Solution Manual for Manufacturing Processes for Engineering Materials, 6th Edition

INSTRUCTOR’S RESOURCE MANUAL A LGEBRA FOR C OLLEGE S TUDENTS EIGHTH EDITION Robert Blitzer Miami Dade College Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Reproduced by Pearson from electronic files supplied by the author. Copyright © 2017, 2013, 2009 Pearson Education, Inc. Publishing as Pearson, 501 Boylston Street, Boston, MA 02116. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ISBN-13: 978-0-13-418175-2 ISBN-10: 0-13-418175-1 www.pearsonhighered.com Instructor’s Resource Manual with Tests Algebra for College Students, Eighth Edition Robert Blitzer TABLE OF CONTENTS MINI-LECTURES (per section) Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Mini-Lectures Answers ML-1 ML-1 ML-11 ML-19 ML-25 ML-32 ML-43 ML-52 ML-61 ML-67 ML-75 ML-83 ML-88 Included at end of section ADDITIONAL EXERCISES (per section) AE-1 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Additional Exercises Answers AE-1 AE-46 AE-76 AE-115 AE-151 AE-196 AE-250 AE-295 AE-292 AE-367 AE-399 AE-421 AE-457 GROUP ACTIVITIES (per chapter) A-1 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Group Activities Answers A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-10 A-11 A-12 A-13 Copyright © 2017 Pearson Education, Inc. TEST FORMS CHAPTER 1 TESTS (6 TESTS) T-1 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-1 T-3 T-6 T-9 T-11 T-14 CHAPTER 2 TESTS (6 TESTS) T-17 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-17 T-21 T-25 T-29 T-34 T-39 CUMULATIVE REVIEW 1-2 (2 TESTS) T-44 Form A (FR) Form B (MC) T-44 T-47 CHAPTER 3 TESTS (6 TESTS) T-51 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-51 T-54 T-57 T-60 T-64 T-68 CHAPTER 4 TESTS (6 TESTS) T-72 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-72 T-76 T-80 T-84 T-89 T-94 CUMULATIVE REVIEW 1-4 (2 TESTS) T-99 Form A (FR) Form B (MC) T-99 T-102 CHAPTER 5 TESTS (6 TESTS) T-107 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-107 T-110 T-113 T-116 T-119 T-122 CHAPTER 6 TESTS (6 TESTS) T-125 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-125 T-127 T-129 T-131 T-134 T-137 CUMULATIVE REVIEW 1-6 (2 TESTS) T-140 Form A (FR) Form B (MC) T-140 T-143 Copyright © 2017 Pearson Education, Inc. CHAPTER 7 TESTS (6 TESTS) T-148 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-148 T-151 T-154 T-157 T-161 T-164 CHAPTER 8 TESTS (6 TESTS) T-167 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-167 T-171 T-175 T-178 T-182 T-186 CUMULATIVE REVIEW 1-8 (2 TESTS) T-190 Form A (FR) Form B (MC) T-190 T-193 CHAPTER 9 TESTS (6 TESTS) T-197 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-197 T-200 T-203 T-206 T-210 T-214 CHAPTER 10 TESTS (6 TESTS) T-218 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-218 T-222 T-226 T-230 T-235 T-239 CUMULATIVE REVIEW 1-10 (2 TESTS) T-242 Form A (FR) Form B (MC) T-242 T-245 CHAPTER 11 TESTS (6 TESTS) T-249 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-249 T-251 T-253 T-255 T-257 T-259 CHAPTER 12 TESTS (6 TESTS) T-273 Form A (FR) Form B (FR) Form C (FR) Form D (MC) Form E (MC) Form F (MC) T-273 T-275 T-277 T-279 T-281 T-283 FINAL (2 TESTS) T-286 Form A (FR) Form B (MC) T-286 T-292 Copyright © 2017 Pearson Education, Inc. TEST ANSWER KEYS T-299 Chapter 1 Chapter 2 Cumulative Review 1-2 Chapter 3 Chapter 4 Cumulative Review 1-4 Chapter 5 Chapter 6 Cumulative Review 1-6 Chapter 7 Chapter 8 Cumulative Review 1-8 Chapter 9 Chapter 10 Cumulative Review 1-10 Chapter 11 Chapter 12 Finals T-299 T-301 T-305 T-306 T-309 T-313 T-315 T-317 T-318 T-319 T-321 T-324 T-325 T-327 T-331 T-332 T-336 T-338 Copyright © 2017 Pearson Education, Inc. Mini Lecture 1.1 Algebraic Expressions, Real Numbers and Interval Notation Learning Objectives: 1. Translate English phrases into algebraic expressions. 2. Evaluate algebraic expressions. 3. Use mathematical models. 4. Recognize the sets that make up the real numbers. 5. Use set-builder notation. 6. Use the symbols ∈ and ∉. 7. Use inequality symbols. 8. Use interval notation. Examples: 1. Write each English phrase as an algebraic expression. Let x represent the number. a. Three less than five times a number. b. The product of a number and six, increased by four. 2. Evaluate each algebraic expression for the given value or values of the variable(s). a. x 2 + 5 x + 3 , for x = 2 b. x 2 + 2( x + y ) , for x = 3 y = 4 3. Use the roster method to list the elements in each set. a. {x | x is an integer between 4 and 9} b. {x | x is an even whole number less than 10} 4. Use the meaning of the symbols ∈ and ∉ to determine whether each statement is true or false. a. 3 ∈ {x | x is a natural number} b. 9 ∉ {1, 3, 5, 7} 5. Write out the meaning of each inequality. Then determine whether the inequality is true or false. c. 3 ≥ 3 d. 2 ≤ –5 a. –10 > –8 b. – 2 ≤ 0 6. Express the interval [-5, ¥) in set builder notation and graph. Teaching Notes: • Be sure to go over important vocabulary for the section including: variable, algebraic expression, constant, exponential expression, equation, formula, natural numbers, whole numbers, integers, rational, irrational and real numbers. • Brainstorm the many words that translate to the four basic operations. Ex: increased addition. • b n = b· b· … b (b appears as a factor “n” times). • Order of operation rules include: 1. First, perform all operations within grouping symbols. 2. Evaluate all exponential expressions. 3. Do all multiplication and division in the order in which they occur, working from left to right. 4. Last, do all additions and subtractions in the order in which they occur, working from left to right. • is read “greater than” • ≤ is read “less than or equal to”, ≥ is read “greater than or equal to” Copyright © 2017 Pearson Education, Inc. ML-1 Answers: 1. a. 5x – 3 b. 6 x + 4 2. a. 17 b. 23 3. a. {5, 6, 7, 8} b. {0, 2, 4, 6, 8} 4. a. true b. true 5. a.–10 greater than –8, false b.–2 is less than or equal to 0, true c. 3 is greater than or equal to 3, true d. 2 is less than or equal to –5, false 6. { x | x ³ -5} Copyright © 2017 Pearson Education, Inc. ML-2 Mini Lecture 1.2 Operations With Real Numbers and Simplifying Algebraic Expressions Learning Objectives: 1. Find a number’s absolute value. 2. Add real numbers. 3. Find opposites. 4. Subtract real numbers. 5. Multiply real numbers. 6. Evaluate exponential expressions. 7. Divide real numbers. 8. Use the order of operations. 9. Use commutative, associative, and distributive properties. 10. Simplify algebraic expressions. Examples: 1. Find the absolute value. a. – 8 2. 5 3 – 8 8 Evaluate. 2 a. (– 8) e. – 4. 3 4 c. – 6.24 d. 12 b. – 3 1 +– 4 3 c. 15 – (–10) d. 6.8 – 12.32 f. –52 + 52 g. –32 – (–38) h. 4.2 – (–8.1) b. – 8 2 c. (– 3) d. – 3 4 b. (15) (–1) (–4) c. – Add or subtract. a. –14 + 25 3. b. – Multiply or divide. 3 8 a. – ÷ 5 20 e. (6 )(7 )(0 )(− 2 ) f. 0 18 4 24 0 g. − 8 ÷ d. (–3.3) (1.2) −2 3 h. 3 − 14 ⋅ 7 15 5. Use the distributive property and simplify. a. 6(x –2) b. –3 (6 – y) c. –4(x – 5 – y) 6. Rewrite to show how the associative property could be used to simplify the expression. Then simplify. a. 6(–4x) b. (x + 124) + 376 7. Simplify using the order of operation. 6(–2) – 5(2) a. 5 ⋅ 20 ÷ 4 + 6 b. 15 – 2 2 c. 6 (3 x – 2) – 3 x Copyright © 2017 Pearson Education, Inc. ML-3 Teaching Notes: • Remind students that absolute value measures distance from zero, and for that reason, it is always positive. • Opposites and additives inverses are just different names for the same thing. • Students need to be reminded often that a negative sign is only part of the base if it is inside parentheses with the base. • When opposites are added, the result is zero. • Make sure students understand what is behind subtraction – why subtraction can be changed to addition of the opposite. • Never, never, never multiply the base and the exponent together! Students are often tempted to do this. 3 13 1 c. 6.24 d. 12 2. a. 11 b. – or – 1 c. 25 d. –5.52 e. –1 f. 0 4 12 12 3 g. 6 h. 12.3 3. a. 64 b. –64 c. 81 d. –81 4. a. – b. 60 c. undefined d. –3.96 e. 0 f. 0 g. 12 2 −2 5. a. 6x – 12 b. –18 + 3y c. –4x + 20 + 4y 6. a. (6 ⋅ –4) x = –24 x h. 5 b. x + (124 + 376) = x + 500 7. a. 31 b. –2 c. 15x – 12 Answers: 1. a. 8 b. Copyright © 2017 Pearson Education, Inc. ML-4 Mini Lecture 1.3 Graphing Equations Learning Objectives: 1. Plot points in the rectangular coordinate system. 2. Graph equations in the rectangular coordinate system. 3. Use the rectangular system to visualize relationships between variables. 4. Interpret information about a graphing utility’s viewing rectangle or table. Examples: 1. Plot the following point in a rectangular coordinate system. A. (–2, 3) B. (–4, 0) C. (1, 5) D. (–1, –4) E. (3, –3) F. (0, 2) 2. Complete the table of values for y = x – 3 , then graph the equation. x –2 –1 0 1 2 3. (x, y) Complete the table of values for y = 2 – x 2 , then graph the equation. x –3 –2 –1 0 1 2 3 4. y=x–3 y = 2 – x2 (x, y) Complete the table of values for y = x – 1 , then graph the equation. x –4 –3 –2 –1 0 1 2 3 4 y = |x –1| (x, y) Copyright © 2017 Pearson Education, Inc. ML-5 Teaching Notes: • The point plotting method is one method for graphing equations. • The graph of a linear equation is a line. • The graph of a quadratic equation is a parabola. • The graph of an absolute value equation is a “V” shape that can shoot upward or downward. Answers: 1. 2. (–2, –5) (–1, –4) (0, –3) (1, –2) (2, –1) 3. (–3, –7) (–2, –2) (–1, 1) (0, 2) (1, 1) (2, –2) (3, –7) 4. (–4, 5) (–3, 4) (–2, 3) (–1, 2) (0, 1) (1, 0) (2, 1) (3, 2) (4, 3) Figure for Answer 3 Figure for Answer 4 Copyright © 2017 Pearson Education, Inc. ML-6 Mini Lecture 1.4 Solving Linear Equations Learning Objectives: 1. Solve linear equations. 2. Recognize identities, conditional equations, and inconsistent equations. 3. Solve applied problems using mathematical models. Examples: Solve each equation. If fractions are involved, you may want to clear the fractions first. 1. a. 5 x + 7 = 22 b. 32 + 3 x = 7 x c. 3 + 2 x = –9 2. a. 5(3 x + 2) = 4(2 x – 1) 3. a. b. 5( x – 4) – (2 x – 6) = 5( x – 4) x 4x –7= 5 10 b. c. 25 – x = 3( x – 5) a +1 2 – a 5 – = 8 3 6 y – 4 3y – 1 1 1 – =1 d. (2 x + 8) = (3x – 5) 5 5 3 3 Solve and determine whether the equation is an identity, a conditional equation, or an inconsistent equation. c. 4. a. 5 x + 3 = 2( x – 4) + 3x b. 6(2 x – 4) + 8 = 8 x + 4( x + 4) c. 3a + 2(a + 4) = 5(a + 1) + 3 d. 6(4 y + 4) = 8(3 y + 3) e. 0.6 x – 10 = 1.4 x – 14 f. 6( x – 1) + 3(2 – x) = 0 Teaching Notes: • Students may need to be reminded there is no “right” or “wrong” side of the equation. Some students have a problem when the variable ends up on the right side of the equation. • Students need to practice clearing equations of fractions by multiplying each term (whether it is a fraction or not) by the least common denominator of all the terms. Answers: 1. a. 3 b. 8 c. –6 2. a. –2 b. 3 c. 10 3. a. 10 b. 3 c. –8 d. –55 4. a. Inconsistent ; No Solution b. Inconsistent; No Solution c. Identity; infinitely many solutions d. Identity; infinitely many solutions e. 5; conditional equation f. 0; conditional equation Copyright © 2017 Pearson Education, Inc. ML-7 Mini Lecture 1.5 Problem Solving and Using Formulas Learning Objectives: 1. Solve algebraic word problems using linear equations. 2. Solve a formula for a variable. Examples: Solve the following using the five step strategy for solving word problems. 1. When 12 is subtracted from three times a number, the result is 36. What is the number? 2. 15% of what number is 255? 3. In a triangle, the measure of the third angle is twice the measure of the first angle. The measure of the second angle is twenty more than the first. Find the measure of each angle. 4. The dog run is six feet longer than it is wide and the perimeter measures 32 feet. Determine the measurements of the length and width of the dog run. 5. A new automobile sells for $28,000. If the mark-up is 25% of the dealer’s cost, what is the dealer’s cost? Solve each formula for the specified variable. 1 6. V = Bh for B 3 8. S = 180(n – 2) for n 10. Ax + By = C for A 7. A = P(l + rt ) for P 9. f = KMm for M d2 Teaching Notes: • Use the five step strategy for solving word problems. • Read the problem carefully. Let a variable represent one of the quantities in the problem. • If necessary, write an expression for any other unknown quantities in the problem in terms of the same variable used in step 1. • Write an equation to describe the conditions of the problem. • Solve the equation and answer the problem’s question. • Remind students to always check to make sure their answer makes sense. Answers: 1. 3 x – 12 = 36; x = 16 2. 0.15 x = 255; x = 1700 3. x + ( x + 20) + 2 x = 180; 40  , 60  , 80  4. 2 x + 2( x + 6) = 32; x = 5 feet wide, x + 6 = 11 feet long A 3V S 5. x + 0.25 x = 28,000; x = $22,400 6. B = 7. P = 8. n = +2 180 h l + rt fd 2 C – By 10. A = 9. M = Km x Copyright © 2017 Pearson Education, Inc. ML-8 Mini Lecture 1.6 Properties of Integral Exponents Learning Objectives: 1. Use the product rule. 2. Use the quotient rule. 3. Use the zero-exponent rule. 4. Use the negative exponent rule. 5. Use the power rule. 6. Find the power of a product. 7. Find the power of a quotient. 8. Simplify exponential expressions. Examples: Simplify. Final answers should not contain any negative exponents. 1. a. y 4 ⋅ y 3 b. (4a)(3a 4 ) y 10 2. a. 4 y 25a 6 b. 5a 2 1 b. – 3 5 3. a. 4 –2 c. ( xy 4 z )(–4 xyz 3 ) d. ( 12 m 3 n 5 )(6m 3 n –3 ) 18 x 4 y 6 z d. 6x 2 y 4 z –2 x3 x –6 c. c. 6 0 d. 6x 0 e. 3x –2 y –3 f. 7 x 4 y –5 g. a –2 a –8 h. 15 –1 4. a. (x 10 ) 2 b. ( y –6 ) –3 c. (4 2 ) –1 d. (a 4 ) 2 2 0 2 –3 5. a. (3a b ) b. (5 x y ) 6. a. (5 x 6 y –3 )(4 x 2 y ) 2 b. –2  6x 2  d.  – 3   2y  2 5x 4 y 3 (2 x – 3 y ) – 2  1  d.  4 6  a b  –3 2 c.   3 –2 –2 ( 2 x 3 y –2 ) –2 (6 x – 4 y 4 ) – 2 c. Teaching Notes: • Exponent rules are very easy as presented – one at a time. Students often become confused when several rules are used in one problem. Constant reinforcement and lots of practice will help. • Remind students that when a variable appears to have no exponent – there is an invisible exponent of one. • Never, never, never multiply a base and an exponent together. • Always (exception: scientific notation) write final answers with positive exponents only. Answers: 1. a. y 7 b. 12a 5 c. – 4 x 2 y 5 z 4 d. 3m 6 n 2 2. a. y 6 b. 5a 4 c. x 9 d. 3 x 2 y 2 z 3 c. 161 3. a. 161 b. 125 c. 1 d. 6 e. d. a 8 5. a. 9a 4 b. x6 y 4 25 c. 9 4 3 x2 y3 f. 7 x4 y5 d. 9 x 4 y 6 6. a. 80 x10 y b. Copyright © 2017 Pearson Education, Inc. ML-9 4. a. x 20 b. y 18 g. a 6 h. 151 9 y12 x14 c. 20 y 5 x2 d. a 12 b18 Mini Lecture 1.7 Scientific Notation Learning Objectives: 1. Convert from scientific to decimal notation. 2. Convert from decimal to scientific notation. 3. Perform computations with scientific notation. 4. Use scientific notation to solve problems. Examples: 1. Write each number in decimal notation. – b. 2.015 × 10 4 a. – 3.4 × 10 5 2. Write each number in scientific notation. a. 32,500,000,000 b. –0.00417 c. 9432 × 10 4 3. Perform the indicated computations, writing the answers in scientific notation. 6.8 × 10 4 b. a. (2.4 × 10 3 )(8 × 10 –5 ) 4 × 10 – 2 4. In Central City, the population is 176,000. Express the population in scientific notation. Teaching Notes: • A number is written in scientific notation when it is expressed in the form a × 10 n with 1 ≤ | a | < 10 and “n” is an integer. • When multiplying terms written in scientific notation (a × 10 n )(b × 10 m ) = (a × b) × 10 n + m . • • • • a × 10 m a = ×10 m – n. b ×10 n b When multiplying or dividing is complete, make sure the final answer is in scientific notation. Students need to be reminded that a number must be written as a number between 1 and 10 to be in scientific notation. The sign of a number has nothing to do with the sign of the power when a number is written in scientific notation. When dividing terms written in scientific notation Answers: 1. a. –340,000 b. 0.0002015 2. a. 3.25 × 1010 b. 4.17 × 10 –3 c. 9.432 × 10 7 3. a. 1.92 × 10 –1 b. 1.7 × 10 6 4. 1.76 × 10 5 Copyright © 2017 Pearson Education, Inc. ML-10 Mini Lecture 2.1 Introduction to Functions Learning Objectives: 1. Find the domain and range of a relation. 2. Determine whether a relation is a function. 3. Evaluate a function. Examples: 1. Find the domain and range of the relation. a. {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25)} b. {(1, -1), (0, 0), (-5, 5)} 2. Determine whether each relation is a function. b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)} a. {(5, 6), (5, 7), (5, 8), (5, 9), (5, 10)} 3. Find the indicated function value. a. f (3) for f ( x) = 3 x – 2 c. h(–1) for h(t ) = t 2 – 3t + 2 4. b. g (–2) for g ( x) = 2 x 2 – x + 4 d. f (a + h) for f ( x) = 2 x + 3 Function g is defined by the table x 0 1 2 3 4 g (x) 2 4 6 8 10 Find the indicated function value. a. g(2) b. g (4) Teaching Notes: • A relation is any set of ordered pairs. • The set of all first terms “x-values” of the ordered pairs is called the domain. • The set of all second terms “y-values” of the ordered pairs is called the range. • A function is a relation in which each member of the domain corresponds to exactly one member of the range. • A function is a relation in which no two ordered pairs have the same first component and different second components. • The variable “x” is called the independent variable because it can be assigned any value from the domain. • The variable “y” is called the dependent variable because its value depends on “x”. • The notation f(x), read “f of x” represents the value of the function at the number “x”. Answers: 1. domain {1, 0, -5} range {-1, 0, 5} 2. a. not a function b. function 3. a. 7 b. 14 c. 6 d. 2a + 2h + 3 n 4. a. 6 b. 10 Copyright © 2017 Pearson Education, Inc. ML-11 Mini Lecture 2.2 Graphs of Functions Learning Objectives: 1. Graph functions by plotting points. 2. Use the vertical line test to identify functions. 3. Obtain information about a function from its graph. 4. Identify the domain and range of a function from its graph. Examples: State the domain of each function. 1. Graph the function f ( x) = 3x and g ( x) = 3 x + 1 in the same rectangular coordinate system. Graph integers for x starting with –2 and ending with 2. How is the graph of g related to the graph of f ? 2. 3. Use the vertical line test to identify graphs in which y is a function of x. a. b. c. 6 Use the graph of f to find the indicated function value. a. f(2) b. f(0) c. f(1) Y 4 2 X -6 -2 0 -2 -4 -4 -6 4. Use the graph each function to identify its domain and range. a. b. 6 Y 6 4 (-5 ,1) Y 4 (1, 1) 2 (-2, 1) (4, 1) 2 X -6 -4 -2 0 -2 2 4 X 6 -6 -4 -2 0 -2 -4 -4 -6 -6 Copyright © 2017 Pearson Education, Inc. ML-12 2 4 6 2 4 6 Teaching notes: • The graph of a function is the graph of the ordered pairs. • If a vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Answers: 1. The graph of g is the graph of f shifted up 1 unit. 2. a. yes b. no c. yes 3. a. 0 b. 4 c. 1 4. a. Domain: {-5, 1, 1.4} Range: {1} b. Domain: [ 0, ¥) Range: [ 2, ¥) Copyright © 2017 Pearson Education, Inc. ML-13 Mini Lecture 2.3 The Algebra of Functions Learning Objectives: 1. Find the domain of a function. 2. Use the algebra of functions to combine functions and determine domains. Examples: State the domain of each function. 1. a. f ( x ) = 3 x -1 b. g ( x ) = 4x x-2 c. h ( x ) = x + 2 6- x d. p ( x ) = 1 7 + x + 5 x -9 2. Let f ( x ) = x 2 − 2 x and g ( x ) = x + 3 . Find the following; a. ( f + g )( x ) 3. Let f ( x ) = b. the domain of f + g c. ( f + g )(− 2 ) 5 6 and g ( x ) = . Find the following; x+2 x -1 a. ( f + g )( x ) b. The domain of f + g 4. Let f ( x ) = x 2 + 1 and g ( x ) + x = 3 . Find the following; a. ( f + g )( x ) b. ( f + g )(− 2 ) d. ( f − g )(0 ) f  e.  (− 2) g c. ( f − g )( x ) Teaching Notes: • Students need to be reminded that division by zero is undefined. The value of “x” cannot be anything that would make the denominator of a fraction zero. • Students often exclude values from the domain that would make the numerator zero, warn against this. • Show students why the radicand of a square root function must be greater than or equal to zero. This is a good place to use the graphing calculator so students can “see” what happens. Answers: 1. a. (-¥, ¥) b. (-¥, 2) or ( 2, ¥) c. (-¥, 6) or (6, ¥) 5 6 + x + 2 x -1 d. (-¥, – 5) or (-5, 9) or (9, ¥) 2. a. x 2 – x + 3 b. (-¥, ¥) c. 3 b. (-¥, – 2) or (-2, 1) or (1, ¥) 4. a. x 2 + x − 2 b. 0 c. x 2 − x + 4 d. 4 e. -1 Copyright © 2017 Pearson Education, Inc. ML-14 4. a. Mini Lecture 2.4 Linear Functions and Slope Learning Objectives: 1. Use intercepts to graph a linear function in standard form. 2. Compute a line’s slope. 3. Find a line’s slope and y-intercept from its equation. 4. Graph linear functions in slope-intercept form. 5. Graph horizontal or vertical lines. 6. Interpret slope as rate of change. 7. Find a function’s average rate of change. 8. Use slope and y-intercept to model data. Examples: 1. Use intercepts and a checkpoint to graph each linear function. Name the x-intercept and the y-intercept. a. 2 x + 5 y = 10 b. x – 2 y = 4 2. Find the slope of the line passing through each pair of points. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. a. (2, 5) and (–6, 3) b. (5, 0) and (1, 3) d. (2, 4) and (-6, 4) c. (3, 0) and (3, – 4) 3. a. Find the slope and y-intercept for the line whose equation is 3x + 4 y = 12 and then graph the equation. 1 b. Find the slope and y-intercept for the linear function f ( x) = x + 3 and then graph the 2 function. 4. Graph the linear equations. a. x = 2 b. 3 y = –12 Teaching Notes: • The standard form of the equation of a line is Ax + By = C , as long as A and B are not both zero. • A x-intercept will have a corresponding y coordinate of 0. • A y-intercept will have a corresponding x coordinate of 0. ( rise ) run . • The slope of a line compares the vertical change to the horizontal change • Slope formula is: m = y 2 – y1 . • • • • • A line that rises from left to right has a positive slope. A line that falls from left to right has a negative slope. A line that is horizontal has zero slope. A line that is vertical has an undefined slope. The slope-intercept form of the equation of a line is y = mx + b where m is the slope and b is the y-intercept. x 2 – x1 Copyright © 2017 Pearson Education, Inc. ML-15 Answers: 1. a. b. 1 3 , rises b. m = , falls c. undefined, vertical d. 0, horizontal 4 4 3 3 3. a. y = – x + 3 m = – y – intercept (0, 3) 4 4 2. a. m = b. m = – 4. a. 1 y-intercept (0,3) 2 b. Copyright © 2017 Pearson Education, Inc. ML-16