Backward Working backward is one of many problem-solving strategies that you can use to solve problems. To work backward, start with the result given at the end of a problem and undo each step to arrive at the beginning number. ( ) 4x Solve 3 _ + 1 = 15 by backtracking. 3 Example 1 Think of x as the input and 15 as the output. Make a flowchart to show the operations needed to get from the input to the output. X ·4 ÷3 +1 ·3 Input 15 Output To backtrack, start from the output and work backwards, undoing each operation, until you find the input. The solution is 3. 3 ·4 12 4 +1 ·3 ·3 5 -1 15 ÷3 Solve the equation 12x - 3 = 4x + 13 by doing the same thing to both sides. Example 2 12x - 3 = 4x + 13 Write the equation. 12x - 4x - 3 = 4x - 4x + 13 8x - 3 = 13 8x - 3 + 3 = 13 + 3 8x = 16 x=2 Exercises Subtract 4x from each side. Simplify. Add 3 to each side. Simplify. Mentally divide each side by 8. Solve each equation by backtracking. 1. 8(g – 3) = 24 2. 5(x + 3) = 25 ( 2c ) 3. 7 _ -5 =7 3 Solve each equation by doing the same thing to both sides. 4. 2x + 1 = x + 11 18 Chapter 7 5. a + 2 = 5 + 4a 6. 7y + 25 = 2y Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ÷4 ÷3 Name Date Lesson 7.2 Study Guide and Intervention Inequalities An inequality is a mathematical sentence that contains one of the symbols <, >, ≤, or ≥. Words Symbols m is greater than 7. m>7 r is less than -4. t is greater than or equal to 6. y is less than or equal to 1. Example 1 v+3< 5 - 3 = -3 v < 2 t≥6 y≤1 Solve v + 3 < 5. Then graph the solution. Write the inequality. Subtract 3 from each side. Simplify. The solution is all numbers less than 2. Example 2 r < -4 -4 -3 -2 -1 0 1 2 3 4 Graph y ≤ -3x - 2 y Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Graph the equation y = -3x - 2. Since y ≤ -3x - 2 is the same as “y < -3x - 2 and y = -3x - 2,” the boundary is included in the solution set and the graph should be drawn as a solid line. O x Select a point in each half-plane and test it. Choose (0, 0) and (-2, -2). y ≤ -3x - 2 y ≤ -3x - 2 0 ≤ -3(0) - 2 -2 ≤ -3(-2) - 2 0 ≤ -2 is false. -2 ≤ 4 is true. The half-plane that contains (-2, -2) contains the solution. Shade that half-plane. Exercises Graph each inequality on the number line or grid. 1. c < 5 1 3. y < - _ x-3 2 y -2 -1 0 1 2 3 4 5 6 7 8 2. x ≥ 10 O x 7 8 9 10 11 12 13 14 15 16 17 IMPACT Mathematics, Course 3 19 Name Date Lesson 7.3 Study Guide and Intervention Solve Systems of Equations A set of two or more equations is called a system of equations. Solving a system of equations means finding an ordered pair that is a solution of all the equations. You can solve a system of equations by graphing. If you graph the equations on the same coordinate plane, the point where the graphs intersect is the solution of the system of equations. Example 1 Solve the system y = x - 1 and y = -2x + 5 by graphing. Both equations are in slope-intercept form. Use the slope and y-intercept of each equation to graph the two equations. The graphs appear to intersect at (2, 1). Check this by substituting the coordinates into each equation. The solution of the system of equations is (2, 1). y y = -2x + 5 (2, 1) y=x-1 0 x Another way of solving a system of equations is by substitution. And a third way of solving a system of equations is elimination. Example 2 Use substitution to solve the system of equations. y = 2x 4x - y = -4 Use elimination to solve the system of equations. x - 3y = 7 3x + 3y = 9 4x - y = -4 4x - 2x = -4 2x = -4 x = -2 y = 2x y = 2(-2) y = -4 x - 3y = 7 (+) 3x + 3y = 9 Add to eliminate y. 4x = 16 Divide by 4. x=4 Solve for x. x - 3y = 7 First equation. 4 - 3y = 7 Substitute 4 for x. y = -1 Solve for y. The solution is (4, -1). The solution is (-2, -4). Use elimination to solve each system of equations. 1. x + y = -4 x-y=2 2. 2x + 2y = -2 3x - 2y = 12 Use substitution to solve each system of equations. 3. y = 4x 3x - y = 1 20 Chapter 7 4. x = 2y -1 x -3y = -4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Second equation Substitute 2x for y. Combine like terms. Divide by 2 and simplify. First equation Substitute -2 for x. Simplify. Example 3 Name Date Lesson 8.1 Study Guide and Intervention Use Graphs and Tables to Solve Equations If an object is tossed or projected upward, the formula h = vt - 16t2 approximates the object’s height in feet h above its starting point after t seconds when projected at an initial velocity v in feet per second. Example Suppose a tennis ball bounces upward at an initial velocity of 32 feet per second from the ground. The height of the ball is given by the equation h = 32t – 16t2. a. How high is the ball after one second? h = 32(1) - 16(1)2 Substitute 1 for t. h = 32 - 16 = 16 feet Simplify. b. What is the maximum height the tennis ball reaches? After how much time does it reach this height? Graph y = 32x - 16x2 on your calculator. Height (ft) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 1: 16 14 12 10 8 6 4 2 0 y x 1 2 Time (s) Step 2: Find the coordinates of the highest point on the graph. The highest point is (1, 16). The maximum height is 16 feet. This occurs after 1 second. Exercises Use the example for Exercises 1–4. 1. How high is the tennis ball after 0.5 seconds? 2. How long does it take the ball to hit the ground? 3. Estimate the time it takes for the ball to first reach a height of 8 feet. 4. How could you use the graph to estimate the solution of 32t - 16t2 = 4? IMPACT Mathematics, Course 3 21 Name Date Lesson 8.2 Study Guide and Intervention Quadratic Relationships A simple quadratic equation is of the form y = ax2. The graph is a symmetric U-shaped curve called a parabola. Example 1 Example 2 The equation y = 2x2 is a quadratic equation. The graph is a parabola. The line of symmetry is the vertical axis. To draw the graph, complete the table and then plot the points. x y -2 8 -1 2 10 8 6 4 2 0 0 -3 -2-1 0 1 2 2 8 -4 -6 y 1 2 3 4 5x The table gives the area A of an n × n array of squares with one additional square. n 1 2 3 4 5 6 7 A 2 5 10 17 26 37 50 Exercises In Exercises 1–4, tell whether each equation is a quadratic equation. 1. y = 3x2 2. y = 2x + 1 3. y = 2x3 4. y = 0.5x2 Use the following patterns in Exercises 5–6. Stage 1 Stage 2 5. Write a formula for the area in square units A given n. 6. What is the area in square units when n = 4? 22 Chapter 8 Stage 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A formula for the area in square units of any figure following this pattern is A = n2 + 1. Name Date Lesson 8.3 Study Guide and Intervention Families of Quadratics Quadratic Relationship A relationship described by an equation of the form y = ax2 + bx + c, where a ≠ 0 Example: y = 2x2 + 3x + 8 The parent graph of the family of quadratic fuctions is y = x2. Graphs of quadratic relationships have a general shape called a parabola. A parabola opens upward and has a minimum point when the value of a positive. A parabola opens downward and has a maximum point when the value of a is negative. Example 1 Example 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Use a table of values to graph y = x2 - 4x + 1. Use a table of values to graph y = -x2 - 6x - 7. y x y -1 6 -6 -7 0 1 1 -2 -5 -2 -4 1 2 -3 3 -2 4 1 O x -3 -2 y y x O x 2 1 -1 -2 0 Graph the ordered pairs in the table and connect them with a smooth curve. -7 Graph the ordered pairs in the table and connect them with a smooth curve. Exercises Use a table of values to graph each equation. 1. y = x2 + 2 2. y = -x2 - 4 y y y O O 3. y = x2 - 3x + 2 x x O x IMPACT Mathematics, Course 3 23 Name Date Lesson 8.4 Study Guide and Intervention Inverse Variation When two variables have a constant nonzero product, they are said to be inversely proportional. A relationship in which two variables are inversely proportional is called an inverse variation. Example Suppose you drive 200 miles without stopping. The time it takes to travel a distance varies inversely as the rate at which you travel. Let x = speed in miles per hour and y = time in hours. Graph the variation. The equation xy = 200 can be used to represent the situation. Use various speeds to make a table. y x y 10 20 20 10 20 30 6.7 10 40 5 50 4 30 O 20 40 60 x 60 3.3 1. Complete the table for xy = 12. x -12 -6 -4 -3 -2 -1 1 2 3 4 6 y 2. Graph xy = 12. 10 8 6 4 2 3. Louisa wants to fence in a rectangular pen for her pigs. If the area A of the pen is to be 400 square feet, complete the table to show the possible lengths ℓ and widths w of the pen. Width (ft) 5 10 20 Length (ft) 4. Write an equation for the relationship in Exercise 3. 24 Chapter 8 -6 -4-2 0 y 2 4 6 8 10x -4 -6 40 50 12 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises Name Date Lesson 8.5 Study Guide and Intervention Conjectures A conjecture is an educated guess or generalization that has not yet been proven correct. Example Find the differences of the y values and the differences of the differences in this table for y = x2 + x + 1. Is the relationship linear, quadratic, or neither? x 1 2 3 4 5 6 y 3 7 13 21 31 43 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Differences of y - values: 4 Differences of differences: 6 2 8 2 10 2 12 2 The second differences are constant. In a linear relationship, the first differences are constant. In a quadratic relationship, the second differences are constant. Exercises Use the method of constant differences to make a conjecture about whether the relationship between x and y is linear, quadratic, or neither. Explain how you decide. 1. 3. x 1 2 y 4 10 18 28 40 54 x 1 2 3 4 y 4 2 -2 -8 3 4 5 2. 6 5 6 -16 -26 4. x 1 2 3 4 5 6 y -1 2 5 8 11 14 x 1 2 3 5 6 y 2 9 28 65 126 217 4 IMPACT Mathematics, Course 3 25 Name Date Lesson 9.1 Study Guide and Intervention Backtracking Example 1 Use backtracking to solve the equation √ 5x + 6 = 4. This is the flowchart. +6 ·5 x 4 Use backtracking as shown. ·5 2 10 +6 16 4 square -6 ÷5 The solution is x = 2. Example 2 Solve (a + 3)2 = 36 by backtracking. This is the flowchart. +3 a square 36 +3 -9 or 3 -6 or 6 square 36 -3 The solutions are a = 3 and a = –9. Exercises Solve each equation by backtracking. 5 1. _ x = 2.5 (4 + a) 2. _ = 5 3 3. √2x + 10 = 4 6–x 4. _ = 2 4 5. (c – 7)2 = 16 6. 2(y + 3)2 = 128 26 Chapter 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Use backtracking as shown. Name Date Lesson 9.2 Study Guide and Intervention Factoring To factor a trinomial of the form x2 + bx + c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. x2 + bx + c = (x + m)(x + n), where m + n = b and mn = c. Example 1 Factor each trinomial. a. x 2 + 7x + 10 In this trinomial, b = 7 and c = 10. Factors of 10 Sum of Factors 1, 10 11 2, 5 7 b. x2 - 8x + 7 In this trinomial, b = -8 and c = 7. Notice that m + n is negative and mn is positive, so m and n are both negative. Since -7 + (-1) = -8 and (-7)(-1) = 7, m = -7 and n = -1. x2 - 8x + 7 = (x - 7)(x - 1) Since 2 + 5 = 7 and 2 · 5 = 10, let m = 2 and n = 5. x2 + 7x + 10 = (x + 2)(x + 5) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. To solve an equation by factoring , make one side zero. Use the property that if ab = 0, then a = 0 or b = 0. Example 2 Solve x2 + 6x = 7. x2 + 6x = 7 x2 + 6x - 7 = 0 (x - 1) (x + 7) = 0 x -1 = 0 or x + 7 = 0 x=1 x=-7 Exercises Original equation Rewrite equation so that one side equals 0. Factor. Zero Product Property Solve the equation. Factor each trinomial. 1. x2 + 4x + 3 2. m2 + 12m + 32 3. r2 - 3r + 2 4. x2 - x - 6 5. x2 - 4x - 21 6. x2 - 22x + 121 Solve each equation by factoring. 7. y2 - 5y + 4 = 0 9. x2 = x + 2 11. p2 = 9p - 14 8. m2 + 10m + 9 = 0 10. x2 - 12x + 36 = 0 12. a2 = 11a - 18 IMPACT Mathematics, Course 3 27 Name Date Lesson 9.3 Study Guide and Intervention Completing the Square Since few quadratic expressions are factorable, the method of completing the square is better for solving some quadratic equations. Use the following steps to solve an equation by completing the square. Write the equation in the form x2 + bx + c = 0. 2 ( ) b Add and subtract ( _ ) , the same number, on the left side of the equation. 2 b Find the square root of x + bx + ( _ ) , and then solve for x. 2 b b Find _ and then _ . 2 2 2 2 2 Example Solve x2 + 6x – 7 = 0 by completing the square. x2 + 6x – 7 = 0 x2 + 6x + 9 – 16 = 0 (x + 3)2 – 16 = 0 (x + 3)2 = 16 x + 3 = ±4 x = –7, 1 Solve each equation by completing the square. 1. t2 – 4t + 3 = 0 2. y2 + 10y + 9 = 0 3. y2 – 8y – 9 = 0 4. x2 – 6x – 16 = 0 5. p2 – 4p – 5 = 0 6. x2 + 4x – 12 = 0 7. c2 + 8c – 20 = 0 8. p2 – 3p – 4 = 0 9. x2 + 20x + 19 = 0 10. x2 – 5x – 14 = 0 11. a2 = 22a + 23 12. m2 – 8m = –7 13. x2 + 10x = 24 14. a2 – 18a = 19 15. b2 + 6b = 16 16. 4x2 = 24 + 4x 17. m2 + 2m + 1 = 4 18. 4k2 = 40k + 44 28 Chapter 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises Original equation Add and subtract 3 2 on the left side. Factor x2 + 6x + 9. Add 16 to both sides. Take the square root of each side. Simplify. Name Date Lesson 9.4 Study Guide and Intervention The Quadratic Formula To solve the standard form of the quadratic equation, ax2 + bx + c = 0, use the Quadratic Formula. -b ± √ b2 - 4ac The formula x = __ gives the solutions of ax2 + bx + c = 0, where a ≠ 0. 2a Example 1 Solve x2 + 2x = 3 by using the Quadratic Formula. Rewrite the equation in standard form, subtract 3 from each side, and simplify. x2 + 2x = 3 x2 + 2x - 3 = 3 - 3 x2 + 2x - 3 = 0 Now let a = 1, b = 2, and c = -3 in the Quadratic Formula. Example 2 Solve x2 - 6x - 2 = 0 by using the Quadratic Formula. For this equation a = 1, b = -6, and c = -2. -b ± √ b2 - 4ac x = __ 2a 6 ± √ (-6)2 - 4(1)(-2) __ = 2(1) -b ± √ b2 - 4ac x = __ 2a 6 ± √ 44 =_ 2 2 -2 ± √(2) - 4(1) (-3) 2(1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. = __ = 3 + √ 11 or 3 - √ 11 -2 ± √ 16 =_ 2 -2 + 4 x=_ 2 =1 or ≈ 6.3 or -0.3 x=_ 2 -2 - 4 The solution set is {6.3, -0.3}. = -3 The solution set is {-3, 1}. Solve each equation by using the Quadratic Formula, Round to the nearest tenth if necessary. Exercises 1. x2 - 3x + 2 = 0 2. m2 - 8m = -16 3. 16r2 - 8r = -1 4. x2 + 5x = 6 5. 3x2 + 2x = 8 6. 8x2 - 8x - 5 = 0 7. -4c2 + 19c = 21 8. 2p2 + 6p = 5 9. 48x2 + 22x - 15 = 0 10. 8x2 - 4x = 24 11. 2p2 + 5p = 8 12. 8y2 + 9y - 4 = 0 13. 2x2 + 9x + 4 = 0 14. 8y2 + 17y + 2 = 0 15. 3z2 + 5z - 2 = 0 IMPACT Mathematics, Course 3 29 Name Date Lesson 10.1 Study Guide and Intervention Functions A function is a relationship between an input variable and an output variable in which there is only one output for each input. The set of allowable inputs to the function is called the domain of the function. Example 1 Determine whether the relation {(6, −3), (4, 1), (7, −2), ( −3, 1)} is a function. Explain. Example 2 Use the function machine. If the input is 5, find the output. Since each element of the domain is paired with exactly one output element, this relation is a function. +3 ×2 First 5 is multiplied by 2, giving 10. Then 3 is added. Since 10 + 3 = 13, 13 is the output. In the function rule f(x) = 2x – 1, the variable x represents the input, f is the name of the function, and f(x) represents the output. The symbol f(x) is read “f of x.” Example 3 If f(x) = 3x - 4, find each value. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a. f(3) b. f(-2) f (3) = 3(3) - 4 =9-4 =5 x=3 Multiply. Simplify. f (-2) = 3(-2) - 4 = -6 - 4 = -10 x = -2 Multiply. Simplify. Exercises Determine whether each relationship is a function. 1. {(4, 2), (2, 3), (6, 1)} 2. {(–3, –3), (–3, 4), (–2, 4)} Determine the output from each function machine. 3. The input is 2. +4 4. The input is 3. ×2 -7 +3 If f(x) = 2x - 4 and g(x) = x2 - 4x, find each value. 5. f(4) 6. g(2) 7. f(-5) IMPACT Mathematics, Course 3 30 Name Date Lesson 10.2 Study Guide and Intervention Graphs of Functions Example The graph of a quadratic equation in the form f(x) = (x – b)2 + c is a translation of the graph of f(x) = x2. The translation is located b units to the right and c units up. y Start with a graph of f(x) = x2. 8 Slide this graph to the right 4 units: f(x) = (x – 4)2 6 4 Then slide it up 3 units: f(x) = (x – 4)2 + 3 2 -4 -2 O 2 4 6 8x -2 All the possible output values of a function f are called the range of the function. For the function graphed above in Example 1, the range is f(x) ≥ 3. No matter what you substitute for x, the value of f(x) will be greater than or equal to 3. The turning point, or vertex, is the point on the graph in Example 1 where x = 4. f(x) = (4 – 4)2 + 3 = 0 + 3 = 3, so the vertex has coordinates (4, 3). Graph each equation 1. f(x) = x2 + 1 2. f(x) = (x – 1)2 y 10 6 O 11 10 9 8 7 6 5 4 3 2 1 6 2 -2 y 8 4 -4 3. f(x) = (x + 1)2 + 3 4 2 4x 2 -2 -4 -2 O 2 4x -2 -4-3 -2 0 y 1 2 3 4x -2 4. Find the range of each function in Exercises 1–3. 5. Find the coordinates of the vertex of each function in Exercises 1–3. 31 Chapter 10 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises Name Date Lesson 11.1 Study Guide and Intervention Counting Strategies A probability is a number between 0 and 1 that indicates how likely something is to happen. Often, to find the probability that something will occur, you need to find all the possible outcomes. Example 1 A certain type of watch comes in brown or black and in a small or large size. Find the number of color-size combinations that are possible. Make an organized list. brown/small brown/large black/small black/large In 2 of the 4 outcomes, the watch is brown. The probability that the watch is 2 1 brown is _ , or _ . 4 2 Suppose you can set up a stereo system with a choice of video, DVD, or laser disk players, a choice of cassette or graphic equalizer audio components, and a choice of single or dual speakers. Draw a tree diagram to show the sample space. Example 2 Player Audio cassette vidoe graphic equalizer graphic equalizer cassette laser disk graphic equalizer Outcome VCS VCU VGS VGU DCS DCU DGS DGU LCS LCU LGS LGU The tree diagram shows that there are 12 ways to set up the stereo system. If chosen at random, the probability that the stereo system has a cassette 3 1 and dual speakers is _ , or _ . 4 12 Exercises In Exercises 1-3, a pizza can be ordered with a choice of sausage, pepperoni, or mushrooms for toppings, a choice of thin or pan for the crust, and a choice of medium or large for the size. 1. Draw a tree diagram to show the sample space. 2. How many different kinds of pizza are possible? 3. What is the probability that the pizza has mushrooms on it? 4. In how many ways can you arrange 4 boxes of cereal on a shelf? 32 Chapter 11 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. cassette DVD Speaker single dual single dual single dual single dual single dual single dual Name Date Lesson 11.2 Study Guide and Intervention Modeling with Data Example 1 Judy was comparing prices of admission at several movie theaters. She found these data. Find the mean price at each theater. Which theater gives the best mean price? Theater Colonial Discount Magic Colonial Discount Magic Colonial Discount Magic Time noon noon noon 4 P.M. 4 P.M. 4 P.M. 7 P.M. 7 P.M. 7 P.M. Price $4.00 $3.50 $5.00 $6.00 $5.75 $6.00 $8.00 $7.00 $9.00 (4 + 6 + 8) The mean price at the Colonial is _ = $6.00. The mean price 3 (3.5 + 5.75 + 7) at the Discount is __ = $5.42. The mean price at the 3 (5 + 6 + 9) Magic is _ = $6.67. The Discount Theater offers the best mean price. 3 A box-and-plot divides a set of data into four parts using the median and quartiles. Each of these parts contains 25% of the data. first quartile median third quartile upper extreme, or greatest value Example 2 Make a box-and-whisker plot of the data below. 12, 23, 6, 17, 9, 19, 7, 11, 15, 12 Step 1 Order the data from least to greatest. Step 2 Find the median and the quartiles. Step 3 Draw a number line, graph the median, the quartiles, and the extremes as points above the line. 6, 7, 9, 11, 12, 12, 15, 17, 19, 23 { { Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. lower extreme, or least value Step 4 Draw the box and whiskers. median: 12 first quartile: third quartile: median of lower half = 9 median of upper half = 17 Exercises 5 10 15 20 25 Draw a box-and-whisker plot for each set of data. 1. {17, 5, 28, 33, 25, 5, 12, 3, 16, 11, 22, 31, 9, 11} 2. {$21, $50, $78, $13, $45, $5, $12, $37, $61, $11, $77, $31, $19, $11, $29, $16} IMPACT Mathematics, Course 3 33 Name Date Lesson 12.1 Study Guide and Intervention Work with Algebraic Fractions An algebraic fraction involves division. The denominator cannot equal zero. The expression is undefined for any values of the variable that make the denominator zero. To simplify an algebraic fraction, factor the numerator and denominator. Then divide each by the greatest common factor. (4m – 8) Example 1 For what value of m is the expression _ undefined? (m + 2) m+2=0 The expression is undefined when the denominator equals zero. m+2–2=0–2 Subtract 2 from each side. m = –2 Simplify. Therefore, m cannot equal –2. 54x3 Simplify _. 24xy Example 2 2 (6x)(9x ) 54x3 _ =_ 24xy (6x)(4y) (6x)(9x2) =_ (6x)(4y) 2 9x =_ 4y Divide the numerator and denominator by 6x. Simplify. Example 3 3 =_ 2 Factor denominator. Divide the numerator and denominator by 5n. Simplify. Exercises Find the values of the variable that make the expression undefined. x2 - 2 2b 12 - a 1. _ 2. _ 3. _ 32 + a x+4 b-8 Simplify each expression. 7n3 12ab _ 5. 4. _ a2b2 21n8 x+2 6. _ x2 - 4 Find each product or quotient. mn2 4 n n 8. _ ÷_ 7. _ · _ m mn 3 4 x+2 x-4 9. _ · _ x-4 x-1 34 Chapter 12 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3n 5 _ Multiply _ n · 10 5 _ 5 _ 3n _ 3n _ n · 10 = n · 5 · 2 5 _ 3n =_ n · 5·2 The GCF of the numerator and denominator is 6x. Name Date Lesson 12.2 Study Guide and Intervention Add and Subtract Algebraic Fractions To add algebraic fractions with like denominators, add the numerators and then write the sum over the common denominator. Then, if possible, simplify the result. 5n 7n Example 1 Find _ + _. 15 15 5n + 7n 7n _ 5n _ _ Add the + = 15 15 15 numerators. 12n =_ 15 4n =_ 5 To add algebraic fractions with unlike denominators, factor each denominator to find the LCD. Example 2 n+3 _ 8n - 4 Find _ n + 4n . n = n, 4n = 4 · n, LCD = 4n n+3 _ Only n needs to be renamed. 4(n + 3) _ n+3 _ 8n - 4 _ _ + 8n - 4 n + 4n = 4n 4n Combine like terms. 4n + 12 8n - 4 =_+ _ 4n 4n Simplify. 12n + 8 =_ 4n 3n + 2 =_ n Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises Find the sum. 4 3 _ 1. _ a+a x2 x 2. _ + _ 8 8 2x 3x 3. _ + _ x+5 x+5 Find the difference. 5 3 _ 4. _ a-a 5x x 5. _ - _ 8 8 8t 3t 6. _ - _ w+6 w+6 Find a common denominator. Then find the sum. 7 3 1 1 _ 7. _ 8. _+ _ a + 3a 6x 8 2 3 9. _ + _ 3c 4c Find a common denominator. Then find the difference. 9 3 3 1 1 1 _ 10. _ 11. _ - _ 12. _ -_ m a - 4a 9x 8 4 Solve each equation using any method you like. Make sure your answer does not cause any denominator to be zero. (s + 2) x-2 3 m 13. _ + m = 14 14. _ = x - 4 15. _ = _ 3 6 (s - 1) 2 IMPACT Mathematics, Course 3 35