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

Preview Extract

INSTRUCTORโS
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 Sรฃo 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 ยฉ 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
Instructorโs 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 ยฉ 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 ยฉ 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 ยฉ 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 ยฉ 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 โ and โ.
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 โ and โ to determine whether each statement is true or
false.
a. 3 โ {x | x is a natural number}
b. 9 โ {1, 3, 5, 7}
5. Write out the meaning of each inequality. Then determine whether the inequality is true
or false.
c. 3 โฅ 3
d. 2 โค โ5
a. โ10 > โ8
b. โ 2 โค 0
6. Express the interval [-5, ยฅ) in set builder notation and graph.
Teaching Notes:
โข 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.
โข Brainstorm the many words that translate to the four basic operations. Ex: increased addition.
โข b n = bยท bยท โฆ b (b appears as a factor โnโ times).
โข 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.
โข is read โgreater thanโ
โข โค is read โless than or equal toโ, โฅ is read โgreater than or equal toโ
Copyright ยฉ 2017 Pearson Education, Inc.
ML-1
Answers: 1. a. 5x โ 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.โ10 greater than โ8, false b.โ2 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 โ5, false 6. { x | x ยณ -5}
Copyright ยฉ 2017 Pearson Education, Inc.
ML-2
Mini Lecture 1.2
Operations With Real Numbers and Simplifying Algebraic Expressions
Learning Objectives:
1. Find a numberโs 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. โ 8
2.
5 3
โ
8 8
Evaluate.
2
a. (โ 8)
e. โ
4.
3
4
c. โ 6.24
d. 12
b. โ
3
1
+โ
4
3
c. 15 โ (โ10)
d. 6.8 โ 12.32
f. โ52 + 52
g. โ32 โ (โ38)
h. 4.2 โ (โ8.1)
b. โ 8 2
c. (โ 3)
d. โ 3 4
b. (15) (โ1) (โ4)
c. โ
Add or subtract.
a. โ14 + 25
3.
b. โ
Multiply or divide.
3 8
a. โ รท
5 20
e. (6 )(7 )(0 )(โ 2 )
f.
0
18
4
24
0
g. โ 8 รท
d. (โ3.3) (1.2)
โ2
3
h.
3 โ 14
โ
7 15
5.
Use the distributive property and simplify.
a. 6(x โ2)
b. โ3 (6 โ y)
c. โ4(x โ 5 โ y)
6.
Rewrite to show how the associative property could be used to simplify the expression.
Then simplify.
a. 6(โ4x)
b. (x + 124) + 376
7.
Simplify using the order of operation.
6(โ2) โ 5(2)
a. 5 โ
20 รท 4 + 6
b.
15 โ 2 2
c. 6 (3 x – 2) – 3 x
Copyright ยฉ 2017 Pearson Education, Inc.
ML-3
Teaching Notes:
โข Remind students that absolute value measures distance from zero, and for that reason, it
is always positive.
โข Opposites and additives inverses are just different names for the same thing.
โข Students need to be reminded often that a negative sign is only part of the base if it is
inside parentheses with the base.
โข When opposites are added, the result is zero.
โข Make sure students understand what is behind subtraction โ why subtraction can be
changed to addition of the opposite.
โข 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. โ
or โ 1
c. 25 d. โ5.52 e. โ1 f. 0
4
12
12
3
g. 6 h. 12.3 3. a. 64 b. โ64 c. 81 d. โ81 4. a. โ
b. 60 c. undefined d. โ3.96 e. 0 f. 0 g. 12
2
โ2
5. a. 6x โ 12 b. โ18 + 3y c. โ4x + 20 + 4y 6. a. (6 โ
โ4) x = โ24 x
h.
5
b. x + (124 + 376) = x + 500 7. a. 31 b. โ2 c. 15x โ 12
Answers: 1. a. 8 b.
Copyright ยฉ 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 utilityโs viewing rectangle or table.
Examples:
1.
Plot the following point in a rectangular coordinate system.
A. (โ2, 3)
B. (โ4, 0)
C. (1, 5)
D. (โ1, โ4)
E. (3, โ3)
F. (0, 2)
2.
Complete the table of values for y = x โ 3 , then graph the equation.
x
โ2
โ1
0
1
2
3.
(x, y)
Complete the table of values for y = 2 โ x 2 , then graph the equation.
x
โ3
โ2
โ1
0
1
2
3
4.
y=xโ3
y = 2 โ x2
(x, y)
Complete the table of values for y = x โ 1 , then graph the equation.
x
โ4
โ3
โ2
โ1
0
1
2
3
4
y = |x โ1|
(x, y)
Copyright ยฉ 2017 Pearson Education, Inc.
ML-5
Teaching Notes:
โข The point plotting method is one method for graphing equations.
โข The graph of a linear equation is a line.
โข The graph of a quadratic equation is a parabola.
โข The graph of an absolute value equation is a โVโ shape that can shoot upward or
downward.
Answers: 1.
2. (โ2, โ5) (โ1, โ4) (0, โ3) (1, โ2) (2, โ1)
3. (โ3, โ7) (โ2, โ2) (โ1, 1) (0, 2) (1, 1) (2, โ2) (3, โ7)
4. (โ4, 5) (โ3, 4) (โ2, 3) (โ1, 2) (0, 1) (1, 0) (2, 1) (3, 2) (4, 3)
Figure for Answer 3
Figure for Answer 4
Copyright ยฉ 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 = โ9
2.
a. 5(3 x + 2) = 4(2 x โ 1)
3.
a.
b. 5( x โ 4) โ (2 x โ 6) = 5( x โ 4)
x
4x
โ7=
5
10
b.
c. 25 โ x = 3( x โ 5)
a +1 2 โ a 5
โ
=
8
3
6
y โ 4 3y โ 1
1
1
โ
=1
d. (2 x + 8) = (3x โ 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 โ 4) + 3x
b. 6(2 x โ 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 โ 10 = 1.4 x โ 14
f. 6( x โ 1) + 3(2 โ x) = 0
Teaching Notes:
โข Students may need to be reminded there is no โrightโ or โwrongโ side of the equation.
Some students have a problem when the variable ends up on the right side of the
equation.
โข 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. โ6 2. a. โ2 b. 3 c. 10 3. a. 10 b. 3 c. โ8 d. โ55 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 ยฉ 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 dealerโs cost, what
is the dealerโs cost?
Solve each formula for the specified variable.
1
6.
V = Bh for B
3
8.
S = 180(n โ 2) for n
10.
Ax + By = C for A
7.
A = P(l + rt ) for P
9.
f =
KMm
for M
d2
Teaching Notes:
โข Use the five step strategy for solving word problems.
โข Read the problem carefully. Let a variable represent one of the quantities in the problem.
โข If necessary, write an expression for any other unknown quantities in the problem in
terms of the same variable used in step 1.
โข Write an equation to describe the conditions of the problem.
โข Solve the equation and answer the problemโs question.
โข Remind students to always check to make sure their answer makes sense.
Answers: 1. 3 x โ 12 = 36; x = 16 2. 0.15 x = 255; x = 1700 3.
x + ( x + 20) + 2 x = 180; 40 ๏ฏ , 60 ๏ฏ , 80 ๏ฏ 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 โ By
10. A =
9. M =
Km
x
Copyright ยฉ 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 โ
y 3
b. (4a)(3a 4 )
y 10
2. a. 4
y
25a 6
b.
5a 2
1
b. โ 3
5
3. a. 4 โ2
c. ( xy 4 z )(โ4 xyz 3 )
d. ( 12 m 3 n 5 )(6m 3 n โ3 )
18 x 4 y 6 z
d.
6x 2 y 4 z โ2
x3
x โ6
c.
c. 6 0
d. 6x 0
e. 3x โ2 y โ3
f. 7 x 4 y โ5
g.
a โ2
a โ8
h. 15 โ1
4. a. (x 10 ) 2
b. ( y โ6 ) โ3
c. (4 2 ) โ1
d. (a 4 ) 2
2
0 2
โ3
5. a. (3a b )
b. (5 x y )
6. a. (5 x 6 y โ3 )(4 x 2 y ) 2
b.
โ2
๏ฆ 6x 2 ๏ถ
d. ๏ง๏ง โ 3 ๏ท๏ท
๏จ 2y ๏ธ
2
5x 4 y 3
(2 x โ 3 y ) โ 2
๏ฆ 1 ๏ถ
d. ๏ง 4 6 ๏ท
๏จa b ๏ธ
โ3
๏ฆ2๏ถ
c. ๏ง ๏ท
๏จ3๏ธ
โ2 โ2
( 2 x 3 y โ2 ) โ2
(6 x โ 4 y 4 ) โ 2
c.
Teaching Notes:
โข Exponent rules are very easy as presented โ one at a time. Students often become
confused when several rules are used in one problem. Constant reinforcement and lots of
practice will help.
โข Remind students that when a variable appears to have no exponent โ there is an invisible
exponent of one.
โข Never, never, never multiply a base and an exponent together.
โข Always (exception: scientific notation) write final answers with positive exponents only.
Answers: 1. a. y 7 b. 12a 5 c. โ 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 ยฉ 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.
โ
b. 2.015 ร 10 4
a. โ 3.4 ร 10 5
2.
Write each number in scientific notation.
a. 32,500,000,000
b. โ0.00417
c. 9432 ร 10 4
3.
Perform the indicated computations, writing the answers in scientific notation.
6.8 ร 10 4
b.
a. (2.4 ร 10 3 )(8 ร 10 โ5 )
4 ร 10 โ 2
4.
In Central City, the population is 176,000. Express the population in scientific notation.
Teaching Notes:
โข A number is written in scientific notation when it is expressed in the form a ร 10 n with
1 โค | a | < 10 and โnโ is an integer.
โข When multiplying terms written in scientific notation (a ร 10 n )(b ร 10 m ) = (a ร b) ร 10 n + m .
โข
โข
โข
โข
a ร 10 m a
= ร10 m โ n.
b ร10 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. โ340,000 b. 0.0002015 2. a. 3.25 ร 1010 b. 4.17 ร 10 โ3 c. 9.432 ร 10 7
3. a. 1.92 ร 10 โ1 b. 1.7 ร 10 6 4. 1.76 ร 10 5
Copyright ยฉ 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 โ 2
c. h(โ1) for h(t ) = t 2 โ 3t + 2
4.
b. g (โ2) for g ( x) = 2 x 2 โ 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:
โข A relation is any set of ordered pairs.
โข The set of all first terms โx-valuesโ of the ordered pairs is called the domain.
โข The set of all second terms โy-valuesโ of the ordered pairs is called the range.
โข A function is a relation in which each member of the domain corresponds to exactly one
member of the range.
โข A function is a relation in which no two ordered pairs have the same first component and
different second components.
โข The variable โxโ is called the independent variable because it can be assigned any value
from the domain.
โข The variable โyโ is called the dependent variable because its value depends on โxโ.
โข The notation f(x), read โf of xโ represents the value of the function at the number โxโ.
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 ยฉ 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 โ2 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 ยฉ 2017 Pearson Education, Inc.
ML-12
2
4
6
2
4
6
Teaching notes:
โข The graph of a function is the graph of the ordered pairs.
โข 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, ยฅ) Range: [ 2, ยฅ)
Copyright ยฉ 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 โ 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 )(โ 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 )(โ 2 )
d. ( f โ g )(0 )
๏ฆf ๏ถ
e. ๏ง๏ง ๏ท๏ท(โ 2)
๏จg๏ธ
c. ( f โ g )( x )
Teaching Notes:
โข Students need to be reminded that division by zero is undefined. The value of โxโ cannot
be anything that would make the denominator of a fraction zero.
โข Students often exclude values from the domain that would make the numerator zero,
warn against this.
โข 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 โseeโ what
happens.
Answers: 1. a. (-ยฅ, ยฅ)
b. (-ยฅ, 2) or ( 2, ยฅ)
c. (-ยฅ, 6) or (6, ยฅ)
5
6
+
x + 2 x -1
d. (-ยฅ, – 5) or (-5, 9) or (9, ยฅ)
2. a. x 2 – x + 3 b. (-ยฅ, ยฅ) c. 3
b. (-ยฅ, – 2) or (-2, 1) or (1, ยฅ)
4. a. x 2 + x โ 2 b. 0 c. x 2 โ x + 4 d. 4 e. -1
Copyright ยฉ 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 lineโs slope.
3. Find a lineโs 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 functionโs 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 โ 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 (โ6, 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 = โ12
Teaching Notes:
โข The standard form of the equation of a line is Ax + By = C , as long as A and B are not
both zero.
โข A x-intercept will have a corresponding y coordinate of 0.
โข A y-intercept will have a corresponding x coordinate of 0.
( rise
)
run .
โข
The slope of a line compares the vertical change to the horizontal change
โข
Slope formula is: m = y 2 โ y1 .
โข
โข
โข
โข
โข
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 โ x1
Copyright ยฉ 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 = โ x + 3 m = โ
y – intercept (0, 3)
4
4
2. a. m =
b. m = –
4. a.
1
y-intercept (0,3)
2
b.
Copyright ยฉ 2017 Pearson Education, Inc.
ML-16

## Document Preview (23 of 298 Pages)

You are viewing preview pages of the document. Purchase to get full access instantly.

-37%

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

$18.99 ~~$29.99~~Save:$11.00(37%)

24/7 Live Chat

Instant Download

100% Confidential

Store

##### Harper Davis

0 (0 Reviews)

## Best Selling

The World Of Customer Service, 3rd Edition Test Bank

$18.99 ~~$29.99~~Save:$11.00(37%)

Chemistry: Principles And Reactions, 7th Edition Test Bank

$18.99 ~~$29.99~~Save:$11.00(37%)

Test Bank for Hospitality Facilities Management and Design, 4th Edition

$18.99 ~~$29.99~~Save:$11.00(37%)

Solution Manual for Designing the User Interface: Strategies for Effective Human-Computer Interaction, 6th Edition

$18.99 ~~$29.99~~Save:$11.00(37%)

Data Structures and Other Objects Using C++ 4th Edition Solution Manual

$18.99 ~~$29.99~~Save:$11.00(37%)

2023-2024 ATI Pediatrics Proctored Exam with Answers (139 Solved Questions)

$18.99 ~~$29.99~~Save:$11.00(37%)