Download ALGEBRA 2 and TRIGONOMETRY

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Page i
Teacher’s Manual
with Answer Key
ALGEBRA 2
and
TRIGONOMETRY
Marilyn Davis
AMSCO
AMSCO SCHOOL PUBLICATIONS, INC.
315 HUDSON STREET, NEW YORK, N.Y. 10013
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Page ii
Author of the Teacher’s Manual
Marilyn Davis
Math Consultant
Portions of this book were adapted from the following Amsco Publication:
Teacher’s Manual/Mathematics B
by Edward P. Keenan, Ann Xavier Gantert, and Isidore Dressler
Please visit our Web site at: www.amscopub.com
When ordering this book, please specify: N 159 CD or TEACHER’S
MANUAL/ALGEBRA 2 AND TRIGONOMETRY
ISBN: 978-1-56765-704-3
Copyright © 2009 by Amsco School Publications, Inc.
No part of this Teacher’s Manual may be reproduced in any form without written permission from
the publisher except by those teachers using the AMSCO textbook ALGEBRA 2 AND
TRIGONOMETRY, who may reproduce or adapt portions of the manual in limited quantities for
classroom use only.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10
14 13 12 11 10 09
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Page iii
Contents
About the Teacher’s Manual
v
Assessment
vi
Suggested Time Outlines
Chapter 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The Integers
viii
1
The Rational Numbers
17
Real Numbers and Radicals
30
Relations and Functions
45
Quadratic Functions and Complex Numbers
63
Sequences and Series
79
Exponential Functions
91
Logarithmic Functions
104
Trigonometric Functions
118
More Trigonometric Functions
133
Graphs of Trigonometric Functions
145
Trigonometric Identities
162
Trigonometric Equations
172
Trigonometric Applications
183
Statistics
199
Probability and the Binomial Theorem
217
Summary of Formulas and Trigonometric Relationships
232
Table of Measures
242
Holistic Scoring Rubric
243
Conversion Chart
245
iii
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Contents
Answer Keys
For Enrichment Activities
246
For Extended Tasks
255
For Suggested Test Items
261
For SAT Preparation Exercises
269
For Textbook Exercises
iv
Chapter 1
271
Chapter 2
274
Chapter 3
277
Chapter 4
282
Chapter 5
291
Chapter 6
299
Chapter 7
303
Chapter 8
308
Chapter 9
312
Chapter 10
319
Chapter 11
324
Chapter 12
334
Chapter 13
343
Chapter 14
345
Chapter 15
349
Chapter 16
359
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About the Teacher’s Manual
This manual parallels the organization of the text and provides:
✔ suggestions for assessment.
✔ aims for each chapter.
✔ commentary on each individual section of the text.
✔ leading questions to stimulate classroom discussion.
✔ a variety of approaches to promote flexibility in problem solving.
✔ suggestions to maximize the effectiveness of specific examples and
exercises.
✔ techniques for dealing with difficulties that students may encounter.
✔ reproducible Enrichment Activities and suggestions for more challenging
aspects of topics in the text.
✔ appropriate hands-on activities.
✔ suggestions for Extended Tasks and investigations to be undertaken by
students.
✔ supplementary material that reflects current thinking in mathematics
education.
✔ a set of SAT Preparation Exercises for each chapter.
✔ questions to form the basis for chapter tests.
✔ an answer key for the Enrichment Activities, the Extended Tasks, the
Suggested Test Items, the SAT Preparation Exercises, and the text.
v
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Page vi
Assessment
As a student studies the content of the text associated with this manual, it is
important that the student, teacher, parents, and administrators have tangible
evidence of the student’s progress toward established goals. That evidence can
be obtained in a variety of ways. Some of these tangibles, and the features of the
student text and teacher’s manual that can aid in assembling that evidence, are
listed.
Homework Assignments
The need for consistent reflection on and practice with the content of each day’s
lesson has always been accepted as a necessary factor in promoting the student’s
understanding and skills. Each day’s homework assignment should enable the
student and the teacher to recognize progress as well as identify needs. The text
provides both routine and challenging exercises at the end of each section for
this purpose.
Cumulative Reviews
Cumulative reviews in the text allow the student to evaluate his/her skills within
the format of the Regents Examination in Algebra 2 and Trigonometry. Use the
Holistic Scoring Rubric in this manual to provide the student with insight into
the number of credits his/her responses merit. Guide the student in formulating
responses that will receive full credit on the examination. Measure student
progress by scaling cumulative review scores using the Conversion Chart.
Portfolio
A portfolio is a record of a student’s progress. The selection of materials to be
included in the portfolio presents opportunities for the student to reflect upon
what he/she has done, improve faulty work, and take pride in work that is well
done and in insightful information that has been gained. Significant examples of
homework assignments, tests, and independent research or readings, as well as
completed Enrichment Activities, Hands-On Activities, Bonus Questions, or
Extended Tasks from the student text or this manual, are excellent materials for
inclusion.
Tests and Quizzes
Tests give the student opportunities to demonstrate that he/she understands
concepts and has the ability to make use of that understanding. This manual
provides Suggested Test Items for each chapter. These questions can be used by
the teacher when constructing a test based on the instructional goals established
for the class.
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Independent and Group Projects
Mathematical understanding and enthusiasm are increased by opportunities to
explore in-depth topics of special interest to the students. Some of these topics
may be suggested by the students themselves. Others are given as Explorations
in the Cumulative Reviews of the chapters of the text or as Enrichment Activities in this manual. These Explorations and Activities lend themselves to either
individual or group study.
Journal
A journal can be a useful tool to promote a studied reflection upon day-by-day
progress. A journal may be a record of feelings, understandings, fears, insights,
and questions. Putting these into words can help students to clarify ideas and
identify concepts that need further explanation and exploration. The content of
the journal can be the choice of the student, a response to a teacher’s question
or prompt, or an explanation based on questions from the Writing About Mathematics section of the exercises.
You may ask students to:
• describe characteristics they have noticed about problems that can be
solved using a particular method.
• explain or justify steps for a computation or proof.
• explain key properties or concepts in their own words.
• write their own problems for a particular concept or method.
• explain the possible results for a given situation (such as solving a quadratic
equation).
• give examples of how certain concepts are applied in the real world.
• explain how they organized their work for more complex Applying Skills
problems.
• create a flowchart or other diagram for solving a particular problem.
• explain why a given answer must be incorrect based on properties, number
sense, or estimation.
• explain why they chose one method of proof rather than another.
• make a list of common errors they must watch for in their own work.
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Suggested Time Outlines
Since teaching from a new text can create time problems, the timetable below is offered to assist you
in planning your work. If a state or national test is given before the end of the school year, it may be
difficult to cover the content for the year before the test is administered. To use time most effectively, check if any of the lessons will not be tested. Consider skipping non-tested lessons and returning to them after the test. You may also wish to identify lessons that are predominantly review of
concepts from an earlier grade and reduce the amount of class dedicated to these lessons by assigning them as homework only.
CHAPTER
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1 Year
1.5 Years
2 Years
Days
Days
Days
The Integers
4–5
10–12
15–16
The Rational Numbers
4–5
10
15–16
Real Numbers and Radicals
4–6
10–11
15–16
Relations and Functions
13–15
19–20
25–26
Quadratic Functions and Complex Numbers
13–15
19–20
25–26
Sequences and Series
13–15
15–20
24–25
Exponential Functions
12–14
17–20
22–24
Logarithmic Functions
12–14
17–20
22–24
Trigonometric Functions
13–15
19–20
24–25
More Trigonometric Functions
8–9
15
20–22
Graphs of Trigonometric Functions
8–9
15
20–22
Trigonometric Identities
12–14
18–20
23–25
Trigonometric Equations
12–14
16–21
23–25
Trigonometric Applications
9–10
15–16
20–22
Statistics
9–10
15
22–24
Probability and the Binomial Theorem
9–10
15
20–22
155–180
245–270
335–360
Total
viii
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Page 1
CHAPTER
1
THE
INTEGERS
Aims
• To review operations and properties of
integers.
• To perform operations with polynomial
expressions.
• To solve absolute value equations and
inequalities.
• To factor polynomial expressions.
• To solve quadratic equations and inequalities by factoring.
This chapter reviews properties and skills
that should be familiar to students from previous mathematics courses. Integer solutions to
all exercises in this chapter provide a built-in
check for students looking to refresh their
skills. Factoring is essential for success in algebra, and since the process of factoring reverses
the process of multiplication, multiplication of
polynomials is reviewed before factoring is
explored. Factoring is applied to solve quadratic equations and inequalities. The procedure for inequalities will be new for many
students and the need to consider the signs of
the factors should be discussed in depth.
CHAPTER OPENER
In the golf example, par is equivalent to zero
on a number line, that is, the reference point
from which distances above and below are
determined. Rosie’s standing is found by evaluating the expression 22 1 (22) 1 1 5 23 or
3 below par for the course. Nancy’s standing is
21 1 (21) 1 (21) 5 23, so she is also 3
below par. If par is 100 strokes, then both players shot 97. If par is 95 strokes, then both players shot 92.
1-1
WHOLE NUMBERS,
INTEGERS, AND THE
NUMBER LINE
The number line, introduced with the set of
counting numbers, is extended to include 0
and then all of the integers. Basic concepts
including opposites, subtraction, and absolute
value are reviewed. Emphasize that 2a means
“the opposite of a,” and that the opposite of a
negative number is a positive number.
For Exercise 2, students can write b 2 a 5
2(a 2 b). Using substitution,
a 2 b 5 2(a 2 b), so a 2 b 5 b 2 a.
1-2
WRITING AND
SOLVING NUMBER
SENTENCES
Properties of equality and inequality are
reviewed and applied to solve problems.
Emphasize that to maintain the order of
inequality when multiplying or dividing an
inequality by some number c, c must be posi-
1
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Chapter 1
tive. Illustrate this multiplication and division
property of inequality as follows.
Given: 7 . 4
Let c 5 0.
Is 7(0) . 4(0) or
0 . 0?
No; 0 5 0.
Given: 7 . 4
Let c 5 25.
Is 7(25) . 4(25) or
235 . 220?
No; 235 , 20.
The examples will help students to remember to reverse the inequality when multiplying
or dividing by a negative number.
Example 2 illustrates the approach needed
to solve an equation involving absolute
value. If 6x 2 3 5 u, then the equation becomes u 5 15 and, by definition, u can equal
either 15 or 215. This is the source of the two
derived equations. When solving absolute
value equations, students should be aware that
the absolute value term has to be alone on one
side of the equation. For example, the steps for
solving 2x 1 3 2 4 5 7 are:
2x 1 3 2 4 5 7
2x 1 3 2 4 1 4 5 7 1 4
2x 1 3 5 11
2x1 3 5 11
2x1 323 5112 3
2x1 3 5 211
or
2x1 323 5 2112 3
2x 5 8
2x 5 214
x54
2x 5 27
Ask students if 3x 2 8 5 25 has a solution. (No; 3x 2 8 represents some number and
absolute value cannot be negative.)
1-3
ADDING
POLYNOMIALS
When reviewing the addition of polynomials
in this section, address common errors that
students make, such as writing 3x2 2 x2 5 3 or
failing to write the opposite of each term in an
expression being subtracted. Have students
recall the distributive property as they work
with expressions:
3x2 2 x2 5 3x2 2 1x2
5 (3 2 1)x2
5 2x2
2(4c3 2 2c2 1 5) 5 2(4c3) 2 (22c2) 2 (5)
5 24c3 1 2c2 2 5
For Exercise 25, the solution is found by
using cents to represent the values of the
coins and the toll. If x is the number of quarters, then x 1 3 is the number of dimes and
25x 5 10(x 1 3).
1-4
SOLVING ABSOLUTE
VALUE EQUATIONS
AND INEQUALITIES
To isolate the absolute value expression in an
equation, the same steps are used as for isolating a variable. During this process, the
absolute value expression is treated as a single
entity. To help students see the form, let A represent 4a 1 2 so that 4a 1 2 1 7 5 21
becomes A 1 7 5 21. The quantity A must be
isolated before the equation can be solved, so
A 1 7 2 7 5 21 2 7 and A 5 14.
Call attention to the fact that an absolute
value inequality of the form x 2 a 5 k, k . 0,
has two solutions that separate the number
line into three regions: The graph of the solution set of x 2 a , k is the portion of the number line between the solutions of x 2 a 5 k;
the graph of the solution set of x 2 a . k is
the union of the other two regions, that is, the
portion of the number line to the left of the
smaller solution and the portion of the number line to the right of the larger solution.
To summarize for students, explain that to
solve an absolute value inequality, you must
change from absolute value to plain inequality.
The way to handle the change depends on
which direction the inequality points with
respect to the absolute-value term. To solve
for x:
If ax 1 b * c, solve 2c * ax 1 b * c.
If ax 1 b + c, solve ax 1 b + c and
ax 1 b * 2c.
For Exercise 29, be sure students understand that “as much as” means “less than or
equal to” or “no more than.”
200 2 c # 28
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The Integers
1-5
MULTIPLYING
POLYNOMIALS
1-6
When a polynomial is multiplied by a polynomial, each term of the first factor is multiplied
by each term of the second factor. Thus, when
multiplying a binomial by a binomial, there
are 2 3 2 5 4 multiplications. The product, in
simplest form, has four terms when no two
terms of the product are like terms:
(3a 2 1)(2b 1 1) 5 6ab 1 3a 2 2b 2 1
The product, in simplest form, has three
terms when two terms of the product are like
terms:
(3a 2 1)(4a 1 1) 5 12a2 1 3a 2 4a 2 1
5 12a2 2 a 2 1
The product, in simplest form, has two
terms when two terms of the product are
opposites:
2. Keep in mind that the difference of two
squares is the only binomial form that can
be factored into two binomials. If the factoring is to be done with respect to the set
of integers, the squares must be the
squares of integers or even powers of the
variables.
3. To factor a trinomial into binomial factors, write the terms of the trinomial in
order of descending or ascending powers
of the variables.
ax2 (b) 5 2bx(2ax)
a2 1 a 2 3
3a3 1 3a2 2 9a
2a2(a2 1 a 2 3) 5 2a4 1 2a3 2 6a2
3
1. Always look for a common monomial factor first.
2 a2 2 a 1 3
Suggest that to multiply a polynomial by a
polynomial, students write each polynomial in
standard form, that is, in descending order
according to the values of their exponents.
Some students may benefit from using the vertical form of multiplication, writing similar
terms one under the other. For example,
4
To factor efficiently, students must understand
the use of the distributive property in the multiplication of polynomials. Emphasize the following:
3 2a2 1 3a 2 1
5 9a2 2 4
3a(a2 1 a 2 3) 5
FACTORING
POLYNOMIALS
Example 2 illustrates the technique of factoring by grouping, which may be useful when
four or more terms are given. To group, we
take the terms two at a time and look for common factors for each of the pairs on an individual basis. Factoring by grouping works only
if a new common factor appears, the exact
same one in each term. As noted in the text,
a polynomial of four terms can be factored
into two binomials if the product of the first
and last terms equals the product of the two
middle terms. For example, in the polynomial
ax2 2 bx 2 ax 1 b:
(3a 1 2)(3a 2 2) 5 9a2 1 6a 2 6a 2 4
21(a2 1 a 2 3) 5
3
2
2a 1 5a 2 4a 2 10a 1 3
In Enrichment Activity 1-5: On the Ins
and Outs, students use products of polynomials to analyze a number pattern and then use
algebra to generalize the pattern.
?
abx2 5 abx2 ✔
However, the pairs of terms ax2 2 bx and
2ax 1 b do not yield a common binomial factor. We must change the order of the middle
terms.
ax2 2 ax 2 bx 1 b 5 ax(x 2 1) 2 b(x 2 1)
5 (x 2 1)(ax 2 b)
Factoring is an opposite operation of multiplication. Therefore, we can factor a trinomial
into the product of two binomials by reversing
the steps of the multiplication process.
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Chapter 1
Multiplication:
(2x 2 3)(3x 1 5) 5 2x(3x 1 5)
2 3(3x 1 5)
degree polynomials and provides additional
practice that leads to a general formula for
determining the sum and difference of two
cubes.
5 6x2 1 10x 2 9x 2 15
5 6x2 1 x 2 15
1-7
Factoring:
6x2 1 x 2 15 5 6x2 1 10x 2 9x 2 15
5 2x(3x 1 5) 2 3(3x 1 5)
5 (3x 1 5)(2x 2 3)
Note that the only step that is not easily
reversed is the last step in the multiplication.
There is a unique sum of 10x and 29x, but
there is not a unique way of writing 1x as the
sum of two terms. To choose the correct pair of
terms, we must observe that the product
6x2(215) is the same as the product 10x(29x).
Therefore, the first step of this factoring
method is to find two terms whose sum is the
middle term and whose product is the same as
the product of the first and last terms.
Method 2 presents the approach that uses
checking pairs of factors. Students may find
these tips helpful:
1. If the last term of the trinomial is positive,
the two binomials contain the same operation.
• If the middle term is positive, the two
binomials each contain addition as their
operation.
• If the middle term is negative, the two
binomials each contain subtraction as
their operation.
2. When the last term of the trinomial is
negative, the binomials have opposite
operations.
In Exercises 37 and 38, suggest that students substitute P for the expression in parentheses to identify the factoring technique.
Factor in terms of P, then substitute and
simplify.
Enrichment Activity 1-6: Factoring the
Sum and Difference of Two Cubes applies the
method of finding a binomial factor to higher-
QUADRATIC
EQUATIONS WITH
INTEGRAL ROOTS
The procedure for solving a quadratic equation by factoring rests on the multiplication
property of 0, which allows us to apply the
principle ab 5 0 if and only if a 5 0 or b 5 0.
Students may sometimes try incorrectly to
apply the principle in solving an equation such
as x2 2 6x 5 7 by factoring to get x(x 2 6) 5 7,
and then setting each factor equal to 7.
Emphasize that in order to apply the principle,
one side of the equation must be 0. Thus, the
first step in solving a quadratic equation is to
remove all terms from one side of the equation. Then, combine all like terms before factoring.
Point out that when solving an equation
whose highest-power term has a negative coefficient, we generally make it positive by
multiplying both sides of the equation by 21.
This makes the factoring process easier. For
example:
2x2 1 5x 1 6 5 0
21(2x2 1 5x 1 6) 5 21 ? 0
x2 2 5x 2 6 5 0
(x 2 6)(x 1 1) 5 0
x–650
x56
x1150
x 5 21
The numbers 6 and 21 both satisfy the
original equation 2x2 1 5x 1 6 5 0.
In Examples 2 and 3, point out that since
the 2 that was factored out is an expression not
containing a variable, we do not have to set it
equal to zero.
In Exercise 2, students are asked to justify
the extension of the zero-factor property to
three factors.
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The Integers
1-8
QUADRATIC
INEQUALITIES
When a quadratic inequality is solved algebraically by factoring, two cases must always
be considered. If the product is positive
(. 0), then the factors are either both positive or both negative. If the product is negative (, 0), then either the first factor is
negative and the second factor is positive or
the first factor is positive and the second factor is negative.
If (ax 1 b)(cx 1 d) . 0, then:
(ax 1 b) , 0 and (cx 1 d) , 0
or (ax 1 b) . 0 and (cx 1 d) . 0.
If (ax 1 b)(cx 1 d) , 0, then:
2(ax 1 b) , 0 and (cx 1 d) . 0
or (ax 1 b) . 0 and (cx 1 d) , 0.
Another common method of finding the
solution set of a quadratic inequality is to indicate the signs of the factors of the polynomial
using a number line. The sign graph for the
inequality x2 2 2x 2 15 , 0 of Example 1 is
shown below. The sign of each factor in each
interval is considered.
(x 2 5)
(x 1 3)
23
5
2 2 2 2 2 2 2 2 2 0 1 1
2 0 1 1 1 1 1 1 1 1 1 1
The product is negative if one factor is negative and one factor is positive. The graph
shows this to be the case for 23 , x , 5.
Graphing quadratic inequalities on the
coordinate plane and determining the solutions from the graphs will be examined in
Chapter 5.
Have students practice solving quadratic
inequalities using their calculators. Note that
the inequality symbols can also be entered
from the TEST menu.
5
EXTENDED TASK
For the Teacher:
This task is intended to reinforce students’
understanding of distance and absolute value.
The definition of “walking distance” models
distance in a city with a grid of perpendicular
streets and dense construction that does not
allow for diagonal shortcuts. Students may
need to review the midpoint and distance formulas and the property of the perpendicular
bisector of a segment.
SAT PREPARATION
EXERCISES
Students should be given experience that will
promote ease in dealing with unusual problems presented in a variety of ways, thus
increasing their chances for success in standardized testing. This experience should be
made available throughout the students’
mathematics education. In keeping with the
philosophy that the classroom teacher is
uniquely qualified to offer such training on an
ongoing basis, this Teacher’s Manual provides
a set of SAT Preparation Exercises for each of
the 16 chapters of the book.
These exercises, which are of the type and
level of difficulty students will encounter in
actual SATs, include Multiple-Choice and
Student-Produced Response Questions. Each
set of problems reflects the basic concepts of
the chapter, but many of these problems are
quite challenging. Students may be asked to
attempt individually to solve two or three
problems each night, or students may work in
groups. No matter how these problems are
assigned, students should not be discouraged if
they find them to be difficult. In time, students
will begin to learn how to approach these
exercises and how to find correct solutions.
Teachers may wish to adapt the following
answer sheet for use with the SAT Preparation
Exercises to provide students with practice
gridding answers.
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Chapter 1
SUGGESTED TEST ITEMS
A set of test questions is provided in this
Teacher’s Manual for each chapter of the text.
Because of the variables in testing situations,
such as the ability level of a class and the length
of a class period, these sets of questions need
not be used as formal tests. However, they are
formatted as such to be readily reproducible.
The Bonus items vary in nature, some
requiring special insight into text material,
others needing a fresh and unorthodox
approach to problem solving. Students should
have the opportunity to work on these problems outside of class.
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Name
Class
Date
Answer Sheet
SAT Preparation Exercises (Chapter ____ )
Use a No. 2 pencil. Fill in the circle completely. If you erase, erase completely. Incomplete erasures may be read as answers.
I. MULTIPLE-CHOICE QUESTIONS
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
A B C D E
II. STUDENT-PRODUCED RESPONSE QUESTIONS
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ENRICHMENT ACTIVITY 1-5
On the Ins and Outs
Look at this sequence of consecutive integers:
4, 5, 6, 7
The product O of the outside pair is 4 3 7 5 28.
The product I of the inside pair is 5 3 6 5 30.
The difference between the inside product and the outside product is I 2 O 5 2.
1. Find I 2 O for each of these sequences.
a. 9, 10, 11, 12
b. 18, 19, 20, 21
c. 30, 30 1 1, 30 1 2, 30 1 3
d. x, x 1 1, x 1 2, x 1 3
2. What pattern did you observe in Exercise 1?
3. Find I 2 O for each of these sequences.
a. 7, 10, 13, 16
b. 20, 23, 26, 29
c. 101, 104, 107, 110
d. x, x 1 3, x 1 6, x 1 9
4. What pattern did you observe in Exercise 3?
5. What pattern would you expect to see in the differences of inside and outside
products for numbers that differ by 4?
6. Find I 2 O for each of these sequences.
a. x, x 1 2, x 1 4, x 1 6
b. x, x 1 5, x 1 10, x 1 15
c. x, x 1 6, x 1 2(6), x 1 3(6)
d. x, x 1 k, x 1 2k, x 1 3k
7. Using the answer to Exercise 6d, explain how the difference between the inside
and outside products is related to the numbers in the sequence.
8. If the difference between the inside and outside product is 162, what is the
difference between consecutive terms?
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ENRICHMENT ACTIVITY 1-6
Factoring the Sum and Difference of Two Cubes
We know that when the sum and the difference of the same two terms are multiplied, the product is a binomial because two of the terms of the product have a sum of
0. For example:
(2x 1 5)(2x 2 5) 5 4x2 2 10x 1 10x 2 25 5 4x2 2 25
In a similar way, the product of a binomial and a trinomial whose product has six
terms can be written as a binomial when two pairs of similar terms have a sum of 0.
For example:
(x 1 2)(x2 2 2x 1 4) 5 x3 2 2x2 1 4x 1 2x2 2 4x 1 8 5 x3 1 8
We notice that the product is the sum of two cubes: x3 is the cube of x and 8 is the
cube of 2. In two of the terms whose sum is 0, 2 and 22 are the coefficients of x2, and
4 and 24 (the square of 2 and its opposite) are the coefficients of x. The pattern that
we observe in the product shown above can help us to factor both the sum of two
cubes and the difference of two cubes.
Example
Factor x3 1 27.
Method 1
1. Since 27 is the cube of 3, use 3 and 23 as the coefficients of x2, and 9 and 29 as
the coefficients of x:
x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27
2. Factor out the greatest common factor of each group of three terms:
x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27
5 x(x2 2 3x 1 9) 1 3(x2 2 3x 1 9)
3. Factor out the common trinomial factor (x2 2 3x 1 9):
x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27
5 x(x2 2 3x 1 9) 1 3(x2 2 3x 1 9)
5 (x2 2 3x 1 9)(x 1 3)
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Method 2
1. Again, use 3 and 23 as the coefficients of x2, and 9 and 29 as the coefficients of x,
but arrange the terms in a different order:
x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27
2. Factor out the greatest common factor of each pair of terms:
x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27
5 x2(x 1 3) 2 3x(x 1 3) 1 9(x 1 3)
3. Factor out the common binomial factor (x 1 3):
x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27
5 x2(x 1 3) 2 3x(x 1 3) 1 9(x 1 3)
5 (x 1 3)(x2 2 3x 1 9)
The difference of two cubes can be factored in a similar way.
Exercises
In 1–5, factor each binomial.
1. x3 2 8 5 x3 2 23 5 x3 2 2x2 1 2x2 2 4x 1 4x 2 8
2. x3 1 64 5 x3 1 43 5 x3 1 4x2 2 4x2 2 16x 1 16x 1 64
3. x3 2 64 5 x3 2 43 5 x3 2 4x2 1 4x2 2 16x 1 16x 2 64
4. x3 1 125 5 x3 1 53 5 x3 1 5x2 2 5x2 2 25x 1 25x 1 125
5. x3 2 8y3 5 x3 2 (2y)3 5 x3 2 x2(2y) 1 x2(2y) 2 x(2y)2 1 x(2y)2 2 (2y)3
6. Derive a pattern for the factors of a3 1 b3.
7. Derive a pattern for the factors of a3 2 b3.
In 8–11, use the patterns derived in Exercises 6 and 7 to factor each binomial.
8. 8x3 1 y3
9. x3 2 8y3
10. 125 2 27d3
11. 64x3 1 27y3
12. Factor a4 2 b4 5 a4 2 a3b 1 a3b 2 a2b2 1 a2b2 2 ab3 1 ab3 2 b4 by first finding
the common factor of each pair of terms. Use the same method to factor the
resulting polynomial factor of four terms.
13. Use the factors of the difference of two squares twice to factor a4 2 b4.
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EXTENDED TASK
Going for a Walk
Assume the walking distance between two points on the coordinate plane is the
length of the shortest path between the points that consists only of horizontal and
vertical segments.
1. Find at least three ways to travel from (1, 1) to (7, 5). What is the walking distance
for each?
2. Find the walking distance between each pair of points.
a. (2, 3) and (11, 9)
b. (25, 0) and (8, 26)
c. (1, 27) and (14, 215)
3. a. Find all the integral points that are a walking distance of 6 units from the
point (6, 6). Graph the points.
b. If all the points from part a were connected, what shape is formed?
c. Describe the set of all points whose walking distance from (6, 6) is greater than 6.
d. Describe the set of all points whose walking distance from (6, 6) is less than 6.
4. a. Use the distance formula to find the straight line distance from the origin to (6, 6).
b. What is the walking distance from the origin to (6, 6)?
c. Which distance is greater?
5. a. When are the straight-line distance and walking distance between two points equal?
b. When is the straight-line distance greater than the walking distance?
6. Graph the set of all the points that are the same walking distance from (8, 9) and
(6, 5). Explain how to locate these points.
7. How would you locate all of the points that are the same straight-line distance
from (8, 9) and (6, 5)?
8. Use absolute value notation to write the distance between each pair of points.
a. (7, 3) and (2, 3)
b. (5, 22) and (24, 22)
c. (x1, y) and (x2, y)
9. Use absolute value notation to write the distance between each pair of points.
a. (3, 5) and (3, 20)
b. (24, 28) and (24, 15)
c. (x, y1) and (x, y2)
10. Use absolute value notation to write the walking distance between each pair
of points.
a. (2, 4) and (10, 10)
b. (25, 1) and (211, 7)
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c. (x1, y1) and (x2, y2)
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Algebra 2 and Trigonometry:
Chapter One Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–4, find the value of each given expression.
1. 6 1 213
2. 214 1 (2(29))
3. 10 2 23 2 3
4. 2(25 1 8)
5. Find all integers n such that:
a. n 5 2n
b. n 5 n
c. n 5 24
In 6–12, solve each equation or inequality. Each variable is an element of the set
of integers.
6. 5x 1 8 5 43
7. 19 2 3c 5 31
8. 27 2 4y 5 29
9. 2x 1 6 5 20
10. 6m 1 9 5 15
11. 8k 1 7 . 23
12. 5 , 4x 1 1 # 37
In 13–15, write the sum or difference of the given polynomials in simplest form.
13. (n3 2 5n2 1 2n) 1 (22n2 2 n 1 2)
14. (3c2d 2 2cd2 1 5) 1 (c2d2 2 3c2d 1 2cd2)
15. (10x3 2 3x2 2 2x) 2 (5x2 2 7x 2 3)
16. Subtract 2y2 2 y 1 1 from the sum of y 1 2 and y2 2 4y 1 3.
17. Luis spent 28 minutes more on his math homework than his science homework.
If the total time he spent on both subjects was 152 minutes, how much time did
he spend on each subject?
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18. A 40-foot rope is cut into two pieces, one of which is 2 feet shorter than 6 times
the length of the other. How long is each piece of rope?
19. Mrs. Rudin is five times as old as her daughter Kate. In 9 years, Mrs. Rudin will
be three times as old as Kate. How old is each now?
In 20–22, write the solution set of each inequality if the variable is an element of the
set of integers.
20. 4 2 x 1 1 # 3
21. 2x 2 3 . 5
22. 2x 1 3 2 5 # 10
23. Describe the solution set for the inequality 2x 1 5 1 6 $ 24 if x is an element
of the set of integers.
24. Describe the solution set for the inequality 3x 2 5 1 4 , 2 if x is an element of
the set of integers.
In 25–29, perform the indicated operations and write the result in simplest form.
25. 22x(x 1 3)2
26. 2y [y 2 (3y 1 2)]
27. 3x2(x 2 2)(3x 2 5)
28. 2x 2 [x 1 3(x 2 1) 2 5]
29. 5x 1 4(x2 2 x 1 4)
30. The length of a rectangle is 3 more than twice the width x.
a. Express the area of the rectangle in terms of x.
b. Find the area if the rectangle is 21 inches long.
In 31–36, factor each polynomial completely.
31. 5a 1 ab 1 5b 1 b2
32. 10x2 2 3x 2 18
33. x2 2 9y2
34. 4n3 2 5n2 1 n
35. 8x5 2 2x3
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36. x3 2 3x2 1 4x 2 12
In 37–40, solve and check each equation.
37. x2 1 x 5 30
38. y2 2 3y 2 4 5 0
39. x2 1 5x 2 14 5 0
40. (x 1 4)2 5 16
41. The product of two consecutive positive integers is 132. Find the integers.
42. If the sides of a square are each increased by 4 inches, the area becomes
121 square inches. Find the perimeter of the original square.
In 43–46, write the solution set of each inequality if x is an element of the set
of integers.
43. x2 2 7x 1 12 . 0
44. x2 1 2x # 3
45. x2 2 6x . 0
46. x2 , 2x 1 35
Bonus I: Show that a2 2 b2 2 c2 1 2bc can be factored as (a 2 b 1 c)(a 1 b 2 c).
Bonus II: Solve (x 2 2)(x 1 3)(x 1 5) , 0.
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SAT Preparation Exercises (Chapter 1)
I. MULTIPLE-CHOICE QUESTIONS
In 1–14, select the letter of the correct
answer.
1. If x 2 y 5 8, then which of the following
must equal 64?
(B) x2 1 y2
(A) x2 2 y2
2
2
(C) x 2 2xy 1 y
(D) x2 1 2xy 1 y2
(E) none of these
2. From (x 2 1)2 subtract (x 2 3)(x 2 1).
The result is
(A) 2(x 2 1)
(B) 22(x 2 1)
(C) 4(x 2 1)
(D) 24(x 2 1)
(E) x(x 2 1)
3. If 2x 2 5 5 78, what is the value of
2x 1 7?
(A) 71
(B) 80
(C) 85
(D) 90
(E) 95
4. For what integer n is it true that
4(n 2 15) 5 n?
(A) 220
(B) 212
(C) 0
(D) 5
(E) 20
5. If ▫x▫ 5 x 1 2 and a 5 3b 1 1, then ▫a▫ is
equivalent to
(A) 3b 1 1
(B) 3b 1 1 1 2
(C) 3b 1 1 1 2
(D) 3b 1 3
(E) 3b 1 3
6. What is the sum of the solutions to the
equation 25x 2 5 5 35?
(A) 214
(B) 26
(C) 22
(D) 2
(E) 14
7. At the Kids Carnival, children won a prize
for guessing within 10 the number of jellybeans in a jar. If the jar contained 86 jellybeans, how many winning guesses are a
multiple of 3?
(A) 4
(B) 5
(C) 7
(D) 9
(E) 10
Copyright © 2009 by Amsco School Publications, Inc.
8. If 4 2 x , 4x 1 4, which of the following
must be true?
(A) x . 0
(B) x 5 0
(C) x , 0
(D) x . 22
(E) x , 22
9. What are all the values of x for which
x2 1 9x 1 18 , 0?
(A) 26 , x , 3
(B) 26 , x , 23
(C) x , 23
(D) x , 3
(E) x , 26
10. Which of the following sets of x-values
gives all the solutions of x2 2 12x 5 220?
(A) {210, 22}
(B) {210, 2}
(C) {10, 22}
(D) {10, 2}
(E) {220, 1}
11. What are all the values of w for which
w2 1 2w $ 24?
(A) 26 # w # 4
(B) w # 24 or w $ 6
(C) w # 26 or w $ 4
(D) 24 # w # 6
(E) w $ 24 or w $ 6
12. Which equation has two distinct integral
solutions?
(A) x2 1 25 5 0
(B) (x 2 5)2 5 0
2
(D) x 2 5 5 25
(C) (x 1 5) 5 0
(E) x2 2 25 5 0
13. If x 5 2 is one solution of x2 1 5x 2 c 5 0,
then the other solution is
(A) 214
(B) 27
(C) 7
(D) 10
(E) 14
14. If az 1 bz 5 2z, then what is the value of
a 1 b?
(A) 21
(B) 0
(C) 1
(D) 2
(E) 3
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II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 15–20, you are to solve the problem.
2
15. If x 2 3x 1 1 5 0, then what is the value
of 2x2 2 6x 1 19?
16. A number increased by 6 gives the same
result as that number multiplied by 19.
What is the number?
17. If x 2 y 5 16 and x2 2 y2 5 96, then what
is the value of x?
Copyright © 2009 by Amsco School Publications, Inc.
Date
18. If x2 1 y2 5 34 and xy 5 15, what is the
value of (x 2 y)2?
19. What is the sum of the solutions to the
equation 113 2 m 5 9?
20. A certain number is increased by 17 and
the result is doubled. The final answer is 6
more than 3 times the original number.
What was the original number?
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CHAPTER
2
THE
RATIONAL
NUMBERS
Aims
• To perform arithmetic operations with
rational expressions and write expressions
in lowest terms.
• To simplify complex fractional expressions.
450
hour 5 12 hour 1 18 hour 5 30
60 1 3,600 where
450 is the number of seconds in 18 hour. Since
450 seconds 5 7 minutes 30 seconds, add 7 to
30
the minutes expression to get 37
60 1 3,600. Ask
students to express 35 4 8. Instead of 438, they
30
should find 4 1 22
60 1 3,600.
5
8
• To solve rational equations and inequalities.
• To use ratio and proportion to solve
applied problems.
This chapter defines a rational expression
and applies the rules for addition, subtraction,
multiplication, and division of fractions to
these operations on rational expressions.
Throughout the chapter, attention is called
to the need to delete from the domain of any
expression the values of the variable that
make the denominator equal to 0. Students
will understand the importance of this detail
only if they are asked to state all restrictions
when working with a fraction that contains a
variable in the denominator. This careful
approach to operations with rational expressions will help students to understand the
meaning of an extraneous root when it is
introduced at the end of the chapter.
CHAPTER OPENER
The result for 21 4 8 can be made clear
to
students
by
showing
that
2-1
RATIONAL NUMBERS
We need rational numbers to represent parts
of a whole. You may wish to show how to
locate the rational number ba on a number line
by first dividing the line segment from 0 to 1
into b equal parts and then marking off a of
these parts. This construction will help you to
emphasize the conditions that a and b must be
integers and that b cannot be 0. Since it is
impossible to divide a line segment into 0
equal parts, the symbol 20 is not a representation of a rational number. In fact, 20 is not a
symbol for any number.
Neither a calculator nor a computer will
give an exact representation of a repeating
decimal value. All these devices work with
infinitely repeating decimals rounded to the
number of digits allowed by the operating
system.
A repeating decimal is sometimes called a
periodic decimal. Emphasize that the if and
17
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Chapter 2
only if statement before Example 2 is equivalent to the two statements:
• Every rational number can be written as
an infinitely repeating decimal.
• Every infinitely repeating decimal can be
written as a rational number.
Students should become fluent with the
method presented in this section for writing a
repeating decimal as a rational number. An
alternate method is to write the repeating
decimal as a sum and then use the formula for
the sum of an infinite geometric series (Sec2
tion 6-7). After students find 0.18 5 11
, have
them find the fraction for 0.18 5 17
90 . Other
similar comparisons can be assigned with
regard to the exercises. For example, Exercise
19 asks students to convert 0.83 to the fraction
5
6 ; have students compare the result for
0.83 5 83
99.
Note that a sufficient number of the
repeating digits must be entered in a calculator before the decimal-to-fraction function will
return the desired fraction. For the example
0.18 above, the required entry is .188888888888.
2-2
SIMPLIFYING
RATIONAL
EXPRESSIONS
Because variables represent real numbers,
operations with rational numbers and rational
expressions are very similar. To write a fraction in simplest form, divide both the numerator and the denominator by their greatest
common factor (GCF). A similar approach is
applied for rational expressions, also called
algebraic fractions.
Students should understand that cancellation produces the same result as using the
inverse of the indicated operation. When the
problem involves multiplication, canceling one
of the factors is a division. When the problem
involves addition, canceling one of the terms is
a subtraction.
15
10
3(5)
5 2(5) 5 32
x175317
x53
Canceling a term from the numerator and
the denominator of a fraction is a common
error. If this error occurs when algebraic fractions are being simplified, offer numerical
examples that are comparable.
Algebraic:
y 1 1
y 1 1
y 1 2
1
is in simplest form, and y 1 2 2 2.
Numerical:
5
5
2
2
1
1
1
1
1
2
1
2
1
1
5 67, but 55 1
1 2 2 2.
1
1
5 34, but 22 1
1 2 2 2.
Each cancellation shown above and to the
right must be incorrect because each yields the
value 12, which is not equivalent to the known
value.
We can cancel only factors common to both
the numerator and the denominator. In
5 1 1
5 1 2
5 67 , observe that 5 is not a factor of
3
1 1
either 6 or 7. In 22 1
2 5 4, although 2 is a factor of 4, note that 2 is not a factor of both 3 and
y 1 1
4. In y 1 2, note that y is not a factor of either
y 1 1 or y 1 2.
Any non-zero number divided by itself is 1.
But 00 is indeterminate. Throughout this chapter, call attention to the conditions under
which the original rational expression and the
reduced rational expression are equivalent.
Emphasize the method for factoring out
21 in the numerator or denominator to help
simplify rational expressions. Finally, point out
that one of the reasons for reducing rational
expressions is that numerical substitutions are
much easier when the expression is simplified.
For example,
x2 1 13x
x 1 13
x(x 1 13)
5 x 1 13 5 x
and, to find the value of the original expression for x 5 297.4653, it is clearly easier to
work with the simplified form.
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The Rational Numbers
2-3
MULTIPLYING AND
DIVIDING RATIONAL
EXPRESSIONS
In practice, students will generally find it easier to use Method 2, factoring and cancelling
common factors before multiplying resulting
numerators and denominators.
When simplifying computation by using
cancellation of a factor in the numerator and
denominator of a fraction, students need to be
reminded that:
1. When a cancellation represents the division of a non-zero number by itself, the
resulting factor is 1.
x 2 3
x 2 3
5 1 where x 3.
2. Only in the multiplication of two fractions
can the same factor be canceled from the
numerator and the denominator of two
different fractions.
y
x
y
x
? z 5 z where x 0, z 0.
y
x
In x 1 z , the x’s cannot be canceled.
3. When two fractions are correctly multiplied and the product is reduced to lowest
terms, the simplified result is equivalent
to the product of the given fractions only
for values of the variable for which the
given fractions are defined.
2
x 2 3
x
? x x2 3 5 x where x 0, 3.
4. When all the visible factors have been
canceled from a numerator and denominator, there still exists a factor of 1. By
writing 1 as a factor, we see that
1
1
3 (x 2 1)
3 (x 2 1) (x 2 2)
1
1
5x22
1
By not writing 1, a student may factor
and cancel correctly but overlook the
numerator:
3(x 2 1)
3(x 2 1)(x 2 2)
2x22
Emphasize that before any factors can be
canceled, a division problem must be changed
to multiplication by the reciprocal of the divisor. In the following example, x 2 2 cannot be
canceled when the fractions are written with
the division sign between them, but x can be
19
canceled when the division is changed to multiplication by the reciprocal.
x
x 2 2
4
x(x 2 2)
4
x
5 x 2 2 ? x(x 42 2)
4
5
2 (x 2 0, 2)
(x 2 2)
2-4
ADDING AND
SUBTRACTING
RATIONAL
EXPRESSIONS
When finding the least common denominator
of two or more algebraic fractions, encourage
students to write each polynomial denominator in the same order (ascending or descending). It is easier to recognize the relationship
of binomials such as x 2 2 and 2 2 x if they are
written as x 2 2 and 2x 1 2 or as x 2 2 and
21(x 2 2).
In Example 1, emphasize the use of the distributive property in the numerator when
x2 2 6x 1 9 is subtracted from x2 2 3x; students often use the opposite of only the first
term and will incorrectly write
x2 1 3x2(x2 26x1 9) 5 x2 1 3x2x2 26x1 9.
In the text example on page 54, the numerator is expressed as a binomial (a 1 1), but the
denominator is left as the product of factors
(2(a 1 1)(a 2 1)). When the numerator is simplified, it may contain a factor common to the
denominator and, thus, the rational expression
can be further simplified.
Ask students which approach they think is
easier for Example 2. Most will choose the
first solution since the common denominator
for each expression is convenient to work with
and fewer steps are required to complete the
problem.
2-5
RATIO AND
PROPORTION
A ratio is a comparison by division. Explain
that every rational number is a ratio, but since
the terms of a ratio do not have to be integers,
not every ratio is a rational number. For example, in a right triangle with legs of length 2 and
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Chapter 2
3 and hypotenuse of length !13, the ratio of
2
the shortest side to the longest is !13
, which is
not rational.
A key idea of this section is that equal
ratios are formed when each term of a ratio is
multiplied by the same positive number. In
Example 2, the ratio of the length of the side
to the length of the base is 5 : 2 and if x is any
positive number, then 5x : 2x is an equivalent
ratio.
The discussion of proportion with students
should include the point that if the proportion
a
c
b 5 d is true, then the reciprocal proportion
b
a
5 dc is also true. This idea is useful for solving
5
1
an equation such as x 2
2 5 4. After you invert the proportion, the denominator of the
left term is 1. At that point, all you need to do
is add 2 to each side.
1
x 2 2
2
5 54 S x 2
5 45
1
x 2 2 5 45
x 5 245
Proportions are a specific type of fractional
equation. In Exercises 16–19, cross multiplication will produce quadratic equations that
must be solved. Emphasize that both solutions must be tested in the original proportion.
There are no extraneous roots in these
exercises.
Use Enrichment Activity 2-5: Investigating Ratios and Growth Rate in Leaves to give
students the opportunity to connect mathematics to science and the real world. Have students work in small groups or individually.
Students may need to look up the meaning of
leaf blade and petiole.
2-6
COMPLEX RATIONAL
EXPRESSIONS
Continue to call attention to the fact that the
denominator of a fraction cannot be 0. In a
complex fraction, three of the four possible
parts cannot be zero. Thus:
a
In bc , b, c, and d cannot be 0.
d
For example, in the fraction
x24
x
x21
x22
x, x 2 1, and x 2 2 cannot be equal to 0. Therefore, x cannot be equal to 0, 1, or 2.
In Exercise 1, students need to simplify the
fraction to
a21
a
a2 2 1
a2
and consider carefully the circumstances for
which the three key parts described above will
be zero. In addition to excluding 0 and 1, they
must also realize that 21 will also result in
a2 2 1 equaling 0.
In Exercises 23 and 24, remind students to
follow the correct order of operations.
2-7
SOLVING RATIONAL
EQUATIONS
Students often confuse the procedures
involved in simplifying the sum of two fractions and those used to solve an equation that
has the sum of two fractions on one side.
In simplifying the sum of two fractions,
each fraction is changed to an equivalent fraction by multiplying by a fraction equal to 1.
1
x
1
1x2
1 5
5
1
x
1
? xx 2
2 1 1 x 2 1?x
x 2 1
x
x(x 2 1) 1 x(x 2 1)
1
x
2x 2 1
5 x(x
2 1)
In solving an equation that has the sum of
two fractions on one side, the equation is
changed to an equivalent one in which neither
side contains fractions.
3
1
1x2
1 5 2
3
1
2x(x 2 1) A x1 1 x 2
1 B 5 2x(x 2 1) A 2 B
2(x 2 1) 1 2x 5 x(x 2 1)(3)
1
x
4x 2 2 5 3x2 2 3x
When both sides of an equation are multiplied by the same non-zero number, the resulting equation is equivalent to the original; that
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The Rational Numbers
is, it has the same roots as the original. Therefore, 4x 2 2 5 3x2 2 3x has the same roots as
3
1
1
x 1 x 2 1 5 2 if x 0 and if x 2 1 0 or x 1. Since the roots of 4x 2 2 5 3x2 2 3x are not
0 or 1, the roots of both equations are the
same, 2 and 13 . Summarize for students by
explaining that whenever a variable appears in
any denominator, you must check the possible
solutions in the original equation. When
checking, if a possible solution makes any
denominator equal to 0, that value is not a
solution to the equation. Such values are
extraneous solutions introduced by multiplying by a LCD that contains a variable.
Example 5 illustrates the solution of a standard “motion” problem.
time
time
total
traveling 1
traveling
5 time for
at first speed
at second speed
trip
Motion problems may also state that two
parts of a trip were different distances traveled
at different speeds, but the time for each part
was the same. All the information in the problem is usually needed to construct a solution,
so students should read carefully, look for
needed facts, and make sure they answer the
question that was asked.
2-8
SOLVING RATIONAL
INEQUALITIES
Students may find the second method used in
the examples preferable since it does not
require thinking about when an expression is
positive or negative. The method involves
these steps:
21
1. Identify the excluded values.
2. Solve the corresponding equation.
3. Use the excluded values and solutions to
partition the number line.
4. Test values in each interval.
Method 1 also requires that the student
take into account both the interval on which
the expression is positive or negative and the
interval that yields the proposed solution in
each case. In Example 1, the solution a . 4
applies if a . 0, so a . 4 is the interval where
both conditions are satisfied. The solution a ,
4 applies if a , 0, so a , 0 is the interval where
both conditions are satisfied.
You may wish to suggest that students
work at least one exercise from 8–14 using
both methods.
EXTENDED TASK
For the Teacher:
This extended task is intended to let students see that rational equations are applied in
the workplace. This is a task where students
could work independently or in small groups.
Since electronics is a technical discipline, this
task allows students to see the connections
between mathematics and other disciplines. It
would be possible to use this as a science/math
integrated activity where students could construct circuits modeling each of the five circuits given. Assessment of students’ work
should include the ability to follow directions,
neatness and accuracy of diagrams, understanding of circuits, accuracy of arithmetic
computations, solution of rational equations,
and the ability to express themselves in written form.
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Date
ENRICHMENT ACTIVITY 2-5
Investigating Ratios and Growth Rate in Leaves
As you study mathematics, you should see that it has many connections with the
real world. This investigation is designed to help you see how rational numbers are
connected to science and to nature. As you know from your previous studies, a
rational number is one that can be expressed in fractional form where both the
numerator and the denominator are integers and the denominator cannot be zero. If
the numerator is zero, then the fraction is equal to 0. If the denominator is 1, the fraction is equal to a whole number. If the numerator and denominator are both non-zero
and the denominator is not equal to 1, the fraction may be expressed as a terminating
or repeating decimal.
This investigation requires you to find some measures, to write some ratios from
these measures in both fractional and decimal form, to make a scatter plot of these
ratios, and to draw a conclusion about the growth rate of leaves. Follow the procedure
outlined below to complete this task.
1. Select 10 to 15 leaves from a single tree or bush. Be sure to choose leaves that
are different sizes, from the smallest ones to the largest ones that you can find.
Pick them from different branches, but from the same tree or bush. Do not
damage the tree or bush in removing the leaves you select.
2. Lay one of your leaves flat and, with your ruler, measure the longest part of the
blade of the leaf. Do not include the petiole. Express your measure to the nearest
millimeter or to the nearest tenth of a centimeter. Fold the leaf in half, making the
point of the blade match the base of the blade. Again, ignore the petiole. Crease
carefully, unfold, and measure the width of the leaf at the crease.
3. Express the ratio of the length to the width of your leaf in fractional form. Using
your calculator, express each ratio as a decimal. Round off appropriately. This
length/width ratio is a rational number.
4. Repeat steps 2 and 3 for the rest of your leaves.
5. Construct a data table and record your measurements and ratios.
6. What can you say about your ratios?
7. Make a scatter plot of the length vs. width of the leaves you measured. What
shape do your points make (for example, a tight ball, a pencil shape, and so on)?
8. Now find the average length-to-width ratio of all the leaves you selected and
measured.
9. Using the data table, your scatter plot, and the average length-to-width ratio,
what can you conclude about the growth rate of the leaves on the tree or
bush you selected?
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Date
EXTENDED TASK
Electronic Technician:
Applying Rational Equations in the Workplace
In previous mathematics classes, you have solved and
applied various types of equations. You have developed techniques that enable you to do this algebraically. Many of these
same algebraic techniques are used in the solution of rational
equations; that is, equations involving rational expressions.
Rational equations occur frequently in the workplace. In this
extended task, you will see how rational equations are used in
the electronics industry.
The Setting
Ed works as a technician for a small electronics firm. He repairs, tests, and builds
prototypes of electronic equipment and products. He needs to use his knowledge of
resistors, capacitors, and inductors, as they are essential components used in electronic circuits. He often needs to build resistors to reduce current in a circuit to a
desirable level. Sometimes he connects the resistors in series and other times he
needs to connect them in parallel. On occasion it is necessary for him to combine
them into series-parallel groupings. There are also times when Ed needs to combine
these three basic units to form complete electronic circuits. Resistance in a circuit is
measured in ohms. In electronics, the values of resistance normally encountered are
quite high. Often, thousands, and occasionally even millions, of ohms are used.
The Task
a. Use any resource available to you to find out the difference between parallel
and series circuits. Explain this difference in words.
b. Draw a diagram of a circuit containing two or more resistors connected
in series.
c. Draw a diagram of a circuit containing two or more resistors connected
in parallel.
d. Locate the formulas for the total resistance in a series circuit and the total
resistance in a parallel circuit. State these formulas in algebraic form, and
explain, in your own words, the meanings of these formulas.
e. In each of the diagrams on the following page:
(1) State whether the circuit is a series circuit, a parallel circuit, or a
combination series-parallel circuit.
(2) Using the values given for RT, the total resistance of the circuit, and the
values for all resistors in the circuit except one, compute the missing
resistance.
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1.
Date
RT 5 15,000 ohms
R1
R1 5 5,000 ohms
R2 5 3,000 ohms
R2
R3 5 ?
R3
RT 5 8 ohms
2.
R1
R1 5 15 ohms
R3
R2
R2 5 ? ohms
R3 5 40 ohms
RT 5 400 ohms
3.
R2
R1
R3
R4
R1 5 1,000 ohms
R2 5 1,000 ohms
R3 5 3,000 ohms
R4 5 ?
4.
RT 5 6,500 ohms
R1
R1 5 2,000 ohms
R2
5.
R3
R5
R2
R3 5 18,000 ohms
RT 5 15.8 ohms
R1 5 1.6 ohms
R4
R1
R2 5 ? ohms
R2 5 4.0 ohms
R3 5 6.0 ohms
R3
R6
R4 5 3.8 ohms
R5 5 10.0 ohms
R6 5 ?
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Date
Algebra 2 and Trigonometry:
Chapter Two Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
5
1. Express 12
as a repeating decimal.
In 2–4, write each decimal as a common fraction.
2. 0.625
3. 0.4
4. 0.12
In 5–8, write each rational expression in simplest form and list the values of the variables for which the fraction is undefined.
2x3y2z
5. 4x2yz2
3x 1 xy
6. y 1 3
2
19x 2 14
7. 3x 12 2
3x
y2 2 10y 1 24
8. y2 2 5y 1 4
In 9–17, perform the indicated operations and write the result in simplest form. In
each case, list any values of the variables for which the fractions are not defined.
2a 1 2b
9. a 5a
1 b ? 15a2
2 3
x 2 1
10. x2 3x
2 2x 1 1 ? 3x
12c2
11. c 8c
1 2 4 c2 1 3c 1 2
3
1
1
1 12a
2 3a
12. 4a
3
1
13. a2 2
9 4 4a 1 12
1 5
x 1 4
14. 3x
2x 1 2 2 3x 1 3
1 6
1
15. 4x
x2 2 1 1 x 1 1
4y
5
16. y2 2 25 2 y 1
5
17.
A 3 2 x1 B A 1 1 9x2 12 1 B
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2 2
18. The length of the base of a triangle can be expressed as 10x
x2 2 9 feet. The altitude
3
to that base can be expressed as xx 1
2 1 feet. Express the area of the triangle in
terms of x.
In 19–22, solve each proportion for the variable.
19. 4x 61 5 5 72
3
20. a5 5 a 2
2
1 7
21. 6x 10
5 2x 61 9
2
x 1 1
22. xx 2
1 4 5 x 1 10
23. At a picnic, the ratio of adults to children was 7 : 4. There were 15 more adults
than children. How many people were at the picnic?
In 24–27, simplify each complex rational expression. In each case, list any values of
the variables for which the fractions are not defined.
24.
1
x2x
1 2 x12
3
25. ab
1
b2
1
2
26. x 2
2
x2 2 4
27.
x21
4
x
4
11x1
x
In 28–31, solve each equation and check.
28. x1 1 13 5 x5
3
3x
29. x 2
1 2 x 1 1 5 1
30. x 1 12
x 5 27
8
2
4
31. x 2
3 2 x 1 3 5 x2 2 9
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32. An express train travels 150 miles in the same time that a freight train travels
100 miles. If the express goes 20 miles per hour faster than the freight, find the
rate each train travels.
5
. Find the
33. The sum of two numbers is 10. The sum of their reciprocals is 12
numbers.
In 34–36, solve and check each inequality.
34.
5y 2 3
7
.
15y 2 2
28
2
4
35. b 2
, bb 2
b
2 6
x 1 3
36. 2x
.2
3 2
6
Bonus: Simplify each complex fraction.
1
a.
2x 1
1
1
2x 1 2x
1
b.
m1
1
m 1 m 11 1
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SAT Preparation Exercises (Chapter 2)
8. A
I. MULTIPLE-CHOICE QUESTIONS
In 1–14, select the letter of the correct
answer.
y
3
, then BC equals
If AC 5 23 and BD 5 2x
1 9
5
(A) 3 1 2x
(B) 4x6x
2 x
1 3
(C) 2x3x
(D) 3 2x
(E) x1 1 y1
1
2. If x 5 (1 2 x)(5
2 x) , then which of the
following is the largest number?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
3. If x 1 y 5 10 and x 2 y 5 5, then
x2 2 y2
2x
D
C
1 16
1
1. If x1 1 y1 5 1, then x 1
y equals
1
(A) xy
(B) xy
(C) xy
(D) x
B
(E) 1 13 2x
9. D
y
x
C
x
y
is
equal to
(A) 10
3
(D) 5
(B) 2
(E) 10
y
(C) 3
x
A
B
1
4. If x2 22 1 5 18 2 10
and x . 0, then x is
(A) 18
(B) 9
(C) 4.5
(D) 3
(E) 2
The ratio of the shaded area to the area of
square ABCD is
1
1
5. The average of x 1
2 and x 2 2 is
(A) (x 1 y) 2
(A) x1
(B) x12
(D) x2 x2 4
(E) x2 2x
2 4
7.
(A) 0
(B) 1
(D) x1
y
x
y 2 x
(E) not determinable
(E)
(x 2 y) 2
xy
(x 1 y) 2
xy
x
x2 1 y2
(B) (x 1 y) 2
x2 2 y2
2xy
(C) x2 1 y2
(D) (x 1 y) 2
(E) not determinable
10.
(C) 2
x12
l
equals
(A) 21
(C)
2xy
(C) x2 12 4
6. If xy 5 1, then 2x 12 y 1 y 21 2x is
y
2
(B) xy
(D)
(x 1 y)(x 2 y)
xy
x11
If the volume of the rectangular box is
x2 1 4x 1 3, then the dimension l is
1
(A) x 1
3
3
(C) xx 1
1 2
(E) x 1 2
Copyright © 2009 by Amsco School Publications, Inc.
(B) x 1 3
2
(D) xx 1
1 3
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3
11. Solve 13 1 m62 5 m
.
(A) 23 and 26
(C) 3 and 26
(E) 3 and 6
(B) 29 and 22
(D) 2 and 29
12. If x4 1 4 1 x4 1 4 1 x4 1 4 5 0, then the
value of x is
(A) 24
(D)
1
4
(B) 22
(C) 21
(E) 1
x
x
13. If A 100
2 1 B A 100
1 1 B 5 kx2 2 1, then
the value of k is
(A) 0.1
(D) 0.0001
(B) 0.01
(E) 0.00001
(C) 0.001
n
14. If 36
5 36
n , then which of the following
could equal 72?
(A) n2
(B) 72n
(D) 2n 1 36
(C) n2 4 36
(E) 3n 2 36
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 15–22, you are to solve the problem.
15. If x 1 x1 5 30, then x2 1 12 is equal to
x
what number?
Copyright © 2009 by Amsco School Publications, Inc.
Date
16. If a 24 3b 5 b2, what is the value of ba?
17. The base of a triangle is 5 inches less than
the altitude. The area of the triangle is 42
square inches. Find the base in inches.
18. A fraction is equivalent to 34. If 3 is subtracted from the numerator and 3 is subtracted from the denominator, the new
fraction formed equals 23. Find the numerator and denominator of the original fraction.
19. Lucia rode her bicycle at a constant rate
along a flat road 15 miles long. When the
road became a hill, she reduced her speed
by 4 mph for the last 3 miles of her trip.
The trip took Lucia 2 hours. What was her
rate of speed, in mph, on the flat part of
the road?
20. If kxl 5 1 2
1
X22Y ?
x
, what
1
12x
is the value of
5
2
21. If 2x 32 1 2 3x
4 5 6, what is x ?
22. The ratio of boys to girls at the basketball
game was 7 : 5. There were 12 more boys
at the game than girls. How many boys
and girls were at the game?
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CHAPTER
3
REAL
NUMBERS AND
RADICALS
Aims
• To define the set of irrational numbers and
the set of real numbers.
• To graph inequalities.
• To simplify radical expressions.
• To add, subtract, multiply, and divide radical expressions.
• To rationalize a denominator containing a
radical.
• To solve radical equations.
While most numbers students deal with on
a daily basis are rational, and even though
there is an infinite number of rationals, mathematicians have proved that most of the points
on the number line correspond to irrational
numbers. In this chapter, the primary focus is
on irrational numbers that are radicals.
However, students should be aware that
many other irrationals belong to the set of real
numbers.
CHAPTER OPENER
There are many variations of the proof that
!2 is not a rational number, but the basic
approach is similar to that used by Euclid:
assume the opposite of what is to be proved
30
and show that it leads to a contradiction. That
p
is, assume !2 is a rational number, so !2 5 q .
p
p
Reduce q so that q 5 ba is in simplest form;
thus, a and b are not both even integers. If
2
2
!2 5 ba, then A !2B 2 5 A ba B and 2 5 ba2. Then
2b2 5 a2 and 2b2 is even since it has a factor of
2. Since a2 is equal to 2b2, then a2 is even. However, if the square of an integer is even, then
this integer is even (this result must be previously proven). Thus, a is even. Since a is even,
we may write a 5 2k, where k is some integer.
Then,
2b2 5 (2k) 2 By substitution
2b2 5 4k2
2
2
b 5 2k
2
By squaring 2k
By dividing each side by 2
Since b is equal to 2k2, then b2 is even. Again,
using the fact that if the square of an integer is
even, then the integer is even, it follows that b
is even. This contradicts the assumption that a
and b were not both even. Therefore, the
assumption that there exists a rational number
whose square is 2 has led to a contradiction.
Thus, the real number whose square is 2 cannot be a rational number.
A more advanced version of the proof uses
the result that any perfect square must have an
even number of prime factors and any number
that is twice a perfect square must have an odd
number of prime factors.
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Real Numbers and Radicals
3-1
THE REAL NUMBERS
AND ABSOLUTE
VALUE
From the definition of an irrational number,
we may conclude that a number is irrational if
and only if it can be expressed as an infinite
nonrepeating decimal. Students should realize
that it is often difficult to demonstrate that a
decimal expansion does not repeat. The fact
that p, which has been calculated to millions
of decimal places, has not yet been shown to
repeat does not mean the decimal expansion
will not begin to repeat at a later decimal
place. Therefore, proving that a number is irrational usually requires reasoning that is independent of decimal expansions.
Students have graphed inequalities in previous courses and have solved absolute value
inequalities in Chapter 1. The geometric
approach used to graph x , 3 and x . 3 provides a basis of understanding from which to
proceed to the more formal algebraic procedure.
Call attention to the fact that an absolutevalue equality of the form x 5 k, k . 0,
has two solutions that separate the number
line into three regions. The graph of the solution set of x , k is the region between the
solutions of x 5 k, and the graph of x . k
is the union of the other two regions, that
is, the region to the left of the smaller solution and the region to the right of the larger
solution.
If k is positive, then the equation x 5 k
and the inequalities x . k and x , k have
solutions that are proper subsets of the set of
real numbers. These cases are explained and
summarized in the text. After students understand these concepts, ask them to describe the
solution set for each of the three cases where k
is negative. (If k , 0, then x . k is true for all
real numbers since x is non-negative and
every non-negative number is greater than
every negative number. If k , 0, then x 5 k
and x , k have no real solution, since no nonnegative number is equal to or less than a negative number.)
3-2
31
ROOTS AND
RADICALS
Students are familiar with square-root radicals
and with the use of the calculator to find a
rational approximation when such a radical is
irrational. This section extends the study of
radicals to include other roots and explains
how to evaluate radicals using a graphing calculator. In Chapter 7, students will learn that
nth roots can also be found using a fractional
exponent.
Radical expressions that have indexes of 2,
4, 6, or any even number are said to be even
roots. Radical expressions that have indexes
of 3, 5, 7, or any odd number are said to be
odd roots. When the index is even, the radicand must be non-negative for the radical to
be a real number. However, an odd root of a
positive number is a positive number, and an
odd root of a negative number is a negative
number.
Students should understand that the solution of x 2 5 49 is different from that of
x 5 !49:
x2 5 49
x 5 6 !49
x 5 67
x 5 !49
x57
There are two square
roots of 49, and both are
real numbers.
x is the principal square
root of 49.
The solutions of the following equations
are the same:
x3 5 8
3
x5 !
8
3
x5 !
8
x52
x52
There are three cube roots of 8, but only
one is a real number.
When "x2 is evaluated for any non-zero
real number, the sign of the answer is positive.
For example,
x 5 2: "x2 5 "22 5 !4 5 2
x 5 22: "x2 5 " (22) 2 5 !4 5 2
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Chapter 3
Recall that the absolute value of any real
number x, or x, is also a positive number for
any non-zero value of x. We can conclude that
"x2 5 x for any real number x.
Enrichment Activity 3-2: A Square-Root
Algorithm presents a method that was used to
approximate square-root radicals. The activity
will help students to appreciate the power and
convenience of a calculator.
3-3
SIMPLIFYING
RADICALS
When simplifying radicals, students often have
difficulty finding the largest perfect-square
factor. If this happens, encourage them to try
factors that are not squares. Start dividing the
radicand by its smallest natural-number factors and test to see whether the other factor is
a perfect square. For example:
3-4
ADDING AND
SUBTRACTING
RADICALS
Any two numbers that are elements of the
same set of numbers can be added. In the set
of real numbers, the sum of 5 and !2 is
5 1 !2. Note that the real number 5 1 !2 is
in simplest form; it cannot be written as a
monomial. If we wish to simplify a number or
an algebraic expression that is written as the
sum of two terms, we must be able to express
each term as the product of a rational number
and a common factor.
Compare the addition of two fractions to
the addition of like monomial terms and radicals. Point out how the distributive property is
used in each case.
1. Fractions:
3
7
108 5 2 ? 54
1 27 5 17 (3) 1 17 (2)
5 17 (3 1 2)
5 17 (5) 5 57
108 5 3 ? 36
Since 54 is not a perfect square, 2 ? 54 is not
a useful pair of factors. However, 3 ? 36 gives us
the largest perfect-square factor, and !108 in
simplest form is 6 !3. Point out that it is helpful to know the square of the integers from
1 to 25 and the cubes of the integers from 1
to 10.
Fractional radicands will be reconsidered
in Section 3-7 as an application of rationalizing
the denominator. The approach shown in this
section, utilizing the idea of equivalent fractions, enables students to simplify fractional
radicands before the multiplication rule for
radicals is introduced. Point out that to
simplify #q , we multiply the radicand
np
qn21
qn21 ,
p
q
by
which results in the rational denom-
inator q.
2. Like monomials:
3a 1 2a 5 a(3) 1 a(2)
5 a(3 1 2)
5 a(5) 5 5a
3. Like radicals:
3 !2 1 2 !2 5 !2(3) 1 !2(2)
5 !2(3 1 2)
5 !2(5) 5 5 !2
4. Unlike radicals that can be simplified:
!12
2
1 4 !3 5 !42 ? 3 1 4 !3
2 !3
2
1
5
1
1 4 !3
5 (1 1 4) !3 5 5 !3
In Example 1, the terms !12 and #13 can
be simplified and combined; likewise, the
terms !20 and 2!45 can be simplified and
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Real Numbers and Radicals
combined. However, the resulting terms 73 !3
and 2!5 do not share a common radicand
and cannot be combined.
Similar procedures are used with subtraction.
Students often make the mistake
!a 1 b 5 !a 1 !b . A simple counter
example is provided by !16 1 !9.
!16 1 !9 5 4 1 3 5 7
!16 1 9 5 !25 5 5 and 7 2 5.
Students may also use their calculators to
check examples that do not contain perfect
squares, such as !19 1 !31 2 !19 1 31.
3-5
MULTIPLYING
RADICALS
Emphasize that two radicals to be multiplied
can be combined under a single radical sign
only if they have the same index. Additionally,
the multiplication is valid only if the radicals
represent real numbers.
Point out the use of the multiplication
properties and compare the operations with
similar operations involving variable quantities.
Expression with variables:
3x ? 2y
5 3(x ? 2)y
5 3(2x)y
5 (3 ? 2)(xy)
5 6xy
Associative prop. of mult.
Commutative prop. of mult.
Associative prop. of mult.
Closure under mult.
Expression with radicals:
3!2 ? 2 !5
5 3A !2 ? 2B !5
Associative prop. of mult.
5 3A2 !2B !5
Commutative prop. of
mult.
5 (3 ? 2)A !2 ? !5B Associative prop. of mult.
5 6!10
Closure under mult.
Call attention to Example 2, part c. In general, if a and b are rational numbers, then
A !a 1 !bB A !a 2 !bB 5 a 2 b , a rational
number. (Students may recognize the similarity to the method for factoring the difference
33
of two squares.) This method of obtaining a
rational number is the basis for the work in
Section 3-7.
In Enrichment Activity 3-5: A Radical
Sequence, students add and multiply radical
expressions to investigate the properties of a
sequence that is both Fibonacci-like and geometric. If 1, k, k 2 , . . . is a Fibonacci-like
sequence, then 1 1 k 5 k2 or k2 2 k 2 1 5 0.
Using the quadratic formula in Chapter 5, the
solutions to the equation are 1 62 !5. This
activity shows the relationship between the
golden ratio and the Fibonacci sequence.
3-6
DIVIDING RADICALS
Demonstrate that in a radical of the form #ba ,
the numerator and denominator of the fraction may be simplified separately or together.
For example,
!2
5 !25
5 52 !2
5 52
!4 !2
!2
25
5
or #50
8 5 #4 5 2
50
#8 5
!50
!8
Both approaches are shown in Examples 1 and
2. In Section 3-7, students will see that a third
method can also be applied:
50
#8 5
5
5
5
!50 !8
?
!8 !8
!50 !8
8
!400
8
20
5
8 5 2
Students should practice the different
methods to become fluent, but in future
work they can choose the approach they find
easiest.
3-7
RATIONALIZING A
DENOMINATOR
To rationalize a denominator is to remove all
radicals from the denominator. Before calculators, denominators were rationalized
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Chapter 3
because it is easier to divide by an integral
value than by a radical. Today we rationalize
denominators to have a standard way of
expressing values within the mathematics and
science communities. If a denominator is
composed of a single radical term, then to
rationalize a denominator, multiply both the
numerator and the denominator of the fraction by:
• the denominator, or
• a radical that will result in the radicand in
the denominator having a power equal to
the index.
The second method is illustrated in Example 3. The first method could be used in this
case, but the work is more tedious:
6
!8
6 !8
5 !8
? !8
5 6 !8
8
5 6 ? 28!2
5 128!2 5 3 !2
2
When the denominator of the rational
expression is a binomial that contains a radical, we rationalize the denominator by multiplying the numerator and the denominator of
the fraction by the conjugate of the denominator. The conjugate of a binomial is a binomial
having the same two terms with the sign of the
second term changed. When a binomial is multiplied by its conjugate, the products of the
outer and inner terms will sum to zero.
In Example 4, a linear equation with irrational coefficients is solved. The equation is
not a radical equation (Section 3-8) because
there are no terms with a variable in the radicand.
Exercises 36 and 38 present a higher level
of difficulty. You may wish to assign these
exercises as group work or use them as bonus
questions on a test.
3-8
SOLVING RADICAL
EQUATIONS
When solving a radical equation, the equation
derived by squaring both sides is not equiva-
lent to the given equation. In other words, the
derived equation does not always have the
same solution as the original equation. If a
number is a root of the original equation, that
number must also be a root of the derived
equation, but the converse of this statement is
not true.
It is helpful to demonstrate why an extraneous root occurs. In Example 3, the derived
equation x2 2 2x 1 1 5 15 2 7x is the result
of squaring both sides of the equation
x 2 1 5 !15 2 7x . However, the same
derived equation is also the result of squaring
the equation 1 2 x 5 !15 2 7x. The roots of
x2 2 2x 1 1 5 15 2 7x are 27 and 2; 2 is the
root of x 2 1 5 !15 2 7x, and 27 is the root
of 1 2 x 5 !15 2 7x.
A common mistake made by students is to
incorrectly square the left side of
(x 2 1) 2 5 A !15 2 7xB 2
Some students will incorrectly write
x2 2 1 5 15 2 7x
or
x 2 1 5 15 2 7x
Remind them of the FOIL method and have
them write out (x 2 1)2 5 (x 2 1)(x 2 1).
You may wish to show the solution of an
equation involving two radicals in which
squaring both sides does not eliminate the radical at first. For example:
!2x 1 3 5 !x 1 1 1 1
1st Squaring:
2x 1 3 5 x 1 1 1 2 !x 1 1 1 1
2x 1 3 5 x 1 2 1 2 !x 1 1
x 1 1 5 2 !x 1 1
2nd Squaring:
x2 1 2x 1 1 5 4(x 1 1)
x2 1 2x 1 1 5 4x 1 4
x2 2 2x 2 3 5 0
(x 2 3)(x 1 1) 5 0
x2350
x53
x1150
x 5 21
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Real Numbers and Radicals
Since both roots make the given equation
true, the solution set is {21, 3}.
EXTENDED TASK
For the Teacher:
This extended task is designed to help students make connections among various
branches of mathematics. Geometry, arithmetic operations, and algebra are all
addressed in the task. Students are expected to
know the following from previous work:
1. The relationship that the altitude drawn
to the hypotenuse of a right triangle is the
mean proportional between the segments
formed on the hypotenuse.
35
2. The locus of the vertices of a right triangle with a given segment as hypotenuse is
a circle with the given hypotenuse as a
diameter.
3. The construction to find the midpoint of a
line segment.
4. The construction of a line perpendicular
to a given line at a given point on the line.
5. How to solve problems involving percent.
6. How to measure the length of a line segment to a stated accuracy.
7. How to convert fractions to decimals.
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ENRICHMENT ACTIVITY 3-2
A Square-Root Algorithm
An algorithm is a process or a method used in calculation. Follow the steps of the
algorithm shown in the example below to approximate the square root of 2,210.
.
"22 10 .
STEP 1. Write the number, placing a decimal point after the number
and above the number. Starting at the decimal point, separate
the number into pairs of two digits. Treat any number with an
odd number of digits as a number with a leading 0; for example, treat 729 as 07 29 .
XX
XX
4
STEP 2. Find the largest perfect square that is less than or equal to the
first pair of digits. Write that square below the first group of
digits and its square root above the digits.
STEP 3. Subtract the square from the first pair of digits, and bring down
the next pair of digits.
STEP 4. Double the 4 above the radicand and write this value with a
blank line next to it as shown.
STEP 5. Find the largest single digit d so that 8d times d is less than or
equal to 609.
87 3 7 5 609
88 3 8 5 704 ✘ (Too large)
.
"X
22 X
10 .
16
4 .
"22 10 .
216
610
4 .
"22 10 .
216
(8_)
610
4 7.
"22 10 .
216
(87)
610
XX
XX
XX
Write 7, the value of d, on top of the blank and above the pair
X
10 .
STEP 6. Calculate 87 3 7 and write the product below 610. Subtract
and bring down the next pair of digits. (In this case the next
pair of digits is 00 .)
X
4 7.
"22 10 .00
216
610
(87)
2609
100
XX X
4 7.
STEP 7. Double the 47 above the radicand and write this value with a
blank line next to it as shown.
X.00
X
"X
22 10
216
610
2609
(94_)
100
(87)
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4 7. 0
STEP 8. Find the largest single digit d so that 94d times d is less than
or equal to 609.
X.00
X
"X
22 10
216
610
2609
(940)
100
(87)
940 3 0 5 0
941 3 1 5 941 ✘ (Too large)
Write 0, the value of d, on top of the blank and above the
pair 00 .
STEP 9. Calculate 940 3 0 and write the product below 100. Subtract
and bring down the next pair of digits.
X
XX XX
(87)
(940)
XX XX
STEP 10. Double the 470 above the radicand to obtain 940. The digit d
such that 9,40d 3 d is less than or equal to 10,000 is 1.
(87)
(940)
(9401)
To the nearest tenth, !2,210 < 47.0.
Note: If at any time the difference is 0 and all pairs of digits of the original number
have been used, the process is complete. In such a case, the exact square root has been
found:
3 5.
"1,225.
29
→ 352 5 1,225
(65)
325
2325
0
XX
Exercises
In 1–4, find the exact square root of each given number using the square-root
algorithm.
1. 3,249
2. 5,184
3. 8,281
4. 1,521
In 5–8, find the square root of each given number to two decimal places and round
the result to the nearest tenth.
5. 7
6. 17
Copyright © 2009 by Amsco School Publications, Inc.
7. 35
8. 88
4 7. 0
"22 10 .00 00
216
610
2609
100
20
10000
4 7. 0 1
"22 10 .00 00
216
610
2609
100
20
10000
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ENRICHMENT ACTIVITY 3-5
A Radical Sequence
A Fibonacci-like sequence is one in which each term is the sum of the previous
two terms. A geometric sequence is one in which the ratio of consecutive terms (common ratio) is a non-zero constant. Can a sequence have both properties? Consider
the sequence below:
1, 1 12 !5, 3 12 !5, . . .
1. Show that the sum of the first two terms is equal to the third term of the
sequence.
2. Find the fourth term by adding the second and third terms.
3. Find the fifth and sixth terms of the sequence. Write your answers in simplest
form.
4. Why is the sequence 1, r, r2, r3, . . . a geometric sequence? What is the common
ratio?
5. In the sequence above, what is the ratio of the second term to the first? Call this
ratio r.
6. Show that the third term of the sequence is r 2.
7. Show that the fourth term of the sequence is r 3.
8. Show that the fifth term of the sequence is r4.
9. Show that the sixth term of the sequence is r 5.
10. Use your calculator to find an approximate value of 1 12 !5. Use the library or
Internet to research the meaning of this ratio. What is the name of this ratio?
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EXTENDED TASK
Finding Square Roots Geometrically
Our real number system is composed of two major subsets, the set of rational
numbers and the set of irrational numbers. The word rational is derived from “ratio.”
Therefore, a rational number is any number that can be expressed as the quotient, or
ratio, of two integers. The rational numbers include all integers, zero, and all positive
and negative fractions. Fractional rational numbers may be expressed as either terminating or non-terminating decimals. In contrast, an irrational number cannot be
expressed as the ratio of two integers. Therefore, irrational numbers do not have
exact values. They can be approximated by a rational number and that approximation
can be stated as precisely as we choose.
In a right triangle, the measure of the altitude drawn to the hypotenuse from the
vertex of the right angle is the mean proportional between the measures of the two
segments of the hypotenuse. Therefore, in right triangle ACD below, if DB is the altitude drawn to the hypotenuse, then (DB)2 5 (AB)(BC) or DB 5 !(AB)(BC).
Therefore, if AB 5 1 and BC 5 3, then DB 5 !3.
D
A
B
C
You can use this knowledge to construct the square root of any whole number.
Exercise I
Follow the steps given in the procedure below to construct the square root of 3. Use
a straightedge and compass to make your constructions.
1. Draw line segment ABC, such that AB 5 1 inch and BC 5 3 inches.
a. What is the sum of AB and BC?
b. What is the product of AB and BC?
2. Find, by construction, the midpoint, M, of AC.
3. Construct the locus of the vertices of all right triangles that have AC as
hypotenuse.
4. Construct a line perpendicular to ABC at B. Label the point D where the
perpendicular intersects the locus in step 3.
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5. Is DB 5 !3 inches? Explain your response.
1
6. Use a ruler to find the length of DB to the nearest 16
of an inch expressed as
a decimal.
7. Use your calculator to find !3.
8. Find the percent of difference between your constructed !3 and the calculator
value for !3.
9. What accounts for this difference in step 8?
Exercise II
1. Use the same procedure to find the geometric representation of !5 and of !7.
1
2. Measure each line segment representation to the nearest 16
of an inch and
express as a decimal.
3. Find the ratio !5 : !7.
4. Express !5
as a fraction with a rational denominator.
!7
5. Use your calculator to find the value of the numerical expression you obtained
in step 4.
6. How close is your result in step 3 to the value obtained in step 5?
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Algebra 2 and Trigonometry:
Chapter Three Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–5, tell whether each number is rational or irrational.
1. 37
2. 0.14
3. p 1 6
3
4. !216
5. 0.01002000300004 . . .
In 6–9, find and graph the solution set of each inequality.
6. x , 5
7. a 2 3 $ 7
8. 2x 2 6 1 4 # 12
9. 3x 1 1 1 2 $ 28
In 10–13, evaluate each expression.
10. !441
3
11. 2!264
81
12. #121
4
13. !0.0016
In 14–17, write each radical in simplest radical form. All variables represent positive
numbers.
14. !180
15. "147b4
3
16. "
56a6
17. #25x5
72y
3
In 18–32, perform the indicated operation and simplify.
18. !80 1 !45
19. 4!24 1 !54
20. !75 2 !3
21. 26!50 1 4 !98
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22. A2 1 !3B A2 2 !3B
23. A5 1 !2B 2
24. 2!6A3!6B
25. 3!2A3 1 !2B
26. !2A !18 2 !8B
27. 4 1 2!20
28. 123 !18
!2
29. 3 12 !5
6
30. !5 2
!2
2 1
31. !2
!2 1 1
32. !x 42 y
In 33–36, solve each equation and check.
33. !2a 2 3 5 5
34. x !7 5 6 2 x
35. !3b 1 1 5 b 2 1
36. 2x 5 2!3x 2 2
4
4
37. Determine whether # 59 is less than, greater than, or equal to !330. Justify your
answer without the aid of a calculator.
38. The perimeter of a right triangle is 12 meters. The length of the longer leg is
3 times the length of the shorter leg. Find, to the nearest tenth, the measure
of each side of the triangle.
39. The length of a rectangle is x and the width is !3x 1 1. If the width is 1 less
than the length, what are the dimensions of the rectangle?
Bonus: Each of two congruent circles drawn in the interior of
a square is tangent to two adjacent sides of the square
and to the other circle, as shown in the diagram. If the
measure of a side of the square is 16, find the radius of
each circle.
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SAT Preparation Exercises (Chapter 3)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If
(x 2 3) 2
"x2 2 6x 1 9
(A) 1
(D) 5
7. If 1 2 x , x, then
(A) 212 , x , 0
(C) x . 12
(E) x , 22 or x . 12
5 2 and x . 0, then x 5
(B) 3
(E) 6
(C) 4
2. In a storage room for vegetables, the
Fahrenheit temperature F is controlled
so that it does not vary from 45° by
more than 6°. Which of the following
best expresses the possible range of temperatures?
(A) F 2 45 # 6
(B) F 2 45 . 6
(C) F 2 45 5 6
(D) F 2 6 # 45
(E) F # 39 or F $ 51
3. If 0 , y , x, which statement must be
true?
(A) !x 2 !y 5 !x 2 y
(B) !y 1 !y . !2y
(C) x !y 5 y !x
(D) x !x . y !y
(E) !x 1 !y , !x 1 y
4. If m is a positive number and m2 5 7, then
m3 is equal to
(A) !7
(B) 3 !7
(C) 14
(D) 7 !7
(E) 21
5. How many solutions does the equation
!x 1 10 5 4 !x have?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
6. If !x 1 6 5 x, then which of the following statements is true?
I. x 5 3 is a root.
II. x 5 9 is a root.
III. The sum of the roots is 13.
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I and III only
Copyright © 2009 by Amsco School Publications, Inc.
(B) 0 , x , 12
(D) x , 212
8. If 2 !x 1 3 5 8, then !4x 1 12 is
(A) 1
(B) 5
(C) 8
(D) 13
(E) 15
9.
4 5 16 if
1
(A) x 5 !x 1 !x
(B) x 5 !x 2 x2
1
(C) x 5 x2 1 !x
(D) x 5 x 2 8
(E)
10.
x 5 x 2 10
A
E
D
C
B
Equilateral ADE has sides of !2 and
AC 5 !3. What is the perimeter of
DEBC?
(A) 3 !3 2 2 !2
(B) 3 !3 2 2 !2 1 2
(C) 2 !3 2 3 !2 1 3
(D) 3 !3 2 !2 1 3
(E) !3 2 2 !2 1 3
11. If !x 1 !y 5 !10 and x 2 y 5 5 !2,
then !x 2 !y is
(A) !10
(B) 2 !10
(C) !5
(D) 2 !5
(E) 5 !2
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14.
15.
If AB 5 !3 and BC 5 !5, then the area
of circle O is
(A) 2p
(B) 2!2p
(C) 4p
(D) 8p
(E) 4!2p
13.
A
R S
is a “product-sum
x
4
x 1
is a “product-sum
tandem” if x equals
(A) 1
(B) 2
(D) 4
(E) 5
(C) 3
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. If 4 !2x 1 1 5 20, then what is the value
of x?
B
D
4
tandem” if R and S are
(A) 2 and 6
(B) 4 and 4
(C) 2 and 2
(D) 2 and 4
(E) 1 and 7
C
O
2
17. If x 2 12 # 4 and y 2 8 # 5, what is the
greatest possible value of x 2 y?
C
The diameters AC and BD are perpendicular. If AC 5 2!3 and
mA 5 mB 5 mC 5 mD 5 30,
then the perimeter of the polygon is
(A) 8 1 2 !3
(B) 8 1 4 !3
(C) 4 1 4 !3
(D) 16
(E) 12
Questions 14 and 15 refer to the following
definition.
A “product-sum tandem” is a figure like the
one below in which A !PB A !QB 5 !R 1 !S.
P
Q
For example, 3 6
R
8 2
S
is a
“product-sum tandem” because
A !3BA !6B 5 !8 1 !2.
Copyright © 2009 by Amsco School Publications, Inc.
18. What is the positive solution for
x 5 3 1 !9 2 x?
3
19. If x 5 A !8 B 2, what is the value of x?
20. Evaluate A8 2 !2B A8 1 !2B .
21. If x 2 y 5 16 and !x 2 !y 5 2, determine the value of !x 1 !y.
22. The diagonal of a square is 10 !2 units.
How many units long is the perimeter of
the square?
23. Evaluate !12 1 !27 2 !75.
3
24. If x 5 " !729, what is x 1 5?
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CHAPTER
4
RELATIONS
AND
FUNCTIONS
Aims
CHAPTER OPENER
• To identify relations and functions and
determine the domain and range of a function.
If the trucker must drive for 8 hours a day,
then d 5 8r. When r 5 50, d 5 400. When
r 5 60, d 5 480. The ratio dr is constant.
• To use functional notation and evaluate
functions for given values.
• To find the composition of functions and
the inverse of a function.
• To perform transformations with functions.
• To determine the center-radius form for
the equation of a circle.
• To understand direct and inverse variation.
The concept of a function is basic to the
study of advanced mathematics. This chapter
presents formal definitions of relation and
function and illustrates these ideas with mathematical and real-world examples. The Cartesian graph is the most useful tool for
understanding the properties of different
functions. Several familiar algebraic functions
are reviewed in this chapter and more complex relationships are also presented. The key
idea of function composition is explored and
mastery of this topic is essential for future
work in calculus.
400
50
5 480
60 5 8
As the rate of travel increases, the distance
also increases. The variables are directly
related.
If a trucker must drive 600 miles a day, then
600 5 rt. When r 5 50, t 5 12. When r 5 60,
t 5 10. The product rt is constant.
50 3 12 5 60 3 10 5 600
As the rate of travel increases, the time
required to travel a fixed distance decreases.
The variables are inversely related.
A similar relationship occurs when a fixed
amount of work must be completed. In the
classic example, the more workers painting a
house, the fewer number of hours required to
complete the job.
4-1
RELATIONS AND
FUNCTIONS
When a relation is finite, all ordered pairs can
be tested to describe the relation. When a relation is infinite, it is useful to use an equation,
an inequality, or a graph to indicate how
45
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Chapter 4
ordered pairs of the relation may be found or
how it can be determined whether or not a
given ordered pair is a member of the relation.
Every function is a relation. However, not
every relation is a function. A many-to-one
relation such as
{(1, 5), (2, 5), (3, 5), (4, 5)}
is a function, but a one-to-many relation such
as
{(1, 1), (1, 21), (4, 2), (4, 22)}
is not a function. Ask students to suggest some
real-world relations and decide if they are
functions. For example, the relation that
makes the correspondence between a child
and its mother is a function, but the relation
that makes the correspondence between a
mother and her children is not (one mother
can correspond to several children).
Be sure students understand the definition
of onto. A function is onto if each element of B
is mapped by at least one element of the
domain A.
When the relation is finite, inspection of
the ordered pairs will determine whether
there are two pairs with the same first element. When the relation is infinite, it is not
possible to list all of the ordered pairs for
inspection. Therefore, the vertical line test is
one useful way of determining whether a given
relation is a function. Since the coordinates of
all points on a vertical line have the same first
element, a relation having more than one
point on the same vertical line has more than
one point with the same first element and
therefore is not a function.
When determining the domain of an algebraic function, it is helpful to recall that the
sum, difference, or product of any two real
numbers is a real number. Therefore, any function for which the function value is defined
using only addition, subtraction, or multiplication can have as its domain the set of real
numbers. The domain of any polynomial function, for example, is the set of real numbers.
Often the domain can easily be determined by
identifying the real numbers that must be
excluded. If a function {(x, y) : y 5 f(x)} defines
y in terms of x, it is necessary to eliminate from
the domain the values of x that make:
1. the denominator of a fraction equal to 0.
2. the radicand of an even root less than 0.
4-2
FUNCTION NOTATION
Six common function notations are given in
this section. The notation y 5 f(x) is used to
show that y is a function of the variable x or
that the value of y depends on the value of x.
For this reason, the x-value is referred to as the
independent variable and the y-value is
referred to as the dependent variable.
Be sure students understand that the
parentheses in the symbol f(x) do not indicate
multiplication. For the function f(x) 5 2x 1 3,
f(2) 5 7 or y 5 7. Note that f(2) 5 7 tells us
both elements of the pair, (2, 7), but y 5 7 tells
us just the second element.
Explain that x is a “dummy” variable that
can be replaced by any symbol or expression
that can represent a number in the domain of
the function. For example:
f(x) 5 2x 1 3
f(a) 5 2a 1 3
f(a 1 2) 5 2(a 1 2) 1 3 5 2a 1 7
f(3x 2 1) 5 2(3x 2 1) 1 3 5 6x 1 1
You may wish to have students work Exercise 16 using the sales tax rate that applies to
your locality. For Exercise 17, students should
plan the size of the intervals they will use on
the x-axis (number of muffins) and y-axis
(profit) before they start to draw.
4-3
LINEAR FUNCTIONS
AND DIRECT
VARIATION
Be sure students understand that all linear
equations, other than equations of the form
x 5 a, will be functions. The graph of x 5 2 is
shown at the top of the next page. Note that it
is not a function since it does not pass the vertical line test. Equations of the form y 5 a are
functions since they pass the vertical line test.
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y
1
4-4
y
1
O
1
x
O
1
x
One-to-one functions are important
because they are the only functions with
inverses (Section 4-8). A function is one-toone if there is exactly one input for every output. In other words, if two output values are
the same, the two input-values must be the
same. The function y 5 x3 is one-to-one; the
function y 5 x3 2 x is not since the output 0,
for example, could result from an input of 1 or
an input of 21. The horizontal line test shows
which functions are one-to-one. It is helpful to
point out that any function with all even exponents cannot be one-to-one.
Students should be familiar with transformations of linear functions from previous
algebra courses. An alternative way of describing the appearance of the graph of af(x) from
that stated in the text is that as |a| increases,
the steepness of the graph increases. However,
the terminology “stretching” and “compression” are consistent with the transformation
approach.
A direct variation y 5 kx is a relationship
that indicates that one quantity is a multiple of
the other. The graph of a direct variation is
always a line through the origin. An equation
of a direct variation is a special case of the
slope-intercept form of a line, y 5 mx 1 b.
When m 5 k and b 5 0, y 5 mx 1 b becomes
y 5 kx.
In Hands-On Activities 1 and 2, students
explore transformations by working directly
from given graphs. Have students work individually or in pairs for these activities, then
lead a discussion of the work. Ask students to
explain how they used the given graph to
stretch the graph of f(2x), 2f(x), and so on.
Point out that Exercises 12 and 14 in HandsOn Activity 2 have two possible answers.
(Both 2f(x) and g(x) can be described as the
graph of f(x) reflected in the x-axis or the
y-axis.)
47
ABSOLUTE VALUE
FUNCTIONS
When x is positive or zero, the absolute value
function looks like the graph of y 5 x. When x
is negative, the absolute value function looks
like the graph of y 5 2x.
Point out that absolute value equations of
the form x 5 ay 1 b are not functions. For
example, the equation x 5 y contains two
points whose coordinates are (3, 3) and
(3, 23), indicating that an x-value corresponds
to more than one y-value.
Be sure students understand the graphing
method used in Example 2. Ask them what
graphs they would draw to solve 2x 1 4 $ 8.
(y 5 2x 1 4, y 5 8)
Exercises 13–16 have students explore the
transformations of the graph of y 5 x. If students understand the relationship between the
algebraic form and the graphic form, they will
be able to draw many absolute value graphs
directly from the given equation without having to plot ordered pairs. Similarly, they will be
able to give the equation of an absolute value
graph by inspection. These skills will provide
considerable advantage for the SAT and other
standardized tests.
4-5
POLYNOMIAL
FUNCTIONS
Emphasize that the coefficients an, an21, . . .
represent real numbers; the coefficient an is
non-zero, and the exponents of the variable
terms are all whole numbers. The degree of a
polynomial is determined by the greatest
exponent of the variable terms.
The section reviews the properties of second-degree polynomials; that is, quadratic functions. Students should understand that the
graphs of y 5 x2, y 5 (x 2 h)2, y 5 x2 1 k all
have the same shape. The difference is their
position. When the equation of a parabola is
expressed in the vertex form y 5 (x 2 h)2 1 k,
the vertex is (h, k) and the equation of the axis
of symmetry is x 5 h. As the values of h and k
change, the graph of y 5 (x 2 h)2 1 k is the
graph of y 5 x2 translated |h| units left or right
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and k units up or down. These transformations
are further explored in Exercises 21 and 22.
The x-intercept of a polynomial function is
the x-coordinate of the point where the graph
crosses the x-axis. The y-intercept is the
y-coordinate of the point where the graph
crosses the y-axis. A polynomial function has
exactly one y-intercept, but it can have several
x-intercepts depending on the degree of the
sign polynomial. The x-intercepts are also
called roots, zeros, or solutions of the polynomial, depending on the context of the problem.
The x-intercepts are often where the graph
goes from positive (above the x-axis) to negative (below the x-axis) or vice versa. Evendegree functions of the form y 5 x n are
tangent to the x-axis at the origin. Other evendegree functions may or may not intersect the
x-axis, depending on its location in the coordinate plane. An odd-degree function always
crosses the x-axis at least once, which means
that an odd-degree function always has at least
one real root. In the section, it is pointed out
that if a is a root of f(x), then (x 2 a) is a factor
of f(x); the converse is also true.
A turning point is where the graph changes
direction. The maximum number of turning
points for a function of degree n is n 2 1. The
maximum number of x intercepts for a function of degree n is n, but there may be fewer
intercepts than n depending on the function.
Have students graph y 5 x8 1 1 on their calculators. They will see the graph has no intercepts and only one turning point.
If y 5 f(x) is a polynomial function such
that f(a) , 0 and f(b) . 0, then the function
has at least one real root between a and b.
Intuitively, students should understand that
the only way for a polynomial function to
change from positive to negative is to go
through 0. Polynomial functions are continuous; the domain of any polynomial is the set of
real numbers, so nothing is omitted that would
allow a jump.
Although it is easy to locate the real roots
of higher-degree polynomial functions using a
calculator, students should be aware that for
many years, mathematicians such as Descartes
worked to establish methods for identifying
the roots of polynomial functions. Some of
their conclusions will be explored in the
Enrichment Activities. The calculator can
determine rational roots and very close
approximations of irrational roots because of
the work done by these mathematicians.
Additional concepts about roots, factors,
and remainders will be considered in Chapter
5 in the context of quadratic functions.
Enrichment Activity 4-5: The Method of
Finite Differences considers how a polynomial
function can be constructed if we are given the
values of the function for consecutive values
of x.
4-6
THE ALGEBRA OF
FUNCTIONS
Many functions arise as combinations of other
functions. For example, suppose an airplane is
flying over the Grand Canyon. Then at any
time, t, the height, h(t), of the airplane above
the bottom of the canyon is the sum of the
height, f(t), of the airplane above the rim of
the canyon and the depth, g(t), of the canyon
directly below the airplane and rim.
h(t) 5 f(t) 1 g(t)
Let f and g be functions. The domains of
f 1 g, f 2 g, and fg consist of those values of x
where both f(x) and g(x) are defined, that is,
the intersection of both domains. Because
division by 0 is excluded, the quotient gf is
the function whose domain consists of all
numbers x in the domains of both f and g
where g(x) 0.
Emphasize that the domain of a combined
function must be determined from the separate domains of f and g, rather than from the
rule for the combined function. Consider this
example:
Let f(x) 5 "4 2 x2 and g(x) 5 !x 2 1.
The domain of f is 22 # x # 2 and the domain
of g is x $ 1. Therefore, the domain of the
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product fg is the set of numbers in both
domains, or 1 # x # 2. The rule for fg is:
fg(x) 5 f(x)g(x) 5 "4 2 x2"x 2 1
5 "(4 2 x2)(x 2 1) (1 # x # 2)
However, note that the expression
"(4 2 x2)(x 2 1) is also meaningful for
2` , x # 22 . This is true because
(4 2 x2)(x 2 1) $ 0 for x # 22.
The domain must be based on the original
functions, not the combined function. The
same is true for f 1 g, f 2 g, and gf .
Note that any polynomial g(x) 5 anxn 1
an 2 1xn 2 1 1 c 1 a1x 1 a0 can be thought of
as a combination of constant functions and the
identity function I(x) 5 x (see Section 4-8).
g(x) 5 a fI(x)g n 1 a fI(x)g n21 1 c
n
n21
1 a1I(x) 1 a0
or
g 5 anIn 1 an21In21 1 c 1 a1I 1 a0
In the last part of the section, transformations in the plane are interpreted by applying
function arithmetic.
4-7
COMPOSITION OF
FUNCTIONS
A new function can be formed by using the
output of one function as the input to a second
function. This is called function composition.
Two ways of writing the composition of two
functions are given in the text. The symbol
f(g(x)) indicates more clearly than (f + g)(x)
the order in which the functions are to be evaluated. (In this case, first g, then f.) Emphasize
to students that we start with a value of x and
proceed to the left, first evaluating the function
written closer to x. With the notation f(g(x)),
indicate that we work from the inside out.
Students should be aware of domain
restrictions inherent in function composition.
Consider, for example, the composition
f(g(x)), where f(x) 5 x2 and g(x) 5 !x. Since
f(g(x)) 5 fA !xB 5 A !xB 5 x
2
49
students might erroneously conclude that the
domain of the composition is the set of real
numbers. It is, in fact, the set of non-negative
real numbers. The domain of f + g must be a
subset of the domain of g. In this example, the
domain of the composition is the same as the
domain of g.
As illustrated in Examples 1 and 3,
function composition is not a commutative
operation; that is, it is not always true that
f(g(x)) 5 g(f(x)).
Use Enrichment Activity 4-7: The Difference Quotient to prepare students for later
work finding derivatives. The difference
quotient can be thought of as the composition
of some function f(x) and another function
g(x) 5 x 1 h.
4-8
INVERSE FUNCTIONS
Be sure that students understand that f 21
denotes the inverse of the function f and does
not mean 1f.
To reinforce students’ understanding of the
meaning of an inverse function, use different
one-to-one functions to illustrate the concept.
Explain that the function f(x) 5 2x 2 4 is
formed by multiplying the variable x by 2 and
then subtracting 4. The inverse function is
found by performing the inverse operations, in
the opposite order. The inverse f21(x) is found
by adding 4 (the inverse of subtracting 4) to x
and then dividing the result by 2 (the inverse
of multiplying by 2):
4
5 12x 1 2
f21 (x) 5 x 1
2
The composition of a function and its
inverse function is the identity function, I(x) 5
x. Show this by applying the composition using
specific values:
f21 (f(x)) 5 I(x) 5 x
f
f21
h 12 (2) 1 2 5 3
f
f21
h 12 (26) 1 2 5 21
f
f21
h 12 (24) 1 2 5 0
3 h 2
21 h 26
0 h 24
f21
x h 2x 2 4 h 12 (2x 2 4) 1 2 5 x
f
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Chapter 4
f(f21 (x)) 5 I(x) 5 x
21
f
f
h 5
h 2(5) 2 4 5 6
f21
f
h 0
h 2(0) 2 4 5 24
f21
f
h 2
h 2(2) 2 4 5 0
21
f
f
h 12x 1 2 h 2 A 12x 1 2 B 2 4 5 x
The sets of ordered pairs f 5 {(23, 9),
(22, 4), (21, 1), (0, 0), (1, 1), (2, 4), (3, 9)} and
g 5 {9, 23), (4, 22), (1, 21), (0, 0), (1, 1), (4, 2),
(9, 3)} are inverse relations. Two relations are
inverse relations if and only if whenever one
relation contains the element (a, b), the other
relation contains the element (b, a). Have students explain in their own words the difference between inverse functions and inverse
relations.
Students are already familiar with the formula for converting Fahrenheit temperatures
to Celsius, C(x) 5 59 (x 2 32) , and Celsius to
Fahrenheit, F(x) 5 95x 1 32. Have students
verify that these are inverse functions.
6
24
0
x
4-9
CIRCLES
The distance and midpoint formulas are
reviewed in this section since they must be
applied to problems involving circles. Call
attention to the fact that the equation of a circle can be written if any one of the following
are given:
1. the coordinates of the center and length
of the radius.
2. the coordinates of the center and the
coordinates of one point on the circle.
3. the coordinates of the endpoints of a
diameter.
Next, they must complete the square twice,
once for each variable. As shown, work first
with the variable x, then with the variable y.
Make sure that the same constants are added
to both sides of the equation each time.
For Exercise 28b, students must determine
if the height of the arch at a distance of 3 feet
from the center is at least 6 feet.
4-10 INVERSE VARIATION
When two quantities vary inversely, it means
that as one quantity increases, the other quantity decreases, and vice versa. Refer to the situation described in the chapter opener. When
rate of travel is constant, distance varies
directly as time. When the distance is constant,
rate varies inversely as time. You may wish to
refer to some of the relationships in Section
4-3. By keeping a different quantity constant,
many of the same examples that were used to
illustrate direct variation can be used to illustrate inverse variation.
x
y
The proportion x1 5 y2 is only one of sev2
1
eral true proportions that can be formed given
x1 y1 5 x2 y2. Ask students to suggest others.
Note that in many applications of inverse
variation, only positive values are acceptable
replacements. However, when graphing the
function xy 5 c, which expresses inverse variation, both positive and negative values are
used. The resulting hyperbola is called a rectangular hyperbola.
Hands-On Activity
Instructions:
1. Draw the graph of xy 5 6.
Example 3 illustrates the method for writing the center-radius form of the equation of a
circle given the standard form. Suggest that
students should begin by placing all terms containing like variables together:
2. Fold the paper so that one branch of the
hyperbola coincides with the other. Open
the paper.
x2 1 3x 1 y2 2 4y 2 14 5 0
4. On a separate piece of graph paper, draw
the graph of xy 5 26 and repeat the
directions.
Then, move the constant to the right side of
the equation:
x2 1 3x 1 y2 2 4y 5 14
3. Fold the paper again so that each branch
coincides with itself.
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Discoveries:
1. The folds are lines of symmetry for the
graphs. The equations of these lines are
y 5 x and y 5 2x.
2. When the constant is positive and the axis
of symmetry is y 5 2x, the image of a
point on one branch of the hyperbola is a
point on the other branch.
3. When the constant is positive and the axis
of symmetry is y 5 x, the image of a point
on either branch of the hyperbola is a
point on the same branch.
4. When the constant is negative and the
axis of symmetry is y 5 x, the image of a
point on one branch of the hyperbola is a
point on the other branch.
5. When the constant is negative and the
axis of symmetry is y 5 2x, the image of a
point on either branch of the hyperbola is
a point on the same branch.
6. The graphs of both xy 5 6 and xy 5 26
have point symmetry. The origin is the
point of symmetry.
51
EXTENDED TASK
For the Teacher:
This activity is intended to help students
develop an understanding of inverse variation
and see that the hyperbola is the graphic representation of an inverse variation. Have students work in groups of three on this activity.
One student can pull the spring balance, a second can observe and read the measurements,
and the third can record the information.
Accuracy and neatness should be encouraged
throughout the activity.
You may wish to do this activity in connection with the science teacher at the same time
the science class is studying the concept of the
balance.
Materials for this inquiry might be
obtained from the science teacher or from an
elementary school teacher.
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Class
Date
ENRICHMENT ACTIVITY 4-5
The Method of Finite Differences
If you are given the function f(x) 5 x2 1 2x 1 4, you can find the values of the
function for different values of x:
f(1) 5 1 1 2 1 4 5 7
f(2) 5 4 1 4 1 4 5 12
f(3) 5 9 1 6 1 4 5 19
f(4) 5 16 1 8 1 4 5 28
f(5) 5 25 1 10 1 4 5 39
Suppose you are given the values of a polynomial function, but not the function
itself. The method of finite differences can be used to construct the original function.
Follow these steps:
1. Write the function values with some
space between values.
2. Find the differences between successive
terms. This is called the first difference. If
these differences are not all equal to the
same value, repeat to find the next set of
differences.
12
7
19
7
5
2
28
9
2
39
11
2
first difference
constant
3. Stop when the differences are constant. In this case, the second difference is
constant. The function is of the same degree as the constant difference, so the
function we are looking for is a second-degree function of the form:
f(x) 5 ax2 1 bx 1 c
If constant differences did not appear until the third line, then we would be
finding a third-degree function.
4. Use the given information to write three equations in three unknowns: a, b, and c.
Since f(1) 5 7,
a 1 b 1 c 5 7 (I.)
f(2) 5 12, 4a 1 2b 1 c 5 12 (II.)
f(3) 5 19, 9a 1 3b 1 c 5 19 (III.)
The number of equations is one more than the degree of the function. For
example, if the function were a third-degree polynomial, we would need to solve
a system of four equations in four unknowns.
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5. Solve the system using any method. For example, using elimination:
II–I.
4a 1 2b 1 c 5 12
2 a1 b1c57
3a 1 b
55
So
III–II.
9a 1 3b 1 c 5 19
2 4a 1 2b 1 c 5 12
5a 1 b
57
5a 1 b 5 7
2 3a 1 b 5 5
2a 5 2
a51
Substituting the value for a in the equations above gives:
b52
c54
The function is f(x) 5 x2 1 2x 1 4, which we know is correct.
Note that to use the method, the given function values must be successive terms.
In 1–4, the values given correspond to f(1), f(2), f(3), f(4), and f(5) of some polynomial function. Use the method of finite differences to find the function.
1. 6, 6, 8, 12, 18
2. 4, 15, 30, 49, 72
3. 216, 211, 0, 17, 40
4. 1, 7, 21, 49, 97
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ENRICHMENT ACTIVITY 4-7
The Difference Quotient
The expression below is called the difference quotient.
f(x 1 h) 2 f(x)
h
The difference quotient is the basis of an important concept in calculus. Graphically, the difference quotient represents the slope of a secant line for the function f(x).
y
e
line
f(x 1 h)
in
tl
en
f(x 1 h) 2 f(x)
h
ng
5
ta
change in f(x)
change in x
sec
ant
f(x) 5 x
2
f(x)
x x1h
It is reasonable to think that as h gets smaller and
smaller, the slope of the secant line approaches the
slope of the line tangent to the curve at f(x).
1. a. Form the difference quotient for f(x) 5 x2 and simplify it.
b. Evaluate the difference quotient if h 5 0.
2. a. Write the expression in simplest form: (x 1 h)3
b. Form the difference quotient for f(x) 5 x3 and simplify it.
c. Evaluate the difference quotient if h 5 0.
3. a. Write the expression in simplest form: (x 1 h)4
b. Form the difference quotient for f(x) 5 x4 and simplify it.
c. Evaluate the difference quotient if h 5 0.
4. a. Write the expression in simplest form: (x 1 h)5
b. Form the difference quotient for f(x) 5 x5.
c. Evaluate the difference quotient if h 5 0.
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x
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5. a. Summarize your results by completing the table below.
Function
Difference Quotient when h 5 0
f(x) 5 x2
f(x) 5 x3
f(x) 5 x4
f(x) 5 x5
b. Describe any patterns you observe in the table.
6. Predict the difference quotient when h 5 0 for each function:
a. f(x) 5 x6
b. f(x) 5 x9
c. f(x) 5 xn
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Date
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EXTENDED TASK
The Inverse Variation Hyperbola
This extended task is designed to help you understand
the concept of inverse variation by having you carry out an
experiment that illustrates this type of variation. You
should work in groups of three for this task. You will need
the following materials:
50
100
150
200
250
• a numbered balance (see diagram) containing pegs
equally spaced from the fulcrum from which weights
can be suspended
• several weights (for example, 50 g, 200 g, 500 g)
• a spring balance
Activity 1
1. Measure the distance, in centimeters, between pegs on the numbered balance
and between the fulcrum and the first peg.
2. Suspend a weight of any size on one of the pegs to the left of the fulcrum, and
determine its distance from the fulcrum.
3. Place your spring balance on the first peg to the right of the fulcrum, and exert
a force on the spring sufficient to balance the weight.
4. Repeat this procedure for each peg to the right of the fulcrum on your
numbered balance.
5. Organize and record your data in a table, showing the force required to balance
the given weight and the distance from the fulcrum for each peg to the right of
the fulcrum.
6. Examine your data to determine the relationship between the distance from the
fulcrum and the force you exerted on the spring in order to balance the given
weight.
Activity 2
1. Repeat this entire experiment using a different weight on the left-hand side of
the numbered balance. You may place the new weight on the same peg or on a
different one.
2. Does a similar relationship exist?
Activity 3
State in words and then write an algebraic expression for the relationship that you
found in these two experiments.
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Activity 4
1. On the same set of axes, construct a graph for each set of data you collected.
2. Describe the curves that you obtained in step 1.
Activity 5
1. Using the construction in Activity 4, reflect your curves over the line y 5 2x.
2. Describe in writing the reflected curves drawn for part 1.
3. What is the equation for the object and image curves?
This task illustrates inverse variation. In your own words, define inverse variation
and summarize, in writing, what you have learned from performing this task.
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Algebra 2 and Trigonometry:
Chapter Four Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–5, in each case: a. State the domain of the relation. b. State the range of the
relation. c. State whether the indicated set of ordered pairs defines y as a function
of x. If not, explain.
1. {(22, 3), (21, 2), (0, 1), (1, 2), (2, 3)}
2. x2 1 y2 5 9
3. y 5 !9 2 x
4.
y
3
3 x
O
5.
y
1
O
1
x
6. If f(x) 5 3x2 2 2x, find:
a. f(21)
b. f(5)
8. If h(x) 5
7. If g(x) 5 2x 12 1, find:
a. g(0)
b. g(23)
1
4x 1 1, what
real number is not an element of the domain of h?
9. A constant function is defined by the equation y 5 26. What are the domain
and range of this function?
10. x varies directly as y. If x is 6 when y is 24, find x when y is 40.
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11. The cost of roses, c, is directly proportional to the number of roses purchased, r.
One rose costs three dollars.
a. Write an equation for the relationship described.
b. Is the relationship a direct variation?
c. Find the cost of nine roses.
12. Let f(x) 5 8x. The graph of g(x) is the graph of f shifted 4 units down and
6 units to the right. Write an expression for g(x).
13. A polynomial function of degree three intersects the x-axis at (26, 0),
(22, 0), and (3, 0) and intersects the y-axis at (0, 236). If y 5 p(x), find p(x).
14. If f(x) 5 x2 1 x 2 6 and g(x) 5 x 2 2, find the following:
a. (f 1 g)(x)
b. (g 2 f)(x)
c. (fg)(21)
15. If f(x) 5 x 1 1 and g(x) 5 !x, find each of the following:
a. f(g(9))
b. g(f(9))
c. What is the smallest integer in the domain of g(f(x))?
16. If f(x) 5 x2 and g(x) 5 3x 2 1, find the rule for each composition:
a. f + g(x)
b. g + f(x)
1
17. If r(x) 5 2x 2 1 and s(x) 5 x 1
2 , find each of the following:
a. r(s(3))
b. s(r(3))
In 18–20, write the inverse function in the form y 5 mx 1 b.
18. y 5 2x 2 1
19. y 5 x 1 7
20. y 5 5 2 12x
21. Assume that f(x) has an inverse.
a. If the graph of f(x) lies in the first quadrant, in which quadrant does the
graph of f21 (x) lie?
b. If the graph of f(x) lies in the second quadrant, in which quadrant does the
graph of f21(x) lie?
22. For each of the following, determine if the given function has an inverse.
a. f(x) 5 {(22, 24), (3, 27), (5, 3), (26, 0)}
b. f(x) 5 {(24, 2), (5, 3), (0, 2), (3, 7)}
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23. a. Find the center-radius form of the circle whose equation is
x2 1 6x 1 y2 2 2y 1 6 5 0.
b. Determine the center and radius of the circle.
24. Write an equation for a circle that has its center at (24, 3) and passes through
(21, 7).
25. If y varies inversely as x and y 5 4 when x 5 12, find y when x 5 20.
In 26–29, draw each required graph on a separate piece of graph paper.
26. Use a graph to solve the following:
a. 2x 2 1 5 5
b. 2x 2 1 , 5
c. 2x 2 1 . 5
27. a. Graph y 5 x2 2 5x 1 4.
b. Use the graph to find the solution set of x2 2 5x 1 4 $ 0.
28. a. Draw the graph of y 5 x2 1 2.
b. On the same set of axes, draw the reflection in the line y 5 x of y 5 x2 1 2.
c. Is the reflection drawn in part b a function?
29. a. Write an equation for the relationship between x and y if y varies inversely
as x and y 5 3 when x 5 2.
b. Sketch the graph.
Bonus: For what values of x does f(x) 5 f21(x) if f(x) 5 x 21 1.5?
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SAT Preparation Exercises (Chapter 4)
I. MULTIPLE CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If f is a one-to-one function, f(g(4)) 5 8
and f(4) 5 8, then g(4) equals
(A) 1
(B) 2
(D) 8
(E) 16
(C) 4
x2 2 4
2. For which values of x is f(x) 5 1 2
x 2 5
not defined?
(A) 1 only
(B) 22 and 2
(C) 21 and 4
(D) 24 and 6
(E) 26 and 5
3.
y
1
O 1
x
If the solid curve is the graph of f(x) 5 |x|,
then the broken curve could be the graph
of the function
(A) g(x) 5 x 1 3
(B) g(x) 5 x 2 3
(C) g(x) 5 x 2 3
(D) g(x) 5 x 1 3
(E) g(x) 5 3x
4. If f(x) 5 2x2 2 x2 , which has the smallest
value?
(A) f(22)
(B) f(21)
(C) f A 212 B
(E) f(1)
(D) f A 12 B
1
5. If f(x) 5 x 2
2 and g(x) 5 2x 1 1, then
f(g(3)) 2 g(f(3)) equals
(A) 0
(B) 3
(D) 5
(E) 7
(C) 4
Copyright © 2009 by Amsco School Publications, Inc.
6. If f(x) 5 x 1 1, then f(f(f(x))) equals
(A) x 1 3
(B) 3x 1 3
(C) x3 1 3
(D) 3x 1 1
(E) x3 1 1
7. A circle is tangent to the y-axis and has its
center at (28, 5). The equation of the circle is
(A) x2 1 y2 5 25
(B) (x 1 8)2 1 (y 2 5)2 5 64
(C) (x 2 8)2 1 y2 5 25
(D) (x 2 8)2 1 (y 1 5)2 5 64
(E) x2 1 64 1 y2 1 25 5 0
8. If f 5 {(22, 6), (21, 1), (0, 8), (1, 21),
(2, 4)}, which statement is true?
(A) f(1) 5 f(21)
(B) f(0) 5 f(22) 1 f(2)
(C) 2f(2) 5 f(0)
(D) f(f(21)) 5 f(f(1))
(E) f(22) 5 f(0) 2 f(2)
9. If f(x) 5 f(2x) for all x in the domain,
which equation could define f?
(A) f(x) 5 2x
(B) f(x) 5 x2 2 5
(C) f(x) 5 3x 1 1
(D) f(x) 5 x3 1 x
(E) f(x) 5 x2 2 x3
10. If f(x) 5 3x 2 5 and g(x) 5 x 1 10, which
of the following is equivalent to 3x 1 25?
(A) f(x) 1 g(x)
(B) f(x) 2 g(x)
(C) 6g(x) 2 f(x)
(D) f(g(x))
(E) g(f(x))
11. If f(x) 5 3x and g(x) 5 5x 1 1, what is
g(f(4))?
(A) 61
(B) 63
(D) 241
(E) 252
(C) 83
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12.
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
y
C
B
E
1
Date
O
1
16. If f(x) 5 16 2 x2 and the domain is
24 # x # 5, what is the largest value in
the range?
x
D
17. What is the sum of the coordinates of the
center of the circle whose equation is
A
x2 1 y2 2 6x 2 10y 1 30 5 0?
In the figure above, which lettered point,
other than the origin, lies in the interior of
the circle x2 1 y2 5 16?
(A) A
(B) B
(D) D
(E) E
(C) C
13. If it takes 4 workers 6 hours each to pave
a road, how many hours will it take 12
workers to complete the same job?
(A) 2
(B) 3
(D) 8
(E) 18
14.
(C) 6
y
18. If f 5 {(0, 2), (2, 4), (4, 2)}, what is the
value of f(f(f(0))) 1 f(f(f(2)))?
19. If f(x) 5 2 !x, what is the value of
f(2) ? f(18) ?
20. If 4 painters can paint a house in 9 days,
how many days would it take 6 painters to
paint the house?
21. If f(x) 5 3x 2 4, what is the value of
f21(11)?
22. If f(x) 5 x2 1 x3 and g(x) 5 !x, then
f(g(0.01)) is equal to what number?
23. If f(x) 5 2x 1 1 and 3 2 f(x) 5 x, what is
the value of x?
1O
1
24.
y
f(x)
x
g(x)
1O
1
x
Let f(x) be defined by the graph above. If
f(24) 1 f(2) 5 f(k), then k is equal to
(A) 23
(B) 21
(D) 1
(E) 3
(C) 0
15. If f(x) 5 4x 2 2, then the inverse function
f21(x) is
(A) x4 1 2
(B) x 1 12
2
(C) x 1
4
(D) x 2 12
2
(E) x 2
4
Copyright © 2009 by Amsco School Publications, Inc.
Let f(x) and g(x) be defined by the graphs
above. What is the value of 3f(g(3))?
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CHAPTER
5
QUADRATIC
FUNCTIONS
AND COMPLEX
NUMBERS
Aims
• To use the discriminant to determine the
nature of the roots of a quadratic equation.
• To solve quadratic equations using the
quadratic formula.
• To solve linear-quadratic systems.
• To define the imaginary unit i and simplify
powers of i.
• To add, subtract, multiply, and divide complex numbers.
• To apply the formulas for the sum and
product of the roots of a quadratic equation.
• To solve higher-order polynomial equations using factoring and/or quadratic
form.
This chapter introduces a new number,
!21 or i, and the set of complex numbers that
can be formed using i. Students learn to solve
quadratic equations using the completing-thesquare procedure or the quadratic formula.
The discriminant provides information about
the roots of a quadratic equation and, if the
roots are known, an equation with those roots
can be written. Quadratic methods can be
applied to higher-order polynomial equations
that have special forms.
The chapter opener leads into a discussion
of the procedure for completing the square
and the derivation of the quadratic formula.
5-1
REAL ROOTS OF A
QUADRATIC
EQUATION
Students have learned that if x2 5 a, where a is
a real number, then x 5 6 !a. This property
can be used to solve simple quadratic equations such as x2 5 2. The purpose of the completing-the-square procedure is to transform a
given trinomial into an equivalent expression
that is the square of a binomial so that it is
possible to take the square root of both sides
of the equation.
Emphasize that the coefficient of the
squared term must be 1 before the steps for
completing the square are applied.
This section relates the roots of a quadratic
equation, ax2 1 bx 1 c 5 0, to the points where
the graph of y 5 ax2 1 bx 1 c intersects the xaxis. In the coordinate plane, every point represents a pair of real numbers. Therefore, it is
possible to locate graphically only real roots.
The form y 5 a(x 2 h)2 1 k is the most useful way of writing a quadratic function so that
it may be graphed. Students should recognize
that x 5 h is the equation of the axis of
symmetry for the parabola and (h, k) are
the coordinates of the vertex. Note that if k
is positive and the parabola opens upward,
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the graph cannot intersect the x-axis, so
a(x 2 h)2 1 k 5 0 has no real roots.
In Example 4, be sure students observe
that since 1 was subtracted inside the parentheses, the 1 must be multiplied by the coefficient outside the parentheses, 2, when it is
separated from the trinomial that is the perfect square binomial.
In Exercise 37, it may be necessary to
remind students that consecutive odd integers
can be represented as 2n 1 1 and 2n 1 3. Students must remember to divide to make the
coefficient of the n2 term equal to 1 before
completing the square.
5-2
THE QUADRATIC
FORMULA
Emphasize that only quadratic equations with
rational roots can be solved by factoring in the
set of integers. However, any quadratic equation can be solved by using the quadratic formula. Even if a quadratic is factorable, in cases
where the numbers are large and there are
many possible factor pairs, it is often preferable to simply use the formula. When all the
coefficients of a quadratic equation have a
common factor, divide both sides of the equation by the common factor before using the
quadratic formula. Calculation is greatly simplified with smaller values of a, b, and c.
Remind students to be careful when they
use the quadratic formula. The entire numerator of the quadratic formula must be divided
by 2a. Only when both terms in the numerator
and the denominator have a common factor
may that common factor be divided out.
In Example 1, point out that 2 62 !6
are the
exact roots. Students may use their calculators
to find rational approximations of these roots;
the roots are 2.225 and 20.225 to the nearest
thousandth.
In Exercise 25, students explore a graphical
representation of the quadratic formula.
5-3
THE DISCRIMINANT
If a, b, and c, are rational numbers, the roots of
the equation ax2 1 bx 1 c 5 0 are determined
by the value of the discriminant, b2 2 4ac,
according to the table in the text.
If a, b, and c are irrational numbers, however, the table does not apply. For example, the
equation x2 2 2 !2x 1 1 5 0 has an irrational coefficient. Although the discriminant is
4, a positive perfect square, the roots of the
equation are not rational, but are the irrational numbers !2 6 1. If a, b, and c are imaginary numbers, the rules again do not apply.
For example, the discriminant for the equation
x2 2 4ix 2 5 5 0 is 4, but the roots of the equation are the complex numbers 2i 6 1.
The graphs in this section illustrate the
relationship between the nature of the roots
determined by the discriminant and the number of times the graph of the related function
intersects the x-axis.
The example below shows that the same
parabola can be used to solve many different
equations that have real roots or to decide that
the equation has no real roots.
Example: Sketch the graph of
y 5 2x2 2 2x 1 3, and from the graph determine the roots of the following equations:
a. 2x2 2 2x 1 3 5 0
b. 2x2 2 2x 2 1 5 0
c. 2x2 2 2x 1 8 5 0
d. 2x2 2 2x 2 2 5 0
y55
y
y54
y50
O
x
y 5 25
a. The equation 2x2 2 2x 1 3 5 0 can be expressed as the intersection of the parabola
y 5 2x2 2 2x 1 3 and the line y 5 0.
Answer: The roots are 23 and 1.
b. The equation 2x2 2 2x 2 1 5 0 can be
expressed as 2x2 2 2x 1 3 5 4 and, thus,
can be expressed as the intersection of
the parabola y 5 2x2 2 2x 1 3 and the line
y 5 4.
Answer: The root is 21 (two equal roots).
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Quadratic Functions and Complex Numbers
c. The equation 2x2 2 2x 1 8 5 0 can be
expressed as 2x2 2 2x 1 3 5 25 and, thus,
can be expressed as the intersection of the
parabola y 5 2x2 2 2x 1 3 and the line
y 5 25.
Answer: The roots are 24 and 2.
d. The equation 2x2 2 2x 2 2 5 0 can be
expressed as 2x2 2 2x 1 3 5 5 and, thus,
can be expressed as the intersection of
the parabola y 5 2x2 2 2x 1 3 and the line
y 5 5.
There are no points of intersection because
there are no real roots.
Answer: There are no real roots.
The roots are imaginary.
65
number whose square is negative, we must
define a set of numbers to which the terms
positive and negative do not apply, that is, a set
of numbers that are not part of the real number system. The set of imaginary numbers is
such a set.
Call attention to the way in which multiplication is defined for imaginary numbers. For
example, !216 ? !24 2 !64; that is:
!a ? !b 2 !ab if a , 0 or b , 0
!216 5 !16 ? !21 5 4i
But
!24 5 !4 ? !21 5 2i
and
Thus,
!216 ? !24 5 4i ? 2i 5 8i2 5 28
In general:
5-4
THE COMPLEX
NUMBER SYSTEM
Imaginary Numbers
In the study of numbers, our first encounter
was with the set of counting numbers or natural numbers. As our mathematical knowledge
progressed, we asked the question, “Is it possible to have a number less than 0?” and we discovered the set of integers. Later, in answer to
the question, “Is it possible to have numbers
between the integers, for example, between 1
and 2?” we discovered the set of rational numbers. Finally, in response to the question, “Is it
possible to have a number whose square is 2?”
the irrational numbers were defined and were
combined with the rational numbers to form
the real number system.
The study of imaginary numbers can begin
with the question, “Is it possible to have a
number whose square is negative?” Most students will answer “No.” The next question
might be, “What properties of the real numbers make it impossible to have a real number
whose square is negative?” By the trichotomy
property, which is one of the order properties
of the real numbers, all non-zero numbers are
either greater than 0 (positive) or less than 0
(negative). In the set of real numbers, the
product of two positive numbers is positive
and the product of two negative numbers is
also positive. Therefore, the square of any nonzero real number is positive. In order to have a
!2a 5 !a ? !21 5 i !a if a $ 0
Pure imaginary numbers are multiplied by
expressing them in terms of i and then replacing i2 by its equal, 21, after multiplication is
complete.
Complex Numbers
The correspondence between the set of real
numbers and the points on a horizontal line is
familiar to students. If we begin with the real
number line, it is possible to use transformations to motivate the development of the complex number plane.
The product of any real number, a, and 21
is the opposite of a, 2a. On the real number
line, the image of any real number, a, under a
reflection in the origin is the opposite of a, 2a.
Therefore, multiplication by 21 can be associated with a reflection in the origin. A reflection in the origin is also a rotation of 180°
about the origin. Since i ? i 5 21, we might
identify multiplication by i as the transformation that accomplishes in two steps what multiplication by 21 does in one step. Therefore, if
multiplication by 21 can be associated with a
rotation of 180° about the origin, multiplication by i can be associated with a rotation of
90° about the origin.
If a positive real number, a, is rotated 90°
about the origin, its image, a point above the
real number line on the line perpendicular to
the real number line at the origin, should have
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Chapter 5
as its coordinate the product ai. If a negative
real number, 2a, is rotated 90° about the origin, the image, a point below the real number
line on the line perpendicular to the real number line at the origin, should have as its coordinate the product 2ai. The real number
line and the line that is perpendicular to it at
the origin are then the reference lines or axes
that are used to locate any complex number,
a 1 bi.
The point of intersection of the real axis
and the imaginary axis is 0 1 0i. This number is
defined to be the real number 0, not imaginary
and not both real and imaginary. Since 0i is not
considered to be an imaginary number and
since 2ai 1 ai 5 0i 5 0, a real number, the
set of pure imaginaries is not closed under
addition.
Students should be aware that two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are
equal.
5-5
OPERATIONS WITH
COMPLEX NUMBERS
To add or subtract complex numbers, combine
like terms; that is, combine the real parts and
combine the imaginary parts. In the previous
section, students modeled addition of complex
numbers on a coordinate plane. The diagonal
of the parallelogram that has sides representing the two complex numbers represents the
sum of the two complex numbers.
Since the additive inverse of c 1 di is
2c 2 di, subtraction can be defined as it is in
the set of real numbers:
(a 1 bi)(c 1 di)
5 (a 1 bi)(c) 1 (a 1 bi)(di) Distributive
property
5 ac 1 bi(c) 1 adi 1 bi(di)
Distributive
property
5 ac 1 bci 1 adi 1 bdi2
Commutative
property
5 ac 1 bci 1 adi 1 bd(21)
Definition of i
5 ac 1 bd(21) 1 adi 1 bci
Commutative
property
5 ac 1 bd(21) 1 (ad 1 bc)i Distributive
property
5 (ac 2 bd) 1 (ad 1 bc)i
Substitution
After students have observed that the procedure for obtaining the product of two complex numbers is similar to that for the
multiplication of two binomials, they will be
able to compute the product without memorizing the definition.
Transformations can be used to visualize
the product of two complex numbers. For
h
example, to find (2 1 i)(3 1 2i), let OA represent 3 1 2i and the transformations represent
2 1 i:
(2 1 i)(3 1 2i) 5 2(3 1 2i) 1 i(3 1 2i)
STEP 1: 2(3 1 2i) 5 6 1 4i
h
h
2 ? OA 5 OAr
yi
A9
A
x
O
a 1 bi 2 (c 1 di)
5 a 1 bi 1 (2c 2 di)
STEP 2: i(3 1 2i) 5 3 1 2i2
5 3i 1 2(21)
5 fa 1 (2c)g 1 fb 1 (2d)gi
When we define multiplication and division in the set of complex numbers, we want
the familiar properties (commutative, associative, and distributive) to be valid. Therefore, multiplication is defined using these
properties:
5 22 1 3i
h
h
i ? OA 5 OB
yi
A9
B
A
O
x
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Quadratic Functions and Complex Numbers
STEP 3:
(2 1 i)(3 1 2i)
5 2(3 1 2i) 1 i(3 1 2i)
h
h
5 2 ? OA 1 i ? OA
h
h
5 OAr 1 OB
h
5 OC
5 4 1 7i
yi
C
B
A9
A
O
x
Students should be familiar with complex
conjugates. Conjugates are often used to
change a fraction whose denominator is a
complex number to an equivalent fraction
whose denominator is a real number. Also,
beginning in Section 5-6, students will see that,
given a quadratic equation with real coefficients, complex roots occur in conjugate pairs.
In previous discussions, students have
learned that binomials of the form a2 1 b2 cannot be factored in the set of real numbers. Now
we see that these binomials can be factored in
the set of complex numbers:
a2 1 b2 5 (a 1 bi)(a 2 bi)
x2 1 4 5 (x 1 2i)(x 2 2i)
25 1 y2 5 (5 1 yi)(5 2 yi)
The same principle is used to find the multiplicative inverse of a complex number and
the quotient of two complex numbers. To write
a
1 bi
a complex number of the form ca 1
di or c 1 di
in the form x 1 yi, it is necessary to change the
fraction to an equivalent one whose denominator is a real number. This is accomplished by
multiplying numerator and denominator by
the conjugate of the denominator.
Point out that the division process for complex numbers is converted to a multiplication
process. This is analogous to the situation with
67
fractions: division of fractions is converted to
the process of multiplication by the reciprocal
(multiplicative inverse) of the fraction.
Finally, it is interesting for students to
consider cases where a 1 bi and its conjugate
a 2 bi are multiplicative inverses. This
occurs when a 2 1 b 2 5 1. For example,
5
12
A 135 1 12
13i B A 13 2 13i B 5 1. Challenge students
to find other examples.
See Enrichment Activity 5-5: Complex
Number Operations, Vectors, and Transformations. Multiplication by a complex number
involves three basic transformations: two dilations and a rotation of 90°, as shown in the
text. By investigating graphs produced by
powers of i and powers of a 1 bi (a 0, b 0),
students will discover patterns and learn why
multiplication by a complex number is a transformation called a spiral similarity. (This activity sheet may be used at any time after
Sections 5-4 and 5-5 have been taught.)
5-6
COMPLEX ROOTS OF
A QUADRATIC
EQUATION
Any equation of the form ax2 1 bx 1 c 5 0
has, in the set of complex numbers, roots that
may be real or imaginary. Review the conditions under which the quadratic equation has
real roots. If the roots are not real, then they
are imaginary. If the coefficients a, b, and c are
real numbers, the roots will be real numbers
when the discriminant is greater than or equal
to 0 and imaginary when the discriminant is
less than 0. Only when the coefficients are real
numbers can the discriminant be used in this
way to determine the nature of the roots.
Enrichment Activity 5-6: Quaternions can
be assigned anytime after Section 5-6. An
advanced and abstract topic, the extension of
the complex numbers to a four-tuple system is
presented to give students an opportunity to
work with the special rules that apply to this
system. Quaternions satisfy all field postulates
except commutativity.
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Chapter 5
5-7
SUM AND PRODUCT
OF THE ROOTS OF A
QUADRATIC
EQUATION
Students should be familiar with the two
methods, the reverse factoring technique and
the use of the sum and products of the roots,
for determining an equation from the roots.
Notice that in both methods, as demonstrated
in the text, an equation is written whose leading coefficient a is 1. Any multiple of the equation in the text will also be a solution.
For example, to write a quadratic equation
whose roots are 10 and 22, we may use the
reverse factoring technique:
(x 2 10)(x 1 2) 5 0 S x2 2 8x 2 20 5 0
Any constant, however, may also be used as a
factor. In this way:
4(x 2 10)(x 1 2) 5 0 S 4x2 2 32x 2 80 5 0
1
2 (x
2 10)(x 1 2) 5 0 S 12x2 2 4x 2 10 5 0
Any equation of the form kx2 2 8kx 2 20k 5 0
(k 0) is an acceptable quadratic whose roots
are 10 and 22.
Similarly, when using the sum and the
product of the roots, we can choose a to be any
value. In most cases, we let a 5 1 or we choose
the equation with coefficients that are relatively prime. However, there are times when a
value of a larger than 1 will give the simpler
equation.
Example: Write a quadratic equation
whose roots are 21 and 213.
Solution:
The sum of the roots is 2ba 5 12 2 13
5 16
The product of the roots is ac 5 A 12 B A 213 B
5 216
Let a 5 6.
Then 2b6 5 16
Also, 6c 5 216
2b 5 1
or c 5 21
or b 5 21
Answer: A quadratic equation with roots 21
and 213 is 6x2 2 x 2 1 5 0.
5-8
SOLVING HIGHER
DEGREE POLYNOMIAL
EQUATIONS
Cubic equations can be solved with the quadratic formula if a quadratic factor can be
found. Review the special formulas for cubics:
Sum of two cubes:
a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)
Difference of two cubes:
a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)
When the polynomial has three terms, it is
sometimes effective to break up a term so that
the expression can be factored by grouping.
For example, in Exercise 10,
f(x) 5 x4 1 5x2 1 4
5 x4 1 x2 1 4x2 1 4
Rewrite 5x2 as
x2 1 4x2
5 x2(x2 1 1) 1 4(x2 1 1)
5 (x2 1 4)(x2 1 1)
Another method to apply to Exercise 10 is
to use the quadratic form. Equations that can
be written in the form a[f(x)]2 1 b[f(x)] 1 c
are said to be in quadratic form. Then,
f(x) 5 x4 1 5x2 1 4
5 (x2) 2 1 5(x2) 1 4
5 (x2 1 4)(x2 1 1)
Each factor is then set equal to zero and
the imaginary roots are easily found.
x2 1 4 5 0
x2 5 24
x 5 6 !24
x 5 64i
x2 1 1 5 0
x2 5 21
x 5 6 !21
x 5 6i
Sometimes one or both of the quadratic
form factors will again be factorable.
Some students find it easier to identify the
quadratic form if they use a “dummy” variable. In the example above, let u 5 x2, so the
equation becomes
f(a) 5 u2 1 5u 1 4.
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Quadratic Functions and Complex Numbers
When the trinomial is factored and solutions are found, it is necessary to complete the
work by substituting x2 for u and then solving
for x.
Students should be aware that when
synthetic substitution is used and the number
being tested is a root, the non-zero numbers
in the last step are the coefficient of
the reduced polynomial factor. So, for
f(x) 5 x4 2 3x3 1 x2 2 2x 1 3 and f(1), we have
1
23
1
1 22
1
22 21
3 1
21 23
22
23
0
Then, (x 2 1) is a factor and the other factor is (x3 2 2x2 2 x 2 3). Point out that when
synthetic substitution is used, a 0 must be written as the coefficient of any missing term.
5-9
SOLUTION OF
SYSTEMS OF
EQUATIONS AND
INEQUALITIES
Students have solved systems of linear equations graphically and algebraically. Similar
methods can be applied to solve systems
involving quadratic equations. The number of
real solutions of the system is equal to the
number of intersections of the graphs.
The graphical procedure for solving a system of equations may result in an approximate
answer when the coordinates of the intersec-
69
tion points are irrational. Exact answers may
be obtained algebraically using the substitution method for a quadratic-linear system.
When using substitution, the objective is to
obtain a single equation containing only one
variable. Note that students will sometimes
solve for one variable and think that they have
found the answer. Remind them that the solution, if one exists, consists of one or more
ordered pairs.
When a parabola is graphed in the coordinate plane, the points that lie on the
parabola satisfy an equation of the form
y 5 ax2 1 bx 1 c. Points in the solution sets
of the related inequalities, y . ax2 1 bx 1 c
and y , ax2 1 bx 1 c, lie, respectively, above
and below the parabola. The first inequality
specifies that the value of y is greater than
the value of the trinomial. Therefore, such
a point is located above a point with the same
x-coordinate that has a value of y equal to the
trinomial that lies on the parabola.
Students often equate “above the
parabola” with “inside the parabola,” since
both phrases are descriptions of the points of
the solution set when a is positive. When a is
negative, however, the two descriptions are
not the same. The graph of y . 2x2 makes this
clear.
In Example 4, after the solution of
2x2 1 3x 1 4 # 0 is identified, have students
use the graph to solve 2x 2 1 3x 1 4 . 4.
They should look for the interval in which
the parabola is above the line y 5 4, which is
0 , x , 3.
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ENRICHMENT ACTIVITY 5-5
Complex Number Operations, Vectors,
and Transformations
1. a. On one set of axes representing the complex plane, draw each of the
following complex numbers as a vector:
yi
A
h
(1) A 5 5 1 3i (shown as vector OA in the diagram)
(2) B 5 i ? A
(3) C 5 i ? B
(4) D 5 i ? C
(5) E 5 i ? D
O
b. Without performing the operation, predict the value of F if F 5 i ? E.
c. Write at least one observation relating the graph drawn in part a to
geometric transformations of the plane.
d. List as many specific symmetries as possible found in the graph in part a.
A special transformation of the plane called a spiral similarity occurs when any
point on the plane is multiplied by a complex number a 1 bi, where a 0 and b 0.
In effect, this transformation is the sum of two images; one is a dilation of a, and the
other is a rotation of 90° followed by a dilation of b, or Da 1 (Db + R908) . Exercises 2
and 3 involve spiral similarities. (Note that the sum of two images is not the same as
the composition of two transformations.)
2. a. On a sheet of graph paper, draw a set of axes centered on the page. Draw the
h
vector OA, where A 5 3 1 i, and label points O and A.
b. To demonstrate spiral similarity using multiplication by (1 2 i), draw and
label vectors from O to points B, C, D, E, F, G, and H, where:
B 5 (1 2 i) ? A
C 5 (1 2 i) ? B
D 5 (1 2 i) ? C
F 5 (1 2 i) ? E
G 5 (1 2 i) ? F
H 5 (1 2 i) ? G
E 5 (1 2 i) ? D
c. To see the spiral similarity in this graph, draw segments AB, BC, CD, DE, EF,
FG, and GH.
3. Spiral similarity involves powers. Let us start with A 5 1 1 i and multiply
repeatedly by the same complex number, 1 1 i.
a. Draw a set of axes centered on a sheet of graph paper. Then, on this
graph, draw and label the appropriate vectors for each indicated complex
number:
A 5 (1 1 i)
B 5 (1 1 i)2
C 5 (1 1 i)3
D 5 (1 1 i)4
E 5 (1 1 i)5
F 5 (1 1 i)6
G 5 (1 1 i)7
H 5 (1 1 i)8
I 5 (1 1 i)9
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b. On this graph, draw segments AB, BC, CD, DE, EF, FG, GH, and HI.
c. A pattern should be seen on the graph. If the pattern continues, name the
points, using A, B, C, . . . , through Z 5 (1 1 i)26, that will lie on:
(1) the positive ray of the x-axis
(2) the negative ray of the x-axis
(3) the yi-axis
4. Spiral similarity occurs when a point on the plane is multiplied by a complex
number a 1 bi. Consider the following cases:
Case 1: a 5 0, b 5 0
Case 2: a 0, b 5 0
Case 3: a 5 0, b 0
Case 4: a 0, b 0
In each case, describe the transformation that occurs.
What restrictions are needed for a true spiral similarity to exist?
5. Let A 5 2 1 3i, B 5 the conjugate of A, C 5 A 1 B, and D 5 A ? B.
a. Write the complex numbers B, C, and D.
b. On one set of axes representing the complex plane, draw and label vectors
h h h
h
OA, OB, OC, and OD.
c. Of points A, B, C, and D, tell which ones lie on the x-axis, and explain why.
d. If A 5 a 1 bi, where a 0 and b 0, and if B 5 the conjugate of A,
C 5 A 1 B, and D 5 A ? B, which of points A, B, C, and D, if any, would lie
on the x-axis? Explain your answer.
e. If P 5 a 1 bi, where a 0 and b 0, and if Q 5 the conjugate of P, find all
integral values of a and b such that P 1 Q 5 P ? Q.
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ENRICHMENT ACTIVITY 5-6
Quaternions
Extensions of the number system beyond the complex numbers are called hypercomplex numbers. Mathematicians have shown that it is impossible for all the field
properties to be true in such systems. One hypercomplex system, developed by the
Irish mathematician William Rowan Hamilton (1805–1865), is called quaternions or
four-fold numbers. This system satisfies all properties except commutativity of multiplication.
A quaternion is a number of the form (a, b, c, d) where a, b, c, and d are real numbers. Using the ordered four-tuples
1 5 (1, 0, 0, 0)
i 5 (0, 1, 0, 0)
j 5 (0, 0, 1, 0)
k 5 (0, 0, 0, 1)
we can describe a quaternion as follows:
(a, b, c, d) 5 a(1, 0, 0, 0) 1 b(0, 1, 0, 0) 1 c(0, 0, 1, 0) 1 d(0, 0, 0, 1) (expanded form)
5 a?1 1 b?i 1 c?j 1 d?k
5 a 1 bi 1 cj 1 dk (ijk form)
The number a of a quaternion is called its real or scalar part and the sum
bi 1 cj 1 dk is called its vector part. Quaternions have useful applications in fields
such as physics, computer graphics, and electrical engineering.
1. Write each quaternion in ijk form and expanded form.
a. (4, 6, 2, 3)
b. (27, 0, 5, 8)
c. (0, 0, 6, 9)
d. (23, 0, 0, 0)
2. Write the ordered four-tuple that corresponds to each quaternion.
a. 8 1 2j 1 3k
b. 4i 1 7j 2 k
c. 8 1 j
d. 26i 1 2k
To add quaternions, add their corresponding parts:
(a 1 bi 1 cj 1 dk) 1 (w 1 xi 1 yj 1 zk) 5 (a 1 w) 1 (b 1 x)i 1 (c 1 y)j 1 (d 1 z)k
3. Write each sum as an ordered four-tuple.
a. (5 1 7i 2 3j 1 2k) 1 (11 2 3i 1 9j 1 4k)
b. (27 1 0i 1 5j 1 8k) 1 (0 1 4i 1 7j 2 k)
c. (4 1 6i 1 2j 1 3k) 1 (23 2 6i 2 2j 2 3k)
d. (6j 1 9k) 1 (8 1 j)
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Multiplication uses these rules:
i2 5 j2 5 k2 5 21
ij 5 2ji 5 k
ik 5 2ki 5 2j
jk 5 2kj 5 i
It is easy to remember the multiplication rules using the figure on
the right where the points correspond to the quaternions i, j, k. The
product of two adjacent quaternions is equal to the third if the movement from the first to the second is clockwise and is equal to the third
with a minus sign if the movement is counterclockwise. Thus, multiplication is not commutative.
The quaternion (1, 0, 0, 0) or 1 is the multiplicative identity since it
can be shown that for all Q,
i
k
1 ? Q 5 Q ? 1 5 Q.
4. Find each quaternion product.
a. 3i ? 2k
b. 22j ? 5k
c. 4i ? 6j
d. 3k ? 7k
e. i ? j ? k
f. j ? j ? k
g. j ? k ? j
h. i ? j ? i
5. Use the multiplication rules and the distributive property to find the product
(2 1 3i 1 4j 1 5k)(3 1 7i 1 2j 1 8k).
The quaternions a 1 bi 1 cj 1 dk and a 2 bi 2 cj 2 dk are called conjugates.
The product of quaternion conjugates is a real number.
(a 1 bi 1 cj 1 dk)(a 2 bi 2 cj 2 dk) 5 a2 1 b2 1 c2 1 d2
6. What is the sum of a 1 bi 1 cj 1 dk and its conjugate?
What kind of number is this sum?
7. Write the conjugate of each quaternion. Then find the conjugate product.
a. 6 1 7i 1 2j 2 k
b. 3 1 5j 1 2k
c. 8i 1 3j
d. 5i 2 9j 1 4k
8. In the complex number system, the equation x2 1 1 5 0 has two solutions.
How many solutions does x2 1 1 5 0 have in the quaternion system? What are
the solutions?
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Algebra 2 and Trigonometry:
Chapter Five Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct
answer with insufficient work may receive only partial credit. All scratch paper must
be turned in at the conclusion of this test.
1. Without graphing, describe the translation, reflection, and or scaling that must be
applied to y 5 x2 to obtain the graph of y 5 2x2 2 12x 1 6.
In 2 and 3, find the exact solution for each equation by completing the square.
2. x2 2 6x 2 40 5 0
3. x2 1 3x 2 18 5 0
In 4–9, write each number in terms of i, perform the indicated operations, and write
the answer in a 1 bi form.
4. !225 1 2!236
5. !28 1 !232
6. A3 !23BA !227B
7. A5 !22B A22 !240B
8. A4 !25B 2
9. !21A1 1 !21B
In 10–12, simplify each expression.
10. i16
11. i59
12. i45
In 13–18, perform the indicated operation(s) and express the result in a 1 bi form.
13. (3 2 2i)(4 2 i)
14. (3 2 2i) 2 (4 2 i)
15. (3 2 2i)2
16. i(3 1 i) 2 5i
17. (4 2 bi) 4 (1 2 i)
18. 7 1 9i 2 (i3)3
19. Write the multiplicative inverse of 3 1 3i in a 1 bi form.
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In 20–22, express the roots of each equation in a 1 bi form.
20. x2 2 2x 1 10 5 0
21. x2 1 10x 1 29 5 0
22. 2x2 5 2x 2 1
In 23–28, in each case: a. Find the value of the discriminant. b. Using the discriminant,
determine if the roots of the quadratic equation are (1) rational and unequal,
(2) rational and equal, (3) irrational and unequal, or (4) imaginary.
23. 2x2 2 7x 1 1 5 0
2
24. x 2 x 2 2 5 0
2
25. x 2 5x 5 2
a.
b.
a.
b.
a.
b.
2
a.
b.
2
a.
b.
a.
b.
26. 2x 1 2x 1 3 5 0
27. 4x 2 4x 1 1 5 0
2
28. x 1 2 5 2x
2
29. For what value of k will the roots of 4x 1 4x 1 k be equal?
30. Find the sum of the roots of 2x2 2 8x 1 3 5 0.
31. Find the product of the roots of 2x2 2 8x 1 3 5 0.
32. Write a quadratic equation with integer coefficients that has roots 6 and 29.
33. Write a quadratic equation with integer coefficients that has roots 7 2 3i
and 7 1 3i.
34. The product of two consecutive positive odd integers is 323. Find the
integers by writing and solving a quadratic equation.
35. Find the length of the side of a square whose diagonal is 12 feet longer than
the length of a side. Express the answer exactly and to the nearest hundredth.
In 36–38: a. Solve each quadratic equation. b. Check both roots. (Show your work
on a separate piece of paper.)
36. 2x2 2 5x 1 2 5 0
a.
37. x2 2 2x 1 5 5 0
a.
38. x2 5 2x 1 2
a.
In 39–41, for each quadratic-linear system of equations: a. Solve graphically. Draw
each graph on a separate piece of graph paper. b. Solve algebraically.
39. y 5 x2 1 2x 1 1
40. x2 1 y2 5 5
41. y 5 x2 2 x
y5x13
2x 1 y 5 5
y5x22
b.
b.
b.
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In 42–44, for each inequality: a. Write the solution set of the inequality. b. Graph the
solution set on a number line.
42. x2 2 6x . 0
a.
b.
2
43. x 1 x 2 b # 0
a.
b.
2
44. 4x , 1
a.
b.
Bonus I: a. Write a quadratic equation whose roots are !2 6 1.
b. What is the value of the discriminant of the equation whose roots are
!2 6 1?
c. Explain why the equation, whose discriminant is a perfect square, has
irrational roots.
Bonus II: Suppose you picked an integer at random from 1 through 20 as a value
for c in the equation y 5 x2 1 8x 1 c. What is the probability the resulting
equation will have imaginary roots?
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SAT Preparation Exercises (Chapter 5)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. (2i)3(4i)2 equals
(A) (4i)5
(B) (24i)5
7
(D) (2i)
(E) (22i)7
(C) (8i)5
2. (4 1 2i)2 2 (3 1 2i)2 equals
(A) 7 1 4i
(B) 5 1 3i
(C) 2 2 i
(D) 1 1 4i
(E) 1
3. If (3 1 4i)(5 2 2i) 5 k, then
(4 1 4i)(5 2 2i) equals
(A) k 1 1
(B) k 1 4 1 4i
(C) k 1 5 2 2i
(D) k(4 1 4i)
(E) k(5 2 2i)
4. If xy 5 4i 1 6, x2 is a real number, and
y 5 22 1 b, then b could be
(A) 43
(B) 3
(D) 43i
(E) 3i
8. If a 1 bi is the distance from a 1 bi to
0 1 0i in the complex number plane,
then how many complex numbers a 1 bi
satisfy a 1 bi 5 5?
(A) 0
(B) 1
(C) 2
(D) 4
(E) Infinitely many
9. If x is a number in the first quadrant of
the complex number plane, then x2 is in
which quadrant?
(A) I
(B) II
(C) III
(D) IV
(E) Cannot be determined
10. If x25 5 4,096 1 4,096i and
x23 5 2,048 2 2,048i, then x2 is
(A) 2
(B) 22
(C) 2i
(D) 22i
(E) 2 1 2i
11. Which number is included in the shaded
area?
yi
(C) 23i
5. If the average of 3 1 i, 2 2 2i, and x is
1 1 i, then x is
(A) 22 1 4i
(B) 21 1 2i
(C) 1 1 i
(D) 3 1 3i
(E) 5 2 i
6. If x 1 5i is a factor of 2x2 1 kx 1 15, then
k is
(A) 10i
(B) 7i
(C) 3i
(E) 23i
(D) 27i
i
1
(A) 23 1 0.5i
(C) 21 1 2.5i
(E) 3 1 0.5i
x
(B) 21.5 2 4i
(D) 1 1 3.5i
12. Which number is included in the shaded
area?
7. If (a 1 bi)n is an odd integer, which of the
following must be an odd integer?
yi
I. (a 1 bi) n11
i
II. (a 1 bi) 2n
1
x
2
III. (a 1 bi) n
(A) I only
(C) III only
(E) II and III
(B) II only
(D) I and II
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(A) 22 1 2.5i
(C) 1 2 2i
(E) 2 2 2.5i
(B) 21 1 i
(D) 1.5 1 2i
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II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In questions 16–24, you are to solve the
problem.
y
1
O 1
Date
x
16. What is the value of 35i72 1 7i34?
17. What is the value of
(((1 1 i)i 1 1)i 1 1)i 1 1?
18. What is the value of f (25i) 22g 22?
Which equation is best represented by the
graph above?
(A) y 5 2(x 1 3)2
(B) y 5 2(x 2 3)2
2
(C) y 5 x 2 3
(D) y 5 2x2 2 3
2
(E) y 5 2x 1 3
14. Which parabola intersects the x-axis in
two distinct points?
(A) y 5 (x 1 7)2
(B) y 5 (x 2 7)2
2
(C) y 5 x 2 49
(D) y 5 x2 1 49
2
(E) y 5 2x 2 7
15. A parabola with an equation of the form
y 5 ax2 1 bx 1 c has the point (22, 1) as
its vertex. If (24, 5) also lies on the
parabola, which of the following is
another point on this parabola?
(A) (24, 3)
(B) (21, 3)
(C) (0, 5)
(D) (2, 1)
(E) (4, 5)
Copyright © 2009 by Amsco School Publications, Inc.
19. What is the distance between the points
where the graph of y 5 22x2 2 x 1 15
crosses the x-axis?
20. To the nearest hundredth, what is the distance between the points where the graph
of y 5 x2 2 6x 1 7 crosses the x-axis?
21. If 28(2 2 3i)2 5 x(3 1 2i)2, then what is
the value of x?
22. Let a and b be real numbers,
ax 1 by 5 10, x 5 2 1 2i, and y 5 4 1 3i.
Evaluate b 2 a.
23. What is the value of (3 2 i)2(3 1 i)2?
24. If (3 1 7i)(5 2 yi) 5 43 1 23i, then what
is the value of y?
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CHAPTER
SEQUENCES
AND SERIES
6
Aims
• To know and apply sigma notation.
• To identify an arithmetic or geometric
sequence and find the formula for its nth
term.
• To recognize and find the sum of arithmetic sequences and series.
• To recognize and find the sum of geometric sequences and series.
In this chapter, students learn to represent
and analyze patterns using algebraic methods.
Sequences and series are examples of recursive processes and such processes have many
applications. For example, fractals are created
using a recursive geometric process. In addition to the fascinating visual patterns found in
fractals, number patterns can be identified. In
business, the computation of compound interest, amount of annuities, amortization of
debts, and depreciation of assets involve concepts of geometric sequences and series.
f(5) 5 777.51
f(10) 5 322.37
f(6) 5 689.17
f(11) 5 227.21
f(7) 5 599.51
f(12) 5 130.62
f(8) 5 508.50
f(13) 5 32.58
f(9) 5 416.13
f(14) 5 33.07
The purchase will be paid off at the end of
14 months. Point out that as the balance
decreases, less interest is added on, and the
balance decreases more rapidly after each
payment. Ask students to determine the total
amount paid for the computer ($1,333.07).
6-1
SEQUENCES
In the text, a sequence of seven terms is shown
to be a one-to-one correspondence between
the terms of the sequence and the first seven
positive integers. The correspondence
between the terms 8, 4, 2, 1, 21, 41, 18 and the integers 1, 2, 3, 4, 5, 6, 7 can be written as the set
of ordered pairs U (1, 8) , (2, 4), (3, 2), (4, 1),
A 5, 12 B , A 6, 14 B , A 7, 18 B V . Therefore, we can define
CHAPTER OPENER
this finite sequence as a function whose
Have students complete the payment schedule. The balances at the end of each month are:
domain is {1, 2, 3, 4, 5, 6, 7} and whose rule is
f(1) 5 1,118
f(3) 5 950.29
f(2) 5 1,034.77
f(4) 5 864.54
f(x) 5 A 12 B . The positive integer with which
each element of the sequence is paired is indicated by the position of the term in a listing of
the terms of the sequence or by the subscript
of the term name. For example, a3 5 2 indix21
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Chapter 6
cates that (3, 2) is a pair of the function whose
domain is the set of positive integers and
whose rule is an 5 8 A 12 B .
Be sure students understand that a finite
sequence is a function whose domain includes
only the first n positive integers; the domain of
an infinite sequence is all the positive integers.
When discussing Example 1, emphasize the
difference between the general expression for
an and the recursive rule for an. As will be
shown in later sections, a general rule
expresses each term of the sequence using the
first term a1 and the common difference or
common ratio. The recursive definition for a
sequence describes how to find the nth term
from the terms before it and must include an
initial value. For example, the sequence 4, 7,
10, 13, 16, . . . can be described in two ways:
n21
an 5 4 1 (n 2 1)3 General formula
or
an 5 an21 1 3
where a1 5 4
6-2
¶
Recursive formula
ARITHMETIC
SEQUENCES
Have students graph the ordered pairs for any
of the arithmetic sequences in the section.
Mention that the graph represents a discrete
function whose domain consists of distinct values and there is no continuity between those
values. The range is the set of numbers representing the terms of the sequence. For any
arithmetic sequence, the points graphed will
be along a straight line. A linear function can
be written in the form y 5 mx 1 b. Ask students to compare this slope-intercept form
with the expression for the nth term of an
arithmetic sequence, an 5 a1 1 (n 2 1)d. Students should be able to identify that with a
change in the order of the terms and factors,
y 5 an, m 5 d, x 5 n 2 1, and b 5 a1.
Explain that the general formula can be
used to find the nth term of an arithmetic
sequence or to find the position of a given
term in an arithmetic sequence. The formula
can also be used to solve problems such as the
following:
Example: Determine how many numbers
between 8 and 1,621 are divisible by 6.
Solution: The first number between 8 and
1,621 that is divisible by 6 is 12. Let a1 5 12.
Then by repeatedly adding 6, we could list the
successive numbers that are divisible by 6, so
d 5 6.
a1 1 (n 2 1)d # 1,621
12 1 (n 2 1)6 # 1,621
12 1 6n 2 6 # 1,621
6n 1 6 # 1,621
6n # 1,615
n # 269 (remainder 1)
There are 269 numbers between 8 and
1,621 that are divisible by 6.
In Enrichment Activity 6-2: Arithmetic
Sequences students identify arithmetic
sequences after partitioning the set
S[9] 5 {1, 2, 3, 4, 5, 6, 7, 8, 9} and discover an
interesting property of S[9], that it is the
smallest set such that any separation into two
sets yields at least one set with at least one
three-term arithmetic sequence.
6-3
SIGMA NOTATION
A series is the sum of a sequence. A series may
be finite or infinite depending on whether the
sequence it is based on is finite or infinite. At
this time, some students may find the idea of
an infinite sequence having a sum puzzling,
but this concept will be made clear in Section
6-7. Students who feel that an infinite arithmetic series cannot have a sum are correct.
The summation symbol, , which is the
uppercase Greek letter sigma, gives a compact
way to write a series from the general term of
the corresponding sequence. The letter n used
in the summation notation is called the index
of summation or simply the index. In Example
1, the 1 below the summation symbol is called
the lower limit and the 10 above the symbol is
called the upper limit. If a summation symbol
is written without any upper and lower limits,
it means that all the given data are to be
summed. For example, a formula used to find
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Sequences and Series
the arithmetic mean, x– (read x bar), of a set of
data is x 5 Sx
n , where n is the number of data
values.
Students should be aware that different
summation notation can represent the same
6
Example: Find the eighth term of a geometric sequence if a4 5 16 and r 5 0.5.
Solution: First find the value of a1.
a4 5 a1r421
16 5 a1(0.5) 3
4
16 5 0.125a1
series. For example, a (k 2 2) 2 and a k2
k53
16
0.125
k51
both represent the same sum 1 1 4 1 9 1 16.
6-4
ARITHMETIC SERIES
Sn 5
1 an)
Substitute a1 1 (n 2 1)d for an.
Sn 5 n2 fa1 1 (a1 1 (n 2 1)d)g
5 n2 f2a1 1 (n 2 1)dg
This version of the formula can be used when
the value of the last term is not known.
6-5
GEOMETRIC
SEQUENCES
A geometric sequence, like an arithmetic
sequence, is a function whose domain is the set
of positive integers. However, the graph of the
terms is not linear. In Chapter 7, students will
learn to recognize that the points of the graph
follow the same pattern as the exponential
function y 5 abx. (For the general exponential
function, the value of b must be positive, but
for a geometric sequence an 5 a1r n–1, the value
of r may be either positive or negative.)
• If an , an21, then r , 1.
• If an . an21, then r . 1.
• If the signs of a geometric sequence alternate, then r , 0.
The following example shows how to find a
given term when you know the common ratio
and one term of the geometric sequence, but
not a1.
5 a1
128 5 a1
Now use the formula an 5 a1rn21 to find a8.
a8 5 128(0.5) 821
Two versions of the formula for Sn are given
before Example 1. Students should understand how one is derived from the other.
n
2 (a1
81
5 128(0.5) 7
51
An alternative solution is to start with the
given fourth term and multiply by the common ratio to find a5 to a8.
6-6
GEOMETRIC SERIES
In the formula for the sum of a geometric
series, it is necessary to restrict the value of r to
values not equal to 1 so that division by 0 does
not occur. An example of a geometric series
where r 5 1 is 5, 5, 5, . . . . The sum of the first n
terms is 5n. In general, the sum of a geometric
series with r 5 1 is na1.
In Exercise 1, students are asked to justify
the equivalent expression
a 2 ar
Sn 5 11 2 rn .
Since an 5 a1r n21, we can multiply each side
by r to get anr 5 a1r n. By substituting anr for
a1r n in the formula for the sum of a geometric
series, we have
Sn 5
a1 (1 2 rn)
1 2 r
a 2 a rn
5 11 2 r1
a 2 ar
5 11 2 rn
This version of the formula is useful for
finding the sum if the first and last terms are
given, but the value of n is not given. In Exercises 15–22, sigma notation is used to represent the sum of a geometric series. When n is
small, suggest that students find the sum in
two ways: (1) by using the formula and (2) by
writing out all the terms of the sum and performing the actual addition.
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Chapter 6
INFINITE SERIES
All the geometric series that students have
worked with so far have been finite. The series
n11
1
1 1 12 1 14 1 18 1 16
1 c 1 A 12 B
1c
is an infinite geometric series, as indicated by
the three dots at the end that signify the series
continues indefinitely in the same manner.
We can see that the series above
approaches the value 2 as more terms are
added. (The first six partial sums are 1, 1.5,
1.75, 1.875, 1.9375, 1.96875.) Ask students how
we know that the series will never pass 2. Suggest to students that they write the partial
sums as follows:
1 1 12 5 32 5 2 2 12
1 1 12 1 14 5 74 5 2 2 14
1
1 1 12 1 14 1 18 5 15
8 5 2 2 8
Students should be able to see that each
partial sum is equal to the difference of 2 and a
decreasing positive number. For example, the
sum of the first three terms is 74 or 212 less than
1
so that the sum can
2. In general, Sn 5 2 2 2n21
never be greater than or equal to 2.
Emphasize that even though we often call
a1
1 2 r the sum of an infinite geometric series
with r , 1, this is the limit that the sum
approaches, not a sum that is actually reached
by adding terms. A series that approaches a
limit is called a convergent series. Note that
while it is true that an infinite geometric series
has a finite limit if r , 1, there are other types
of series that converge.
The sum of an infinite geometric series
provides an alternate method for expressing a
repeating decimal as a rational number in the
form ba .
Example: Write a fraction equivalent to
0.343434 . . .
Solution: We can write this decimal as
0.34 1 0.0034 1 0.000034
1 c1 (0.34)(0.01) n21 1 c
This is an infinite geometric series with
a1 5 0.34 and r 5 0.01. Since r , 1,
a
S 5 1 21 r
0.34
0.34
34
512
0.01 5 0.99 5 99
The solution can be checked by using a calculator to divide 34 by 99.
In calculus, students will learn that infinite
series can be used to represent many numbers
such as p, !2, sin 5p
2 , or ln 5. A major advantage of the series representation is that it
allows us to approximate irrational numbers
to any degree of accuracy.
The symbol e was first used to represent
the irrational number
2.718281828459045 . . .
by the Swiss mathematician Leonhard Euler.
It has come to occupy a special place in mathematics and its applications, and it will reappear in Chapter 8.
EXTENDED TASK
For the Teacher:
In this exploration of the harmonic series, students see that the nth term converging to 0 is a
necessary but not sufficient condition for the
series to have a limiting sum. Students participate in the demonstration that the harmonic
series is not bounded by working with specific
partial sums and arranging terms in groups
that are then replaced with smaller values that
sum to 12. Since this process can be continued
without end, the value of the sum can always
be increased by another 12 and is, therefore,
unbounded. Since the original sums were
larger, the original series cannot have a limit.
Although the harmonic series diverges, the
partial sums do not grow rapidly. In fact, if j is
very large, it has been shown that Sj is approximately equal to 0.577216 1 ln j.
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ENRICHMENT ACTIVITY 6-2
Arithmetic Sequences
Suppose the set of numbers
Sf9g 5 51, 2, 3, 4, 5, 6, 7, 8, 96
is separated into two smaller sets
A 5 51, 4, 6, 96 and B 5 52, 3, 5, 7, 86 .
Observe that in set B, the numbers 2, 5, 8 form an arithmetic sequence (difference 3)
and the numbers 3, 5, 7 also form an arithmetic sequence (difference 2). Suppose S[9]
is separated differently:
A 5 52, 3, 86 and B 5 51, 4, 5, 6, 7, 96
Then, in set B, the arithmetic sequences 1, 4, 7 or 5, 6, 7 or 5, 7, 9 or 1, 5, 9 can be identified.
Find ten different ways to separate the set S[9] into two sets A and B. Each set
must have at least one element. Identify an arithmetic sequence of at least three
terms in at least one of the sets and indicate the common difference. If you create a
separation such that neither set contains a three-term arithmetic sequence, write NO
SEQUENCE FOUND. Try sets of different sizes.
Set A
Set B
Arithmetic Sequence (Common Difference)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Did you find any pair of sets A and B such that neither contained an arithmetic
sequence? In fact, mathematicians have shown that whenever S[9] is separated into
two sets, it is always true that at least one of the sets contains an arithmetic sequence
of at least three terms. The proof uses cases and a combination of direct and indirect
reasoning to show the existence of the required sequence in every case.
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Name
Next, consider the set
Class
Date
Sf8g 5 51, 2, 3, 4, 5, 6, 7, 86
Does S[8] have the same property as S[9]?
11. Construct three different separations of S[8] such that at least one of the sets
contains a three-term (or greater) arithmetic sequence.
12. Construct at least one separation such that neither set has an arithmetic
sequence.
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EXTENDED TASK
The Harmonic Series
The series
`
1
1
1
c
an5112131
n51
is called the harmonic series.
1. What happens to the nth term of the harmonic series as n approaches infinity?
2. Find the first six partial sums for the series above.
3. As n becomes infinitely large, do you think the partial sums approach a limit?
`
Do you think the infinite series a n1 has a sum? If so, what do you think the
n51
value of the sum is? If not, explain why not.
4. Observe that
S21 5 S2 5 1 1 12
1
2
S22 5 S4 5 1 1 12 1 13 1 14 . 1 1 12 1 14 1 14 5 1 1 22
since 13 . 14. Thus, S4 . 1 1 22.
Use this result and a similar line of reasoning to show that
S23 5 S8 . 1 1 32.
5. Use the result of the previous question to show that S24 5 S16 . 1 1 42.
6. Predict an inequality for the general case S2 j. What conclusion can be drawn
`
about a n1 ? Explain.
n51
Use a calculator for questions 7 and 8.
j
7. Determine the smallest value of j such that a n1 $ 3.
n51
j
8. Determine the smallest value of j such that a n1 $ 4.
n51
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Algebra 2 and Trigonometry:
Chapter Six Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–6: a. Write the next three terms of each sequence. b. Write an expression for the
general term an.
1. 6, 12, 18, 24, . . .
2. 27, 25, 23, 21, . . .
a.
a.
b.
b.
3. 4, 512, 7, 812, c
a.
4. 28, 4, 22, 1, . . .
a.
b.
5.
2 4 8 16
5, 5, 5, 5 ,
b.
c
6. 21, 1, 21, 1, . . .
a.
a.
b.
b.
In 7–12, find the indicated quantity of the arithmetic sequence.
7. a1 5 5, d 5 7; find a13.
9. a1 5 3, a10 5 75; find d.
11. a1 5 225, an 5 82, d 5 211; find n.
8. a1 5 26, d 5 24; find a17.
10. a1 5 4, an 5 148, d 5 9; find n.
12. a1 5 2.37, d 5 0.98; find a44.
13. Find two arithmetic means between 55 and 130.
14. Find four arithmetic means between 18 and 108.
In 15 and 16, find the value of x that makes each sequence arithmetic.
15. 7, 15, 23, 5x 1 1, . . .
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16. x 1 12, 3x 1 7, 3x, . . .
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In 17 and 18: a. Write the terms of each series. b. Find the sum.
6
17. a (k2 1 1)
a.
b.
a.
b.
k51
8
18. a (5k 2 3)
k53
In 19–22, find the indicated partial sum for each arithmetic series described.
19. a1 5 1, a10 5 19, S10
20. a1 5 24, d 5 3, S16
21. a15 5 106, d 5 28, S15
225
22. a1 5 25
6 , a11 5 6 , S11
In 23–26, find the nth term of each geometric sequence.
23. a1 5 3, r 5 2, n 5 8
24. a1 5 4,096, r 5 14, n 5 7
25. a3 5 112, r 5 24, n 5 6
26. a1 5 34, a4 5 29, n 5 5
27. Find three geometric means between 9 and 144.
In 28–31, find the indicated partial sum for each geometric sequence described.
28. a1 5 5, r 5 22, S6
29. 3 2 6 1 12 2 24 1 c, S9
30. 81 1 27 1 9 1 c, S8
31. a1 5 8, r 5 12, S7
32. A certain substance decomposes and loses 20% of its weight each hour.
If there is originally 4,000 grams of this substance, how much remains
after 5 hours?
33. Maria Lopez invests $3,000 at 4% interest compounded annually in a
savings account. How much money is in her account at the end of
3 years?
In 34–36: a. Find a1 and r for each geometric series. b. Find the sum if it exists;
otherwise write “No sum.”
34. 6 1 3 1 32 1 34 1 38 1 c
35. 1.8 1 2.7 1 4.05 1 6.075 1 c
36. 5 2 2 1 4 2 8 1 16 2 c
5
25
125
Copyright © 2009 by Amsco School Publications, Inc.
a.
b.
a.
b.
a.
b.
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Bonus I:
Class
Date
Each swing of a pendulum travels 90% of its previous distance. If the first
swing is 8 feet long, determine the total distance traveled by the pendulum
by the time it comes to rest.
`
Bonus II: a. Write the series a n(n 11 1) as the sum of terms.
n51
b. Is the series geometric?
c. Find S1, S2, S3, S4, S5.
d. Predict the values of S10 and S20.
e. As n approaches infinity, what number does Sn appear to be
approaching?
1
f. Observe that a n(n 11 1) can be written a A n1 2 n 1
1B.
`
`
n51
n51
Write the terms for S5 without combining any terms.
g. Using the results of part f, what conjecture can you make about an
expression for Sn?
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SAT Preparation Exercises (Chapter 6)
I. MULTIPLE CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. Which of the following is not a geometric
sequence?
(A) 96, 48, 24, 12, . . .
(B) 22, 6, 218, 54, . . .
1 1 1 1
(C) 243
, 81, 27, 9, c
(D) 8, 4, 0, 24, . . .
(E) !2, 2, 2 !2, 4, c
2. The 7th term of the geometric sequence
2,000, 200, 20, 2, . . . is
(A) 0.2
(B) 0.02
(C) 0.002
(D) 0.0002 (E) 0.00002
3. Which number is a term of the sequence
3, 7, 11, 15, . . . ?
(A) 45
(B) 63
(C) 85
(D) 110
(E) 120
4. An arithmetic sequence is represented by
3, 14, 3x 1 4, . . . . What is the value of x?
(A) 7
(B) 10
(C) 14
(D) 21
(E) 25
5. The first term of an arithmetic sequence is
8 and each term is 13 more than the previous one. What is the tenth term in the
sequence?
(A) 117
(B) 125
(C) 130
(D) 138
(E) 142
6. What term of the sequence 212, 25, 2, 9,
. . . is 65?
(A) 10th
(B) 11th
(C) 12th
(D) 13th
(E) 14th
7. How many terms are there in the geometric sequence 58, 254, 52, c, 640?
(A) 8
(D) 11
(B) 9
(E) 12
(C) 10
Copyright © 2009 by Amsco School Publications, Inc.
8. In a geometric series, S4 5 89 and a1 5 35.
What is r?
(A) 25
(B) 213
(C) 215
(D) 31
(E) 52
9. The first term of a geometric sequence is
5 and the third term is 7.2. Which of the
following can be the fifth term?
(A) 8.85
(B) 9.4
(C) 10.368
(D) 11.646 (E) 12.4416
10. If 219, 13, 21, . . . are the first three terms of
a geometric sequence, which of the following must be true?
I. The constant ratio r is 3.
II. The next three terms are 3, 29, 27.
III. The ninth term a9 is 729.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
11. Which arithmetic sequence does not have
a term equal to 60?
(A) 4, 11, 18, . . .
(B) 23, 6, 15, . . .
(C) 220, 25, 10, . . .
(D) 230, 212, 6, . . .
(E) 81, 79.5, 78, . . .
12. By how much does the arithmetic mean
between 1 and 49 exceed the positive geometric mean between 1 and 49?
(A) 0.5
(B) 7.5
(C) 12.6
(D) 17
(E) 18
13. What is the sum of the first eleven terms
of the arithmetic sequence 3, 512, 8, . . . ?
(A) 28
(D)
17012
(B) 14312
(E) 175
(C) 154
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14. For the arithmetic sequence shown below
w, 25, x, y, 4,z, c
what is the value of w 1 x 1 y 1 z?
(A) 22
(B) 0
(C) 8
(D) 10
(E) 15
15. The sum of the first n terms of an
arithmetic series is 1,445, a1 5 13, and
an 5 157. What is the third term of the
series?
(A) 23
(B) 31
(C) 33
(D) 40
(E) 41
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. If 3, 6, 12, 2y 1 6 is a geometric sequence,
what is the value of y?
17. What is the 15th term of the sequence
213, 29, 25, 21, . . . ?
Copyright © 2009 by Amsco School Publications, Inc.
Date
18. What is the sum of the two arithmetic
means between 3 and 60?
19. If the eighth term of a geometric
3
sequence is 38 and the ninth term is 16
,
what is the third term of this sequence?
20. If a1 5 2 and an 5 2n 1 an–1, what is the
sum of the first five terms?
21. The first term of a geometric sequence is
5 and the third term is 80. What is the
quotient when the eighth term is divided
by the fifth term?
22. What is the sum of the two geometric
means between 40 and 135?
23. What is the sum of a19 1 a8 for the
sequence 293, 282, 271, . . . ?
24. What is the sum of the infinite series
4,900 1 1,400 1 400 1 c?
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CHAPTER
7
EXPONENTIAL
FUNCTIONS
• To apply the rules of exponents with
rational exponents.
notational system; for example, the notation
should be unambiguous, easy to use, and applicable in all instances. Present one or more
polynomials and have students use the various
methods shown in the chapter opener to
express them. Ask students to create a system
of their own.
• To change from a radical to an exponential
expression and from an exponential
expression to a radical expression.
7-1
Aims
• To review the laws of exponents.
• To define zero and negative exponents.
• To define and graph exponential functions.
• To solve exponential equations.
Students have used exponential notation in
many of their previous mathematics courses.
This chapter reviews the laws of exponents
and uses these laws to motivate the definitions
of zero, negative, and fractional exponents.
From an exponential function and its graph, an
intuitive understanding of the use of any real
number as an exponent can be developed. The
study of exponential equations begun in this
chapter will be extended when logarithms are
explored in the next chapter. Applications of
exponential functions including compound
interest and radioactive decay are examined.
CHAPTER OPENER
It is interesting for students to see how exponential notation evolved. Lead a brief discussion about the features that are desirable in a
LAWS OF EXPONENTS
Students may be familiar with the rules or laws
of exponents presented in this section. A careful review, however, will reinforce their understanding and help them continue the study of
exponents with confidence.
Call attention to the changes in value that
may result from the use of parentheses. For
example, 262 (26)2 and (3x)3 3x3.
Apply the rules to powers with numerical
as well as literal bases. A common student
error is to write the product 33 ? 34 as 97. Point
out that the product is obtained not by multiplying 3 times 3, but rather by counting how
many times 3 is to be used as a factor.
7-2
ZERO AND NEGATIVE
EXPONENTS
The question “What would happen if . . . ?”
can motivate new ideas. When dividing powers
with like bases, ask, “What happens if the
exponent of the denominator is equal to or
91
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Chapter 7
larger than the exponent of the numerator?”
In these cases, we find that the quotients have
zero or negative exponents. Although zero and
negative exponents do not allow us to define
xa as “x used a times as a factor,” powers with
zero and negative values are consistent with
the rules for operations with powers. An
example such as
1
x2
x5
1
2
5 xx ?? xx ? x ? x ? x 5 x13 and xx5 5 x225 5 x23
1
1
serves to establish that negative exponents are
for
x 2 0,
another way of expressing reciprocals. The
definition of a negative exponent can also be
demonstrated as follows:
x 2 0, xa ? x2a 5 x0 5 1
If the product of two numbers is 1, the
numbers are reciprocals or multiplicative
inverses. Therefore, x29 is the reciprocal of x9,
and xa is the reciprocal of x2a (see Example 2).
By demonstrating that powers with zero
and negative exponents produce consistent
results when the rules for multiplying or dividing powers of like bases are used, we show the
validity of the new definitions.
Emphasize to students that a factor can be
moved from a numerator to a denominator or
from a denominator to a numerator simply by
changing the sign of the exponent.
7-3
FRACTIONAL
EXPONENTS
The definition of fractional exponents can be
introduced by using the rules for multiplying
powers with like bases as shown in the section.
Since the nth root of x is defined to be one of
the n equal factors whose product is x, and
n
1
since the product of n factors of xn is xn or x1,
1
then xn satisfies the definition of the nth root
of x. For example,
1
1
1
1
4
x4 ? x4 ? x4 ? x4 5 x4 5 x
1
Therefore, x4 is one of the four equal factors
whose product is x and is a symbol for the
1
fourth root of x. The power x4 can be used as a
n
symbol for !
x because it satisfies the definition of the nth root of x.
1
n
In the text, xn or !
x is defined only for
x . 0. One unfortunate consequence is that
1
3
(28) 3 does not exist even though !
28 is well1
defined (and equals 22), that is, xn is undefined for negative x when n is odd even though
n
the corresponding principal root !
x exists.
1
You may wish to define xn in the following
way:
• If n is a positive integer such that the prin1
n
cipal nth root of x exists, then xn 5 !
x.
To avoid the problems arising when working with negative numbers, also give the following definition:
n m
n
• "
x 5 A!
x B m 5 x n only when m
n is in
lowest terms.
m
This results in an alternative approach that
also satisfies the rules of exponents.
As a practical matter, writing terms with
fractional exponents allows you to perform
operations on terms more easily when they
have the same numerical or literal base.
4 3
3 2
Compare 7"
x ? 9"
x with the expression
3
2
7x4 ? 9x3 . Emphasize that when changing a
radical expression to exponential form, the
power is placed in the numerator, and the
index or root is placed in the denominator of
the fractional exponent. When the procedure
is reversed to convert exponential expressions
to radical expressions, the numerator of the
fractional exponent is the power and the
denominator is the index or root. Point out
2
3 2
3
that 63 may be written as either "
6 or A !
6 B 2.
To evaluate an expression such as
3
4
4
4
16 5 "
163 5 A !
16 B 3, it is often easier to find
the root first and then raise it to the power.
Since powers and roots are on the same level
in order of operations, this procedure is valid.
Before students begin work on the section
exercises, review the conditions for an expression to be simplified:
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Exponential Functions
1. It has no negative exponents.
2. The expression is not a complex fraction.
3. The index of any remaining radical is the
smallest possible number.
Use Enrichment Activity 7-3: Factoring
Expressions with Rational and Negative
Exponents to extend students’ skills. This topic
is important for students going on to courses
in calculus.
7-4
EXPONENTIAL
FUNCTIONS AND
THEIR GRAPHS
Emphasize that the definition of an exponential function y 5 bx excludes 1 as a value for b
because y 5 1x is not a one-to-one function.
x
Note that the graphs of y 5 2x and y 5 A 12 B
are both one-to-one functions. In the figures,
both graphs may appear to be horizontal near
the x-axis, but in fact they are not.
Exponential functions of the form y 5 bx
will have a shape similar to y 5 2 x when
x
b . 1 and a shape similar to y 5 A 12 B when
0 , b , 1.
When sketching the graph of y 5 2x , in
addition to positive integers, we used 0, negative integers, and rational fractions as values of
x. We evaluated y according to the definitions
of these exponents and located points on the
graph using the x- and y-values as coordinates.
When we draw the smooth curve through
these points, we are assuming that the expression 2x is defined for all values of x that are
real numbers. Having made that assumption,
we can read from the graph values of 2x for
which x has irrational values, that is, values
such as 2!3 and 2!5. More rigorous definitions
of the values of such powers are derived in
advanced mathematics courses.
Students often fail to appreciate the difference between a linear and an exponential
function in a practical situation. The following
93
example illustrates the difference between linear growth and exponential growth.
• Exponential growth: If $1 is invested at 8%
interest over a period of x years, the value
y of the investment is y 5 (1.08)x.
• Linear growth: If $1.08 is added to an initial investment of $1 each year over a
period of x years, the value y of the investment is y 5 1 1 1.08x.
Use a graphing calculator to sketch
the graphs of these functions. First set the
values of x from 0 to 100 with a scale of 10,
and the values of y from 21 to 300 with a
scale of 30.
ENTER:
WINDOW 0 ENTER
100 ENTER
10 ENTER 21 ENTER 300 ENTER
30 ENTER
Now enter the equations.
ENTER:
Yⴝ 1.08
^
X,T,⍜,n
1 ⴙ 1.08 X,T,⍜,n
ENTER
GRAPH
DISPLAY:
Notice that, throughout the period of 100
years shown on the graph, the y-values of the
linear function rise steadily, and the y-values
of the exponential function rise slowly at first
but then more and more rapidly. For about the
first 53 years, the y-values of the linear function are greater than the y-values of the exponential function. Thereafter, however, the
y-values of the linear function are smaller than
the y-values of the exponential function.
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Chapter 7
7-5
SOLVING EQUATIONS
INVOLVING
EXPONENTS
To solve equations where the variable is the
base of a power, the goal is to make the exponent equal to 1 so the variable can be isolated.
If time allows, have students suggest how
they might estimate the value of x if 5x 5 60.
(Since 52 5 25 and 53 5 125, the value of x is
between 2 and 3. The value of x should be
much closer to 2 than 3. Alternatively, they
could graph y 5 5x on a calculator and locate
the approximate value of x for which y 5 60.)
If A x3 B 5 x,
2
a
2
then x3a 5 x1 and 32a 5 1 or a 5 32,
where x 0, 1.
So A x3 B 2 5 x.
2 3
Therefore, to solve an equation such as
2
x3 5 4,
it is necessary to raise both members of the
equation to the 32 power in order to obtain an
equivalent equation that has x as one side.
2
Note that x3 means that the cube root of x is
squared. Therefore, we use the inverse operation of taking the cube root, which is raising
the number to the third power, and the inverse
operation of squaring the number, which is
taking the square root of the number.
A x3 B 2 5 42
2 3
3
7-7
APPLICATIONS OF
EXPONENTIAL
FUNCTIONS
This section examines several interesting
applications of exponential increase (growth)
and decrease (decay).
With regard to the discussion of compound
interest, it is reasonable to assume that the
more frequently the interest is compounded,
the larger the compounded amount will be.
However, the calculations show that although
the compound amount increases, it does
not increase dramatically. Continuing the
calculations:
Paid hourly for 1 year:
0.05
A 5 100 A 1 1 8,760
B
< 105.1270947 or 105.13
8,760
3
x 5 42
x 5 A !4B 3 5 23 5 8
Remind students that variable expressions
with fractional exponents are defined only for
non-negative variables.
7-6
SOLVING
EXPONENTIAL
EQUATIONS
This section demonstrates a relatively simple
method of solving exponential equations. The
equations solved here are special cases, since
both sides of the equation can be expressed as
powers of the same base. Their solutions provide additional practice with exponents.
The general procedure for solving exponential equations is discussed in the next
chapter, after the introduction of logarithms.
Paid each minute for 1 year:
0.05
A 5 100 A 1 1 525,600
B
< 105.1271107 or 105.13
525,600
Note that the expression A 1 1 k1 B can be
interpreted as the compounding of an investment of $1 at a 100% interest rate compounded k times per year. Thus, no matter how
often the interest is compounded, the compound amount will never exceed $2.72. In general, if Ao is invested for t years at a rate r
compounded continuously, then the compound amount A is given by the exponential
function
k
A 5 Aoert
This function gives the general form for
growth and decay.
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Exponential Functions
EXTENDED TASK
This activity is intended to introduce students
to the exponential function through a handson activity. Students may work on this task in
small groups or individually.
Some students may see the patterns for the
total number of holes before making all of
the folds, punches, and counting the holes.
They should be encouraged to extrapolate but
should complete the activity to verify their
predetermined patterns.
95
Some students may have difficulty expressing a number (like 24) as a constant times a
power of 2. If so, they should be encouraged to
look at the common factors of those numbers
or to write the prime factorization of each
number. This activity is a good example of
connecting one topic in mathematics to
another.
Part IV of the extended task helps students
relate pure mathematics (the exponential
function) to real-world situations (examples of
growth and decay).
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ENRICHMENT ACTIVITY 7-3
Factoring Expressions with Rational
and Negative Exponents
One reason that both rational and negative exponents are introduced in algebra is
that you will need to understand and use them in more advanced courses such as calculus. Recall that you learned how to factor a monomial from a polynomial. Polynomials must have terms with nonnegative integer exponents. To factor x3 1 x7, you
factor out the common variable with the smallest exponent.
x3 1 x7 5 x3 (1 1 x4)
The exponent on the variable term in parentheses was found by subtracting the
lesser exponent from the greater exponent; 7 2 3 5 4. The same procedure is used to
remove a common factor with rational or negative exponents from expressions that
are not polynomials.
Example 1
2
8
Factor x5 2 x5.
Solution: The smaller exponent is 25.
x5 2 x5 5 x5 A 1 2 x5 2 5 B
8
2
8
2
2
5 x5 A 1 2 x5 B
6
2
The answer can be checked using the distributive property and the laws of
exponents.
Example 2
Factor x23 1 x27 and then write the answer without negative exponents.
Solution: Since 27 is less than 23, x27 is the term with the smaller exponent.
x23 1 x27 5 x27 (x232(27) 1 1)
5 x27 (x4 1 1)
4
5 x x17 1
Simplify using positive exponents.
Example 3
2
1
Factor x3 2 7x3 1 12.
Solution: Observe that x3 5 Ax3 B 2. Let u 5 x3, so the expression becomes a
trinomial.
2
1
1
2
1
x3 2 7x3 1 12 5 u2 2 7u 1 12
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Factor the trinomial.
u2 2 7u 1 12 5 (u 2 4)(u 2 3)
1
Substitute x3 for each u.
x3 5 7x3 1 12 5 A x3 2 4 B A x3 2 3 B
2
1
1
1
Multiply to check.
In 1–10, factor each expression. Write the answer without negative exponents.
5
1
1
3
4
2
1. y2 1 y2
2. x3 2 x3
3. x25 1 x5
4. x2 2 x21
5. b21 2 b
6. 2w212 1 w27
5
1
3
7. x22 1 x22 1 x2
9
4
1
8. c25 2 3c25 1 c5
9. 2x24 2 6x25
10. 5x 2 10x21
In 11–16, factor each expression.
1
11. x 1 6x2 1 9
2
1
14. 2y7 1 y7 2 3
2
1
12. x5 1 4x5 2 5
1
15. 8b 1 2b2 2 1
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1
1
13. x2 1 x4 2 20
1
1
16. 15x3 2 14x6 1 3
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EXTENDED TASK
Holes, Holes, and More Holes:
An Exponential Investigation
In this extended task, you will investigate the pattern created when a folded piece of
paper has holes punched in it. For this task you will need about 20 small squares of
thin paper, such as tissue paper or origami paper, approximately 5 3 5, and a hole
punch. In this activity, you will perform three similar tasks, tabulate data about the
tasks, and examine the data for patterns.
Part I
Task 1: Punch a hole in one paper square. Take a second paper, fold it in half, and
punch a hole. Fold a third paper in half twice and punch a hole. Fold a fourth paper in
half three times and punch a hole. Fold a fifth paper in half four times and punch a
hole. Fold a sixth paper in half five times and punch a hole. Open up your papers and
record the results of the task in the table below, listing the number of holes and then
expressing that number as a power of 2. The first two entries in the table have been
done for you.
# of folds
0
1
# of holes
1
2
# of holes expressed
as a power of 2
20
21
2
3
4
5
a. What pattern do you observe?
b. If you fold the paper n times, how many holes will you have?
c. Express h, the total number of holes in the paper, as a function of n, the
number of times you fold the paper.
Task 2: Repeat Task 1, but punch two holes in the paper each time rather than just
one hole. Record your data in the table below.
# of folds
0
1
# of holes
2
4
# of holes expressed
as a power of 2
21
22
2
3
4
5
a. Do you observe a similar pattern this time?
b. Express the numbers in row 3 as the constant 2 times a power of 2; that is,
21 5 2 3 20 and 22 5 2 3 21.
c. Express h, the total number of holes in the paper, as a function of n, the
number of times you fold the paper.
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Task 3: Repeat Task 1 again, but this time punch 3 holes in the paper each time.
Record your data in the table below and express the number of holes in row three as
3 times a power of 2.
# of folds
0
1
# of holes
3
6
# of holes expressed
as a power of 2
2
3
4
5
3 3 20 3 3 21
Express h, the total number of holes that you will have, as a function of n, the
number of times you fold the paper.
Part II
On graph paper, or using a graphing calculator, graph the functions written in the
three tasks of Part I and discuss the nature of your graphs.
Part III
The type of function investigated in this extended task is referred to as the exponential function. Do you think this is an appropriate title? Explain your answer.
Part IV
Exponential functions are often associated with growth and decay. Research
growth and decay functions and give at least two real-world examples of this type of
function.
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Algebra 2 and Trigonometry:
Chapter Seven Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–8, evaluate each expression.
1
1
1. 273
3.
A 15 B
2. 14(25) 2
23
2
4. (28) 3
1
5. 2120
6. 0.1622
8. A 814 B 2
3
3
7. 3225
In 9 and 10, evaluate each function for the given value.
9. g(x) 5 2x 1 222x; g(3)
10. f(x) 5 A xx9 B
6
22
; f(4)
In 11–16, simplify each expression and write the answers with positive exponents.
All variables represent positive numbers.
x3y21
1
11. xy23
12.
1
13.
(25c22d4) 2
(5c2d) 22
15.
a
1
x4y3
b
1
x2 y
2
9ab2
3
3a21b2
2(x3y)
14. 2(x23y) 21
16. a
y4 23
2b
y25
In 17 and 18, write each expression as a power with positive exponents and express
the answer in simplest form. All variables represent positive numbers.
4
17. "4x2
Copyright © 2009 by Amsco School Publications, Inc.
3
18. "64x6y7
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In 19 and 20, write each power as a radical expression in simplest form. All variables
represent positive numbers.
3
1
20. (25x10y) 2
19. (3x) 4
21. a. Sketch the graph of y 5 A 45 B for the interval 22 # x # 2.
b. Sketch the graph of the reflection in the y-axis of the graph drawn in part a.
x
c. Write an equation for the function graphed in part b.
In 22–27, solve each equation.
1
22. 2x2 5 8
23. x22 1 1 5 5
24. 4 2 3x21 5 x
25. 511x 5 125
26. 9x 5 27x21
27. 42x 5 8
28. Jackie invested $3,000 at 4% per year compounded monthly. Find the value
of Jackie’s investment after 2 years.
29. A tractor that cost $28,000 depreciates in value by 15% after each year. Find
the expected value of the tractor after 7 years.
30. A five-year CD pays 4.5% compounded continuously. Find the value of a
$10,000 investment at the end of the five years.
a11
a
Bonus I: Simplify: 222 21202
3
3
Bonus II: Determine if " !x 5 " !x, x $ 0. Justify your answer.
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SAT Preparation Exercises (Chapter 7)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. The product 3 ? 321 is not equal to
(B) 1
(C) 33
(A) 9–1
(E) 90
(D) 30
2. If 4x 5 8y, then the ratio of y to x is
(A) 232
(B) 223
(C) 31
(D) 23
(E) 32
3. If x 5 x2 1 x22, then 21 is equal to
(A) 0
(B) 1
(D) 414
(E) 16
(C) 214
4. If 44x25 5 64, then what is the value of x?
(A) 22
(B) 212
(C) 21
(D) 32
(E) 2
4
4
4
5. If 2 1 22x 1 2 5 6, then what is the value
of x?
(A) 22
(B) 2
(C) 3
(D) 4
(E) 7
x
6. If 9.9999 equals 10 2 10 , then x is
(A) 21
(B) 22
(C) 23
(D) 24
(E) 25
7. If 10,000 1 6,000 5 2x1y ? 5x, then y is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
8. If 2x13 5 10, then 2x equals
(A) 7
(B) 2
(C) 35
(D) 10
(E) 54
7
Copyright © 2009 by Amsco School Publications, Inc.
9. The perimeter of a square with an area of
24x is
(A) 22x11
(B) 22x12
(C) 24x1 1
4x12
8x
(D) 2
(E) 2
10. If 4x1k is 25% of 4x, then k is
(A) 21
(B) 212
(C) 41
(D) 12
(E) 1
11. The reciprocal of A 12 B 22 can be written as
each of the following except
1
1
1
(A) (2) 22
(B) (2) 2
(C) A 12 B 2
1
(D)
1
1
(2) 2
(E) #12
x
x
x
x
12. 2 1 2 2x11 12 1 2 5
(A) 0
(B) 1
(D) 3
(E) 4
(C) 2
13. What is the average of 3x, 3x11, 3x12, and
3x13?
4x 1 6
(A) 14x16
(B) 34x15
(C) 3 4
(D) 10(3x)
(E) 30x
14. If " !x 1 !x 1 !x 1 !x equals !2,
then x is
(A) 2
(B) 12
(C) 14
1
(D) !2
15. 10116 2 10117 5
(A) 210917
(D) 10917
(E) 2!2
1
(B) 210
1
(E) 10
(C) 10117
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II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In questions 16–24, you are to solve the
problem.
16. If 3n 5 27, what is the value of 6n21?
17. If 5y2 5 25, then what is the value of 5y22?
n
6
23
n
18. If 4 5 16 , then what is the value of n?
n23
19. If 7 5 7 ? 7
of n?
, then what is the value
Copyright © 2009 by Amsco School Publications, Inc.
Date
20. For what non-negative integer value of x
will A 235 B
2x 1 1
be the smallest?
515
n
1
21. If 5 5 125
, what is the value of n?
22. If N* means NN23, then what is the value
of (3*)*?
10
23. If 33x 1 3 5 27 3 , what is the value of x?
24. If f(x) 5 2x, what is the value of
f(102) ? 2f(2100)?
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CHAPTER
8
LOGARITHMIC
FUNCTIONS
Aims
• To define a logarithmic function as the
inverse of an exponential function.
• To apply the properties of logarithms to
write equivalent logarithmic expressions.
• To evaluate logarithmic expressions in any
base.
• To solve exponential and logarithmic
equations.
• To graph logarithmic functions.
This chapter emphasizes the relationship
between exponential and logarithmic forms
and develops the skills students need to solve
exponential and logarithmic equations. Students use their calculators to obtain the
logarithmic values needed for their computations.
CHAPTER OPENER
Logarithms were first developed as an aid in
computation and as such made possible the
advancement of sciences such as astronomy
that require long and tedious computations.
Today, the development of high-speed
computers has rendered obsolete the use of
logarithms for computational purposes. Nevertheless, an understanding of logarithms
104
and the computational rules that govern their
use is essential for the study of advanced
mathematics.
To impress upon students how their studies
have been simplified by the availability of
powerful hand calculators, you may wish to
display an old table of logarithms. Explain that
learning how to use this table required several
lessons and involved terms such as mantissa
and characteristic.
8-1
INVERSE OF AN
EXPONENTIAL
FUNCTION
Under a reflection in the line whose equation
is y 5 x, the image of the graph of an exponential function is the graph of the inverse of the
exponential function. Since under this reflection the image of (x, y) is (y, x), the equation of
the function that is the inverse of y 5 bx can be
written as x 5 b y . However, the equation
defining a function is usually written in a form
that expresses y in terms of x. To solve x 5 by
for y in terms of x, new notation is introduced.
In the equation x 5 by, y is the value of the
exponent and x is the value of the power. To
solve this equation for y, we describe y in
words and then replace the words with symbols and abbreviations that maintain the
essential idea of the description. Encourage
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Logarithmic Functions 105
students to identify the core idea of the statement 3 5 log2 8 as “3 is the log” or “3 is the
exponent.” Then the other parts can be added
so that this statement is read as “3 is the exponent that must be used with the base 2 to give
the power 8.”
Call attention to the domains and ranges of
the exponential and logarithmic functions.
Since these are inverse functions, the domain
of one is the range of the other. Since the base,
b, of an exponential function is always positive, the function value, bx, is always positive.
In other words, the range of the exponential
function y 5 bx is the set of positive real numbers (y . 0), and the domain of the logarithmic function y 5 logb x is the set of positive
real numbers (x . 0). Note that the domain of
the exponential function and the range of the
logarithmic function are unrestricted; both are
the set of all real numbers.
Since y 5 bx and y 5 logb x are inverses,
their composites are the identity function. If
f(x) 5 logb x and g(x) 5 bx, then
f(g(x)) 5 logb bx 5 x
and
g(f(x)) 5 blogb x 5 x
Explain to students that to
x
graph a logarithmic function such
1
as y 5 log3 x, they should change it
9
to exponential form, x 5 3y. Using
1
3
x 5 3y, they can construct a table
1
of values. The table is easier to
develop if values are selected for y
3
and then corresponding x-values
9
are found.
Alternatively, they can find the
ordered pairs for y 5 3x and then
interchange x and y in each ordered pair.
y
22
8-2
LOGARITHMIC FORM
OF AN EXPONENTIAL
EQUATION
This section continues to focus on recognizing the meaning of an expression such as
log 5 25 5 2. After selecting the key idea
(log 5 2), the student will more easily recognize the base as 5 and the value of the power
as 25 in order to write the equivalent expression 52 5 25. Students usually consider exponential form to be easier than logarithmic
form because exponents are more familiar to
them, and will initially want to change the log
equations into the more familiar exponential
equations. Encourage them to become familiar, through repeated practice, with the log
form as well.
Some students may ask why they cannot
use the LOG key on their calculators to do
the exercises in this section. Explain that this
key gives logarithms to the base 10 only. In
Section 8-4 they will learn to use this key, and
in the Hands-On Activity of Section 8-5 they
will derive a formula that can be used to find
logs to any base on a calculator.
As students prepare to begin the exercises,
remind them that they can find the value of a
variable in a logarithmic equation logb x 5 y
when values for two of the variables are
known.
21
0
1
2
8-3
LOGARITHMIC
RELATIONSHIPS
The product rule, the quotient rule, and the
power rule are derived in this section. Call
attention to the fact that these rules are simply
restatements of the laws of exponents. Compare the operations involved by making a
chart such as the following:
Operations Using Powers
Operations Using Logs
Multiplication
Addition
Division
Subtraction
Raising to a power
Multiplication
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106 Chapter 8
Point out that the identity logb an 5 n logb a
states that the log of a power is the product of
two logs, logb a and n. Since n is an exponent, it
is a log. For example,
log2 323 5 3 log2 32
5 3(5)
5 15
Here it was necessary to recall that 25 5 32,
but 3 is already a log since it is an exponent.
Another helpful example is to have stu2
dents compare the expanded forms of log5 x3
2
and log5 A x3 B .
2
log5 x3 5 log5 x2 2 log5 3
Quotient rule
5 2 log5 x 2 log5 3
If time allows, have students find the log of
14 and the log of 140.
log 14 < 1.1461
log 140 < 2.1461
They should notice that log 14 and log 140
have the same decimal part, 0.1461, called the
mantissa, but different integer parts, 1 and 2.
The integer part of a common logarithm of a
number is called its characteristic and indicates
the magnitude of the number. The characteristic is the exponent of 10 when the original
number is expressed in scientific notation. So,
14 5 1.4 3 101 and 140 5 1.4 3 102. Also,
log (140) 5 log (14 3 10) 5 log 14 1 log 10
5 1.1461 1 1
Power rule
However,
2
log5 A x3 B 5 2 log5 A x3 B
Power rule
5 2.1461
Have students predict values for log 0.014
and log 1,400,000.
5 2flog5 x 2 log5 3g Quotient rule
8-5
5 2 log5 x 2 2 log5 3
The examples and the exercises in this section give students practice in applying the
rules for logarithms before they make use of
calculators to evaluate logs.
8-4
COMMON
LOGARITHMS
This section introduces the student to the use
of the calculator to find the common logarithm of a number and to find the number, or
antilogarithm, when the common logarithm of
that number is known.
An alternative way of verifying the log of 8
that does not require storing the value is:
ENTER:
LOG
8 ENTER
2nd
10x
2nd
ANS
ENTER
DISPLAY:
NATURAL
LOGARITHMS
Since e is the base for natural logarithms, ln
e 5 1. All properties of logarithms that students have learned apply to natural logarithms
as well. The antilogarithm in base e is often
denoted antiln x.
Students should be aware that when they
change a natural logarithm to exponential
form, the base of the exponential expression
must be e. If y 5 ln x, then ey 5 x.
Since y 5 ln x is the inverse of y 5 ex, it follows that:
ln ex 5 x
eln x 5 x
The general change of base formula is usually
stated as follows:
• For all a ., b . 0, and x . 0, a 1, b 1,
log x
loga x 5 logb a
b
log(8
.903089987
^(Ans
8
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Logarithmic Functions 107
In the change of base formula, 10 is often
used in place of base b because we know how
to find common logarithms. Replacing base b
with 10, we get
log x
log x
loga x 5 log10 a or loga x 5 log a
10
Since e is a smaller base than 10, for x . 1 it
follows that ln x . log x. The graph of y 5 ln x
is above the graph of y 5 log x for x . 1.
Enrichment Activity 8-5: Finding e presents a complex fraction that approximates e.
Many other expressions have been used to
represent this important number.
8-6
EXPONENTIAL
EQUATIONS
When both sides of an exponential equation
cannot be written as a power of the same base,
we often begin by taking the logarithm of both
sides of the equation, as illustrated in the solution of 5x 5 32.
Point out that while Example 1 could be
solved using either common or natural logs,
the solution to Example 3 should use natural
logs because the formula involves e.
Exercises 15–20 illustrate some of the
many applications for exponential equations
in business and the sciences.
In Enrichment Activity 8-6: State Population Growth, students use researched data
about their home state to create linear and
exponential models. They use their models to
predict population size in the future.
8-7
LOGARITHMIC
EQUATIONS
Emphasize that before the rule
logb A 5 logb C S A 5 C can be used to
solve a logarithmic equation, it is necessary to
write each side of the equation as a single log
with the same base.
Two methods are shown for Example 1. A
computational method is a third alternative.
Evaluate ln 12 2 ln 3 using a calculator. Then
use the result as the exponent for ex.
ENTER:
LN 12
ⴚ
LN 3
2nd
ANS
)
)
ENTER
2nd
ex
ENTER
DISPLAY:
ln(12)-ln(3)
1.386294361
e^(Ans
4
The answer 4 will be displayed. Note that
the parenthesis must be inserted after the 12.
Example 2 shows that logarithmic equations may have extraneous solutions. When
examining solutions, if you obtain the logarithm of a non-positive number, that answer is
not a solution to the equation.
EXTENDED TASK
For the Teacher:
This activity attempts to show students
how a topic in pure mathematics was applied
by a scientist (Richter) to a real-world phenomenon. Although the activity does not
require students to use logarithms directly, it
does demonstrate to students the relationship
of mathematics (logarithms) to the real world.
Students should be encouraged to research
earthquakes and the Richter scale to better
understand the theory behind the nomogram.
There are many references to earthquakes in
the library and on the Internet.
Students might also research other realworld phenomena that apply logarithms.
These include, but are not limited to: astronomy (brightness of a star), compound interest,
sound (decibels), musical scales, f-stop on a
camera, and atmospheric pressure.
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108 Chapter 8
Some of the major twentieth-century
earthquakes that were recorded at 7.0 or
above on the Richter scale are listed in the
table below. Your students may find others.
Year
Location
Size
Year
Location
Size
1906
San Francisco
8.3
1992
Indonesia
7.5
1933
Japan
8.9
1993
Japan
7.7
1964
Alaska
8.5
1994
Sumatra
7.0
1976
China
8.0
1995
Russia
7.5
1978
Iran
7.7
1995
Mexico
7.6
1980
Italy
7.2
1996
China
7.0
1985
Mexico
8.1
1996
Indonesia
7.5
1989
San Francisco
7.1
1997
Pakistan
7.3
1990
Iran
7.7
1997
Iran
7.5
1990
Philippines
7.8
1998
Indonesia
7.8
1991
India
7.0
1999
Mexico
7.5
1992
California
7.5
For each 0.3 difference on the Richter
scale, an earthquake is 100.3 or almost twice as
powerful. For example, the 1997 earthquake in
Pakistan had almost twice the magnitude of
the 1996 earthquake in China, and the 1933
earthquake in Japan had 100.6 or almost four
times the magnitude of the 1906 earthquake in
San Francisco.
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ENRICHMENT ACTIVITY 8-5
Finding e
One method of representing the irrational number e is with the complex fraction
shown below.
1
21
1
11
1
21
1
11
1
11
1
41
1
11
1
11
1
61
11
1
1
11 811...
1. Evaluate the part of the fraction shown by entering it into a calculator as
follows:
2 1 1 4 (1 1 1 4 (2 1 1 4 (1 1 1 4 (1 1 1 4 (4 1 1 4 (1 1 1
4 (1 1 1 4 (6 1 1 4 (1 1 1 4 (1 1 1 4 (8 1 1)
2. Write the number displayed.
3. Does this fractional expression appear to give a good approximation of e?
4. Research the history of mathematicians’ development of understanding of e.
Include other representations, such as infinite series, equations, and formulas
that have been used to approximate e.
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ENRICHMENT ACTIVITY 8-6
State Population Growth
Use a reference book or Internet source to find the following information for the
state in which you live.
Name of state:
Population in 1960:
Population in 2000:
1. What was the change in population from 1960 to 2000? Be sure to indicate if
the change was positive or negative.
2. What was the percent change from 1960 to 2000?
3. If the growth pattern from 1960 to 2000 was linear, what was the
average yearly change in population for the state?
4. Write a linear equation to model the population change.
5. Use the linear model to predict the state population for:
a. 2010
b. 2025
c. 2100
6. Suppose the population grew continuously during the period from
1960 to 2000.
a. Write an exponential equation that can be used to find the yearly
rate of growth.
b. Find the growth rate to the nearest hundredth of a percent.
7. If the growth rate remains the same, predict the population for each year
using the exponential model.
a. 2010
b. 2025
c. 2100
8. Compare your predictions from the linear and exponential models.
9. If your state had a positive growth rate, how many years at that rate would it
take for the 2000 population to double? If the growth rate was negative, how
many years would it take for the 2000 population to be halved?
10. What rate of growth is needed for a population to double in 10 years?
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Extended Task
Calculating the Magnitude of an Earthquake: A
Mathematical Application
Two measures are usually associated with earthquakes. The first of these, intensity,
has to do with the degree of damage done to structures, with the amount of disturbance to the surface of the ground, and with the extent of animal reaction to the shaking. Hence, intensity depends upon the density of the population, the type of
construction, the type of ground, and so on.
In order to compare earthquakes worldwide, a measure is needed that does not
depend upon the variables associated with intensity. The term magnitude is used for
this measure. In California in 1936, Charles Richter developed a strictly quantitative
scale to measure the magnitude that can be applied to earthquakes worldwide,
whether the area is inhabited or uninhabited. The scheme was to use wave amplitude
measured by a seismograph. Although there have been some modifications, the
Richter scale, developed to measure the magnitude of an earthquake, is still in use
today.
Because the size of quakes varies enormously, the amplitudes of the waves differ
by factors of thousands from earthquake to earthquake. It is therefore most convenient to compress the range of wave amplitudes by using some mathematical concept.
Richter defined the magnitude of a local earthquake as the logarithm to the base 10
of the maximum seismic wave amplitude (in thousandths of a millimeter) recorded
on a standard seismograph at a distance of 100 kilometers from the earthquake’s epicenter. This means that every time the magnitude goes up by 1 unit, the amplitude of
the earthquake waves increases by more than 10 times.
A nomogram can be used to find the magnitude of a quake on the Richter scale.
The nomogram uses the difference in arrival times of the S- and P-waves and the
amplitude to find the magnitude of the quake. The P-wave is the primary or fastest
wave traveling away from a seismic event through the rocks. A P-wave compresses
and expands the material in the direction it is traveling. The S-wave is the secondary
seismic wave, traveling more slowly than the P-wave. An S-wave compresses and
expands the material perpendicular to the direction it is traveling. The amplitude is
the height of the maximum wave motion on the seismogram. To use the nomogram:
1. Find the difference in arrival times of the S- and P-waves by calculating the
difference between the end of the P-wave and the end of the S-wave.
2. Measure the height of the maximum wave motion on the seismogram.
3. Place a straightedge between appropriate points on the distance and amplitude
scales to obtain the magnitude.
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The diagram below illustrates this process for an amplitude of 23 millimeters
and a 24-second difference in the arrival times of the S- and P-waves.
P
S
0 10 20
S – P = 24 sec
500
400
300
200
50
40
20
5
2
100
50
20
10
5
2
1
0.5
5
20
10
8
6
4
Amplitude = 23 mm
6
30
100
60
40
10 20 30
Nomogram
4
3
2
0.2
0.1
1
0
Magnitude
0
Distance S – P
(km) (sec)
Amplitude
(mm)
The magnitude of this earthquake is 5 on the Richter scale. The magnitude is
obtained by drawing a line from the amplitude and the difference in arrival times on
the scales. Notice also that this earthquake is about 200 kilometers away.
Use the nomogram to find the magnitude of the earthquakes shown in the table
below. The table gives the distance in kilometers and the amplitude in millimeters.
Distance
(kilometers)
Amplitude
(millimeters)
1.
5
50
2.
60
3.
100
100
4.
40
50
5.
300
1
0.5
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Magnitude
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Algebra 2 and Trigonometry:
Chapter Eight Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct
answer with insufficient work may receive only partial credit. All scratch paper must
be turned in at the conclusion of this test.
In 1–3, solve each equation for y in terms of x.
1. x 5 7y
2. x 5 82y
3. x 5 log6 y
In 4–9, in each case: a. Write the expression in exponential form. b. Solve for x.
4. 0 5 log6 x
a.
b.
a.
b.
6. 4 5 logx 4
a.
b.
7. x 5 log4 32
a.
b.
1
8. log3 27
5x
9. logx 0.04 5 22
a.
b.
a.
b.
3
2
5. log25 x 5
10. If f(x) 5 log4 x, find f(1,024).
11. If h(x) 5 log32 x, find h(16).
12. Write the expression 1022 5 0.01 in logarithmic form.
In 13–18, for each equation, write log x in terms of log a, log b, and log c.
13. x 5 ab2
!a
14. x 5 bc
3
15. x 5 c#ba5
3
16. x 5 A ac
b3 B
2
3 2
17. x 5 "
ab
18. x 5 a2b !c
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In 19–22, for each equation, write N in terms of a, b, and c.
19. log N 5 log a 1 2 log b 2 log c
20. log N 5 12 log a 2 2(log b 1 log c)
21. log N 5 12 (log a 1 2 log b) 2 log c
22. log N 5 2 log a 1 13 (log b 2 log c)
23. If log 463 5 b, write an expression for log 46.3 in terms of b.
24. Solve for x: log4 (x 2 2) 1 log4 (x 1 5) 5 2 log4 (x 1 1)
25. Solve for x: 2 log5 x 2 log5 (2x2 2 1) 5 2
26. a. Evaluate each expression to four decimal places.
(1) 3 log 50
(2) log (3 3 50)
(3) log 503
(4) log 3 1 log 50
b. Make two observations based on the evaluation of the expressions in part a.
In 27–29, solve each equation for x to the nearest hundredth.
27. 27x 5 156
28. 3x11 2 12 5 24
x
29. 332 5 15
In 30–32, evaluate each expression to four decimal places.
30. ln 4.52 1 ln 0.085
31. ln 10,000 2 ln 999
32. (ln 0.1278)2
33. At the end of each year, employees of the Plumtree Software Company
receive a pay increase of 2.5%. Devon had a starting salary of $24,000. For
how many years has Devon worked for Plumtree Software if after his last
wage increase his salary is over $58,000?
34. The Chans are saving money to go on a trip to Europe for their twenty-fifth
wedding anniversary. They have five years to save $9,000 for the trip. If the
five-year CD they buy now pays 514% interest compounded continuously,
how much do they need to invest now in order to have $9,000 for the trip?
(Use A 5 Pert.)
35. Find log9 22 to the nearest thousandth.
36. Find log8 77 to the nearest thousandth.
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In 37–39, find the value of each logarithm using the values log10 2 5 0.3010 and
log10 5 5 0.6990. Show your work on a separate piece of paper.
37. log10 0.4
38. log10 2.5
39. log10 25
40. a. Sketch the graph of y 5 log5 x in the interval 0.04 # x # 25 on a separate
piece of graph paper.
b. On the same set of axes, sketch the graph of y 5 5x in the interval 22 # x # 2.
c. Under what transformation is the graph drawn in part b the image of the
graph drawn in part a?
Bonus: Solve for x: logx 4 1 logx 9 5 2
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SAT Preparation Exercises (Chapter 8)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If V 5 pr 2h, then log V equals
(A) 2p(log r 1 log h)
(B) p log 2r 1 log h
(C) 2 log pr 1 log h
(D) log p 1 2 log r 1 log h
(E) (p log r2)(log h)
8. The average of log 2 and log 18 is
(A) log 6
(B) log 9
(C) log 10
(D) log 20
(E) log 36
9. If x 5 log2 5, then 4x is
(A) 2
(B) !5
(D) 10
(E) 25
10.
2. If log 7 5 m, then log 490 equals
(A) 2m 1 1
(B) m2 1 1
(C) 10m
(D) 10m2
(E) 70m
log2 8
If the area of the rectangle is 1, then the
height is
(A) log2 4
(B) log2 3
(C) log2 13
3. If log x 5 0.3721, then
(A) 21 , x , 0
(B) 0 , x , 0.1
(C) 0.1 , x , 1
(D) 1 , x , 10
(E) 10 , x , 100
4. If 4x 5 8y, then the ratio of x to y is closest
to
(A) 2
(B) 32
(C) 34
(D) 32
(E) 12
5. If log x 1 log (x 1 3) 5 1, then which of
the following could be a value of x?
(D) log2 18
(E) log8 2
11. If log2 x is 25% of log2 y, then y, expressed
in terms of x, is
(A) 14x
(B) 4x
(C) x4
4
(E) !x
(D) x4
12.
c
I. 25
(C) !10
b
II. 2
III. 5
(A) I only
(C) III only
(E) II and III only
6.
(B) II only
(D) I and II only
2(2a)
2aq 2b
In the division exercise above, b equals
(A) 2a
(B) 2a 1 1
(C) 2a 2 1
(D) 2a 1 2
(E) 2a 2 2
7. If 2 log x 5 log 2x, then x is
(A) 0
(B) 1
(C) 2
(D) 4
(E) any number
Copyright © 2009 by Amsco School Publications, Inc.
a
For a, b, and c shown above, 2 log c equals
(A) log 2a 1 log 2b
(B) 2 log a 1 2 log b
(C) 2 log (a 1 b)
(D) log (a2 1 b2)
(E) log (2a 1 2b)
13. If the volume of a cube is 63a, then the
surface area of the cube is
(A) 62a+1
(B) 6(32a)
(C) 6(26a)
3a
2a+1
(E) 3
(D) 6(2 )
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14. If loga b2 5 21, then a, expressed in terms
of b, is
(A) 2b
(B) b2
(C) 22b
(D) 22
(E) b2
b
1
15. log xy
5
(A) log x 1 log y
(B) log x 2 log y
(C) 1 2 log xy
(D) 2(log x 1 log y)
(E) 1 2 log x 2 log y
Date
16. If y 5 log2 x, then what is the value of x
when y 5 3?
17. Solve for z: log4 (log2 16) 5 z
18. Solve for y: log y 5 14 log 16 1 12 log 81
19. If log (0.1) ? log x 5 3, then what is the
value of x?
20. What is the exact value of log5 25 !5?
21. Solve for w in the equation
2w(1,000) 5 2,048,000.
22. Solve for x: logx 27 1 logx 3 5 2.
23. Solve the equation for m:
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
Copyright © 2009 by Amsco School Publications, Inc.
log5 32 2 4 log5 m 5 3 log5 12
24. If log x3 2 log y 5 4 and
log x 1 log y 5 2, what is the value
of log xy?
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CHAPTER
9
TRIGONOMETRIC
FUNCTIONS
Aims
• To define the sine, cosine, and tangent
functions.
• To learn the reciprocal trigonometric functions: secant, cosecant, and cotangent.
• To use a calculator to find trigonometric
function values, when defined, for angles
of any measure.
CHAPTER OPENER
Eratosthenes used the fact that at noon on the
summer solstice, the longest day of the year, at
Syene (the ancient name for Aswan), there
were no shadows because the sun was directly
overhead. However, at the same time on the
same day in Alexandria, some 500 miles to the
north, the angle formed by the top of a tower
and the edge of its shadow was 7.2°.
• To use a calculator to find the degree
measure of an acute angle in Quadrant I
when given a trigonometric function value.
7.2°
Alexandria
7.2°
• To use reference angles to determine
trigonometric function values for angles in
Quadrants II, III, and IV.
• To derive basic trigonometric identities.
In this chapter, another type of function,
the trigonometric function, is defined in terms
of the length of a line segment associated with
the unit circle.
The trigonometric functions relate measures associated with circular motion to measures associated with linear motion. This
relationship is a common one in everyday life
and is applied in many types of machinery. For
example, in an automobile, the linear motion
of the pistons is transformed into the circular
motion of the wheels. The trigonometric functions describe mathematically this relationship
of circular motion and linear motion.
118
Syene
Center
of Earth
Assuming the sun’s rays are parallel, the
angle at the center of Earth is congruent to the
angle of the shadow (alternate interior
angles). Then, using the fact that there are 360°
in a full rotation, the following proportion can
be written to find Earth’s circumference:
500
7.28
9-1
x
5 3608
and x 5 25,000 miles
TRIGONOMETRY OF
THE RIGHT TRIANGLE
This section reviews the trigonometry of the
right triangle, which most students will have
studied in previous courses.
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Trigonometric Functions 119
If students are familiar with sine, cosine, and
tangent expressed as ratios of the measures of
the sides of a right triangle, it may be useful to
review these ratios and to show how they are
special cases as the more general definitions are
developed.
Exercise 1 focuses on the cofunction relationship sin u 5 cos (90 2 u). In Example 2 and
Exercises 12 and 13, the special characteristics
of the 45-45-degree right triangle and the 3060-degree right triangle are used to find
trigonometric function values. For Exercises
18 and 19, remind students that an angle of
depression or elevation is the angle formed by
the line of sight and a horizontal line.
9-2
ANGLES AND ARCS
AS ROTATIONS
In previous courses, students have worked
only with angles whose degree measure was
limited to values between 0 and 180. This section, by considering an angle as a rotation,
introduces the use of any rational number as
the degree measure of an angle. Intuitively, it
is clear that any real number can be the degree
measure of an angle. However, this is not formally established until Section 10-4, after arc
length has been introduced.
Just as we find it convenient to associate
linear distance with the real number line, so we
find it convenient to associate angle measure
with the coordinate plane. On the number line,
any point to the right of 0 is associated with a
positive number and any point to the left of 0 is
associated with a negative number. Similarly,
from the non-negative ray of the x-axis (the
initial side of the angle), any angle measured
counterclockwise has a positive measure and
any angle measured clockwise has a negative
measure.
Coterminal angles are angles in standard
position that have the same terminal side.
Since the location of a point can be described
by using a simpler coterminal angle measure,
this concept will play a major role in simplifying and finding trigonometric function values.
For example:
sin 7508 5 sin (7508 2 3608) 5 sin 3908
5 sin (3908 2 3608) 5 sin 308
9-3
THE UNIT CIRCLE,
SINE, AND COSINE
This section uses the definitions of the sine
and cosine as ratios of the measures of the
sides of a right triangle in order to motivate
the definitions of the sine and cosine functions.
As the cases for the four quadrants are
examined, emphasize that the ratios of the xcoordinate and y-coordinate of a point on the
terminal side of an angle in standard position
to the distance of the point to the origin do
not depend upon the particular point but,
rather, depend entirely on the angle. Thus, we
can define functions from the set of angles in
standard position to the sets of all such ratios.
These ratios we call the sine and cosine of the
angle, respectively.
Since the distance d from the origin to the
point (p, q) on the terminal side of any angle u
is "p2 1 q2, we have
q
q
or d
"p2 1 q2
p
p
or d
cos u 5
2 1 q2
p
"
sin u 5
For the unit circle "p2 1 q2 5 1, so
q
sin u 5 1 5 q
p
cos u 5 1 5 p
Therefore, the point at which the terminal
side of an angle in standard position with
measure u intersects the unit circle has the
coordinates (cos u, sin u).
Encourage students to memorize the quadrants where sine and cosine are positive and
where they are negative. Be sure students
understand that the sign of the function value
is determined by the quadrant where the terminal side lies, not by the sign of the measure
of the angle.
9-4
THE TANGENT
FUNCTION
The tangent function assigns to every angle
measure u a unique value t that is the y-coordinate of the point where the terminal side of
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120 Chapter 9
the angle intersects the line tangent to the unit
circle at point (1, 0):
tangent
uht
The illustrations for Case 5 show that the
tangent is undefined for u 5 90° and u 5 270°
because there is no intersection. The proof
sin u
that tan u 5 cos
u is then given in Example 4.
Since division by 0 is undefined, it therefore
follows that the tangent is undefined for
angles that have a sine of 0, that is, angles that
have their terminal sides on the x-axis, namely
90° and 270°.
The diagram below can help students
remember the quadrants where the function
values are positive:
S
Sine positive
A
All function values positive
T
Tangent positive
C
Cosine positive
Students can use the mnemonic All Students
Take Calculus. They may also enjoy making up
other phrases having words with the relevant
first letters and, in so doing, will remember the
key ideas.
9-5
THE RECIPROCAL
TRIGONOMETRIC
FUNCTIONS
Three new functions are defined in this section. For each function defined here, the function value is the reciprocal of the function
value of a previously defined trigonometric
function. For example:
sec u 5 cos1 u (cos u 2 0)
The domain of the cosine function is the set
of all real numbers. The domain of the secant
function is the set of all real numbers such that
the cosine value for that real number is not 0.
Therefore, the domain of the secant function is
the set of all real numbers that are not 90 plus
a multiple of 180 (90 1 180k, for any integer
k).
For every real number u that is in the
domain of the secant function, if (u, c) is a pair
of the cosine function, then A u, 1c B is a pair of
the secant function. Notice that the second
elements are reciprocals when the first elements are the same.
Similar relationships exist between the
domain and the range of the sine and cosecant
functions and of the tangent and cotangent
functions. Since sin u and tan u are 0 for any
value of u that is a multiple of 180°, these values are excluded from the domain of the cosecant and cotangent functions.
tan u is undefined for real numbers that differ from 90° by a multiple of 180°. By using the
u
definition cot u 5 cos
sin u , we can define cot u as
0 for these values of u:
cot p2 5
cos p
2
sin p
2
5 01 5 0
Notice that the secant, cosecant, and cotangent functions can be defined in terms of the
sine, cosine, and tangent functions or in terms
of the lengths of line segments associated with
the unit circle.
Finding values for reciprocal functions
with a calculator is covered in Section 9-7.
9-6
FUNCTION VALUES OF
SPECIAL ANGLES
Although the derivations given in this section
can be used whenever it is necessary to find an
exact value of a trigonometric function that
corresponds to an angle of 30°, 45°, or 60°,
encourage students to memorize these function values, since they will be used frequently
in the applications of the trigonometric functions.
In this section, exercises are limited to
trigonometric functions of angles with measures of 30°, 45°, and 60° and to function values
of quadrantal angles.
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Trigonometric Functions 121
9-7
FUNCTION VALUES
FROM THE
CALCULATOR
The calculator can be used to find the values
of trigonometric functions for angles other
than 30°, 45°, and 60° and to find the measure
of an angle when one of its function values is
known.
Students should be made aware that, in
some situations, a solution may require an
exact function value. For this reason, it is
important that students memorize function
values for 30°, 45°, and 60° angles as presented
in Section 9-6. In other situations and for angle
measures other than those special values, we
use the rational approximations displayed as
function values by the calculator. Approximate function values are often rounded to the
nearest ten-thousandth (four decimal places).
Students should be taught how to determine function values of angles: first, with integral degree measures, such as 47° and 238°;
second, with measures containing a portion of
a decimal degree, such as 62.5°; third, with
measures given in degrees and minutes, such
as 75° 169.
An angle measure of 25 degrees 9 minutes
27 seconds is written as 25° 99 270. Some calculators, such as the TI-831/841, can display
angle measures in this format, called
degree/minute/second (or DMS) mode.
• To convert an angle in degrees to
DMS mode, enter the angle, then
4 .
press 2nd ANGLE
• To write an angle in DMS mode, use
° (degree) and 9 (minutes) from the
ANGLE menu, and 0 (seconds) or
ⴖ . For example, to write
ALPHA
25° 99 270,
ENTER:
22 2nd
9 2nd
ANGLE
ANGLE
27 ALPHA
1
2
ⴖ
Not all calculators have this feature, however, and students must learn how to enter a
measure using degrees and minutes into any
graphing calculator.
Since there are 60 minutes of angle measure in 1 degree, and 60 seconds of angle measure in 1 minute, it follows that
9
27
258 9r 27s 5 A 25 1 60
1 3,600
B 5 25.15758
In the text, by disregarding seconds, we
limit angle measures to degrees and minutes.
Therefore, a measure such as 37° 459 is entered
into a calculator as a mixed number of
degrees, that is, as (37 1 45 4 60).
For Example 2, sec 54° can also be found
using the reciprocal key on the calculator:
8
ENTER:
COS
54
)
x ⴚ1
ENTER
Examples 4 and 5 show how to use the
inverse capabilities of a calculator to find the
measure of an angle when one of its function
values is given. Students will learn more about
inverses in Section 10-5.
Positive function values, as shown in this
section, lead to angle measures between 0°
and 90°, or angles in Quadrant I. These acute
angles can be regarded as reference angles
that relate to all quadrants, as we will see in
Section 9-8. For this reason, trigonometric
function values are restricted here to positive
numbers.
If students attempt to enter negative function values on a calculator, they will discover
that:
• For sin21 and tan21, u is displayed as a negative angle between 0° and 290°, that is, as
an angle in Quadrant IV.
• For cos21, u is displayed as a positive angle
between 90° and 180°, that is, as an angle
in Quadrant II.
Negative trigonometric function values will be
studied in Chapter 10 as trigonometric graphs
are drawn and special functions are defined.
Enrichment Activity 9-7: Reflection and
Refraction explains the physical laws that
describe the behavior of light. The activity provides an application of finding trigonometric
functions of a given angle and finding the
angle when its sine is known.
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122 Chapter 9
9-8
REFERENCE ANGLES
AND THE
CALCULATOR
The use of reflections in the axes or the origin
simplifies the derivation of the reference angle
formulas, helps to reinforce students’ understanding of transformations, and demonstrates
a practical application of transformations.
Again, emphasize that the sign of the function value is dependent on the quadrant in
which the terminal side of the angle lies, not on
the sign of the angle measure.
Be sure students are aware of the differences in the two methods presented in Example 4. As discussed in Section 9-7, the
calculator returns a negative angle measure
between 0° and 290° when the arcsine of a
negative value is requested. This negative
angle is the reflection of a positive first-quadrant reference angle. This positive angle must
be used to find the third-quadrant angle
required in the example.
EXTENDED TASK
For the Teacher:
This extended task helps students connect
mathematics to the real world. You might wish
to have students do some research on pilot
rules and regulations for flying.
The extended task provides an opportunity
for some career education. If you live in the
proximity of an airport, you may also wish to
invite a pilot or air traffic controller into your
classroom to discuss the role that mathematics
plays in his or her job.
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ENRICHMENT ACTIVITY 9-7
Reflection and Refraction
When light passes from one medium (such as air) into another
medium (such as water), part of the incident light is reflected and
part is refracted. Reflection of light at any interface follows a simple law—the angle of incidence (u1) of the incident ray equals the
angle of reflection (u19) of the reflected ray.
The refracted ray is the ray that enters the new medium. The
angle relationship between the incident ray and the refracted ray
is given by Snell’s law:
incident
ray
normal
u1 u19
air
water
u2
n1 sin u1 5 n2 sin u2
where n1 and n2 are the indices of refraction of the two media, u1
is the angle of incidence, and u2 is the angle of refraction. The
index of refraction of a medium is the ratio of the speed of light c
in a vacuum to the speed of light v in the indexed medium:
Indices of Refraction
Benzene
1.501
Diamond
2.419
Dry Air
1.000
Ethyl Alcohol
1.362
Plate Glass
1.523
Example 1
Quartz
1.458
A light ray passes from air to water with an angle of incidence
equal to 30°. Find the angle of refraction (to the nearest degree).
Solution: According to Snell’s law,
Water
1.333
n 5 vc
The table gives the index of refraction for some common
materials. Since the index of refraction of air is very close to 1, it
is usually treated as such in computations.
n1 sin u1 5 n2 sin u2
In this case, n1 5 1.000, sin u1 5 sin 30° 5 0.5, and n2 5 1.333.
(1.000)(0.5) 5 1.333 sin u2
0.5
1.333
5 sin u2
0.3751 5 sin u2
u2 5 228
Use a calculator to find the angle
that has a sine equal to 0.3751.
(nearest degree)
Copyright © 2009 by Amsco School Publications, Inc.
reflected
ray
refracted
ray
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Exercises
1. The angle of incidence of a light ray passing from air to plate glass is 60°.
Find the angle of refraction to the nearest degree.
2. The angle of incidence of a light ray passing from air to a diamond is 45°. Find
the angle of refraction to the nearest degree.
3. The speed of light is about 186,000 miles per second. To the nearest mile per
second, what is the speed of light in:
a. Benzene
b. Quartz
c. Water
4. The angle of incidence of a light ray passing from air to a different medium
is 30°. The angle of refraction is 17.6°. Find the index of refraction, to the nearest
thousandth, of the medium.
Total internal reflection occurs when there is no refracted
light––all the incident light is reflected. In order for total internal
reflection to take place, the following two conditions must be met:
• The index of refraction of the medium where the light ray starts
is greater than the index of the medium the light ray approaches.
(The starting medium is more dense than the second medium.)
normal
n2
n1
• The angle of incidence is greater than the critical angle, uc.
uc
The critical angle occurs when the angle of refraction is 90° and
depends on the medium. Different media have different critical
angles. To find the critical angle for a medium with index n1 traveling
into a less dense medium with index n2, substitute u2 5 908 into Snell’s law:
n1 sin uc 5 n2 sin 908
n
sin uc 5 n2 ? 1
1
n
sin uc 5 n2
1
For any incident angles that are greater than the critical angle, all light will be
reflected back into the first medium.
Total internal reflection is the principle on which fiber-optic cable works. Light
injected at the end of the cable will be internally reflected many times over its entire
length because the cable is made of light-transmitting material with an index greater
than 1.
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Example 2
What is the critical angle for water with respect to air?
Solution: The index of refraction for water is 1.333. The critical
angle is the one where the angle of refraction is 90°.
sin uc 5 1.000
1.333
sin uc 5 0.7502
(nearest ten-thousandth)
uc 5 48.68 < 498
normal
air 1.000
water 1.333
Exercises
In 5–9, find the critical angle of each material with respect to air. Round to the
nearest degree.
5. Benzene
6. Ethyl alcohol
7. Diamond
8. Plate glass
9. Quartz
10. Find the critical angle to the nearest degree for a fiber-optic material with
index 1.800.
11. The critical angle for a material with respect to air is 58°. Find the index of
refraction of the material to three decimal places.
Copyright © 2009 by Amsco School Publications, Inc.
uc
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EXTENDED TASK
Trigonometry in Aviation
Trigonometry is an area of mathematics very important to aviation and the airplane pilot. Most commercial planes are equipped with the necessary equipment to
allow them to fly using Instrument Flight Rules when there is restricted visibility. On
the other hand, many private planes do not have this capability. They are required to
fly using Visual Flight Rules, which permit them to fly a plane only if the cloud ceiling
(height of the cloud from the ground) is 1,000 feet or more and if the ground visibility is at least three miles.
A trained eye is usually able to
determine the cloud ceiling and the
cloud
ground visibility during daylight
hours. At night, however, trigonometry is needed to help the pilot determine cloud height. This is
accomplished by using a ground
observer positioned 1,000 feet from a
parabolic reflector light source posiCloud
tioned at the same height as the
height
observer’s eye. The light is directed at
Parabolic
Observer's eye
light
the clouds at a constant angle of 70°.
source
The light from the parabolic light
u
70°
source hits the clouds and is reflected
to the observer’s eye. The angle
1,000 ft
formed by the reflected light and the
Ground
horizontal to the observer’s eye level
is the angle of elevation, whose measure is u.
1. Express the height of the cloud as a function of the angle of elevation of the
cloud from the observer’s eye such that u is the only independent variable in
the equation.
2. Find the cloud height to the nearest ten feet for each angle of elevation. (Since
the distance between the ground and the observer’s eye will vary from observer
to observer, you can ignore it for these calculations. Since it will be approximately
5 feet, it would not be significant in determining the cloud ceiling.)
a. 25°
b. 45°
c. 86°
3. For which of these angles would the cloud height be adequate to permit the pilot
to fly the aircraft?
4. Find, to the nearest degree, the least possible angle of elevation that would
provide a cloud cover that would allow the pilot to fly.
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Algebra 2 and Trigonometry:
Chapter Nine Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial
credit. A correct answer with insufficient work may receive only partial credit.
All scratch paper must be turned in at the conclusion of this test.
1. In PQR, R is a right angle, RP 5 7, and PQ 5 25. Give the value
of each ratio as a fraction.
a. cos P
b. sin P
c. tan P
d. cot P
e. csc P
f. sec P
g. sin Q
h. tan Q
P
25
7
Q
R
2. A fisherman observes that the measure of the angle of elevation of the top of a
120-foot lighthouse to the boat is 6°. How far, in feet, is the boat from the
lighthouse? Round to the nearest tenth.
1
3. In the diagram, QOPT intersects circle O at Q A 22 !2
2 , 23 B
and
h
1
P A 2 !2
3 , 3 B . RT
y
is tangent to circle O at R(1, 0). If
P
mROP 5 u, find:
u
a. sin u
O
b. cos u
Q
c. tan u
d. cot u
e. sec u
f. csc u
g. sin (180° 1 u)
h. cos (180° 1 u)
i. tan (180° 1 u)
j. the coordinates of T
4. The terminal side of A is in Quadrant I and sin A 5 23. Find:
a. cos A
b. tan A
c. csc A
d. sec A
e. cot A
f. sin (180° 2 A)
In 5–7, state the quadrant where the terminal side of u is located.
5. sin u , 0, cos u . 0
6. cos u , 0, cot u . 0
7. sec u , 0, csc u . 0
Copyright © 2009 by Amsco School Publications, Inc.
T
R
x
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In 8–10, state a coterminal angle with a measure of u such that 0 # u , 360.
8. 495°
9. 670°
10. 2900°
In 11–18, find the exact value of each expression.
11. sin 45°
12. tan 135°
13. cos 210°
14. cot (230°)
15. sin 270°
16. sec 240°
17. tan 30°
18. csc (2225°)
In 19–26, find each function value to four decimal places.
19. sin 45°
20. cos 93°
21. sec 285°
22. tan 200.5°
23. cos 126° 529
24. csc 36°
25. tan (–128°)
26. cot 17.5°
In 27–29, for each given function value, express u, where 0° # u , 360°: a. to the
nearest degree b. to the nearest minute.
27. cos u 5 0.5619
a.
b.
28. sin u 5 0.9200
a.
b.
29. tan u 5 4.0511
a.
b.
30. If sin u 5 0.9, find the value of sin (u 1 180°).
31. Find the degree measure of an angle in Quadrant III where sin u 5 cos u.
In 32–35, write each expression as a function of a positive acute angle.
32. tan 320°
33. cos 210°
34. cos 700°
35. tan (2112.5°)
In 36 and 37, solve for x. Round to the nearest hundredth.
36. 2x 1 sin 30° 5 cos 30°
37. 2x 1 sin 45° 5 sin 135°
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In 38 and 39, write each expression in terms of sin u, cos u, or both. Simplify the
expression if possible.
38. sec u cot u
u
39. tan
sec u
Bonus I: Find the image of the point (2, 0) under a rotation of 60° about the origin.
Bonus II: In square WXYZ, the midpoint of side WZ is P. Find the
degree measures of 1, 2, and 3 to the nearest tenth of
a degree.
W
P
Z
Copyright © 2009 by Amsco School Publications, Inc.
2
X
3
1
Y
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SAT Preparation Exercises (Chapter 9)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
7.
1. In right triangle PQR, where R is 90°,
the cosine of /P is 58. What is the sine of
P?
(A) 5 !39
39
(D) !39
(B) !39
8
(E) 8 !39
5
(A) 7 !3
2
(D) 14
(B) 7
(E) 14 !3
P
(C) !39
5
2. If the length of an altitude of an equilateral triangle is 7 !3, then the length of a
side is
(C) 7 !3
8. P
(A) 12
5
5
(B) 13
5
(D) 2 12
(E) 2 12
5
5
(C) 2 13
5. Which expression is equivalent to
cos 310° 1 cos 190°?
(A) 2cos 50° 2 cos 10°
(B) 2cos 50° 1 cos 10°
(C) cos 50° 1 cos 10°
(D) cos 50° 1 cos (210°)
(E) cos 50° 2 cos 10°
6. The expression sin 20° 2 cos 30° 1 sin 40°
2 cos 50° 1 sin 60° 2 cos 70° equals
(A) 0
(B) 1
(C) sin 20°
(D) 2cos 70°
(E) 2cos 30°
Copyright © 2009 by Amsco School Publications, Inc.
Q
T
R
S
In rectangle PQRS, if tan /QPT 5 15 and
tan /TSR 5 12, then tan PQS 5
(B) 80 sin 39°
(D) 80 tan 39°
4. If P(25, 12) lies on the terminal side of
u in standard position, then tan u is
R
S
If QR 5 RS, then which of the following
could be tan P?
(A) 0
(B) 0.8311
(C) 1
(D) 1.312
(E) 2
3. When measured from a point on the
ground that is 80 feet from the base of a
lamppost, the angle of elevation is 39°.
Which of the following equations represents the height of the lamppost?
(A) 80 cos 39°
(C) 80 sin 51°
(E) 80 tan 51°
Q
9
(A) 10
(B) 45
(D) 21
(E) 25
7
(C) 10
9.
x°
x°
x°
x°
x°
x°
Given sin x° 5 12 and x is an acute angle.
If all the rays pictured were extended
indefinitely, how many intersection points
would there be?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 9
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10.
14.
12
13
x
a
Q
y
b
13
Date
12
In the figure above, which of the following must be true?
II. tan x 5 tan a
If mP is greater than mQ, which of
the following is true?
III. cos y 5 sin b
(A) I only
(C) I and II only
(E) I, II, and III
(B) III only
(D) I and III only
I. sin P . sin Q
II. cos P . cos Q
III. sin P . cos Q
(A) I only
(C) III only
(E) I and III only
Questions 11–13 refer to the following figure.
R
15.
1
P
r
(A) r
(B) pr
(D) p1
(E) r
(C) 1r
p
1 sin R
13. The expression sin Pcos
equals
P
(C)
(E)
p 1 r
4
Q
P
Q
12. If sin P 5 k(tan P), then k equals
p 1 r
r
p 1 r
p
(B) II only
(D) I and II only
p
11. The expression (sin P)(sin P) is equivalent to
(B) 1 2 p2
(A) 1 1 p2
2
(C) r
(D) 1 2 r2
2
(E) 1 1 r
(A)
R
P
I. sin y 5 cos b
(B) p 1 r
(D)
p 1 r
2
Copyright © 2009 by Amsco School Publications, Inc.
x°
y°
R
S
If diameter PR measures 1 unit, then the
perimeter of PQRS can be expressed as
(A) (sin x°)(cos x°)(sin y°)(cos y°)
(B) sin x° 1 cos x° 1 sin y° 1 cos y°
(C) (tan x°)(tan y°)
(D) tan x° 1 tan y°
(E) sin(x 1 y)° 1 cos(x 1 y)°
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. In right triangle LMN, where M 5 90°,
N 5 45°, and LN 5 20, what is the area
of the triangle?
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17. If equilateral triangle ABC has a side of
7.6 feet, what is the value, to the nearest
tenth, of an altitude drawn to any side?
Date
23.
8
18. In ABC, C is a right angle and
tan /A 5 57. What is sin B to the
nearest thousandth?
In the figure above, if cos A 5 !3
2 , what is
BC?
24.
S
21. If cos u 5 0.600 and 0° , u , 180°, what is
the value of tan u to the nearest thousandth?
22. If sin x 5 0.655, what is the value of tan x
to the nearest thousandth?
C
A
19. In PQR, R is a right angle and
cos /P 5 49. Find the degree measure of
Q to the nearest tenth.
20. If cos 293° 5 cos B° and 0° # B # 90°,
what is the value of B?
B
t
R
r
s
T
In the figure above, if s is 20% less than t,
and r is 25% less than s, what is the measure of sin R?
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CHAPTER
10
MORE
TRIGONOMETRIC
FUNCTIONS
For all x in S 2 p2 , p2 T , the expressions
Aims
• To define and use radian measure and to
convert radians to degrees and degrees to
radians.
• To derive basic trigonometric identities.
• To define the inverse trigonometric
functions.
• To know and apply the cofunction relationships between trigonometric ratios.
In this chapter, students explore the
domain and range of each trigonometric function. Since the trigonometric functions are not
one-to-one functions, they do not have inverse
functions. However, the domain of a trigonometric function can be restricted so that the
inverse function exists. The chapter also covers
the important Pythagorean identities and the
very useful cofunction relationships.
CHAPTER OPENER
In calculus, it is shown that
3
5
7
sin x 5 x 2 x3! 1 x5! 2 x7! 1 c
`
(21) n
5 a (2n 1 1)!x2n 1 1
n50
2
4
6
cos x 5 1 2 x2! 1 x4! 2 x6! 1 c
`
(21) n
5 a (2n)! x2n
3
5
7
9
11
x
sin x 5 x 2 x3! 1 x5! 2 x7! 1 x9! 2 11!
2
4
6
8
x10
cos x 5 1 2 x2! 1 x4! 2 x6! 1 x8! 2 10!
are accurate to five digits. After students have
learned about radian measure, suggest that
they evaluate the expressions above for common angles such as p4 , p3 , p2 and compare the
values calculated with the actual values of the
sines and cosines. (The work is easily accomplished with a graphing calculator.) The accuracy is impressive. (See Section 10-2,
Hands-On Activity 2.)
10-1 RADIAN MEASURE
Degree measure of an angle is based on an
arbitrary choice of 360° as the measure of one
complete rotation. The 360° division may have
been the invention of astronomers of antiquity, who divided the zodiac into 12 signs, each
having 30 parts. The 360 subdivisions thus correspond roughly to the number of days in the
year. Another conjecture is that the origin lies
in the Mesopotamian use of numeration base
60, a number conveniently divisible by ten different factors.
Radian measure of an angle is based on the
ratio of arc length and measure of the radius
of a circle with center at the vertex of the
angle. An advantage of radians in working
with trigonometric functions is that radian
n50
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134 Chapter 10
measure, as a ratio, is independent of any unit
of measure.
Although a simple proportion enables students to determine the radian equivalent of
any degree measure, they will find it convenient to know the radian equivalents of common angle measures such as 30°, 60°, 90°, and
180°. From these values, the radian measures
of many other commonly used angles can be
easily derived.
Example: Since both 30° and p6 radians are
measures of the same angle, an angle of 150°
or 5(30°) has a radian measure of 5 A p6 B or 5p
6.
10-2 TRIGONOMETRIC
FUNCTION VALUES
AND RADIAN
MEASURE
This section provides practice with function
notation and radian measure. Encourage students to become familiar with radian measure.
Explain that, in advanced mathematics, radian
measure is used almost exclusively.
It is interesting to note that the sine of
small angles can be approximated with the
radian measure of the angles. Take the two
sides of the triangle as equal to r.
Hands-On Activity
Instructions:
1. Draw a circle with a radius of at least 2
inches.
2. Cut a piece of string whose length is equal
to the radius of the circle. Place the string
along the circle and mark points at each
end to locate the endpoints of an arc
equal in length to the radius of the circle.
3. Draw rays from the center of the circle
through the endpoints of the arc, forming
a central angle. Using a protractor, find
the degree measure of this central angle.
4. Repeat steps 1–3 two or three times, using
a circle with a different radius each time.
Discoveries:
1. The measure of the central angle whose
intercepted arc is equal in length to the
radius of the circle is 1 radian or about
57°.
2. The measure of the angle remains
unchanged when the size of the circle
changes.
In Section 9–2, angular speed was briefly
introduced in the homework exercises.
Enrichment Activity 10-1: Angular Speed
and Linear Speed continues its development
with linear speed. The exercises present
opportunities for students to see radian measure in real-world applications.
h
u
s
r
For small angles, s h and this is nearly an
isosceles triangle. The sine of the angle is
sin u 5 hr and the angle in radians is u 5 sr, and
so:
sin u 5 sin sr
5 hr
< sr
That is, for small angles, the sine of the angle is
approximately equal to the angle (in radians).
p
For example, the angle 30
is equal to 6° and
sin 6° 5 0.1045.
p
30
5 0.1047
The error in using the radian measure
rather than the sine is 2 parts in 1,000 or 0.2%.
Remind students that for the radian formula
to work, lengths must be in the same unit. (See
exercise 31.)
Enrichment Activity 10-2: The Angle
Between Two Lines presents a formula for the
tangent of the angle between two lines. Students write lines in slope-intercept form, find
the tangent, find the angle measure in degrees,
and then convert to radians.
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More Trigonometric Functions 135
10-3 PYTHAGOREAN
IDENTITIES
Three new trigonometric identities are introduced in terms of an equation for the unit circle. Since this equation in turn derives from
the Pythagorean relationship, the associated
trigonometric equations are known as the
Pythagorean identities. These, together with
the reciprocal and quotient relationships previously defined, comprise the eight basic identities summarized in this section.
It is often helpful to use simple algebraic
equations to clarify for students the definition
of an identity as an equation that is true for all
values of the variable:
x 1 x 5 2x, x ? x 5 x2, 2(x 1 1) 5 2x 1 2
The identities enable us to change a
trigonometric expression into an equivalent
one involving one or more different functions,
and also to express each of the trigonometric
functions in terms of any of the others. Therefore, if, for a given element of the domain, one
trigonometric function value is known, the values of each of the other trigonometric functions can be found.
10-4 DOMAIN AND RANGE
OF TRIGONOMETRIC
FUNCTIONS
Have students explore the domain and range
for the sine, cosine, and tangent functions
using their calculators. Have them select
twenty values of u between 0 and 2p radians
and evaluate the three trigonometric functions
for each value. Ask them to describe their
results including the least value they found for
each function, the greatest value, and any
angles for which a function was not defined.
Some students may recognize patterns of how
the values of the functions change as u
increases from 0 to 2p. Ask them to predict
the values for the functions from 2p to 4p and
check their predictions. Then have them use
the inverse function capability of the calculator to explore possible range values.
The section carefully examines the domain
and range of each of the six trigonometric
functions, justifies the conclusions geometrically and algebraically, and summarizes the
key results in a table. Understanding the
domain and range for each function will help
students in the next section when they learn
about inverse functions and also in Chapter 11
where the functions are graphed.
10-5 INVERSE
TRIGONOMETRIC
FUNCTIONS
The relation y 5 arcsin x is not a function, but
by selecting some subset of this relation, we
can obtain a function. To do this, we can select
any subset of the domain where all values of
the range occur exactly once (one-to-one) and
the range changes from the minimum value
(21) to the maximum value (1) (onto).
Although there are infinitely many possible
subsets, the one that is commonly selected
includes the values of x smallest in absolute
value, that is, the values closest to 0.
The values in these restricted domains are
sometimes called principal values. In some
texts, capital letters are used to distinguish
trigonometric functions with restricted
domains from the usual trigonometric functions. So, for example, y 5 Sin x if and only if
y 5 sin x and 2p2 # x # p2
In Chapter 11, the concept of inverse functions will again be considered when the
trigonometric functions are graphed. Students
will be able to apply the familiar vertical and
horizontal line tests to determine how the
domain of each trigonometric function must
be restricted. The multiple representations of
these functions contribute to the students’
increased understanding.
Remind students that the composition of a
function with its inverse must be the identity
function; for example,
(sin21 + sin)(x) 5 (sin + sin21)(x) 5 x
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136 Chapter 10
Students may wish to try Example 2 on
their calculators. When correctly entered, the
display will show the approximate value
0.8660254083. Since the solution to this example must be an exact value, student may recognize that the decimal is the approximation for
!3
2 .
10-6 COFUNCTIONS
The derivations of the identity
sin u 5 cos (908 2 u)
and the related identities that follow in this
section state that u is the measure of an acute
angle and (90° 2 u) is the measure of the complement. These are the conditions required for
the definition of cofunctions. However, these
identities are not restricted to first-quadrant
angles but are true for all replacements of the
variables where the function are defined.
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ENRICHMENT ACTIVITY 10-1
Angular Speed and Linear Speed
The basic formula connecting rate R, distance d, and time t is R 5 dt. This rate is a
speed, expressed as a measure of distance divided by a unit of time.
As a point P on a circle moves to a new location P9 on the same circle, changes
occur that result in two types of speed.
• Linear speed (v) is the change in the arc length s divided by
a period of time t. Just as R 5 dt, the formula for linear
P9
P
speed is v 5 st .
R
• Angular speed (v, the Greek letter omega) is the change in
the angle measure u, expressed in radians, divided by a
O
period of time t. The formula for angular speed is v 5 ut .
From Section 10-1, we know that the radian measure u of a central angle, the
length s of the intercepted arc, and the radius r are related by the formula u 5 sr, or
s 5 ru . By substitutions involving this formula, we see that
v 5 st S v 5 rut S v 5 r A ut B S v 5 rv
Linear speed and angular speed are used extensively in engineering, physics, and
technological studies. A simple example will show how to use the formulas.
Example
A minute hand on a clock is 5 inches long.
a. Find the angular speed of this minute hand in radians per hour.
b. Find the linear speed of the endpoint of the minute hand.
Solution:
a. It takes 1 hour for the minute hand to make a full rotation. In a full rotation,
u 5 2p radians. Thus,
5 2p radians>hr
v 5 ut 5 2p radians
1 hr
b. From part a, v 5 2p radians/hr. Thus,
v 5 rv 5 5 in. A 12p
hr B 5 10p in.>hr < 31.4 in.>hr
Answers: a. 2p radians per hour
b. 10p inches per hour
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Exercises
Unless otherwise noted, express answers in terms of p to show an exact value.
1. a. On a clock, indicate the time required for each of the following hands to complete a full rotation: (1) an hour hand (2) a minute hand (3) a second hand
b. Find the angular speed on a clock, expressed in radians per hour, for each of
the hands listed in part a.
c. The lengths of the hands are usually different on a clock. Find the linear
speed, expressed in inches per hour, for: (1) a 6-inch hour hand (2) an 8-inch
minute hand (3) a 10-inch second hand.
For Exercises 2 and 3, assume that the radius of Earth is 6,400 kilometers.
2. Earth completes a full rotation on its axis every 24 hours or once per day.
a. State the angular speed of Earth in: (1) radians per day (2) radians per hour
b. State the linear speed of a point on Earth’s equator: (1) in kilometers per day
(2) to the nearest 100 kilometers per hour
c. State the linear speed of a point at Earth’s North Pole. Explain your answer.
3. A satellite travels 800 kilometers above Earth’s surface and circles the planet
once every 6 hours.
a. Find the angular speed of the satellite in radians per hour.
b. Find the linear speed of the satellite in kilometers per hour.
4. A bicycle has a wheel with a 28-inch diameter. The bicycle is ridden at a steady
pace, resulting in the wheel turning at 120 revolutions per minute.
a. Find the angular speed of the bicycle wheel in radians per minute.
b. Find the linear speed of a point on the wheel in: (1) inches traveled per
minute (2) feet traveled per minute
c. Is 880 feet per minute a good estimate of the linear speed found in part b(2)?
Explain your answer.
d. Express 880 feet per minute as a rate in miles per hour.
5. In a machine, a belt is connected to a wheel that turns at 240 revolutions per
minute. The radius of the wheel is 3 inches.
a. Express the angular speed of the wheel in: (1) radians per minute (2) radians per second
b. Find the linear speed of the wheel in: (1) inches per minute (2) inches per
second
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6. A model plane, attached to a stake in the ground by a string,
flies in circles, making one revolution every 5 seconds. The
radius of the circle made by the plane is 15 feet.
a. Express the angular speed of the plane in radians per
minute.
b. Express the linear speed of the plane in feet per minute.
c. Find, to the nearest foot, the distance traveled by the
plane in 1 minute.
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Date
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ENRICHMENT ACTIVITY 10-2
The Angle Between Two Lines
Suppose l1 and l2 are two intersecting lines. Then we define the
angle from l1 to l2 to be the angle u through which l1 must be
rotated counterclockwise about the point of intersection in order to
coincide with l2. Thus, 0 # u , p.
There is a relationship between u and the slopes of the lines. If
l1 and l2 are two non-vertical lines that are not perpendicular and l1
has slope m1 and l2 has slope m2, then the tangent of the angle u
from l1 to l2 is given by
l1
u
l2
m 2 m
2
1
tan u 5 1 1
m1m2
Exercises
In 1–5, find: a. the tangent of the angle u from l1 to l2. b. the angle u to the nearest
degree. c. the radian equivalent of the answer to b.
1. l1: y 2 2x 5 2
l2: 2y 1 5x 5 17
4. l1: x 1 y 5 6
l2: x 1 2y 5 8
2. l1: y 5 4x 2 2
l2: 3y 5 22x 1 7
3. l1: 2y 5 x
l2: y 2 3x 5 7
5. l1: y 5 2x 2 2
l2: y 5 25x 1 5
6. The equation of line A is x 1 y 5 6 and the equation of line B is 3x 2 4y 5 4. Find, to the nearest degree: a. the angle from line A to line B. b. the angle from line B to line A. (Make sure to
express each angle as a positive angle.)
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Algebra 2 and Trigonometry:
Chapter Ten Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–4, express each given degree measure in radians.
1. 75°
2. 18°
3. 2240°
4. 510°
In 5–8, express each given radian measure in degrees.
5. 3p
5
7.
6. 23p
7p
4
8. 4p
9
9. Find the radian measure of a central angle that intercepts an arc of 12 inches
in a circle whose radius has a length of 5 inches.
10. Circle O has a radius of 1.4 inches. What is the length, in inches, of an arc
intercepted by a central angle whose measure is 3 radians?
In 11 and 12, find, to the nearest ten-thousandth, the radian measure u of a
first-quadrant angle with the given function value.
11. sin u 5 0.4658
13. If f(x) 5 sin 4x 2 cos 2x, find f A p8 B .
12. tan u 5 8.2794
14. Find the value of (sin2 u 1 cos2 u 2 5)2.
1
15. Find the value of (3 sec2 u 2 3 tan2 u 1 3) 2.
In 16–18, write each expression in terms of sin u, cos u, or both. Simplify the expression
if possible.
16. tan2 u 1 1
17. sin u (cot2 u 1 1)
u csc u
18. costan
u
In 19–24, find the exact value of u in radians.
19. u 5 arcsin !2
2
21. u 5 arctan (21)
23. u 5 arctan !3
Copyright © 2009 by Amsco School Publications, Inc.
20. u 5 arccos 0
22. u 5 arcsin (20.5)
24. u 5 arccos A 212 B
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In 25–30, find the exact radian measure u of an angle with the smallest absolute
value that satisfies the equation.
25. u 5 arccot !3
26. u 5 arccsc (22)
29. cot u 5 0
30. csc u 5 22
27. sec u 5 !2
28. u 5 arcsec (21)
In 31–34, find the value of each expression.
7
31. cos A arcsin 25
B
5
32. tan A arccos 13
B
33. sec A arccos 25 B
34. sin (arctan A2!3B B
35. What is the greatest negative integer in the range of the secant function?
36. If cos u 5 sin (u 1 20°) and u is the measure of an acute angle, find the value
of u.
37. If tan 2u 5 cot (2u 1 15) and u is the measure of an acute angle, find the value
of u.
38. A box contains 181 chips. Each chip is marked with one of the integers from
290 to 90. Each integer occurs exactly once. What is the probability that if a
chip is randomly selected, the tangent function will be undefined at the value
indicated by the chip?
u sec u
Bonus: If tan u 5 45, find sincot
u .
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SAT Preparation Exercises (Chapter 10)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. The length of the arc intercepted by a
central angle of 4 radians in a circle with a
radius of 2.5 centimeters is
(A) 0.625 cm
(B) 1.6 cm
(C) 4 cm
(D) 5 cm
(E) 10 cm
2. In a circle whose radius measures 1.8
inches, a central angle intercepts an arc of
length 4.5 inches. The radian measure of
the central angle is
(A) 0.4
(B) 1.25
(C) 2.5
(D) 2.7
(E) 8.1
3. Angle M measures 7p
10 and angle N
measures 5p
12 . The difference in their
measures M 2 N expressed in degrees is
(A) 12°
(B) 20°
(C) 24°
(D) 51°
(E) 75°
4. Which is the radian measure of an angle
that measures 495°?
(A) 7p
2
(B) 9p
3
(D) 13p
6
(E) 15p
12
5. 2 csc p3 1 8 tan p6 5
(A) 4 !3
6.
A tan
(A)
1
9
(D) 2
(B)
1
3
(B)
(D) !3
5p 2
6 B
(E)
10 !3
3
!3
2
(C) 11p
4
7. To four decimal places,
2 5p
sin2 5p
9 1 cos 9 5
(A) 0.8116
(B) 0.9698
(D) 1.0744
(E) 1.7814
8. 1 1 1cos x 1 1 2 1cos x is equivalent to
(A) cos22 x
(B) 2 sin2 x
2
(D) tan2 x
(C) 2 sec2 x
(E) 2 csc2 x
9. Evaluate A 3 sec2 p4 2 3 tan2 p4 1 5 B .
2
(A) 64
(D) 8
(B) 36
(E) 4
(C) 16
10. cos 320° 5
(A) sin 40°
(C) sin 50°
(E) 2cos 40°
(B) cos 50°
(D) cos 60°
11. tan 190° 5
(A) 2tan 10°
(C) tan1108
(E) 2cot 10°
(B) tan 80°
(D) cot 80°
12. Which is not in the domain of y 5 sec x2?
(A) 0
(B) p2
(C) p
(D) 3p
2
(C) 2 !3
(C) 1.0000
(E) 2p
13. Which is not in the range of y 5 csc x?
(A) 217
(B) 23
(C) 0
(D) 1
(E) 4
14. If u 5 arcsin (cos (2p)), then a coterminal angle with u has measure
5
(C) 1
(E) 3
(A) p2
(D)
3p
2
(B) 3p
4
(E) 2p
15. tan (cos21 (20.5)) 5
(A) 2!3
(D) !3
Copyright © 2009 by Amsco School Publications, Inc.
(C) p
(B) 2 !3
2
(E) 3 !3
2
(C) !3
2
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II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. A sector has a radius of 8 centimeters and
a central angle of 13p
36 . To the nearest hundredth, find its arc length.
Date
19. Find the exact value of
A tan p3 2 sec p3 B A tan p3 1 sec p3 B 1 3.
20. If x 5 cos21 A !3
2 B , what is the degree
measure of x?
21. If sin (2x 1 5) 5 cos (7x 2 5), find x.
5
17. Evaluate sin A tan21 12
B as a fraction in
lowest terms.
22. If tan 215° 5 cot u and u is an angle in
Quadrant I, what is the degree measure
of u?
18. Find the exact value of
(5 2 4 sin2 x 2 4 cos2 x)3 when
x 5 10100.
24. If cos u 5 34, find tan u (cot u 1 tan u).
Copyright © 2009 by Amsco School Publications, Inc.
23. If f(x) 5 22 sec 2x, evaluate f A p3 B .
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CHAPTER
11
GRAPHS OF
TRIGONOMETRIC
FUNCTIONS
Aims
• To graph the sine, cosine, and tangent
functions.
• To define amplitude and period and to
study the effects of amplitude and period
on the trigonometric graph.
• To graph the reciprocal functions.
• To write the equation of the trigonometric
function that is represented by a given
periodic graph.
• To graph the inverse trigonometric functions.
In this chapter, students will study the
graphs of the trigonometric functions and
relate characteristics of the graphs to the
equations of the functions. Since the trigonometric functions are periodic, their graphs are
symmetric with respect to a translation in the
horizontal direction.
CHAPTER OPENER
The hertz (Hz) is the unit of measure for frequency; for example, 440 waves per second is
equal to 440 Hz. A healthy human ear can
hear sound frequencies from about 20 Hz to
20,000 Hz. As people get older, they often lose
the ability to detect sounds at high frequencies. Frequencies above 20,000 Hz are called
ultrasonic waves. Most humans cannot hear
ultrasonic frequencies, but bats can detect frequencies as high as 100,000 Hz. Sonar, or
sound navigation, uses ultrasonic sound waves
to estimate the size, shape, and depth of submerged objects. Ultrasound is also used in
many medical diagnostic tools and treatments.
Infrasonic waves have frequencies that are less
than 20 Hz. These waves may be produced by
sources such as thunder or heavy machinery.
Their effects are sensed as a rumble rather
than actually heard.
11-1 GRAPH OF THE SINE
FUNCTION
In previous chapters, students have found the
values of trigonometric functions for angles
measured in degrees and radians. They have
found exact values for special angles and
approximate values using a calculator. As is
often the case, a graph is an effective way of
presenting the data; by studying the graph, key
patterns and characteristics can be recognized.
Graph paper that is divided into 1-centimeter blocks, each of which is further divided into
five 2-millimeter sections, is well suited for student work. A convenient scale uses two large
blocks to represent 1 along the y-axis and one
large block to represent p6 along the x-axis.
See, for example, the graph of y 5 sin x on
page 435 of the text. (Note that the scale used
in the text is slightly smaller than 1 centimeter.)
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146 Chapter 11
Call attention to various symmetries of the
graph. Because the function is periodic, the
graph is its own image under a translation
T 2kp,0 for any integer, k. When k 5 1, 2kp
assumes its smallest positive value, 2p, which
is the period of the function. Point out that this
translational symmetry makes it possible to
sketch the graph of y 5 sin x for all values of x
by simply repeating the curve drawn for the
values from 0 to 2p.
The graph has other line and point symmetries. For example, the graph has point symmetry since it is its own image under a reflection
in the origin or in any point on the x-axis
whose x-coordinate is an integral multiple of
p. Also, the graph has line symmetry since it
is its own image under a reflection in the
line whose equation is x 5 p2 or any other vertical line whose x-intercept differs from p2 by
an integral multiple of p, for example, x 5 3p
2
or x 5 5p
.
2
The Hands-On Activity is a very dynamic
demonstration of the relationship between the
unit circle and the sine and cosine functions.
Because they are both defined using a unit circle, the sine and cosine are often called circular functions.
Amplitude, period, and frequency will be
discussed in greater depth in Section 11-3.
11-2 GRAPH OF THE
COSINE FUNCTION
The graph of y 5 cos x is presented in a manner similar to that used for y 5 sin x.
The graph of y 5 cos x has the same translational symmetry as the sine graph since it has
the same period, 2p.
Like the sine graph, the cosine graph has
point and line symmetries. It is symmetric with
respect to the y-axis or with respect to any vertical line whose x-intercept is a multiple of p.
The symmetry with respect to the y-axis means
that, for all x, cos x 5 cos (2x). This identity
will be proven algebraically in Chapter 12. The
cosine graph has point symmetry with respect
to the point A p2 , 0 B or any other point on the
x-axis whose x-coordinate differs from p2 by an
integral multiple of p, for example A 3p
2 , 0 B or
A 2p2 , 0 B .
Encourage students to describe ways in
which the functions y 5 sin x and y 5 cos x are
the same and ways in which they are different.
Ask students the following question: Suppose
you graphed y 5 sin u and then decided you
wanted to describe the graph using the cosine
function instead. How could you do this?
A If y 5 sin u, then y 5 cos A u 1 p2 B . B
11-3 AMPLITUDE, PERIOD,
AND PHASE SHIFT
Call attention to the careful definition of
amplitude. Students may ask whether amplitude may be alternatively defined as the maximum value of the function. This alternate
definition is correct only for functions where
the minimum and maximum values have the
same absolute value, such as the function
defined here. Notice that the definition is not
correct for a function such as y 5 3 1 2 sin x
that has a maximum value of 5, a minimum
1
value of 1, and an amplitude of 5 2
2 or 2.
The generalization that the function
y 5 a sin x has an amplitude of a can be
extended to y 5 c 1 a sin x, which also has an
amplitude of a.
For the discussion of period, be sure that
students understand the order of operations.
To find values for the graph of y 5 sin 2x, it is
necessary to choose a value of x, multiply that
value by 2, and then find the sine values.
Call attention to the fact that, no matter what
value is assumed by 2x, the sine cannot be
greater than 1 or less than 21. For the interval
0 # x # 2p, the function y 5 sin x attains its
maximum value once and its minimum value
once, y 5 sin 2x attains its maximum value
twice and its minimum value twice, and
y 5 sin 12x attains its maximum value once but
does not attain its minimum value.
In the case of sound and light, waves that
are in phase (crests of the two waves are
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Graphs of Trigonometric Functions 147
together and troughs are together) interfere
constructively and the amplitude of the resulting wave is the sum of the individual amplitudes. Waves that are out of phase interfere
destructively; if two waves of the same amplitude and same frequency are 180° (p) out of
phase, the resultant amplitude is zero and no
sound would be heard.
Sometimes a problem is written so that it
appears as if the period and phase shift are
inside the trigonometric function; for example
y 5 sin (2x 1 p) makes it look like the period
is twice as fast and the horizontal shift is p.
However, all phase shifts must be factored out
to be analyzed, so y 5 sin 2 A x 1 p2 B shows
that the period is twice as fast but the horizontal shift is p2 .
Note that the frequency is the number of
cycles per unit time and is measured in hertz.
The angular frequency, on the other hand, is
the number of cycles in the 2p interval and
is measured in radians per second. If the
|b|
frequency is 2p
, then the angular frequency is
b.
In Enrichment Activity 11-3: Graphing
Combined Functions, students use tables of
values and a graphing calculator to draw and
analyze the graphs of sums and differences of
sine and cosine functions.
11-4 WRITING THE
EQUATION OF A SINE
OR COSINE GRAPH
The steps for writing an equation of a sine or
cosine graph are the same. For an equation
using sine, the basic cycle is found by identifying a portion of the graph that begins at 0,
increases to its maximum, decreases to 0, continues to decrease to the minimum, then
increases to 0. For an equation using cosine,
the basic cycle begins at the maximum value,
decreases to 0, continues to decrease to the
minimum, increases to 0, then increases to the
maximum.
Be sure students understand Step 3 of the
solution to Examples 1 and 2. The interval for
one cycle is found in the previous step by the
visual process described above. The difference
between the endpoints of the interval is the
period.
Since the period of the function was
defined as P 2p
b P , we can find b by equating the
two expressions for the period.
Since the basic sine and cosine curves
begin at 0, the phase shift c is positive if the
cycle on the given graph begins to the left of 0
and negative if the cycle begins to the right of
0. Have students compare this relationship to
the equation of the basic absolute value equation y 5 x or the parabola y 5 x2 that is translated left or right.
If students have difficulty answering Exercise 2, suggest that they graph the two functions with their graphing calculator. They may
also verify their equations for Exercises 3–14
by graphing. It is convenient to use the
trigonometric window, found by pressing
ZOOM 7, for these graphs. In this window,
each tickmark on the x-axis represents p2 and
each tickmark on the y-axis represents 1 unit.
Enrichment Activity 11-4: Polar Coordinates is the first of two activities focusing on
this topic. Students have learned how to associate a pair of coordinates on the unit circle
with the sine and cosine functions, and polar
coordinates are an extension of this idea. The
activity allows students to practice converting
from rectangular to polar coordinates and vice
versa. In the next activity, they explore some of
the interesting polar graphs. Both activities
can be used at any time during the chapter.
11-5 GRAPH OF THE
TANGENT FUNCTION
The graph of the tangent function is similar to
that of the sine or of the cosine function in several respects:
1. It is the graph of a periodic function with
a period of p. Therefore, it has translational symmetry.
2. It has point symmetry since it is its own
image under a reflection in the origin
or in any point on the x-axis whose
x-coordinate is an integral multiple of p2 .
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148 Chapter 11
The graph of the tangent function differs
from that of the sine or of the cosine function
in several respects:
interval; in this case the vertical line test fails,
which indicates that the inverse relation is not
a function for the interval. Have them perform
1. It is discontinuous at p2 and at any value
of x that differs from p2 by an integral multiple of p.
a similar analysis for the interval 2p2 to p2 to
reinforce the reason why the domain of
the sine function is restricted to this interval.
The interval includes all of the values of sin x
from 21 to 1 and the sine is a one-to-one function on this interval, so the inverse is also a
function.
2. It is always increasing.
3. It has no maximum or minimum values
and, therefore, no amplitude.
4. It has no line symmetry.
11-8 SKETCHING
TRIGONOMETRIC
GRAPHS
11-6 GRAPHS OF THE
RECIPROCAL
FUNCTIONS
To draw the graph of y 5 csc x, use the table
that was made to graph y 5 sin x and determine the value of csc x by finding the reciprocal of the given value of sin x. Whenever
sin x 5 0, csc x is undefined. Students may find
it helpful to find values of csc x for values of x
close to those for which csc x is undefined,
p 14p 16p
, 15 , 15 , and 29p
such as 15
15 . Remind them
that the calculator must be in radian mode and
p
1
that csc 15
must be evaluated as
p .
sin 15
ENTER:
1 ⴜ
ⴜ
DISPLAY:
15
SIN
2nd
)
ENTER
p
1/sin(π/15)
4.809734345
The right parenthesis after 15 is optional.
11-7 GRAPHS OF INVERSE
TRIGONOMETRIC
FUNCTIONS
Have students use the horizontal line test with
the graph of y 5 sin x from 0 to p. Since the
horizontal line test fails, they know the function is not one-to-one on this interval. Have
them identify the portion of the graph of the
inverse sine function that corresponds to this
After students have learned how changing the
values of a and b affect the graphs of
y 5 a sin bx and y 5 a cos bx, it should no
longer be necessary for them to sketch a graph
by making a table of values and plotting
points. Emphasize, however, the importance of
locating the points of intersection with the
x-axis, and the maximum and minimum points,
before sketching the curve.
Encourage students to determine the
period of the curve, to mark off intervals equal
to the period on the x-axis, and then to divide
these intervals into four sections. The maximum point(s), minimum point(s), and points
of intersection with the x-axis will occur at the
endpoints of these four sections. An accurate
sketch can be made after these critical points
are located.
If students have been comparing their
graphs with those displayed on a graphing calculator, be sure that they understand that
a graph shown on the calculator varies with
the viewing window. For example, the graph of
y 5 sin x when WINDOW is from
Xmin 5 22p to Xmax 5 2p
Ymin 5 22 to Ymax 5 2
will look exactly like the graph of y 5 2 sin 2x
when WINDOW is from
Xmin 5 2p to Xmax 5 p
Ymin 5 24 to Ymax 5 4
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Graphs of Trigonometric Functions 149
Students should have ample practice drawing the graphs of trigonometric functions on
paper and comparing these graphs with those
displayed on a calculator.
In Exercises 21–23, students may realize
they are solving trigonometric equations. For
example, Exercise 21 can be used to solve
2 sin x 5 cos x graphically. Each side of the
equation is graphed and the points of intersection of the two graphs are the solutions on the
specified interval of the equation. Trigonometric equations are discussed in Chapter 13.
Enrichment Activity 11-8: Graphing
Polar Equations continues the work with
polar coordinates begun in the previous
enrichment activity. Students use their graphing calculators in polar mode to graph the
polar rose family r 5 a sin bu and the cardioid
family r 5 a(1 6 sin u) and r 5 a(1 6 cos u).
They identify how changing values alter the
appearance of the graphs. Students also graph
the Archimedean spiral r 5 au.
EXTENDED TASK
For the Teacher:
The purpose of this activity is to help students see that there are phenomena in the real
world that are modeled by trigonometric
curves. The activity will help reinforce the concepts of the period and amplitude of trigonometric curves.
If desired, you could have the students
research data for their own city rather than
Buffalo, New York. You might also have them
research data for their own city and draw the
corresponding graph and compare it with the
data and graph for Buffalo, New York. A comparison of the Buffalo graph and equation
with similar results for a city in the southern
hemisphere would also provide interesting
discussion material.
An extension of this task is to have them
find another phenomenon besides weather
that can be modeled by a trigonometric graph.
Possibilities include tides, the number of daylight hours, certain predator-prey relationships, or the motion of a Ferris wheel.
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ENRICHMENT ACTIVITY 11-3
Graphing Combined Functions
Trigonometric functions can be combined by addition or subtraction, and the
resulting functions can be graphed. The table below shows values of sin x and cos x
for 0 # x # 2p. Note that to graph combined functions, it is often useful to show many
function values.
x
0
p
6
p
4
p
3
p
2
2p
3
3p
4
5p
6
p
sin x
0
0.5
0.71
0.87
1
0.87
0.71
0.5
0
cos x
1
0.87
0.71
0.5
0
20.5
20.71
20.87
21
7p
6
5p
4
4p
3
3p
2
5p
3
7p
4
11p
6
2p
sin x
20.5
20.71
20.87
21
20.87
20.71
20.5
0
cos x
20.87
20.71
20.5
0
0.5
0.71
0.87
1
sin x 1 cos x
sin x 2 cos x
x
sin x 1 cos x
sin x 2 cos x
1. Complete the two rows of values for y 5 sin x 1 cos x.
2. Sketch the graph of y 5 sin x 1 cos x.
3. a. What is the maximum value of y 5 sin x 1 cos x? Where does this value occur?
b. What is the minimum value of y 5 sin x 1 cos x? Where does this value occur?
c. What is the period of y 5 sin x 1 cos x?
4. Complete the two rows of values for y 5 sin x 2 cos x.
5. Sketch the graph of y 5 sin x 2 cos x.
6. Answer questions 3a–c for y 5 sin x 2 cos x.
7. For what values of x is sin x 1 cos x 5 sin x 2 cos x?
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A graphing calculator is a useful tool for displaying and analyzing the graphs of
combined functions. To see how the combined function was derived, you can graph
the component functions and the combined function on the same screen. Make sure
the calculator is in RADIAN mode and begin by selecting the trigonometric window.
For the examples below, adjust Ymin 5 22 and Ymax 5 2.
8. Enter Y1 5 sin x, Y2 5 0.5 sin 2x, and Y3 5 Y1 1 Y2. Graph Y1, Y2, and Y3 in the
interval 0 # x # 2p. (To view only Y3, deselect Y1 and Y2.)
9. Answer questions 3a–c for y 5 sin x 1 0.5 sin 2x.
10. a. Graph Y1 5 sin x and Y2 5 sin (x 1 p).
b. Predict what you will see on the screen if you graph y 5 sin x 1 sin (x 1 p).
Give reasons for your answer.
c. Verify your prediction by graphing the combined function.
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ENRICHMENT ACTIVITY 11-4
Polar Coordinates
In your study of algebra, you have learned to identify each point in the plane with
its coordinates in a Cartesian coordinate system. However, there are other coordinate
systems, one of the most important being the polar coordinate system.
Any point P in the plane can be identified with two numbers r and u
P(r, u) y
where
r
r 5 distance from the origin to the point
u 5 angle between the positive x-axis and the line drawn from the
origin to the point
x
The pair (r, u) is called a set of polar coordinates for the point P. For
point P there are infinitely many possible choices of u, any two differing by
a multiple of 2p. However, there is only one set of polar coordinates (r, u) such that
r . 0 and 0 # u , 2p. The figures below show several points in the plane along with
their polar coordinates.
y
(2, p4 )
(2, 3p4 )
y
x
y
x
y
x
x
(1, 3p2 )
(1, 5p4 )
Every point in the plane has both Cartesian and polar coordinates, and it is possible to convert from one type of coordinates to the other. Suppose a point P in the
plane has polar coordinates (r, u) and Cartesian coordinates (x, y). From the definition of sine and cosine:
y
r
P
y 5 r sin u
x 5 r cos u
u
x 5 r cos u
y 5 r sin u
x
If the Cartesian coordinates of P are (x, y), then
y
r2 5 x2 1 y2 and tan u 5 x for x 2 0
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u
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Example 1
Find the Cartesian coordinates of the point P having polar coordinates A 3, p4 B .
3 !2
Solution: x 5 3 cos p4 5 3Q !2
2 R 5 2
3 !2
y 5 3 sin p4 5 3Q !2
2 R 5 2
3 !2
The Cartesian coordinates of P are Q 3 !2
2 , 2 R.
Example 2
Find the polar coordinates of the point P having Cartesian coordinates A25, 5 !3B .
Solution: r2 5 (25) 2 1 A5 !3B 2 5 25 1 75 5 100
r 5 !100 5 10
Also,
!3
5 2!3
tan u 5 525
Since A25, 5!3B lies in Quadrant II, tan21 A2!3B 5 2p
3 . (Note that we must be
sure that u lies in the correct quadrant.)
The polar coordinates of P are A 10, 2p
3 B.
In 1–8, find the Cartesian coordinates for each point with the given polar coordinates.
1.
A 3, p3 B
5. (1, 3p
2 B
2.
A 2, p2 B
3. A 4, 3p
4 B
7. A 2, 3p
2 B
6. (5, 0)
4.
8.
A 1, 11p
6 B
A 3, 7p
6 B
In 9–14, find the polar coordinates for each point with the given Cartesian coordinates.
9. (3, 3)
12. (24, 0)
10. (4, 24)
13. A3, 3 !3B
11. (0, 5)
14. A22 !3, 2B
An equation written in Cartesian coordinates can be converted to polar form by
substituting x 5 r cos u and y 5 r sin u in the equation. For instance, in polar form,
the equation y 5 2x 1 2 becomes
y 5 2x 1 2
r sin u 5 2r cos u 1 2
15. Find the polar equation of the circle x2 1 y2 5 a2, where a . 0.
16. Write the equation of the line 2x 1 3y 5 4 in polar coordinates. Express the
answer in the form r 5 f(u).
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ENRICHMENT ACTIVITY 11-8
Graphing Polar Equations
Polar coordinates allow for the graphing of some very interesting figures. To graph
polar equations, you must select Pol graphing mode before you enter values for
window variables or any polar equations. Radians must also be used for angles. ZTrig
7 ) followed by ZSquare ( ZOOM
5 ) works well for most of the
( ZOOM
graphs below; adjust the window variables as needed.
Part I
1. The polar equations r 5 a sin bu and r 5 a cos bu for a . 0 and b , 0 are the
equations of a rose. To explore the appearance of the rose for different values
of b, graph each equation below and make a sketch of the graph for reference.
a. r 5 2 sin 3u
b. r 5 2 sin 2u
c. r 5 2 cos 5u
d. r 5 2 cos 4u
2. Try other values of a and b in the equations r 5 a sin bu and r 5 a cos bu, and
observe how the new values affect the graph.
3. How many petals does the graph have if b is odd? How many if b is even?
4. What type of symmetry do graphs involving the sine have? What type of
symmetry do graphs involving the cosine have?
5. How does the value of a affect the appearance of the rose?
Part II
6. The polar equations r 5 a(1 6 sin u) and r 5 a(1 6 cos u) for a . 0 both graph a
cardiod. To explore the appearance of the cardiod for different equations, graph
each equation below and make a sketch of the graph for reference.
a. r 5 2(1 1 sin u)
b. r 5 2(1 2 sin u)
c. r 5 2(1 1 cos u)
d. r 5 2(1 2 cos u)
7. How are the graphs the same? How are they different?
8. What type of symmetry do graphs involving sine have? What type of symmetry
do graphs involving cosine have?
9. Try different values for a in the equations r 5 a(1 1 sin u) and r 5 a(1 1 cos u),
and explain the effect on the graphs.
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Part III
10. The polar equation r 5 au graphs an Archimedean spiral. For this graph, set the
window variables umin 5 0 and umax 5 6p. Graph each equation below and make
a sketch of the graph for reference.
a. r 5 0.2u
b. r 5 20.2u
11. How do the two spirals differ?
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EXTENDED TASK
Temperature
Trigonometric graphs are periodic, that is, they repeat themselves over an interval,
or period. For example, the sine function produces a curve that is periodic. The graph
of the equation y 5 sin x has a period of 2p radians and an amplitude of 1.
Many familiar phenomena are periodic: sound waves, electrical currents, business
cycles, air pollution levels, and so on. In this extended task, you will graph real-world
data and compare the resulting graph to a trigonometric function that models
the physical situation. For this task, we will use data familiar to all: monthly normal
temperatures
The table below gives the normal monthly temperatures, in degrees F, for Buffalo,
New York.
Normal
Temperature
Month
Normal
Temperature
January
24°
July
71°
February
25°
August
69°
March
34°
September
62°
April
45°
October
51°
May
57°
November
41°
June
66°
December
29°
Month
Graph, on a set of axes, the normal monthly temperatures for Buffalo. Place the
months January through December on the horizontal axis and the temperatures in °F
on the vertical axis. Use an appropriate scale. Use only half the horizontal axis for the
months January through December. Observe the resulting curve and make any statements about it that you can.
Now extend the labeling on the horizontal axis to include another January
through December and graph the data a second time. This will give you a repeat
curve congruent to the first curve drawn.
1. What is the range for the monthly normal temperatures for Buffalo, New York?
2. Find one-half of this range and add this value to the lowest normal monthly
temperature. What is this temperature?
3. Draw a horizontal line on your graph at this temperature.
4. Superimpose a coordinate grid on your graph. Place the x-axis along the
horizontal line drawn in Exercise 3 above and the y-axis at the first point where
this x-axis intersects the temperature curve.
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5. Using a pen of a different color, highlight one cycle of your curve, beginning at
the origin.
6. Based on the placement of the x-axis and y-axis, which trigonometric curve
models the normal monthly temperatures for Buffalo, New York?
7. How many days are there in the period of your curve?
8. What is the amplitude of your curve?
9. Find another set of weather data and graph those data in a similar fashion.
10. Discuss your graph, answering questions such as:
a. Is the graph periodic?
b. Does the graph model one of the trigonometric curves?
c. How does it compare to the normal monthly temperature graph for the
Buffalo data?
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Algebra 2 and Trigonometry:
Chapter Eleven Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–3, for each function: a. Find the amplitude. b. Find the period. c. Sketch the
graph in the interval 0 # x # 2p. Clearly label the points where the graph crosses
the x-axis and where the minimum and maximum values occur.
1. y 5 cos x
a.
b.
2. y 5 3 sin 2x
a.
b.
3. y 5 2 sin 12x
a.
b.
4. a. Sketch the graph of y 5 cos 2x in the interval 2p # x # p.
b. On the same set of axes, sketch the graph of y 5 2sin x in the interval
2p # x # p.
c. Which of the graphs drawn in parts a and b is symmetric with respect to
the y-axis?
d. Which of the graphs drawn in parts a and b is symmetric with respect to
the origin?
e. How many points do the graphs of y 5 cos 2x and y 5 2sin x have in
common in the interval 2p # x # p?
In 5–8, for each function, find: a. the amplitude. b. the period. c. the phase shift.
5. y 5 cos A x 1 p4 B
6. y 5 2 sin A x 2 p3 B
7. y 5 cos 2 A x 1 p2 B
8. y 5 12 sin 4 A x 2 p6 B
a.
b.
c.
a.
b.
c.
a.
b.
c.
a.
b.
c.
In 9 and 10, write the equation of the graph as a: a. sine function b. cosine function. In
each case, choose one cycle with the lower endpoint closest to zero to find the phase shift.
9.
2
10.
y
1
1
23p
2
2p
21
y
x
p
2
p
3p
2
x
25p 24p
3
3
22p 2p
3
3
p
3
21
22
a.
2p
b.
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a.
b.
2p
3
p
4p
3
5p
3
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11. What is the domain of the secant function?
12. What is the range of the cotangent function?
13. In the interval p # x # 3p
2 , how are the values of the function y 5 csc x
changing?
14. What is the value of y 5 cot u when tan u 5 !3
3 ?
In 15–18, find each exact value in degrees.
15. y 5 arcsin !3
2
16. y 5 arccos 12
17. y 5 arctan Q2 !3
3 R
18. As you ride a Ferris wheel, the height that you are above the ground varies
periodically. Consider the height of the center of the wheel to be the starting
point. A particular wheel has a diameter of 46 feet and travels at a rate of
3 revolutions per minute.
a. Sketch a graph in which the horizontal axis represents time and the vertical
axis represents height in relation to the starting point.
b. What is the period?
c. What is the amplitude?
19. Is there an interval where the graphs of y 5 cos x and y 5 sec x are both
increasing? Give the interval or explain why no interval satisfies the condition.
Bonus: a. Sketch the graph of y 5 sin x for the interval 22p # x # 2p.
b. What is the period of the function?
c. Under what transformation(s) is the graph of y 5 sin x its own image?
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SAT Preparation Exercises (Chapter 11)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
4.
P( p6 ,
1. If, for some real number x, x 1 cos x 5 7,
then x 1 2p 1 cos (x 1 2p) is
(A) 7 1 2p
(B) 9p
(C) 7 2 2p
(D) 5p
(E) Not determinable
(C) A p3 , 1 B
(E) A p6 , 0.6 B
x
1
(B) A 5p
6 , 22 B
1
(C) A 5p
6 , 2B
1
(D) A 7p
6 , 22 B
1
(E) A 11p
6 , 22 B
y
2
p
x
The shaded area lies between the x-axis
and the graph of y 5 sin x, 0 # x # p.
Which point lies in the shaded area?
(A) (1, 1)
Q(p, 0)
(A) A 12, p6 B
5.
O
)
If point R (not shown) is the reflection of
P in Q, then the coordinates of R are
y
1
1
2
O
2. The graph of y 5 0.5 intersects the graph
of y 5 sin x, 0 # x # 2p, at how many
points?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
3.
y
(B) A p2 , 2 B
(D) A p4 , 12 B
1
x
O
p
2p
The equation of the graph shown is
(A) y 5 sin (x 1 1)
(B) y 5 1 1 sin x
(C) y 5 sin 2x
(D) y 5 2 1 sin x
(E) y 5 2 sin x
6. For what value of k is y 5 cos 2x the
image of y 5 sin 2x under the translation
Tk,0?
(A) p4
(B) p2
(D) p
(E) 5p
4
(C) 3p
4
7. Which is not in the domain of y 5 sec x2?
(A) 0
(B) p2
(C) p
(D) 3p
2
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8. At which values of x can cot x be equal to
tan x?
(A) 0 and p
(B) 0 and 2p
(C) p4 and p2
(D) p4 and 3p
4
Date
15. The maximum value of y 5 sin x 1 cos x
in the interval 0 # x # p2 is
(A) 0
(D) 2
(E) p and 2p
9. If sin x 5 cos (2p), then x can be equal to
(A) p2
(B) 3p
(C) p
4
(D) 3p
2
(E) 2p
(B) 1
(E) 2 !2
(C) !2
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
10. The graph y 5 arcsin x is obtained by
reflecting the graph of y 5 sin x in the line
(A) y 5 0
(B) y 5 x
(C) y 5 2x
(D) x 5 0
(E) x 5 p
16. What is the amplitude of the function
y 5 3 cos 12x 2 2?
11. The period of y 5 12 sin 3x is
(A) 12
(B) p2
(C) 2p
3
18. What is the period of the curve whose
equation is y 5 13 sin px?
(D) 3
(E) 3p
12. For how many values of x in the interval
0 # x # 2p does cos x 5 2cos x?
(A) 0
(B) 1
(C) 2
(D) 4
(E) An infinite number
13. For how many values of x in the interval
0 # x # 2p does 2 sin 4x 5 1?
(A) 0
(B) 2
(C) 4
(D) 8
(E) 16
14. Which is an increasing function in the
interval p # x # 3p
2?
I. y 5 csc x
II. y 5 cos x
III. y 5 tan x
(A) I only
(C) III only
(E) I, II, and III
(B) II only
(D) I and III only
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17. If x 5 cos21 !3
2 , what is the degree measure of x?
19. If f(x) 5 22 cos 2x, evaluate f A 3p
4 B.
20. To the nearest hundredth, what is the
maximum value of y 5 arcsin x?
21. To the nearest hundredth, what is the
maximum value of y 5 sin x 2 cos x for
0 # x # 2p?
22. What is the exact value of
cot A arcsin A 56 B B ?
23. For how many values of x in the interval
2p # x # p does cos 2x 5 2 sin x?
24. For how many values of x does the graph
y 5 cos nx have a minimum value in the
interval 0 , x , 2p if n is a positive integer?
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CHAPTER
12
TRIGONOMETRIC
IDENTITIES
Aims
• To distinguish between conditional equations and identities.
• To demonstrate methods of proving identities.
• To prove basic identities that use the sum
or difference of two angle measures.
• To prove basic identities that use twice or
half the angle measure.
In previous chapters, students learned
eight basic identities resulting from the definitions of the trigonometric functions. In this
chapter, these basic identities are reviewed
and then used to prove other identities and to
derive relationships that make it possible to
find function values of a given angle using
function values of other angles.
CHAPTER OPENER
After students examine the illustration of the
parking angle, ask them what value of u corresponds to the maximum road space used. (0°)
For u . 0, the length of the space is the
hypotenuse of a right triangle; if that length is
fixed at, say, 10 feet, then the length of the
road space required is 10 cos u and the width
of the space is 10 sin u. As u increases from 0°
to 90°, the length of the road space decreases
and the width increases.
162
12-1 BASIC IDENTITIES
Since the solution set of a conditional equation is a proper subset of the domain, this is
equivalent to saying that there is at least one
element in the domain where the equation is
false. An identity, however, is an equation that
is true for all values of the domain.
To show that an algebraic equation is an
identity, we usually use the properties of the
field of real numbers, such as the commutative
property or the distributive property, and
arithmetic substitutions to transform one side
into the other. To show that a trigonometric
equation is an identity, we usually substitute
one or more of the basic identities in the chart
on page 483 and use the properties of the field
of real numbers.
In Example 1, call attention to the question
mark above the equal sign. The question mark
is used to emphasize that we do not know
whether the two expressions on either side of
the equation are equal. Thus, proving an identity is similar to checking the solution of an
equation.
12-2 PROVING AN
IDENTITY
Some students look upon proving identities as
similar to solving puzzles and enjoy the work,
while other students may find the work
tedious and unrewarding. Explain to students
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Trigonometric Identities 163
that the real importance of trigonometric
identities becomes clear in advanced courses
such as calculus where trigonometric identities
facilitate the derivation and application of key
formulas and processes.
Emphasize that the proof of an identity is
different from the solution of a conditional
equation. In solving an equation, we work on
both sides of the equation simultaneously by
performing the same operation on both sides.
In proving an identity, we transform an expression either on one side of the equality only, or
on both sides independently. There is often
more than one series of steps to establish an
identity. Either side of the equality can be
transformed to prove an identity, although it is
usually most efficient to work with the more
complicated side, if there is one.
Examples 1 through 4 show solutions
obtained by working with only one side of the
equality. It is also a valid method to work with
each side of the equality, transforming each
expression separately into the same form. You
may wish to present an example.
Example: Prove the identity
tan u 1 cot u 5 sec u csc u.
Solution:
?
tan u 1 cot u 5 sec u csc u
sin u
cos u
sin2 u
sin u cos u
u ? 1
1
1 cos
sin u 5 cos u ? sin u
2
u
1
1 sincos
u cos u 5 cos u sin u
?
sin2 u 1 cos2 u ?
1
sin u cos u 5 cos u sin u
1
sin u cos u
5 cos u1sin u ✔
By showing that both sides of the equality
reduce to a common form, we have proved the
identity.
The comment following Example 1 can be
further explained by pointing out that when
we prove a trigonometric identity, we are
really working backward. The last step of the
proof is really the first step in the reasoning
process. Since that step is true, we may conclude that the next-to-last step is also true, and
so on continuing all the way back to the original equation.
12-3 COSINE (A 2 B)
Before presenting the proof of the identity
cos (A 2 B) 5 cos A cos B 1 sin A sin B,
you may wish to review the distance formula
d2 5 (x2 2 x1)2 1 (y2 2 y1)2.
The proof places two angles A and B in
standard position. The terminal sides of the
angles intersect the unit circle at points P and
Q, whose coordinates can be written in terms
of the sine and cosine values of A and B. With
these coordinates, we can find the distance PQ
by using the distance formula. By rotating
OQP through an angle of 2B, we obtain an
angle in standard position whose measure is
(A 2 B) and a distance P9Q9 that is equal
to the distance PQ. Equating these two
expressions for distance enables us to write
cos (A 2 B) in terms of the sine and cosine
values of A and B.
Note that the identities given in this and
the following sections have other possible derivations. The derivations given here were chosen because they are valid for all values of A
and B and because they make use of relationships previously established in this course.
In Example 1, the exact function value for
cos 15° is found by writing 15° as (60° 2 45°)
and using the cosine rule for cos (60° 2 45°).
Ask students whether the same exact function
value can be found by using cos (45° 2 30°);
they should see this approach is valid and will
be applied in Exercise 18. Students should also
recognize that the cosine rule can be used for
angles in degree or radian measure.
12-4 COSINE (A 1 B)
The proof of the identity cos (2u) 5 cos u is
relatively straightforward. However, the proof
of the identity sin (2u) 5 2sin u may be less
obvious to students. We begin by replacing A
in the identity
sin A 5 cos (90° 2 A)
with 2u to obtain the identity
sin (2u) 5 cos (90° 2 (2u)).
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164 Chapter 12
If, at this point, we were to rewrite the
right side of this identity using the formula
for cos (A 2 B) , we would obtain
sin (2u) 5 sin (2u). Therefore, we first express
(90° 2 (2u))
as
The symbol 6 means plus or minus. Note
that in the cosine formula, when the sign on
the left side of the equation is plus, the sign on
the right side is minus; when the sign on the
left side is minus, the sign on the right side is
plus. The signs match each other in the sine
formula.
(u 2 (290°))
to derive the desired relationship
sin (2u) 5 2sin u.
To derive an identity for cos (A 1 B), we
write A 1 B as the difference of two measures
and then use the identity for the cosine of the
difference of two angle measures.
In Example 1, the exact value of cos 105° is
!6
found to be !2 2
. Have students check
4
this result by using rational approximations
found with a calculator; first the approxima!2 2 !6
4
tion for
< 20.2588, and second, the
approximation for cos 105° 20.2588. Ask
students if there is another way to find the
exact value of cos 105° using known values.
(Possible answer: cos (150° 2 45°))
12-5 SIN (A 2 B) AND
SIN (A 1 B)
Ask students to provide examples that show
sin (A 2 B) sin A 2 sin B
sin (A 1 B) sin A 1 sin B
(Possible answers: sin 60° sin 90° 2 sin 30°
and sin 90° sin 60° 1 sin 30°)
In
this
section,
the
identity
sin u 5 cos (90° 2 u) is again used to derive
new identities. By replacing u with A 2 B and
regrouping the variables in the expression
(90° 2 (A 1 B)), we are able to use the formula for the cosine of the sum of two angle
measures to obtain an expression equal to
sin (A 2 B).
The sum and difference of angle formulas
can be summarized as follows. For all values of
A and B,
cos (A 6 B) 5 cos A cos B 6 sin A sin B
sin (A 6 B) 5 sin A cos B 6 cos A sin B
12-6 TANGENT (A 2 B)
AND TANGENT (A 1 B)
Call attention to the fact that, when dividing
A cos B
by the ratio cos
cos A cos B, we are dividing by a ratio
that is equal to 1 if cos A 0 and cos B 0.
If cos A 5 0, then tan A is undefined; if
cos B 5 0, then tan B is undefined. Therefore,
the equation
A 1 tan B
tan (A 1 B) 5 1tan
2 tan A tan B
is true for all values of A and B where both tan
A and tan B are defined. If 1 2 tan A tan B 5 0,
then tan (A 1 B) is undefined.
Similarly, the equation
A 2 tan B
tan (A 2 B) 5 1tan
1 tan A tan B
is true for all values of A and B where tan A
and tan B are defined. If 1 1 tan A tan B 5 0,
then tan (A 2 B) is undefined.
In Exercise 18, this identity is derived in
Section 9-8 for the special case when u is
an acute angle. Here, by applying the sum
formula for tangent, the identity is shown
to be true for all values of u where tan u is
defined.
12-7 FUNCTIONS OF 2A
Of all the identities presented in this chapter,
the identities for sin 2A and cos 2A are, perhaps, the ones most frequently used in
advanced mathematics courses. It is important
that students memorize them or be able to
derive them from the formulas for cos (A 1 B)
and sin (A 1 B).
Point out to students that the form of the
identity chosen for cos 2A depends on what
you are given in a problem and what you are
asked to find. In Exercise 23, the identity can
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Trigonometric Identities 165
be proved by substituting 1 2 sin2 u for cos2 u
in cos2 u 2 sin2 u 5 cos 2u.
12-8 FUNCTIONS OF 1
2A
The identities for the sine, cosine, and tangent
of 12A are unique in that they are the only common identities we have derived so far where it
is necessary to choose the correct sign for the
result. The sign of sin 12A, cos 12A, and tan 12A
must be determined from the quadrant where
1
2A lies.
When students use their calculators to
explore graphs of identities, remind them that
cot x must be entered as 1 4 tan x. Ask if they
can think of other ways to check an identity
graphically. Some students may suggest that if
the difference of the two sides of the identity is
treated as a function, then the screen should
be blank. For example, if sin2 x 1 cos2 x 2 1 is
entered as Y1, no points will be graphed (other
than y 5 0).
See Enrichment Activity 12-8: Forming
Identities where students form identities by
matching trigonometric expressions in one
column with those in a second column. One
approach to find matches is to have students
simplify both columns to simpler expressions.
EXTENDED TASK
In this activity, students are given values for
sin 1° and cos 1° and are asked to construct a
table of sines and cosines for angles measuring
1° to 10° using the formulas for sin (A 1 B)
and cos (A 1 B). They are also asked to find
the values with a calculator and compare the
two sets of values. Students will find that the
values compare very favorably. The activity is
then extended using the sine and cosine values
to obtain tangent values. Again, the values
from the formula computation are close to the
values from a calculator.
This activity will reinforce students’ memorization of the sum identities and give practice
with computation and rounding skills.
Although students have seen the algebraic
proofs of the identities, the work in this activity will provide additional verification of the
formulas. Note that the activity may be
assigned at any time following Section 12-5.
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ENRICHMENT ACTIVITY 12-8
Forming Identities
For each expression in Column I, there is one and only one equivalent expression
in Column II. Match the equivalent expressions so that, for all possible replacements
of the variable whereby each side of the equality is defined, an identity is formed.
Column I
Column II
2u
1. sec2 u 2 cos
cos u
a. sin u cot u
2. sin (908 1 u)
b. cot u
1
3. csc u 2 cot
u cos u
c. sec2 u 2 tan2 u
4. cos (2708 2 u)
d. sec2 u csc2 u
cos2 u 1 sin2 u
5. 1 22tan
u 2 sin 2u
e. 2cos u tan u
sin u
6. sin u A cos u 1 cot
uB
f. tan u csc u
u 1 sin 2u
7. sinsec
u 1 2
g. cos u tan u
8. (sin u 1 cos u) 2 2 (sin u 2 cos u) 2
h. cos2 u
3
u 2 2 cos u
9. (sincos
u 2 1) 2 1 2 sin u
i. sin u cos u
2 cos2 u 2 1
10. cos
2
u 2 sin2 u
j. 2 csc u cot u
2u
11. tan u cot u 2 2sin
cot u
k. 2sin u cot u
2
12. cos u 1 sin u cos u cot u 2 1
l. sin u sec u
13. cos u 2sincos2uu cos 2u
m. csc u
14. sec2 u 1 csc2 u
n. 2 sin 2u
15. 1 2 1cos u 2 1 1 1cos u
o. cos 2u
Answers:
1.
2.
3.
4.
5.
6.
7.
9.
10.
11.
12.
13.
14.
15.
Bonus: On separate paper, find the exact value of each expression in Column I
when u 5 30°.
Copyright © 2009 by Amsco School Publications, Inc.
8.
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EXTENDED TASK
You are given the following information:
sin 1° 5 0.01746
cos 1° 5 0.99985
1. Construct a table of sine and cosine values using the identities for the sine and
cosine of the sum of two angles. For each computation, use 1° as one of the
angles in the sum. Then compare the values you calculated using the formulas
with the values obtained directly from a calculator. Complete the table below
rounded to four decimal places.
By Formula
Angle
Sin
From Calculator
Cos
Sin
Cos
1°
2°
3°
4°
5°
6°
7°
8°
9°
10°
2. How did the values for sine and cosine found using the formulas compare with
the values from the calculator?
sin x
3. Use the identity tan x 5 cos
x to find the tangent values for the angles from 1°
to 10°. Compare with the values from a calculator. Complete the table below
rounded to four decimal places.
Angle
Tan (by formula)
1°
2°
3°
4°
5°
6°
7°
8°
9°
10°
Copyright © 2009 by Amsco School Publications, Inc.
Tan (calculator)
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Algebra 2 and Trigonometry:
Chapter Twelve Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct
answer with insufficient work may receive only partial credit. All scratch paper must
be turned in at the conclusion of this test.
In 1 and 2, write each expression as a single function.
1. cos2 u sec u csc u
2. sin u (1 1 cot2 u)
In 3–6, write true or false if A 5 90°.
3. sin 2A 5 2 sin A cos A
4. cos 2A 5 2 sin A 213 A B
A
5. tan 2A 5 1 22tan
tan2 A
6. cos 12A 5 sin 12A
7. If tan A 5 2 !5
2 and A is in the second quadrant, find:
a. cos 2A
b. the quadrant where 2A lies
8. If x is the measure of an obtuse angle and cos x 5 218, find sin 12x.
9. If B 5 arcsin Q2 !7
4 R , find sin 2B.
In 10–13, use a sum or difference formula to find the exact value of each expression.
10. cos 285°
11. sin 255°
12. cos (2210°)
13. cos 195°
In 14–19, sin x 5 0.28, cos y 5 0.6, and x and y are measures of acute angles. Find
the exact value of each expression.
14. cos x
16. sin
1
2x
18. sin (x 1 y)
Copyright © 2009 by Amsco School Publications, Inc.
15. sin y
17. cos 2y
19. tan (y 2 x)
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20. From the top of a lighthouse 220 feet above a river, the keeper spots a boat
sailing directly toward the lighthouse. The keeper observes that the angle of
depression of the boat is 6° and then, sometime later, observes the angle to
be 14°. To the nearest tenth of a foot, find the distance the boat has sailed
between the times the two observations were made.
In 21–24, in each case: a. Prove that the given statement is an identity. Show your
work on a separate piece of paper. b. Find the domain where the identity is defined.
21. csc x ? tan x 5 sec x
2
2
2
b.
2
22. cos x 1 cos x cot x 5 cot x
b.
sin 2x 2 sin x
sin x
23. cos
2x 1 cos x 5 cos x 1 1
cos x
24. sec x1 1 1 1 sec x1 2 1 5 2sin
2
x
b.
b.
Bonus: In the diagram, BG and CF divide rectangle ADEH into three congruent
squares. The degree measures of DAE, DBE, and DCE are x, y, and z,
respectively. Prove that x 1 y 1 z 5 90°. (Hint: Find tan (x 1 y).)
H
G
y
x
A
Copyright © 2009 by Amsco School Publications, Inc.
F
B
E
z
C
D
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SAT Preparation Exercises (Chapter 12)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If sin u 5 0.438, what is the value of
sin u cos u tan u to the nearest hundredth?
(A) 1.00
(B) 0.71
(C) 0.44
(D) 0.19
(E) Cannot be determined
9. sin2 x 1 3 cos2 x 5
(A) 22 1 cos 2x
(C) 2 1 cos 2x
(E) 2 2 cos 2x
10.
2. cos4 u 2 sin4 u is equal to
(A) 2
(B) 4
(C) cos 2u
(D) sin 2u
(E) cos2 u 1 sin2 u
3. sin A u 1 p6 B 1 cos A u 1 p3 B 5
(A) sin u
(B) cos u
p
(C) sin A u 1 3 B
(D) cos A u 1 p6 B
(E) 1
4. If sin u 5 0.9 and 1 2 cos2 u 5 k, what is
k?
(A) 0.81
(B) 0.405
(C) 0.19
(D) 0.1
(E) Cannot be determined
5. Where defined, tan x sin x cos x csc2 x is
equivalent to
(A) tan2 x
(B) sec x
(C) sin x
(D) cot2 x
(E) 1
6. If sin x cos x 5 52 and tan x 5 12, then
sin2 x 5
(A) 15
(D) 2 !5
5
(B) 52
(C) !5
5
(E) 53
7. The value of (sin x 1 cos x)2 2 sin 2x is
(A) 22
(B) 21
(C) 0
(D) 1
(E) 2
8. If cos 2x 5 45 and cos x 1 sin x 5 k, then
cos x 2 sin x is
(A) 45k
(B) 35k
(D) 35k
4
(E) 5k
(C) 54k
Copyright © 2009 by Amsco School Publications, Inc.
(B) 22 1 sin 2x
(D) 2 1 sin 2x
y
1
Q
O
P
R
S
p x
Rectangle PQRS is inscribed in the graph
of y 5 sin x, 0 # x # p. If PQ 5 sin k,
then PS equals
(A) k
(B) p2 2 k
(C) p 2 k
(D) 2p 2 k
(E) p 2 2k
11.
x°
x°
If each edge of the square has a measure
of 1, then the shaded region has an area of
(A) 12
(B) 1 2 sin x°
(C) 1 2 sin 2x°
(D) 1 2 12 sin x8
(E) 1 2 12 sin 2x8
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12.
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
S
16. In PQR, R is a right angle. What is the
value of sin2 P 1 sin2 Q?
R
x°
x°
P
17. What is the exact value of
(4 sin 2u)(5 sin 2u) 1 (2 cos 2u)(10 cos 2u)?
Q
If PQ 5 2 and RQ 5 1, then PS equals
(A) 2 !5
3
(B) 35
(E) 2 !5
(D) 10
3
(C) 2 !10
3
(A) 12 cos 558 1 !3
2 sin 558
(B) 12 cos 558 2 !3
2 sin 558
22. Angle A is in the second quadrant and
cos A 5 235. Angle B is in the third
(E) !2
2 (cos 408 1 sin 408)
2
14. Where defined, (sin x 1 cos x) 2
(tan2 x 1 cot2 x) 1 sec2 x 1 csc2 x 5
(A) 23
(B) 21
(C) 0
(D) 1
(E) 3
P
S
1
1
R
If PQ 5 2 and QR 5 SR 5 1, then
2!2 cos PQS equals
(A) 1 1 !3
(B) 1 2 !3
(C) !3 2 1
(E) !2 1 1
7
quadrant and sin B 5 225
. What is the
exact value of cos (A 1 B)?
1
23. If tan (A 1 B) 5 217
6 and tan A 5 2 ,
what is the exact value of tan B?
24. If cos 2u 5 !5
4 , what is the exact value of
2
cos x?
Q
2
19. If 6(cos 80° cos 20° 1 sin 80° sin 20°) 5 x,
what is x?
21. If sin u 5 sin 37° cos 12° 2 cos 37° sin 12°
and u is an angle in the second quadrant,
what is the degree measure of 3u?
(C) 12 cos 258 2 !3
2 sin 258
!3
(D) 2 cos 258 1 12 sin 258
15.
18. What is the exact value of
sin4 u 2 cos4 u
cos2 u 2 sin2 u 1 7?
20. If cos u 5 15 and u is an angle in the fourth
quadrant, find the exact value of sin 2u .
13. cos 85° equals
2
Date
(D) !2 2 1
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CHAPTER
13
TRIGONOMETRIC
EQUATIONS
Aims
• To solve first-degree trigonometric equations algebraically.
• To solve higher-order trigonometric equations using factoring and the quadratic formula.
• To develop methods of solving trigonometric equations using substitution of
identities.
• To solve trigonometric equations
graphically.
In this chapter, categories of trigonometric
equations are examined and the appropriate
solution methods for each are developed.
While the same procedures used for solving
algebraic equations can be used to solve
trigonometric equations, students need to
understand how the value of the unknown
leads to all solutions of the equation.
As usual, there is often more than one
way to solve a trigonometric equation.
Although some methods may be more efficient, students should use the approach that
makes sense to them and that they can apply
with confidence.
172
13-1 FIRST-DEGREE
TRIGONOMETRIC
EQUATIONS
Students should be aware that trigonometric
equations are not identities. An identity is true
for any angle in the domain of the function
involved. A trigonometric equation is true for
some specific angles, if at all. For example, the
equation sin x 5 22 has no solution since 22
is outside the range of the sine function.
Emphasize Step 3 of the procedure for
solving a linear trigonometric equation illustrated in the example for 5 cos u 1 7 5 3. Students may not recognize the correct method
for working backward from the value of 143°
to the reference angle and then using this reference angle to find the second solution in the
third quadrant.
Trigonometric equations are usually solved
for values of the variable in the interval
f08, 3608) or f0, 2p) . There are other solutions
outside that interval. These other solutions differ by integral multiples of the period of the
function. In this section, the period of each
function is 360° or 2p radians.
The use of a graphing calculator greatly
simplifies the solving of trigonometric equa-
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Trigonometric Equations 173
tions. The calculator is especially helpful when
the equation involves several different functions, involves multiples of angles, has fractional or decimal coefficients, or has constants
that do not easily lend themselves to standard
algebraic methods.
When 3(sin A 1 2) 5 3 2 sin A is solved by
graphing
Y1 5 3(sin X 1 2)
Y2 5 3 2 sin X
we are forming a system of equations and finding their intersection points. Alternatively,
the equation can be solved by simplifying as
follows:
3(sin A 1 2) 5 3 2 sin A
3 sin A 1 6 5 3 2 sin A
4 sin A 1 6 5 3
4 sin A 1 3 5 0
If Y1 5 4 sin X 1 3 is then graphed, we look
for the zeros of that function. Both methods
will produce the same solutions.
13-2 USING FACTORING
TO SOLVE
TRIGONOMETRIC
EQUATIONS
Review factoring techniques (greatest common factor, difference of squares, factoring
trinomials) and the solution of second-degree
equations before beginning this section. It
is important that students realize that the
equation
3x2 2 5x 2 4 5 0
is a quadratic equation in x, since x is the variable that is squared, but that
2
3 cos x 2 5 cos x 2 4 5 0
is a quadratic equation in cos x, since cos x, not
x, is the variable that is squared. If students
have difficulty with this concept, they may find
it helpful to rewrite equations in a nontrigonometric form before attempting to solve
them.
Example: Find all the values of u in the
interval 0° # u , 360° that satisfy the equation
2 cos2 u 1 cos u 2 1 5 0.
Solution: Let x 5 cos u. The equation can
now be written as
2x2 1 x 2 1 5 0
(2x 2 1)(x 1 1) 5 0
2x 2 1 5 0
x1150
x 5 12
Replace x with cos u.
cos u 5 12
u 5 608
or
x 5 21
cos u 5 21
u 5 1808
or u 5 3008
Answer: {60°, 180°, 300°}
Emphasize that factoring can be used only
if one side of the equation is 0. Example 2
illustrates two solution methods: factoring
using the difference of two squares and the
square root method, an approach used with
quadratics of the form ax2 1 c 5 0 where c is
negative or 0.
Any factoring method may be applied to
trigonometric equations. Factoring by grouping works in some special cases where there
are an even number of terms and common factors can be found for groups of them. For
example,
4 sin x cos x 2 2 sin x 2 2 cos x 1 1 5 0
2 sin x(2 cos x 2 1) 2 1(2 cos x 2 1) 5 0
(2 sin x 2 1)(2 cos x 2 1) 5 0
The solution can then be completed by setting each factor equal to 0 as usual.
13-3 USING THE
QUADRATIC
FORMULA TO SOLVE
TRIGONOMETRIC
EQUATIONS
Factoring can be applied only to cases where
the quadratic equation has rational roots. The
quadratic formula can be used for any seconddegree equation.
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174 Chapter 13
As suggested in Section 13-2, students who
have difficulty recognizing the values for a, b,
and c in the quadratic formula may find it
helpful to change equations to non-trigonometric form. After some practice, they should
be able to forgo this intermediate step and
work directly with the given equation.
cot2 u 5 csc u 1 1
cos2 u
sin2 u
5 sin1 u 1 1
cos2 u 5 sin u 1 sin2 u
One more substitution is needed, that is,
cos2 u 5 1 2 sin2 u.
1 2 sin2 u 5 sin u 1 sin2 u
13-4 USING SUBSTITUTION
TO SOLVE
TRIGONOMETRIC
EQUATIONS
INVOLVING MORE
THAN ONE
FUNCTION
Although various suggestions are given in this
section for solutions of trigonometric equations, here, too, students should realize that
there is often more than one way to proceed.
Careful observation may suggest special techniques for some equations. For instance, some
algebraic operations, such as squaring, may
produce answers that are not solutions of the
original equation. Therefore, it is necessary to
check solutions in the given equation.
In general, if an equation involves two or
more different functions, try replacing each
function using either a ratio identity or a reciprocal identity. In the new expression:
• simplification may be possible, resulting in
an equation that is either factorable or
solvable by the quadratic formula.
• the substitutions may create fractions that
can be combined using an LCD. The products that result from the multiplications
used to create equivalent fractions may be
parts of identities that can be substituted
to simplify the expression.
As students work the exercises, suggest
they first think whether there is an obvious
substitution that will express one function in
terms of the other. If not, as in the case of
Exercise 13, they should try writing all functions in terms of sine and cosine:
Simplifying results in the equation
2 sin2 u 1 sin u 2 1 5 0
which can be solved by factoring.
In Enrichment Activity 13-4: Solving
Trigonometric Inequalities, students use
graphing to identify the intervals where
inequalities involving constants or two functions are true.
13-5 USING SUBSTITUTION
TO SOLVE
TRIGONOMETRIC
EQUATIONS
INVOLVING
DIFFERENT ANGLE
MEASURES
This section extends the concept of using
substitution to solve trigonometric equations
by focusing on identities for multiple angle
measures.
Where an equation containing a function
of a multiple angle can be solved directly, students must know how to obtain all solutions of
x between 0° and 360°. For example,
2 sin 3x 5 1
sin 3x 5 12
3x 5 308 S x 5 108
3x 5 3908 S x 5 1308
3x 5 7508 S x 5 2508
3x 5 1508 S x 5 508
3x 5 5108 S x 5 1708
3x 5 8708 S x 5 2908
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Trigonometric Equations 175
To obtain all solutions of x between 0° and
360°, it is necessary to solve for all solutions of
3x between 0° and (3)360° 5 1,080°. To do this,
add 360° to each of the original solutions two
times and divide each result by 3. In general, if
a solution is found for nu, add 360° or 2p to
each of the original solutions (n 2 1) times
and divide each result by n.
EXTENDED TASK
For the Teacher:
This extended task is designed to give students practice in working with the skills pre-
sented in this chapter as well as those skills
that are prerequisite to working with trigonometric identities and equations. Students could
be encouraged to make up their own grids
with equations, using a variety of expressions.
As well as being very useful, students should
see mathematics as an avenue for recreation.
Hence, this extended task was designed as a
mathematical puzzle.
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ENRICHMENT ACTIVITY 13-4
Solving Trigonometric Inequalities
In Section 5-9, you learned how to use graphs of quadratic inequalities in two variables to solve quadratic inequalities in one variable. Trigonometric inequalities can be
solved in the same way.
Example
Solve the trigonometric inequality sin x . 212 for the interval 0 # x , 2p.
Solution: First, find the solutions to the corresponding equation:
sin x 5 212
11p
The roots of the equation are 7p
6 and 6 and separate the number line into three
intervals:
0 # x , 7p
6,
7p
6
, x , 11p
6 ,
11p
6
, x # 2p
The solutions to the inequality sin x . 212 correspond to the interval(s) where the
graph of y 5 sin x is above the graph of y 5 212.
1
y
0.5
20.5
21
x
p
6
p
3
p
2
2p 5p
3 6
p
7p 4p 3p 5p 11p
6 3 2 3 6
2p
From the graph, we see that this relationship is true in the intervals 0 # x , 7p
6
and 11p
.
,
x
#
2p
6
11p
Answer: 0 # x , 7p
6 or 6 , x # 2p
In 1–4, solve each inequality for 0 # x , 2p.
1. tan x $ 1
3. cos x # 0
2. sec x , 21
4. cot x , !3
5. a. If 0 # x , 2p, for what exact values of x does sin x 5 cos x?
b. Explain how you would solve sin x , cos x by graphing. What are the solutions?
c. For what values of x is the inequality sin x $ cos x true?
6. Solve sin x $ cos x for 0 # x , 2p.
7. Solve cos x , !3
2 for 0 # x ,2p.
8. Solve cot x $ tan x for 0 # x , 2p.
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EXTENDED TASK
Find the Letter: A Trigonometric Puzzle
1. Solve each of the problems a–k and shade in the correct answer in the chart below.
45°, 135°,
225°, 315°
30°
2 !5
5
7p
12
csc x
sin 2x
63
65
12
7 p
cot x
1
sin x cos x
135°
and
315°
135°
cos2 x
240°
sin2 x
1 2 cos x
sin x
100°
!5
5
sec x
p
3
1 2 sin x
cos x
45°
and
225°
cos 2x
33
65
210°
260°
a. The expression 1 2 cos2 x is equal to
.
b. How many degrees are there in 3p
4 radians?
c. Express sec x 2 tan x as an expression in sin x and cos x.
x
d. The expression cos
sin x is equal to
.
e. What is the smallest positive angle which satisfies the equation
2 cos2 x 2 5 cos x 1 2 5 0?
f. If 2 sin x 1 1 5 0 and x is in Quadrant III, what is a possible value for x?
g. The expression sin1 x 5
.
h. If cos A 5 35 and angle A is acute, find the exact value of sin 12A.
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i. In the interval 0° # x # 360°, sin x 5 cos x. Find all values of x.
5
j. If sin A 5 35, sin B 5 13
, and angles A and B are acute angles, what is the value
of cos (A 2 B)?
k. How many radians are in 105°?
2. In the chart below, find the letter that corresponds to each answer that you
shaded on the previous page.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
3. In the blanks below, use the letters you have shaded in Exercise 2 to write an
expression. (Note: Two extra vowels have been supplied to aid you in your task.)
O
I
?
Copyright © 2009 by Amsco School Publications, Inc.
.
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Algebra 2 and Trigonometry:
Chapter Thirteen Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may receive only partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–12, find the exact solution set of each equation if 0° # u , 360°.
1. 8 sin u 1 1 5 23
2. 3 tan u 2 2 5 tan u
3. 2 sin2 u 2 1 5 0
4. 3 tan2 u 2 1 5 0
5. 2 cos2 u 2 7 cos u 5 4
6. 2 sin2 u 1 3 sin u 1 1 5 0
7. sec u 5 sec1 u
8. 2 cos u sin u 2 cos u 5 0
9. sec2 u 2 tan u 2 1 5 0
11. cos 2u 1 cos u 1 1 5 0
10. 3 cos 2u 2 5 cos u 5 1
12. tan u 5 3 cot u
In 13–20, find to the nearest degree all values of u in the interval 0° # u , 360° that
satisfy each equation.
13. 10(cos u 1 1) 5 6
14. 2 cos2 u 2 5 cos u 2 1 5 0
15. tan2 u 5 8 tan u 2 5
16. cos2 u 2 3 sin u 1 2 5 0
17. 6 cos u 1 1 5 5 sec u
18. tan2 u 5 tan u 1 1
19. 3 cos 2u 1 5 cos u 1 2 5 0
20. 2 cos 2u 1 cos u 5 0
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21. Express, in radians, the exact values of u in the interval 0 # u , 2p that are roots
of the equation 4 sin3 u 2 sin u 5 0.
22. Express, in degrees, the exact values of u in the interval 0 # u # p that are roots
of the equation 4 sin2 3u 5 3.
Bonus: Solve the equation sin x 5 cos x for 0 # x , 360°.
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SAT Preparation Exercises (Chapter 13)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If 0° # u , 360°, which of the following
are solutions of
I. 30°
3 tan u 1 !3 5 2 !3?
II. 150°
III. 210°
(A) I only
(C) I and II only
(E) I, II, and III
(B) II only
(D) I and III only
(B) p2
(D) 3p
4
(E) 5p
3
8. Which is not a solution of sin 2u 5 tan u if
0° # u , 360°?
(A) 45°
(B) 105°
(C) 180°
(D) 225°
(E) 315°
9. In the interval 0 # u , 2p, what is the
solution set of
2 sin 2u
cos u
2. One root of the equation
3 cos u 2 1 5 1 2 cos u is
(A) p4
7. Which equation is equivalent to
cos 2u 2 2 sin u 1 2 5 0?
(A) 22 sin2 u 2 2 sin u 1 3 5 0
(B) 22 sin2 u 2 2 sin u 2 1 5 0
(C) 2 cos2 u 1 cos u 1 2 5 0
(D) 2 cos2 u 2 2 sin2 u 1 1 5 0
(E) 2 cos u 2 sin2 u 2 3 5 0
(C) 2p
3
3. To the nearest degree, the measure of the
acute angle that satisfies the equation
csc u 5 3(csc u 2 5) is
(A) 4°
(B) 8°
(C) 15°
(D) 82°
(E) 172°
4. In the interval 0° # u , 360°, what is the
solution set of
cos2 u 2 cos u 2 2 5 0?
(A) {0°}
(B) {90°}
(C) {180°}
(D) {0°, 180°}
(E) {0°, 90°, 180°}
5. The smallest non-negative measure for
which 2 sin2 u 2 3 sin u 1 1 5 0 is
(A) 0°
(B) 30°
(C) 45°
(D) 60°
(E) 120°
6. If 0° # x , 360°, how many real roots
does the equation 16 sin3 x 2 sin x 5 0
have?
(A) 0
(B) 2
(C) 3
(D) 4
(E) 6
Copyright © 2009 by Amsco School Publications, Inc.
2 sin1 u 5 0?
(A) { }
(B) U p3 , 2p
3 V
(C) U p3 , 5p
3 V
(D) Up6 , 5p
6 V
7p 11p
(E) U p6 , 5p
6, 6, 6 V
10. If 0° # u , 360°, how many solutions does
the equation cos u 2 cos u sin2 u 5 0
have?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
11. If cos (u 2 40°) 5 sin 60°, then the degree
measure of u can be
(A) 20°
(B) 30°
(C) 60°
(D) 70°
(E) 90°
12. How many solutions to
cos2 u 2 5 cos u 1 6 5 0
are there in the interval 0° # u , 360°?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
13. Which is not a solution of
tan x 2 sin x
tan x 1 sin x
x 2 1
5 sec
sec x 1 1 ?
(A) p8
(B) 3p
5
(D) 7p
2
(E) 13p
4
(C) 5p
6
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14. If 0° # x , 360°, which of the following
are solutions to sin x 5 1 1 cos x?
I. 908
18. What is the sum of all the degree values
of u in the interval 0° # u , 360° that satisfy the equation 3 tan2 u 2 1 5 0?
19. If A and B are acute angles and
II. 1808
III. 2708
(A) I only
(C) I and II only
(E) I, II, and III
Date
sin (A 1 B) 5 (sin A 1 cos A) sin B,
(B) II only
(D) I and III only
11p
15. If 7p
6 and 6 are all the solutions to
2 sin2 u 2 3 sin u 1 p 5 0
for 0 # u , 2p, then the value of k is
(A) 23
(B) 22
(C) 21
(D) 2
(E) 4
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. In the interval 0 # u , 2p, what is the
radian measure of the greatest angle that
is a solution to 2 sin u 5 tan u?
17. Find, to the nearest degree, the least
positive value of u in the interval
0° # u , 360° that is a solution to
3 cos2 u 1 2 sin u 2 1 5 0.
Copyright © 2009 by Amsco School Publications, Inc.
then what is the degree measure of B?
20. If cos 2x 5 cos2 2x and 0° , x , 90°, what
is the degree measure of x?
21. If twice the product of sin x and cos x
equals the product of tan x and cot x,
what is the least positive value for x, in
degrees?
22. Find x to the nearest tenth of a degree if
3 csc x 2 6 5 csc x and 90° , x , 180°.
23. Find to the nearest tenth of a degree the
smallest positive value of x if
2
sin x 5 4 sin x 1 1.
24. In the interval 2p # x # p, how many
solutions are there to the equation
sin 2x 5 x?
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CHAPTER
14
TRIGONOMETRIC
APPLICATIONS
Aims
• To be able to express any point in the
coordinate plane in terms of its distance
from the origin and the measure of an
angle.
• To derive and apply the Law of Cosines.
• To derive and apply the Law of Sines.
• To determine the area of a triangle or a
parallelogram, given the measures of two
sides and the included angle.
• To recognize the ambiguous case and to be
able to determine the number of possible
solutions.
• To be able to find the missing measures of
the sides and angles of a triangle when sufficient information to determine the triangle is provided.
In the study of triangle congruence, students learned what information is sufficient to
determine the triangle, that is, to establish that
one and only one triangle can be constructed
using the given information.
When it has been decided that a triangle
can be constructed using the given data, the
missing measures of the sides and angles, as
well as the area of the triangle, can be found.
The formulas derived in this chapter make it
possible to compute the desired measures.
CHAPTER OPENER
If a distress signal is received at two coast
guard stations, the distance between the stations and the direction of the ship’s signal at
each station can be used to determine the
ship’s exact location, which coast guard station
is closer, and from which station help can
arrive more quickly. In this situation, the
measure of two angles and the included side of
a triangle are known. The exact position of the
ship may be determined by drawing the triangle with the two coast guard stations and the
ship at the vertices and then using the Law of
Sines to find the distance of the ship from each
coast guard station. The problem is similar to
Exercise 22 in Section 14-5. Analagous problems can be described to apply to aviation (the
distance between two airports and an aircraft)
or forestry (the distance between two ranger
stations and a fire).
14-1 SIMILAR TRIANGLES
This section reviews the conditions that must be
met for two triangles to be similar and then
applies the similarity relationship to derive the
formulas used to write rectangular coordinates
in trigonometric form.Any point can be located
using its distance from the origin and the measure of the angle in standard position. The ideas
in this section are prerequisite for students’
understanding of the derivation of the Law of
Cosines that follows in the next section.
183
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184 Chapter 14
In Example 2, students must be aware of
the correct steps to use once the calculator has
returned a value for the inverse cosine or
inverse sine. The calculator does not give the
reference angle, nor does it give the required
third-quadrant angle. For Exercise 1, students
should explain that R 5 arctan 28.48
25.3 5 588, so
it is not necessary to find OA first.
To summarize the Law of Cosines, explain
that the measure of one side is defined in
terms of the measures of the other two sides
and the angle opposite the first side. Emphasize that the Law of Cosines can be used with
the following given information:
• the measures of two sides and the included
angle (SAS), or
• the measures of three sides (SSS)
14-2 LAW OF COSINES
For the derivation of the Law of Cosines given
in the text, the vertex A of the angle whose
measure is known is placed at the origin of the
coordinate plane and one side whose measure
is known is placed along the x-axis so that a
second vertex, B, is on the x-axis. The derivation would have been essentially the same if
point C had been placed on the x-axis.
To repeat the derivation expressing b in
terms of a, c, and cosine B, position ABC on
the coordinate plane so that the coordinates of
B are (0, 0) the coordinates of A are (c, 0) and
the coordinates of C are (a cos B, a sin B).
An alternate derivation that does not
involve coordinates can be shown using the
diagram below.
C
b
A
h
x
D
a
c2x B
c
Consider ABC with height h and sides
with measures a, b, and c. Let AD 5 x. Then
DB 5 c 2 x. Using the Pythagorean Theorem
for DCB, we have
a2 5 (c 2 x) 2 1 h2
a2 5 c2 2 2cx 1 x2 1 h2
In ADC:
b2 5 x2 1 h2
so:
a2 5 c2 2 2cx 1 b2
Since cos A 5 xb: x 5 b cos A
a2 5 c2 2 2c(b cos A) 1 b2
Then:
a2 5 b2 1 c2 2 2bc cos A
Problems involving the SAS situation are
examined in this section and the SSS case is
considered in the next section.
14-3 USING THE LAW OF
COSINES TO FIND
ANGLE MEASURE
A triangle is determined by the measures of its
three sides; that is, if the measures of three
sides are given, the triangle can have exactly
one size and exactly one shape. The measures
of the three angles can be found using the Law
of Cosines.
Recall that in order to construct a triangle
using three given line segments as sides, the
measure of each side must be less than the
sum of the measures of the other two. For
example, 10, 3, and 4 cannot be the measures
of the sides of a triangle because 10 . 3 1 4. If
we try find A using 10, 3, and 4 as the values
of a, b, and c in the Law of Cosines, the result
is a value for cos A that is less than 21 and
therefore not acceptable:
2
c2 2 a2
cos A 5 b 1 2bc
2
42 2 102
5 3 12(3)(4)
5 9 1 16242 100
5 23.125
Since 21 # cos A # 1, there is no angle
whose cosine is 23.125. This example verifies
our original observation that 10, 3, and 4 cannot be the measures of the sides of a triangle.
Students are asked to consider a similar situation in Exercise 1.
For Exercises 13–18, encourage students to
anticipate the location of the angle before using
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Trigonometric Applications 185
the calculator to find the measure. If
cos A is between 0 and 1, A is an acute angle; if
cos A is between 21 and 0, A is an obtuse angle;
if cos A 5 0, A is a right angle.Although for any
given value of cos A there are infinitely many
values of A, the calculator will always return a
value for A between 0° and 180°, a value that
can be the measure of an angle of a triangle.
14-4 AREA OF A TRIANGLE
Note that in this section the use of A to represent area is avoided since in the discussion A
denotes the vertex of an angle. When presenting the derivation in class, you may wish to use
a letter such as K to denote the area. The letter
b also has two different meanings in this section: (1) the base of a triangle in the formula
for area and (2) a particular length of a side in
the formula using the sine.
The formula derived in this section uses the
measures of two sides and the included angle
(SAS). The derivation of a formula for the
area of a triangle in terms of two angles and an
included side (ASA) is suggested as a bonus
question in the Chapter Test for this chapter.
In Exercise 15, students are asked to find
the area of a particular equilateral triangle.
They can easily derive the general formula for
the area of an equilateral triangle with side s.
Area of equilateral nABC 5 12ab sin /C
5 12 (s)(s) sin 608
5 12s2 !3
2
2
5 s !3
4
The formula for the area of a triangle in
terms of the measures of the three sides of the
triangle is called Heron’s formula. Enrichment
Activity 14-4: Heron’s Formula leads students through the steps needed to derive this
formula.
14-5 LAW OF SINES
The Law of Sines can be applied to a triangle
when the given information is:
• the measures of two angles and the measure of any side (ASA and AAS), or
• the measures of two sides and the angle
opposite one of the sides (SSA)
Note that in the first case, the sum of the
measures of the two angles must be less than
180°. In the second case, a unique solution
does not always exist, and sometimes a solution does not exist at all. The ambiguous case
of SSA is examined in Section 14-6.
The use of the area formula derived in
Section 14-4 simplifies the derivation of the
Law of Sines. This identity indicates that, for a
given triangle, the ratio of the measure of any
side to the sine of the angle opposite is constant. The largest side is opposite the angle
with the largest sine value and, therefore, the
largest angle of the triangle; the smallest side
is opposite the angle with the smallest sine
value and, therefore, the smallest angle of the
triangle.
If C is a right angle, then ABC is a right
triangle where:
• c is the measure of the hypotenuse
• a is the measure of the leg opposite A
• b is the measure of the leg opposite B
Since sin C 5 sin 90° 5 1, the Law of Sines
can be used to prove that sin A 5 ac and
sin B 5 bc (see Exercise 18).
14-6 THE AMBIGUOUS
CASE
Emphasize that when students solve problems
involving triangles, they must analyze the data
to determine whether or not a solution exists
and, if a solution does exist, whether it
includes one or two triangles. The measures of
two sides and the angle opposite one of them
(SSA) are not sufficient to determine a triangle. From such information, one triangle, two
triangles, or no triangle may be constructed.
This section derives a set of rules to determine
which of these possibilities applies for a given
set of data.
Although the rules are summarized at the
end of the section, students should be encouraged to use reasoning rather than memorization as they solve problems. As shown in
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186 Chapter 14
Example 1, the number of solutions can be
found by applying the Law of Sines and then
interpreting the results. In that example, two
possible values of B in ABC are found and
tested with the measure of A, the given
value. Since the sum of the measures of two
angles of a triangle must be less than 180, only
one of the measures of B is a solution and
only one triangle is possible. If the value of
sin B had been greater than 1, no triangle
would have been possible.
It is interesting to examine how the
ambiguous case is reflected when the Law of
Cosines is used to find the possible solutions
for SSA. After substituting in the Law of
Cosines, a quadratic equation is obtained. If
the discriminant of the equation is negative, no
triangle is possible since there are no real
roots. If the discriminant is positive, there may
be zero, one, or two positive values for the
measure of the third side, giving zero, one, or
two possible triangles. An example is given
below.
Example: Use the Law of Cosines to find
two possible measures of c in ABC when
a 5 7, b 5 8, and mA 5 60. Then use the
results to find, to the nearest degree, two possible measures of B in ABC.
Solution: To find c, use the Law of Cosines
with a on the left-hand side. Use this form
because after known values are substituted, it
is the only equation that will have just one
variable to solve for.
a2 5 b2 1 c2 2 2bc cos A
Substitute a 5 7, b 5 8, m/A 5 60:
72 5 82 1 c2 2 2(8)c cos 608
49 5 64 1 c2 2 2(8)c(0.5)
49 5 64 1 c2 2 8c
0 5 c2 2 8c 1 15
The quadratic is factorable:
0 5 (c 2 3)(c 2 5)
So c 5 3 or c 5 5.
Next, let c 5 3. Either the Law of Cosines
or the Law of Sines can be used to find the
measure of B. Using the Law of Cosines:
b2 5 a2 1 c2 2 2ac cos B
64 5 49 1 9 2 2(7)(3) cos B
64 5 58 2 42 cos B
6 5 242 cos B
6
cos B 5 242
6
cos21 A 242
B < 98.218 < 988
Note that mC is easily found using this
result.
m/C 5 1808 2 (m/A 1 m/B)
5 1808 2 (608 1 988) 5 228
A similar calculation using c 5 5 gives
mB 5 82° and mC 5 38°.
In this example, the quadratic equation was
solved by factoring and the two solutions were
obtained. Often, however, the computation
required to solve the quadratic equation
makes this a much lengthier problem than the
approach that begins with the Law of Sines.
Hands-On Activity
Instructions:
Choose any acute A and any line segment of length c as an angle and an adjacent
side of a triangle. In this activity, line segments
of different lengths will be used for possible
sides opposite A. The activity can be done
using either a compass, straightedge, and pencil, or geometry software.
1. For convenience, draw A with one ray
horizontal. Find point B on the oblique
(non-horizontal) ray of A such that
AB 5 c.
2. From B, construct the perpendicular to
the horizontal ray of A. Call the foot of
the perpendicular D. Let BD 5 h. One
right triangle has been drawn, ABD,
with sides of lengths c and h and A
opposite side h. Note that sin A 5 hc or
h 5 c sin A.
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Trigonometric Applications 187
3. Draw a line segment of length a such that
c . a . h. Open your compass to length
a. With the point of the compass at B,
h
draw two arcs that intersect AD. Label
one point of intersection C and the other
C9. Draw BC and BCr. Two triangles have
been drawn, ABC and ABC9, with
sides of length c and a and A opposite
side a.
4. Draw a line segment of length e such that
e . c. Open your compass to length e.
With the point of the compass at B, draw
h
an arc that intersects AD. Label the point
of intersection E. Draw BE. One triangle
has been drawn, ABE, with sides of
length c and e and A opposite side e.
5. Draw a line segment of length f such that
f , h. Open your compass to length f.
With the point of the compass at B, show
h
that no arc can be drawn to intersect AD.
No triangle can be drawn with sides of
lengths c and f and A opposite the side
of length f.
Discoveries:
Given the lengths of two sides, a and c, and
A opposite side a:
1. One right triangle can be drawn if
a 5 c sin A.
2. Two triangles can be drawn if
c . a . c sin A.
3. One triangle can be drawn if a . c.
4. No triangle can be drawn if a # c sin A.
What If . . .
The given angle is obtuse? (One triangle
can be drawn if a . c. No triangle can be
drawn if a # c.)
14-7 SOLVING TRIANGLES
This section summarizes the work of the preceding sections and provides some suggestions
for approaching problems where the measures
that are given and those to be determined are
related as parts of a triangle. The importance
of a well-drawn figure should be constantly
stressed. Although it is incorrect to draw conclusions based on the diagram, the figure can
help students to visualize the relationships and
verify the results.
At each step in the solution process, students should be encouraged to question the
reasonableness of the answers they obtain.
Students should be aware that there are
many other identities that can take the measures of the sides of a triangle and relate them
to an expression involving trigonometric functions. A well-known set of equations is named
for Karl Mollweide, who was an astronomer
and a teacher. Mollweide’s equations involve
all six parts of a triangle:
a 1 b
c
5
a 2 b
c
5
cos 12 (A 2 B)
sin 12 C
sin 12 (A 2 B)
cos 12 C
In Enrichment Activity 14-7: The Law of
Tangents, students will be introduced to this
less commonly used law and will be guided
through the steps for applying the relationship
to problem situations.
EXTENDED TASK
For the Teacher:
This extended task is intended to show students how a person uses trigonometry to do
his or her job. In this case, a land developer (or
surveyor) uses trigonometry to find the number of square feet in the area of a building lot
subdivision. The task also helps students connect to other parts of mathematics, particularly geometry and arithmetic. It might also be
seen as a task involving consumerism.
Students should recognize that they have
two options available to complete the task.
Since ABCD is a trapezoid, they could use the
formula for the area of the trapezoid. They
must first, however, find the length of the altitude. The other option is to draw one of the
two diagonals and find the areas of the two triangles formed using the trigonometric formula for area of a triangle, A 5 12ab sin C .
Whichever option the student chooses, he or
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188 Chapter 14
she must first use the Law of Sines and/or Law
of Cosines to find the measures of needed
sides and/or angles.
In addition to the trigonometric skills
needed to complete the task, the following
skills are necessary: ability to round numbers
to a stated precision, conversion skills, equivalences (1 acre 5 43,560 square feet), arithmetic skills, and finding percent. The task
requires that students draw upon mathematics
they have learned in their previous schooling.
As an extension to this task, you might give
students the option of taking the 18-acre tract
of land that Mr. Kronau purchased and
designing it as a development. Lots could be of
varying sizes, and the infrastructure such as
roads, a park, a pond, and so on could be
included. Other variables can be introduced
into the task to create extensions.
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ENRICHMENT ACTIVITY 14-4
Heron’s Formula
If we know the measures of the three sides of a triangle, the size and the shape of
the triangle are determined. We should be able to find the area of the triangle in
terms of these measures. The formula for this purpose is found in the works of Heron
of Alexandria, although some historians believe that the formula was originally
developed by Archimedes.
To derive Heron’s formula, we will need the following algebraic relationships:
a2 1 2ab 1 b2 5 (a 1 b) 2
a2 2 2ab 1 b2 5 (a 2 b) 2
(a 1 b) 2 2 c2 5 f (a 1 b) 1 cg ? f (a 1 b) 2 cg
The perimeter P of ABC, whose sides have measures a, b, and c, is
P 5 a 1 b 1 c.
The semi-perimeter s is one-half the perimeter, or s 5 a 1 2b 1 c.
2
b2 2 c2
1. Use the Law of Cosines, cos C 5 a 1 2ab
, to show that
1 1 cos C 5
(a 1 b) 2 2 c2
2ab
2 a 1 b 1 c a 1 b 2 c
5 ab
?
?
2
2
2. Show that a 1 2b 2 c 5 s 2 c.
2
? s(s 2 c) .
3. Using steps 1 and 2, show that 1 1 cos C 5 ab
2
2
2
b 2 c
4. Use the Law of Cosines, cos C 5 a 1 2ab
, to show that
1 2 cos C 5
c2 2 (a 2 b) 2
2ab
2 c 1 a 2 b c 2 a 1 b
5 ab
?
?
2
2
5. Show that c 1 a2 2 b 5 s 2 b.
6. Show that c 2 a2 1 b 5 s 2 a.
2
? (s 2 b)(s 2 a) .
7. Using steps 4, 5, and 6, show that 1 2 cos C 5 ab
8. Using steps 3 and 7, show that
9. Show that
2
sin C 5 ab
!s(s 2 a)(s 2 b)(s 2 c).
Area 5 !s(s 2 a)(s 2 b)(s 2 c) (Heron’s formula)
In 10–13, in each case, find the area of the triangle by using the methods
specified.
A
10. In right ABC, mC 5 90, a 5 5, b 5 12, and c 5 13.
13
Find the area of ABC by using:
a. Area 5
1
2 (base)(height)
12
b. Heron’s formula
B
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5
C
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11. In BCD, b 5 21, c 5 13, d 5 20, and h 5 12, where h is the
altitude drawn to base b. Find the area of BCD by using:
a. Area 5 12bh
b. Heron’s formula
B
c 5 13
D
d 5 20
12
C
b 5 21
12. In equilateral RST, the length of each side is 4. Express, in simplest radical form, the area of RST by using:
S
a. Area 5 12bh (First find the height, h.)
b. Area 5 12rs sin T
c. Heron’s formula
13. In ABC, a 5 12 meters, b 5 20 meters, and c 5 28 meters.
Express, in simplest radical form, the area of ABC by using:
a. Area 5 12ab sin C (First use the Law of Cosines to find mC.)
b. Heron’s formula
Copyright © 2009 by Amsco School Publications, Inc.
4
R
4
4
T
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ENRICHMENT ACTIVITY 14-7
The Law of Tangents
Although less commonly used than the Law of Cosines or the Law of Sines, the
Law of Tangents can be applied when two sides of a triangle and the angle between
those sides are known:
• In ABC with sides a, b, c opposite angles A, B, C respectively:
a2b
a 1 b
5
tan 12 (A 2 B)
tan 12 (A 1 B)
or
b2c
b 1 c
5
tan 12 (B 2 C)
tan 12 (B 1 C)
or
c2a
c 1 a
5
tan 12 (C 2 A)
tan 12 (C 1 A)
The problem below illustrates the use of the Law of Tangents in solving a triangle.
Complete each step as indicated.
Problem
A
In ABC, a 5 26, b 5 14, and C 5 80°. Find the remaining
sides and angles of ABC.
1. Use the measures of sides a and b to write an equation relating the tangents of the sum and difference of unknown
angles A and B.
2. Since mC 5 80, what must the sum of the measures of A
and B equal?
14
80°
C
26
3. Substitute the value you found for the sum of the angles into the equation
derived in Step 1 and simplify the result. This will give a numerical value
for tan 12 (A 2 B) . Round to four decimal places.
4. Use the inverse tangent function on your calculator to find 12 (A 2 B) and then
A 2 B. Round to three decimal places.
5. Write and solve a system of equations using the values found for
A 1 B (Step 2) and A 2 B (Step 4). Round to the nearest tenth.
6. Use the values found in Step 5 and the Law of Sines to solve for side c.
Round to the nearest tenth.
7. Summarize the results of ABC.
a5
A5
b5
B5
c5
C5
8. If you did not know the Law of Tangents, how could you have solved ABC?
Describe the steps you would use.
Copyright © 2009 by Amsco School Publications, Inc.
B
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9. In ABC, a 5 6, b 5 2.5, and C is a right angle.
a. What is the sum of A and B?
b. What is the value of tan 12 (A 1 B) ?
c. Use the Law of Tangents to find tan 12 (A 2 B) and then A 2 B to the
nearest tenth.
d. Complete the solution of ABC using the Law of Sines. Round to the
nearest tenth.
e. Verify the values for a, b, and c using the Pythagorean Theorem.
f. Verify the values for A and B using the tangent ratio for a right triangle.
10. Solve ABC by using the Law of Tangents. Round your answers to the
nearest tenth.
A
14
120°
C
21
B
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EXTENDED TASK
Land for Sale: A Trigonometric Investigation
Mr. Kronau is a land developer. He has just purchased an 18-acre tract of land,
which he is subdividing into building lots. One of the lots that lie along the main road
is shown below and labeled ABCD. The measures of the sides are rounded to the
nearest 5 feet and marked on the diagram. The measure of ABC, to the nearest
degree is 95°. The lot is in the shape of a trapezoid with AB CD. (Note: The diagram
is not drawn to scale.)
B
135 ft
95°
A
270 ft
190 ft
C
345 ft
D
Road
1. Find the area of the subdivision to the nearest 100 square feet.
2. What is the approximate size of the lot in acres?
3. If Mr. Kronau makes all of the subdivisions about the same size, how many lots
will he have available to sell?
4. If Mr. Kronau sells the lots for $5,500 per acre, about how much is the cost of
one lot?
5. If Mr. Kronau sells all the available lots, about how much will he receive in all?
6. If Mr. Kronau bought the tract of land for $65,000, what would be his percent of
profit to the nearest whole percent?
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Algebra 2 and Trigonometry:
Chapter Fourteen Test
Write your answers legibly in the space provided below. Show any work on
scratch paper. An incorrect answer with sufficient work may receive partial credit. A
correct answer with insufficient work may only receive partial credit. All scratch
paper must be turned in at the conclusion of this test.
In 1–3, write in simplest radical form the coordinates of each point A if A is on the
terminal side of angle u in standard position.
1. OA 5 3, u 5 45°
2. OA 5 12, u 5 150°
3. OA 5 0.75, u 5 300°
4. The coordinates of point A are (8, 6).
a. Find OA, the distance of point A to the origin.
b. Find the measure, to the nearest degree, of the angle in standard position
whose terminal side contains point A.
5. In ABC, b 5 9, c 5 15, and sin B 5 0.3. Find the measure of acute C.
6. In PQR, q 5 10, r 5 15, and cos P 5 0.23. Find p.
7. In DEF, d 5 12, e 5 3, and f 5 10. Find mD to the nearest degree.
8. In ABC, b 5 10, B 5 32°, and A 5 40°. Find a and c to the
nearest tenth.
9. In ABC, b 5 12, c 5 23, and A 5 100°. Find the area of ABC to the
nearest tenth.
10. Find to the nearest integer the area of a rhombus if the measure of a side is
27 centimeters and the measure of one angle is 65°.
11. A ship sails on a course forming an angle measuring 23.7° with a straight
coastline. After sailing 74.3 nautical miles on this course, the ship changes
heading and sails back toward the coast. After sailing 98.7 nautical miles
on the new course, the ship arrives back at the coast. To the nearest tenth,
how far from its starting point is the ship when it again meets the coast?
12. A sign is placed on the roof of a building that is 50 feet high. From a point on
the ground, the angle of elevation of the bottom of the sign is 35° and of the
top of the sign is 48°. Find the height of the sign to the nearest foot.
13. In ABC, angle A measures 40° and angle B measures 70°. The length of
side a is 10 inches. Find the perimeter of ABC to the nearest tenth.
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14. The measure of diameter BC is 30 centimeters, and the measure of chord AB
of the circle is 12 centimeters. Find to the nearest degree the measure of
minor arc AB.
X
15. To avoid a marshy stretch of a straight path, a hiker walks 200 feet in a
direction that is at an angle of 20° with the path, then turns and walks
300 feet back to the original path.
a. Find, to the nearest degree, the measure of the acute angle where the
path followed by the hiker returns to meet the original path.
b. Find to the nearest 10 feet the distance between the points where the
hiker left and returned to the path.
16. In ABC, A 5 40°, b 5 10, and a 5 8.
a. How many distinct solutions are there for ABC?
b. Show the solution(s) indicated in part a rounded to the nearest tenth.
Bonus: If in ABC we are given the measures of A, B, and c (ASA), use the
Law of Sines and the formula for the area of a triangle (area 5 12bc sin A) ,
to show that:
2
sin A sin B
Area 5 2csin
(A 1 B)
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SAT Preparation Exercises (Chapter 14)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. In ABC, mC 5 100 and
cos A , cos B. Which of the following
is true?
I. m/A , m/B
7.
II. m/A . m/B
III. sin A . sin B
(A) I only
(C) III only
(E) II and III only
6. The sides of a parallelogram measure 6
inches and 8 inches and the smaller angle
is 60°. The length of the longer diagonal of
the parallelogram is
(A) 2 !13 inches
(B) 10 inches
(C) 2 !37 inches
(D) 13 inches
(E) 15 inches
R
(B) II only
(D) I and III only
2. The angles of a triangle are in a ratio of
1 : 3 : 8. The ratio of the longest side of the
triangle to the next longest side is
(A) 8 : 3
(B) !2 : 1
(C) 2 : !3
(D) !6 : 2
(E) !8 : !3
If sin P is 125% of sin Q, then PR is what
percent of RQ?
5. The area of ABC is 35!3. If a 5 7 and
b 5 20, the measure of C could be
(A) 30° only
(B) 30° or 150°
(C) 45° or 135°
(D) 60° only
(E) 60° or 120°
(A) 6623%
(B) 75%
(D) 125%
(E) 150%
4
3
A
If sin A 5 k(cos A), then k is
(A) !7
3
(D) 3 !5
5
(B) 3 !7
7
(E) !21
3
(C) 53
9.
R
P
Q
If the radius of circle P is 1, then the
radius of circle Q is
(A) 2 sin P
(B) 2 cos P
1
(C) 2 sin
P
(E) 2 tan P
Copyright © 2009 by Amsco School Publications, Inc.
(C) 80%
8.
3. In ABC, A 5 30°, b 5 16, and
a 5 8 !2. Which could be the degree
measure of C?
(A) 135°
(B) 105°
(C) 90°
(D) 75°
(E) 60°
4. For which set of measures can more than
one triangle be formed?
(A) p 5 12, q 5 10, mp 5 100
(B) p 5 12, q 5 10, mP 5 30
(C) p 5 10, q 5 14, mP 5 150
(D) p 5 6, q 5 12, mP 5 30
(E) p 5 6, q 5 10, mP 5 30
Q
P
1
(D) 2 cos
P
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10.
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12
P
15. In isosceles triangle ABC, AB 5 AC 5 4
and the area of the triangle is 4. Find the
degree measure of the base angles.
(A) 30°
(B) 60°
(C) 75°
(D) 77°
(E) 81°
R
4
3
Q
In the rectangular box pictured,
sin RPQ equals
(A) 14
(B) 31
5
(D) 13
(E) 21
5
(C) 12
11. If sin A 5 0.5 and A is an obtuse angle,
what is the value of tan A rounded to
three decimal places?
(A) 21.733
(B) 20.577
(C) 0.500
(D) 0.577
(E) 1.733
12. In triangle ABC, the measure of C is 90°
and the measure of A is 40°. If AB 5 7,
what is the value of CB to the nearest
tenth?
(A) 4.5
(B) 5.4
(C) 7.0
(D) 9.0
(E) 10.9
13. In a circle with center O, the length of
radius OA is 13 units. A chord BC that is
24 units long is drawn. If mOBC 5 u,
then sin u is equal to
(A)
5
13
13
(D) 12
(B)
5
12
Date
(C)
12
13
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. The coordinates of the vertices of PQR
are P(6, 8), Q(28, 0), and R(0, 0). Find the
degree measure of R to the nearest
tenth.
17. The sides of a triangle are in the ratio of
4 : 5 : 6. Find the measure of the smallest
angle to the nearest tenth.
18. In isosceles ABC, mC 5 30 and
BC 5 14. What is the least possible area,
in square units, of ABC?
19. Two small airplanes depart from the same
airport at the same time on courses forming an angle measuring 57°. If the planes
fly at speeds of 150 and 180 miles per
hour, respectively, how far apart are the
planes, to the nearest tenth of a mile, at the
end of 30 minutes?
20.
B
110°
(E) 13
5
14. Find the degree measure, to the nearest
tenth of a degree, of the largest angle of a
triangle if the sides measure 10, 12, and
13 units.
(A) 38.7°
(B) 43.1°
(C) 46.9°
(D) 71.8°
(E) 72.1°
Copyright © 2009 by Amsco School Publications, Inc.
20 130°
O
A
C
Triangle ABC above is inscribed in circle
O, mAB 5 110, mBC 5 130, and
BC 5 20. Find the area of ABC to the
nearest integer.
X
X
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21. Triangle ABC is drawn in the coordinate
plane with point A at (22, 0) and vertex B
on the positive ray of the x-axis. The
measure of angle A is 65° and BC 5 20.
Find the x-coordinate of point C to the
nearest hundredth.
22. A triangular plot of land has sides of
lengths 120 feet, 90 feet, and 70 feet. Find,
to the nearest degree, the measure of the
largest angle.
23. In an isosceles triangle, the measure of
the vertex angle is 64° and the base is 12
units. Find the length of the altitude to the
base of the triangle to the nearest tenth.
Copyright © 2009 by Amsco School Publications, Inc.
Date
24.
R
10
O
T
S
In the figure above, two lines are drawn
from point T tangent to circle O at points
R and S. Chord RS is drawn, separating
circle O into two arcs whose measures are
in the ratio of 3 : 1. If RS 5 10, find the
area of RST.
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CHAPTER
15
STATISTICS
Aims
• To understand the differences among various methods for collecting data and to
determine factors that may affect the outcome of a survey.
• To review different methods of organizing
and displaying data.
• To review the measures of central tendency.
• To define various measures of dispersion
and their applications.
• To define and apply the normal distribution.
• To construct scatter plots and determine
the regression model that is most appropriate.
• To interpret the value of the correlation
coefficient as a measure of the strength of
the relationship for a linear regression
model.
• To use the regression model to interpolate
and extrapolate from data.
Modern technology has the capability of
making available vast quantities of information. Intelligent use of information requires an
understanding of its applicability to our lives.
Statistics provides methods of organizing and
analyzing data to enable us to identify trends
and their implications for the future. Understanding statistical methods helps us to be critical consumers of information and to avoid
being misled by invalid conclusions.
The material on statistics in this chapter is
introductory but is intended to help students
to begin to understand some of the statistical
measures used to analyze data and to lay a
foundation for more comprehensive work
with statistics in their advanced studies and
career preparation.
CHAPTER OPENER
Students who are preparing for college have
already taken or will soon take one of the
standardized tests (SAT or ACT) that most
schools require of their prospective students.
The companies that administer these tests provide information about the distribution of
scores on their tests. Discuss with students the
information about test scores given in the
booklets prepared by the test companies.
Explain that companies field-test the questions for standardized tests and use the feedback to determine the level of difficulty of
each question and the percent of students who
are expected to answer each question correctly. Information about standardized tests
can also be obtained from the Internet.
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200 Chapter 15
15-1 UNIVARIATE
STATISTICS
Univariate data (one-variable data) describe a
single characteristic of a population or sample.
For example, for the set of students in a class,
univariate data would include each student’s
height or age. Quantitative data take on
numerical values. Qualitative or categorical
data describe some characteristic of the population, such as eye color or favorite ice cream
flavor.
A census is a very complex, time-consuming, and expensive project. These factors have
motivated a second strategy for data collection: the sample survey. The process of drawing conclusions about the nature of the entire
population based on data from a sample is
called statistical inference and is the basis for
the usefulness of statistics in daily life. The
question of how accurately the sample statistics will represent the actual population begins
with the method used to select the population
sample. The selection process is called sample
design. Some sampling techniques are flawed
because they may be biased. Bias is a systematic error that favors a particular subset of the
population or that tends to encourage only
certain outcomes in the data.
Voluntary response sampling is used by
radio shows that invite listeners to call in their
opinions about a controversial issues. The fact
that people with strong opinions respond
works well for the show, since the discussions
may become quite emotional, but the opinions
are not necessarily representative of the whole
population.
In Example 2, the method of choosing
every 4th person is called systematic sampling.
For systematic sampling to be valid, the
researcher must be careful that the ordering
principle is not connected to the nature of the
population.
In addition to the wording of the question,
survey answers can also be biased by the order
of choices (the first answer tends to receive a
greater number of selections), the appearance
and/or demeanor of the interviewer, and the
honesty of the answers given by the participants (guaranteeing confidentiality is often
used to motivate honest responses). Newspapers and magazines often provide details of
the surveys and studies that are being
reported. Ask students to find examples of
surveys, observational studies, and controlled
experiments and critique the methods used.
Review the construction of stem-and-leaf
diagrams, frequency distribution tables, and
histograms, as these data displays will be used
throughout the chapter.
The Hands-On Activity that begins in this
section will be analyzed using the methods
described in subsequent sections.
15-2 MEASURES OF
CENTRAL TENDENCY
This section reviews the measures of central
tendency (mean, median, and mode), first and
third quartiles, and the box-and-whisker plot.
Graphing calculators and some scientific calculators can be used to find these measures. To
understand what these measures represent,
however, it is important that students have
some initial practice in finding statistical measures with calculator use limited to basic computations. Discuss the circumstances under
which one of these measures may be more
appropriate than the others.
The mode is the easiest average to find but
is the one that tells us the least about the other
data entries. Call attention to the fact that
although, for the sake of simplicity of computation, sets of data with few entries are often
used as exercises, the mode has significance
only when the number of occurrences of the
modal value is significantly higher than the
number of occurrences for the other values of
the data.
The median is the middle value. The number of data entries larger than or equal to the
median is equal to the number of data entries
smaller than or equal to the median. The
median is often considered to be a better
measure of central tendency than the mean
when there are outliers.
The mean is the measure of central tendency that is most frequently used. The mean
is commonly called the average. The sum of
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the absolute values of the differences,
2
a (x 2 x) , for data values smaller than the
mean is always equal to the sum of these differences for data values greater than the
mean. The mean is the average used to find
other statistical values such as the variance
and the standard deviation.
The first quartile is often denoted by Q1,
the second by Q2 and the third by Q3. Emphasize that if the data set contains an odd number of values, then the median is the middle
data value and is not included in the calculation of either the first or third quartiles. If the
number of data values is even, then the
median is the average of the two middle values
and all data values are used in the calculation
of the first and third quartiles.
Be sure students understand the method
described in Example 2b for locating the
median and first and third quartiles on a stemand-leaf diagram and the note below the
example.
The interquartile range as a measure of dispersion is discussed in Section 15-4.
As the Hands-On Activity continues, students use the data collected about estimates of
a minute to prepare the five statistical summary.
15-3 MEASURES OF
CENTRAL TENDENCY
FOR GROUPED DATA
In most serious statistical studies, the data is
usually large, and many values occur more
than once. In such situations, the data is organized into a table where xi represents a value
within the range of the set of data, fi represents
the number of times that value occurs, and xi fi
is the total value of the fi entries having the
value of xi. The mean is found by summing all
the xi fi entries and then dividing by the sum of
the fi entries.
When data is grouped into intervals, it is
convenient to use one number to represent
each interval. For the purpose of finding the
mean of interval data, it is assumed that all values within an interval are represented by the
midpoint of the interval.
In the example involving weights, the
actual values can be any number in an interval
(length, weight, temperature are examples of
continuous variables). Since the intervals are
recorded using whole numbers of pounds, the
weights that would fall in the interval 240–249
extend from 239.5 to 249.5.
Percentiles are useful in providing positional information in large data sets. For example, a score of 510 on a standardized test does
not tell us the position of this score relative to
the entire set of scores. If we are told that the
score of 510 is in the 56th percentile, then we
know the position of the score relative to the
entire set.
In some discussions, percentile is defined
as the percentage of data values below the
specified value. With that definition, the percentile rank of 7 misspelled words would be
based only on the number of essays with fewer
than 7 misspelled words:
87
100
5 87%
The method for approximating the median
for grouped data assumes that the number of
values in an interval are evenly spaced in the
interval. Drawing number lines as shown will
help students determine the position of the
median.
The Hands-On Activity is continued by
having students group their data into intervals
of 5 seconds and then finding the mean using
the methods of this section.
15-4 MEASURES OF
DISPERSION
Although the range is the easiest measure of
dispersion to calculate, it tells us nothing about
how the data is distributed within the limits of
the range. The range is based only on the
extremes and does not account for intermediate values.
The interquartile range (IQR) specifies the
length of the interval that contains approximately the middle 50% of the data. The
interquartile range corresponds to the “box”
of a box-and-whisker plot. Similarly, each
“whisker” of a box-and-whisker plot represents about 25% of the data.
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The interquartile range also provides criteria for identifying outliers in the data. If a data
value is greater than 1.5 3 (interquartile
range) plus the upper quartile or less than the
lower quartile minus 1.5 3 (interquartile
range), that data value is an outlier. Note that
when an outlier is present, the whiskers of a
box-and-whisker plot extend to the most
extreme non-outlier value on either side of the
box as appropriate.
Some graphing calculators allow the option
of a box plot identifying no outliers or a modified box plot indicating outliers. These options
are identified by different icons on the STAT
PLOT menu of the TI-831/841 graphing calculators.
The concepts of this section are applied as
the Hands-On Activity continues.
estimates are better when the formulas use
division by n 2 1 rather than n.
• If the data represents a population:
n
Variance 5 s2 5 n1 a (xi 2 2
x )2
i51
n
Standard deviation 5 s 5
Ç
1
n a (xi
22
x) 2
i51
Strictly speaking, the greek letter m should
be used to denote the mean of a population.
Greek letters are generally used to indicate
population measures, while English letters are
generally used to indicate sample measures.
• If the data represents a sample:
n
1
22
Variance 5 s2 5 n 2
1 a (xi 2 x)
i51
15-5 VARIANCE AND
STANDARD
DEVIATION
Today, the standard deviation is the most frequently used measure of dispersion in statistical studies. Standard deviation is particularly
important because of its relationship to the
normal curve. This relationship is explained in
detail in Section 15-6.
Since standard deviation is the square root
of another statistical measure called variance,
we begin by defining variance. Variance measures the deviation of each data value from the
mean. The variance is the mean of the squares
of the differences of each data value from the
mean of the data values. We square each difference so that positive and negative differences do not cancel each other out. The
standard deviation is the square root of the
variance.
There are two variances and standard deviations that can be calculated for any data set.
If the data represents a population, then the
exact values of the variance and standard deviation can be found. However, if the data represents a sample of a population, then we can
find only estimates for the variance and standard deviation. It has been found that these
Standard deviation 5 s
n
5
Ç
1
n 2 1 a (xi
22
x) 2
i51
Note that s 5 0 only when all the data values are the same; otherwise, s . 0. Moreover,
standard deviation is strongly influenced by
extreme values.
Most graphing calculators can determine
standard deviation. However, students will
have a better understanding of the measure if
they carry out the necessary steps using a calculator at first for basic computation only.
They may then check their results using the
statistical capabilities of the calculator.
15-6 NORMAL
DISTRIBUTION
Distributions of physical measurements (such
as heights and weights) or test scores for very
large data sets often have symmetric, bellshaped graphs. Normal distribution curves
have an infinite base—the long, flat-looking
tails actually extend to infinity. The normal
curve is symmetric about a vertical line
through its highest point. The x-coordinate of
the highest point represents the mean, the
median, and the mode, which coincide in a
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normal distribution. The y-coordinate of the
highest point represents the frequency of the
mean, the median, and the mode. The area
under the standard normal curve is 1, and the
area under a portion of the curve between two
vertical lines represents the probability that
values that fall between the values that are the
x-coordinates of those lines will occur. (This
will be explored in Chapter 16.) There is a
point on each side of a normal curve where the
slope is steepest. These two points are called
points of inflection and the distance from the
mean to either point is equal to one standard
deviation. The greater the standard deviation,
the less steep the sides of the curve are.
before computers and graphing calculators,
this conversion allowed us to use a single standard normal table to determine probabilities
for any normal distribution.
Conversion to z-scores is a combination of
two transformations:
1. By subtracting the mean from each x, we
transform the mean to 0.
2. By dividing each difference x 2 2
x by the
standard deviation, we transform the standard deviation to 1.
The set of z-scores for the data set (assuming it represents the population)
{2, 4, 6, 8, 10, 12, 14, 16}
is
point of
inflection
point of
inflection
x2s
x
x1s
{–1.528, 21.091, 20.655, 20.218, 0.218, 0.655,
1.091, 1.528}.
The mean of the transformed data is 0, the
standard deviation is 1, and each data value
has the same relative position in the set.
Note that given a z-score, we can reverse
the procedure that was used and find the original x-value: x 5 2
x 1 zs.
15-7 BIVARIATE
STATISTICS
Normal curves with the
same mean but different
standard deviations.
Note that while the standard deviation can
be calculated for any distribution, the rule for
percentages of data in the “standard deviation
regions” is reliable only for normal curves.
The z-score allows us to compare the position of a data value relative to the mean and
standard deviation of the data. When you convert a set of data to a set of z-scores, the transformed data has a mean of 0 and a standard
deviation of 1, regardless of the mean and standard deviation of the original data. In the days
When we explore bivariate or two-variable
data, we are interested in determining if an
association exists between the two variables
observed, how this association can be measured, and how it can be expressed mathematically and used to make predictions.
A scatter plot is a technique for visually
displaying a relationship between two quantitative variables. Individual observations are
written in (x, y) form and then plotted. The
resulting scatter plot is examined to see if the
points show a distinct pattern. Once the
strength and direction (positive or negative) of
the association have been determined, algebraic models can be fit to the data for predictive purposes.
The term linear regression refers to the
process of finding an equation of the form
y 5 ax 1 b for a set of bivariate data. This
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linear model is called the regression line.
One purpose of this linear equation is to
enable us to predict a typical value of a
dependent (or response) variable from
a given value of the independent (explanatory) variable.
If yi is the actual or observed value of the
dependent variable and ŷi is the predicted
value (called the “fit”) of the dependent variable, then we wish to choose a “line of best fit”
that minimizes the residual (yi 2 ŷi) at xi for
each data value. The least squares regression
line is one such line of best fit that minimizes
the sum of the square of the residuals or
n
2
a (yi 2 ŷi) . The “squares” in the name
i51
refers to the squares of the residuals.
For example, the linear regression equation
given in the text for the SAT scores is
y 5 0.693x 1 151.03
Using this equation, the predicted reading
score for a math score of 530 is
ŷ 5 0.693(530) 1 151.03
ŷ 5 518.32 < 518
Since the actual reading score for a 530
math score is 530, the residual at this point is
530 2 518 5 12. Similarly, for a math score of
566, the predicted reading score is
ŷ 5 0.693(566) 1 151.03
ŷ 5 543.268 < 543
For this point, the residual is 543 2 543 5 0.
Two important characteristics of the
regression line are:
1. It always includes the point (2
x, 2
y) .
2. The sum of the residuals is 0.
On the TI-83+/841 calculators, when a linear regression is executed, the residuals are
automatically stored in the list RESID. This
list can then be selected from the LIST
NAMES menu.
15-8 CORRELATION
COEFFICIENT
The correlation coefficient r is a number
between 21 and 1 that can be used to determine the strength and direction of the linear
association between the variables of a set of
bivariate data. A negative r implies an inverse
relationship (as one variable increases, the
other decreases) and a positive r implies direct
variation (as one variable increases, so does
the other). When r 5 1, the data pairs, when
plotted on a scatter plot, are collinear. Emphasize that although a value of the correlation
coefficient can always be found, the data
should be plotted to determine if it makes
sense to look for a linear relationship. This
idea will be discussed in more detail in Section
15-9.
In addition to the value of r, many graphing
calculators provide a value of r2. The value r2 is
called the coefficient of determination and provides information regarding the percent of
variation in the dependent variable that is
accounted for by the regression line. Alternatively, we can say that r2 gives the percentage
of variation in y that is predictable from a
knowledge of x. So, in Example 3, the value of
r2 5 0.33 indicates that 33% of the change in
the number of pounds purchased is accounted
for by the cost per pound. More importantly,
we can conclude that 67% of the change in
pounds purchased must be associated with
other variables that are not included in the linear model.
In Enrichment Activity 15-8: Calculating
the Correlation Coefficient, students are given
the formula for the correlation coefficient and
they are guided through the steps for completing the calculations.
15-9 NON-LINEAR
REGRESSION
Often, a straight line is not the best model for
representing a relationship between two variables. Such cases are clearly indicated when
the scatter plot shows a distinctive curved pattern.
Emphasize that students should not make
a judgment about the appropriateness of a
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model based on a comparison of the correlation coefficient of different models. For logarithmic, exponential, and power models, the
correlation coefficient describes the goodness
of fit for transformed data that has been linearized, not the original data. A larger correlation coefficient does not necessarily mean a
better fit.
The following describe the transformations
used to obtain each model:
Logarithmic Model
y 5 a 1 b ln x
Replace ln x by x9
Linear equation: y 5 a 1 bx9
Power Model
y 5 axb
ln y 5 ln a 1 b ln x
Replace ln y by y9, ln a by a9, ln x by x9
Linear equation: y9 5 a9 1 bx9
Exponential Model
y 5 abx
ln y 5 ln a 1 x ln b
Replace ln y by y9, ln a by a9, ln b by b9
Linear equation: y9 5 a9 1 b9x
Students have an opportunity to perform
some of these transformations in the Exploration at the end of the chapter. In that activity, they will work with the data from
Examples 1 and 2.
Note that when the TI-831/841 perform
any of the polynomial regressions, it returns
only the value R2. The absence of a value for r
and the fact that an upper-case R2 is shown
indicates that some technique other than linearization was used to carry out the regression. However, as the coefficient of
determination, R2 still indicates the percentage of variation in y that is accounted for by
the polynomial equation.
15-10 INTERPOLATION AND
EXTRAPOLATION
When we use the regression equation to make
predictions, it is often worthwhile to replace
the x and y variables with the names of the
variables in the context of the problem. So, in
Example 1, we could write
miles 5 32.365 gal 2 2.076
In this example, the coefficients are interpreted as follows:
slope: the number of miles driven is
predicted to increase by
32.365 for each increase of
1 gallon of gas
y-intercept: the number of miles driven is
predicted to be 22.076 when
0 gallons of gas are needed.
As this example shows, sometimes the
y-intercept makes no sense in the context of
the data set. A negative number of miles
driven is meaningless here.
While extrapolation must be applied with
great care, it is often the technique for making
predictions about population growth, product
sales, and other variables.
EXTENDED TASK
For the Teacher:
This activity is intended to provide students with an opportunity to make use of their
knowledge of statistics by designing and conducting a survey on their own. It allows them
to see that statistics is a valuable tool in the
real world. You may want to work cooperatively with the language arts teachers and have
the students present an oral report of their
survey and findings. A four-point scoring guide
is given below as a sample that might be used:
[4] The student independently completes
the task. He/she has a well-defined investigation and has developed three to five questions
that will adequately aid in the investigation.
The population selected for the survey is
appropriate and unbiased. The data are organized and displayed in an orderly manner. The
analysis of the data is thorough. Appropriate
measures of central tendencies and dispersions are accurately computed. The student
conducts the survey with the required number
of participants. The written report is neat and
thorough. It contains conclusions, inferences,
and or predictions that are appropriate from
the collected data.
[3] The student needs some assistance to
complete the task. He/she has a well-defined
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investigation and has developed three to five
questions that will adequately aid in the investigation. The questions, however, are not well
worded. The population selected for the survey is appropriate and unbiased. The data are
organized and displayed, but are not particularly neat or well presented. The analysis of
the data is somewhat difficult to follow.
Appropriate measures of central tendencies
and dispersions are computed but contain
some minor errors. The student conducts the
survey with the required number of participants. The written report is fairly neat but
somewhat sketchy. The conclusions, inferences, and or predictions from the collected
data are somewhat vague.
[2] The student needs considerable assistance from the teacher to complete the task.
His/her investigation is not well defined but
does contain the minimum of three questions.
The population selected for the survey is
either inappropriate or biased. The data are
poorly organized and displayed, making the
analysis difficult and vague. Appropriate
measures of central tendencies and dispersions are either lacking or contain serious
errors. The student fails to conduct the survey
with the required number of participants. The
written report is sketchy, containing few meaningful conclusions, inferences, and or predictions from the collected data.
[1] Even with considerable assistance from
the teacher, the student is unable to complete
the task. The investigation is poorly defined
and flawed. It contains fewer than three good
questions. The student demonstrates a lack of
ability to select an appropriate and unbiased
population. The data lack any organization.
They are either not displayed at all or so
poorly displayed that they are meaningless.
There is little, if any, analysis of the data. Few,
if any, appropriate measures of central tendencies or dispersions are included. The student
fails to obtain the required number of participants. The student summary is omitted or
meaningless, and he/she makes no attempt to
state any significance of the survey.
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ENRICHMENT ACTIVITY 15-8
Calculating the Correlation Coefficient
The correlation coefficient is a number that measures the strength and direction
of linear association of the variables in a set of bivariate data. For a sample data set,
the correlation coefficient, denoted r, can be found using the formula:
n
xi 2 2
x
yi 2 2
y
1
r5n2
1 a A sx B A sy B
i51
The formula is actually the sum of the products of the z-scores of the two variables divided by 1 less than the sample size.
To calculate the correlation coefficient for this sample data set
{(1, 3), (2, 6), (4, 20), (7, 22)}
follow the steps below. Round your answers to three decimal places.
x ).
1. Find the sample mean of the x-values (2
y) .
2. Find the sample mean of the y-values (2
3. Find the sample standard deviation of the x-values (sx).
4. Find the sample standard deviation of the y-values (sy).
5. Use the values you found in steps 1–4 to complete the table.
xi
yi
1
3
2
6
4
20
7
22
xi 2 2
x
sx
Q
yi 2 2
y
sy
xi 2 2
x yi 2 2
y
sx R Q sy R
Total:
6. Divide the total from the table by n 2 1.
7. Compare the value of r calculated using the steps above with the value given by
a calculator.
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8. Complete the table and calculate the correlation coefficient.
xi
yi
0
4
2
7
5
16
9
14
12
8
17
25
23
19
30
53
xi 2 2
x
sx
Q
yi 2 2
y
sy
Total:
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xi 2 2
x yi 2 2
y
sx R Q sy R
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EXTENDED TASK
Taking a Survey: Designing a Statistical Study
Many surveys are conducted in today’s world. There are surveys to find out what
products people prefer, surveys to obtain opinions of political ideas, surveys to determine the effectiveness of a new medicine, and so on. Many economic, political, and
social decisions are made based on the results of surveys. Surveys are conducted by
statisticians who, after they collect data, organize, display, and analyze those data. In
this extended task, you will play the role of a statistician.
Your task is to decide upon some topic that you would like to investigate and
design a study as part of your inquiry. Your investigation must include a survey questionnaire consisting of a minimum of 3 questions and a maximum of 5 questions. The
questions should be in either a multiple-choice format, a Likert scale format, or an
open-ended format.
Examples
Multiple-Choice Format: How many times have you attended a movie at a movie theater during the past 6 months? (A) 4 or more (B) 3 (C) 2 (D) 1 (E) none
Likert Scale Format: How much do you agree/disagree with the following statement:
“All students should be required to take 4 years of mathematics during their high
school career.”
(1) Strongly agree (2) Slightly agree (3) Slightly disagree (4) Strongly disagree
Open-Ended Format: Approximately how many miles do you drive per year?
Select an appropriate and unbiased population that you will use for your survey.
Using the questionnaire that you have designed, survey a total of fifty people from
your selected population. Organize and display your data in an orderly manner that
will make it possible for you to effectively analyze it. Analyze your data using appropriate measures of central tendencies and dispersion. Analyze the data for each question in your survey.
Write a thorough description of your survey, including conclusions, inferences,
predictions, and so on, that you can make as a result of the analysis of your data.
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Algebra 2 and Trigonometry:
Chapter Fifteen Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct
answer with insufficient work may receive only partial credit. All scratch paper must
be turned in at the conclusion of this test.
1. Each employee in a corporation fills out a questionnaire for a survey about
commuting times. What type of data collection strategy does this represent?
2. The owner of an excursion boat recorded the number of passengers on each
of the 12 trips that the boat made one weekend as follows:
20, 19, 22, 17, 22, 23, 23, 22, 21, 20, 21, 22
Find:
a. the mean
b. the median
c. the mode
d. the first quartile
e. the third quartile
f. the range
g. the variance to the nearest hundredth
h. the standard deviation to the nearest hundredth
3. Use the data set below
27, 39, 51, 69, 46, 60, 81, 23, 53, 55, 54, 46
to find:
a. the interquartile range
b. any outliers in the set
4. The records kept by two teachers of the numbers of students absent from their
classes in the past month are shown below.
Mrs. Alvarez: 4, 2, 1, 1, 3, 2, 2, 0, 1, 2, 4, 3, 1, 2, 6, 3, 1, 0, 0, 2
Mr. Kazin: 0, 0, 1, 4, 4, 7, 1, 0, 2, 6, 0, 0, 0, 1, 2, 6, 2, 3, 1, 0
a. Using Mrs. Alvarez’s record, find:
(1) the mean
(2) the median
(3) the mode
(4) the standard deviation to the nearest hundredth
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b. Using Mr. Kazin’s record, find:
(1) the mean
(2) the median
(3) the mode
(4) the standard deviation to the nearest hundredth
c. Which set of data more closely resembles a normal distribution? Explain
your answer.
5. The table shows the recorded high temperatures for the month of September in
New York City. For this set of data, find:
Temperature (°F)
Number of days
90–94
1
85–89
3
80–84
6
75–79
12
70–74
7
65–69
1
a. the mean
b. the standard deviation to the
nearest hundredth
6. The mean of the ages of the employees of a large department store is 42 years
with a standard deviation of 8. Find the z-score of the age of an employee who is
a. 30
b. 62
7. The mean of the math scores on an SAT test is 511 with a standard deviation of
112. In the same year, the mean of the math scores on the ACT test is 20.6 with a
standard deviation of 5.0. Carol scored 650 on the SAT and 28 on the ACT.
a. On which test did Carol have the higher z-score?
b. On which test did Carol do better? Explain.
8. In a set of data that approximates a normal distribution, the mean is 5 and the
standard deviation is 2.1.
a. What percent of the values are expected to be between 2.9 and 5? Round
your answer to the nearest whole percent.
b. What percent of the values are expected to be above 9.2? Round your
answer to the nearest tenth.
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9. The heights of 800 students have a normal distribution with a mean of 65 inches
and a standard deviation of 5 inches. To the nearest tenth, what percent of the
students are between 60 and 72 inches?
10. The mean score on a standardized reading test is 487 and the standard
deviation is 93. If the scores are normally distributed and 9,000 students took
the exam, about how many students had scores from 394 to 580? Round your
answer to the nearest hundred.
In 11 and 12, for each of the given scatter plots, determine whether the correlation
coefficient would be close to 21, 0, or 1.
11. 60
12. 300
50
250
40
30
200
150
20
10
100
50
0
10 20 30 40 50 60 70
1
2 3 4 5
6 7
8
13. The table below lists the twelve sales representatives employed by BestTech, Inc.,
each person’s annual sales for 2008, and the number of years the person has
worked for the company.
Sales Representative
Years with Company
Annual Sales ($1,000s)
Arnold
25
650
Bass
1
80
Chan
5
120
Dern
4
180
Etlis
12
360
4
140
Gold
20
450
Hardy
30
550
7
280
Jackson
17
300
Kendall
18
350
Lamont
8
220
Ferrara
Ives
a. Find an equation for the line of best fit for the data. Let years with company
be the independent variable.
b. Using the equation found in part a, predict the sales for a person who
worked for the company for 10 years. Round to the nearest thousand.
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c. If Hardy stays with the company for another 5 years, to the nearest
thousand, what sales does the model predict for the final year? What method
is used to make this prediction?
14. Several years ago, Janine Lucas opened a sandwich shop called Between the
Bread. As the store became popular, she opened more locations. The table
below shows the years since she opened her first store and the number of
stores she had each year.
Year of Operation
1
2
3
4
5
6
7
Number of Stores
1
2
4
7
15
27
52
a. Make a scatter plot of the data.
b. Identify the type of model that would best fit the data and find the
equation.
c. Using the equation found in part a, how many stores would Janine expect
to have in her ninth year of business?
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SAT Preparation Exercises (Chapter 15)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
1. If x 1 y 5 14, y 1 z 5 11, and z 1 x 5 9,
what is the mean of x, y, and z?
(A) 17
3
(D) 17
(B) 17
2
(E) 34
(C) 34
3
2. What is the arithmetic mean of 39, 315, and
324?
(A) 48
(B) 316
(C) 33 1 35 1 38
(D) 38 1 314 1 323
(E) 347
3. On Tuesday, 20 of the 25 students in an
algebra class took a test and their average
was 86. On Wednesday, the other 5 students took the test, and their average was
81. What was the average for the entire
class?
(A) 81.8
(B) 82
(C) 83.5
(D) 85
(E) 87
4. If the mode, mean, and median for the
values x, 1, 1, 2, 2, 3, 3, 4, 4, 8, and 8 are
integers and mode . mean . median,
then x is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 8
5. Let M be the median and D be the mode
of the following set of numbers:
12, 60, 90, 20, 60, 40
What is the mean of M and D?
(A) 22
(B) 40
(C) 50
(D) 55
(E) 60
Copyright © 2009 by Amsco School Publications, Inc.
6. The number of boys and girls in a tenthgrade class is the same. The average
weight of the boys is 146 pounds. The
average weight of the girls is 122 pounds.
What is the average weight of all the students in this class?
(A) 178 pounds
(B) 168 pounds
(C) 134 pounds
(D) 124 pounds
(E) Cannot be determined
7. If x 5 3 and y 5 4, what is the value of the
median of the following set?
5x 1 2y, 3x 2 y, 3(x 1 y), x 1 4y6
(A) 8
(D) 14
(B) 11
(E) 15
(C) 13
8. If the mean of four different negative integers is 29, what is the least possible value
for any of the four integers?
(A) 236
(B) 233
(C) 230
(D) 218
(E) 23
9. On a standardized test with a normal distribution of scores, the mean is 40 and the
standard deviation is 6.6. Approximately
what percent of the scores fall in the
range 26.8 to 46.6?
(A) 34%
(B) 68%
(C) 81.5%
(D) 95%
(E) 99%
10.
Measure (xi)
Frequency (fi)
27
400
29
350
33
150
35
600
41
300
What is the median of the data shown in
the table above?
(A) 33
(B) 34
(C) 35
(D) 150
(E) 900
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11. The scores of an exam have a normal distribution with a mean of 60.1 and a standard deviation of 7.3. The percentage of
scores that are between 74.7 and 82 is
closest to
(A) 2%
(B) 20%
(C) 40%
(D) 84%
(E) 97.5%
III. There is a very strong association
between daily calorie intake and
blood pressure.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
12. The scores on a test are normally distributed with a mean of 79. If the interval 71
to 87 contains approximately 68% of
scores, which value can be the standard
deviation?
(A) 2
(B) 4
(C) 8
(D) 12
(E) 16
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
13. In a large survey of teenagers regarding
bottled water, the mean number of
bottles consumed per week was 20
with a standard deviation of 3.5. If a
normal distribution is assumed, which
interval represents the total number
of bottles per week that approximately
95% of this group will drink?
(A) 6–34
(B) 13–20
(C) 13–27
(D) 20–27
(E) 20–34
14. Which of the following would you expect
to be true about the correlation between
number of family members and weekly
food budget?
(A) strong and positive
(B) weak and positive
(C) strong and negative
(D) weak and negative
(E) zero
15. Suppose a study finds that the correlation coefficient relating daily calorie
intake with blood pressure is r 5 0.95.
Which of the following are proper
conclusions?
I. A high-calorie diet causes high blood
pressure.
II. Low blood pressure is the result of a
low-calorie diet.
Copyright © 2009 by Amsco School Publications, Inc.
16. Let :x; 5 the greatest integer that is less
than or equal to x. For example, :4.23; 5 4
and :8; 5 8. What is the average of :10.75;
and :23.68; ?
17. To the nearest hundredth, how much
smaller is the population standard deviation of the data set than the sample standard deviation of the same data?
57, 3, 23, 156
18. A packing machine fills cardboard
boxes with oatmeal. Suppose the amount
of oatmeal per box forms a normal distribution with a mean of 16.1 ounces and a
standard deviation of 0.08 ounce. If
2,000 boxes are filled with oatmeal, how
many can be expected to have between
15.94 ounces and 16.26 ounces of oatmeal? Round your answer to the nearest
hundred.
19. Consider the set of points {(2, 7), (3, 10),
(4, 13), (5, s), (8, t), (11, 34)}. What is the
value of s 1 t when there is a perfect
linear correlation between the x- and
y-values?
20. If the mean of p, q, and 14 is 114, what is
the average of p and q?
21. The ages of the members of a women’s
bridge club are 49, 58, 62, 68, 74, 76, 76,
and 89. If Winifred joins the club, the
median will not change. What is
Winifred’s age?
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22. What is the mean of the integers 249,000
to 50,000 inclusive?
23. The mean of 7 positive integers is 405.
Two of the integers are 103 and 199 and
the other numbers are greater than 199.
If all 7 integers are different, what is
the greatest possible value for any of the
7 integers?
Copyright © 2009 by Amsco School Publications, Inc.
Date
24. The mean of the test scores of a class of
n students is 68 and the mean of the
test scores of a class of m students
is 94. The mean score, when both
classes are combined, is 88. What is the
n
value of m
?
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CHAPTER
16
PROBABILITY
AND THE
BINOMIAL
THEOREM
Aims
• To review probability concepts learned in
previous courses.
• To use permutations, combinations, and
the Counting Principle to determine the
number of elements in a sample space.
• To study the probability associated with
events where there are two outcomes.
• To know and apply the binomial probability formula to events involving the terms
exactly, at least, and at most.
CHAPTER OPENER
Have students suggest questions involving
probability that might be of interest to medical
researchers. For example, what is the probability that a person will contract a specific disease? What is the probability that a person’s
condition will improve if a certain medicine is
used? What is the probability that a piece of
diagnostic equipment will fail to detect a physical problem or indicate a problem is present
when the patient is actually healthy (that is, a
false positive result)?
• To use the normal distribution as an
approximation for binomial probabilities.
16-1 THE COUNTING
PRINCIPLE
• To expand powers of binomials using the
Binomial Theorem.
This section reviews the counting principle
and independent and dependent events.
For experiments with a large number of
outcomes, it is inefficient to use a list or tree
diagram to find the total number of outcomes.
The counting principle can be applied to decisions or events that take place in succession.
Emphasize that in using the counting principle, we determine the number of choices available for the second activity by assuming that
the first activity has taken place. So if there 5
different fruits in a bowl and you eat 2, you
have 5 choices for your first selection but only
3 choices left for your second selection (the
selections are dependent; the first selection
affects the second).
Many of the statistical concepts used to
evaluate data in business, medicine, government, technology, and population growth were
first developed to predict results in games of
chance. The study of statistics has, from its
beginnings, been linked with the study of
probability. In this chapter, basic concepts of
probability and combinatorics will be
reviewed and then extended to finding the
probability that a given outcome will occur a
specified number of times in repeated trials of
an experiment. These probability concepts will
then be used to derive the Binomial Theorem.
217
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218 Chapter 16
Point out that the counting principle gives
only the number of outcomes. For some problems, if we want more information about the
nature of each outcome, then a list or tree diagram is more helpful.
Note that many of the exercises preview
the application of the counting principle for
permutations that will be discussed in the next
section.
important. The formula for the number of
combinations of n things taken r at a time
(r # n) is
nCr
P
n
5 Q r R 5 nr!r
A variation of this formula is obtained by
substituting the rule for nPr:
5 (n 2n!r)!r!
By examining the formulas above, students
can see that the number of combinations is
always less than the corresponding number of
permutations by a factor of r!.
Note that nCn 5 1 and nC1 5 n.
nCr
16-2 PERMUTATIONS AND
COMBINATIONS
Review the counting principle and its application to the formula for permutations. The formula for the number of permutations of n
things taken r at a time (r # n)
P 5 n(n 2 1)(n 2 2) c(n 2 r 1 1)
n
r
n!
5 (n 2
r)!
can be considered to be the result of repeated
applications of the counting principle. The elements to be arranged are selected from the
same set without repetition, as that the number of possible selections decreases by 1 each
time. Point out that the question, “How many
different four-digit numbers can be made
using each of the digits 2, 3, 7, 9 exactly once?”
can be answered either by using the counting
principle or permutations. However, the question, “How many different four-digit numbers
can be made using the digits 2, 3, 7, 9 if digits
can repeat?” can be answered only by using
the counting principle.
Note that for n objects taken n at a time,
n!
n!
5 (n 2
n)! 5 0! 5 n! (since 0! 5 1)
Sometimes the set of elements to be
arranged contains a subset whose elements
cannot be distinguished from each other. In
such a case, we first determine the number of
permutations as if the elements are all different, and then we divide by the number of
arrangements of the identical elements among
themselves.
The key difference between permutations
and combinations is that permutations are
used when the order of the objects matters,
and combinations are used when order is not
n Pn
16-3 PROBABILITY
This section reviews the terminology and principles of probability. Combinatorics can be
applied to answer probability questions as
long as we keep in mind that for equally likely
outcomes, the probability of an event E is the
ratio of the number of outcomes in the event,
n(E), to the number of outcomes in the sample
space, n(s).
n(E)
P(E) 5 n(S)
Students may find different solution methods for the problems in the text. Consider the
situation involving Jacob and Emily. Two
approaches are given, but others are possible.
The probability that Jacob is picked is the
probability of two disjoint events: Jacob is chosen on the first pick or Jacob is chosen on the
second pick. We find the probabilities and
then add them.
P(Jacob chosen first) 5 18 ? 77 5 18
P(Jacob chosen second) 5 78 ? 17 5 18
P(Jacob chosen) 5 18 1 18 5 28 5 14
Similarly,
1 11
1
P(Emily chosen first) 5 12
? 11 5 12
1
1
P(Emily chosen second) 5 11
12 ? 11 5 12
1
1
2
1 12
5 12
5 16
P(Emily chosen) 5 12
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Probability and the Binomial Theorem 219
Then,
P(Jacob and Emily chosen)
5 P(Jacob chosen) ? P(Emily chosen)
1
5 14 ? 16 5 24
For Example 2, the probability that both
students have fewer than two absences is the
product of the probabilities that the first and
second students chosen have fewer than two
absences. Using the probability form of the
counting principle for dependent events:
P(first , 2) ? P(second , 2 given first , 2)
1,176
9,730
49 48
84
5 140
? 139 5
5 695
< 0.12
Enrichment Activity 16-3: Chi-Square
(x2) Test for Goodness of Fit presents a simplified introduction to this topic. The problem
of deciding whether or not a set of data fits a
given distribution is discussed and the method
of calculating the chi-square statistic is
explained. The concepts of degrees of freedom
for each distribution and the significance level
are avoided; instead, students are given the
critical value and told that the chi-square statistic must be compared with the given critical
value.
16-4 PROBABILITY WITH
TWO OUTCOMES
The conditions for a binomial probability
experiment are:
• The n trials or observations are independent of each other.
• There are only two outcomes on any trial,
which can be considered success or failure.
• The probability of a success is the same for
each trial or observation.
• The number of trials, n, is fixed in advance.
Every experiment can be considered to
have two outcomes, a favorable outcome and a
non-favorable one that consists of all other
outcomes. When a question asks for the probability of exactly 3 heads on 5 tosses of a coin, it
is asking for the probability of 3 heads and 2
tails. If a question asks for the probability of
exactly 7 juniors on a committee of 10 chosen
at random from the student body, it is asking
for the probability of randomly choosing 7
juniors and 3 students who are not juniors.
Students should be encouraged to reword a
problem in this way before writing the solution since, in the second case, it is necessary to
consider the probability of all 10 outcomes (7
juniors and 3 not juniors) to obtain the correct
probability.
Hands-On Activity
Instructions:
A calculator can be used to simulate an
activity with two outcomes when the probabilities of the component parts are known. The
steps given below describe how to use calculator-generated random numbers to solve the
following problem.
The probability that a customer in a dress
shop will make a purchase is 15 . What is the
probability that if there are 4 customers in the
shop, exactly 2 will make purchases?
1. Seed the random number generator of
your calculator using the rand command.
Randomly press a few digits. Then press
你
STO佡 MATH
ENTER
ENTER .
A sample screenshot is shown below:
7898923547 >rand
7898923547
2. Use the randBin command to simulate a
trial of 4 customers walking into the store.
你
MATH
7 4 , 1 ⴜ
ENTER:
5 ENTER
DISPLAY:
randBin(4, 1/5
0
The randBin command takes as input the
number of trials (4) and the probability of
success A 15 B and returns the simulated
number of successes observed in the 4 trials. Note that your answers will vary
based on the value used for the random
seed in step 1.
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220 Chapter 16
3. Repeat the previous step for a total of
20 times.
4. Of the 20 simulations, count the number
of times there were exactly 2 successes.
Write the ratio of this number to the total
number of simulations, 20. This ratio is the
empirical probability that exactly 2 of 4
customers will make a purchase. How
does this value compare with the theoretical probability?
5. Now combine the results of all the students in the class. Write the ratio of the
total number of times there were 2 successes to the total number of simulations.
How does this value compare with the
theoretical probability?
Discoveries:
1. A calculator can be used to simulate a situation under consideration in a probability study.
2. The greater the number of simulations,
the more closely the empirical probability
matches the theoretical probability.
Enrichment Activity 16-4: Geometric
Probability Distribution explains the difference between a binomial distribution and a
geometric distribution. In a geometric experiment, we are interested in the number of trials
that are needed to achieve the first success.
(Note this is not the geometric probability discussed in Section 16-3.) The general formula is
derived and students practice applying the formula to solve problems. A method for simulating this type of situation is demonstrated using
a calculator.
16-5 BINOMIAL
PROBABILITY AND
THE NORMAL CURVE
Direct students’ attention to the chart preceding Example 1. Ask them to suggest any other
phrases that mean the same as at least or at
most; for example “up to 8 successes” or “a
minimum of 8 successes.” Call attention to
when the endpoint is included.
As shown in Example 1, problems involving at least or at most have two methods of
solution.
Method 1: P(5 on at most seven dice)
5 P(one 5) 1 ? ? ? 1 P(seven 5s)
Method 2: P(5 on at most seven dice)
5 1 2 P(5 on more than seven
dice)
Students should understand both methods
and should be encouraged to use the shorter
solution when possible.
The table for the number of heads when 10
coins are tossed shows that the binomial distribution when p 5 12 is symmetric. For other values of p, binomial distributions will be only
somewhat symmetric.
Before technology became readily available, a normal approximation to the binomial
was the preferred method for finding binomial
probabilities when n was large or when many
cases needed to be considered. For example,
the probability of tossing more than 65 heads
in 100 coin tosses would require a very long
calculation.
The normal approximation method works
best when n is large and when the probability
of success, p, is not close to 0 or 1. Alternatively, some mathematicians recommend the
criteria that np $ 10 and n(1 2 p) $ 10. The
reason for these conditions is that, when they
are met, the binomial distribution is relatively
symmetric and the normal curve provides a
fairly good fit. If either condition is not met,
the binomial distribution will be skewed significantly and the approximation by a normal
curve will not be accurate.
Students should also be aware that the
binomial probability distribution is a discrete
distribution and, by applying the normal curve
approximation, we are finding a discrete probability using a continuous distribution. By
adding or subtracting 0.5 to the endpoints of
the region of interest, we are making a continuity correction to make up for gaps between
the discrete values that are being approximated by a continuous function.
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Probability and the Binomial Theorem 221
So that students may understand how a distribution may be skewed, have them find the
probability of n successes for n 5 0, 1, 2, . . . , 10
when p 5 .1. Ask them to make a probability
plot similar to the one in the text. (See Exercise 2.)
16-6 THE BINOMIAL
THEOREM
Students should note the following patterns in
the expansion of (x 1 y)n:
and 10x2y3 and another two terms are
5x4y1 and 5x1y4.
• The number of terms in the expansion of
(x 1 y)n is n 1 1.
In the sigma notation form of the binomial
theorem, i 5 0 for the first term, i 5 1 for the
second term, and so on. In general, the value of
i is always one less than the number of the
term you want to find. For example, the eighth
term of (x 1 y)12 is given by
5 7
12C7x y
n
• The exponent of (x 1 y) , n, is the exponent of x in the first term and the exponent of y in the last term.
• In successive terms, the exponent of x
decreases by 1 and the exponent of y
increases by 1.
• The sum of the exponents in each term is
n.
• The coefficients are symmetric. The coefficients increase and then decrease from left
to right. Terms where exponents are
reversed have the same coefficients. For
example, in (x 1 y)5, two terms are 10x3y2
5 792x5y7
When the binomial theorem is used to
compute probability, students must think carefully to choose the correct term. For example,
if Sara’s probability of making a basket on any
attempt is p 5 34, then her probability of missing is m 5 14. To find the probability that Sara
will make exactly 5 baskets in 8 attempts, compute the term of (m 1 b)8 where the exponent
of b is 5. Since i is always one less than the
number of the term, find the term where i 5 5
or the sixth term.
3 5
8C5m b
5 56 A 14 B
3
A 34 B
5
1,701
5 8,192 < .2076
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ENRICHMENT ACTIVITY 16-3
Chi-Square (x2) Test for Goodness of Fit
An important question is whether or not a set of observed data values are compatible with a set of expected values. For example, you may want to know if public
opinion on marriage has changed in the past 40 years. (In this case, the expected values would be the results of a 40-year-old survey, and the observed values would be
the results of the same survey done in the present.) A Chi-square test for goodness of
fit allows us to answer such questions.
Chi-square statistic: x2 5 a
(observed 2 expected) 2
expected
The chi-square statistic takes into account the size of the difference between each
observed and expected value relative to the size of the expected value. So an
observed difference of 10 is more significant if it comes from an expected value of 100
than from an expected value of 10,000. The bigger the chi-square value, the more
likely the observed values do not conform to the expected values.
Problem 1
A manufacturer packs a rebate coupon in each box of its cereal and claims that
the probabilities of getting various amounts are as follows:
Coupon Amount
$0.50
$1.00
$2.00
$3.00
.40
.30
.20
.10
Probability
To see if the company’s claim is accurate, a consumer group randomly selects 500
boxes of this cereal and obtains the following data:
Coupon Amount
$0.50
$1.00
$2.00
$3.00
224
162
82
33
Number
Do these data provide evidence that the company is lying? That is, can any differences be due to chance (random variation), or are the differences significant enough
to doubt the company’s assertion?
1. Complete the table. The first row is done for you.
Expected
Sample
Expected
Observed
Amount
Probability
Size
Frequency
Frequency
$0.50
.40
500
.40(500) 5 200
224
$1.00
.30
500
162
$2.00
.20
500
82
$3.00
.10
500
33
Copyright © 2009 by Amsco School Publications, Inc.
(Observed 2 Expected) 2
Expected
(224 2 200) 2
200
5 2.88
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Name
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2. Calculate the chi-square statistic by finding the sum of the numbers in the last
column of the table.
3. To decide how large a calculated chi-square value must be to be significant, we
must compare it with a critical value. You will learn more about finding the critical value in advanced courses. For this problem, the critical value is 7.81. If the
calculated chi-square value is greater than 7.81, then we must reject the company’s claim about the probability of getting various coupon amounts. (Otherwise, the only conclusion we can make is that we have insufficient evidence to
reject the company’s claim.) Moreover, this critical value is chosen so that our
chances of incorrectly rejecting the company’s claim is only 5%. Based on your
calculations, should the claim be rejected?
Problem 2
A die is rolled 600 times and the following results are recorded.
Face
Frequency
1
2
3
4
5
6
113
88
103
117
75
84
Do these data cast doubt on the fairness of the die? To decide, calculate the
chi-square statistic and compare it with the critical value of 11.07.
1. Complete the table below.
Expected
Sample
Expected
Observed
Face
Probability
Size
Frequency
Frequency
1
1
6
600
2
1
6
600
3
1
6
600
4
1
6
600
5
1
6
600
6
1
6
600
2. Calculate the chi-square statistic.
3. Should the claim that the die is fair be rejected? Explain.
Copyright © 2009 by Amsco School Publications, Inc.
(Observed 2 Expected) 2
Expected
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Problem 3
1. Conduct your own experiment. Toss a die 120 times and record the frequency
for each face.
2. Compute the chi-square statistic for your data.
3. Compare your statistic with the same critical value as above (11.07). What conclusion can you draw based on your experiment?
4. Make a list of the chi-square statistics calculated by all members of the class who
performed the experiment. What percent of the class had values that were
greater than the given critical value?
Copyright © 2009 by Amsco School Publications, Inc.
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ENRICHMENT ACTIVITY 16-4
Geometric Probability Distribution
Recall the conditions for a binomial experiment:
• The n trials are independent of each other.
• There are only two outcomes on any trial, which can be considered success or
failure.
• The probability of a success is the same for each trial.
• The number of trials, n, is fixed in advance.
The conditions for a geometric experiment are similar with one important
difference:
• The n trials are independent of each other.
• There are only two outcomes on any trial, which can be considered success or
failure.
• The probability of success is the same for each trial.
• The number of trials is not fixed.
A geometric distribution shows the number of trials needed until the first success
is achieved.
Example
Suppose the probability that a box of cereal has a free movie ticket is .2. Find the
probability that the first free ticket is found in:
a. the 1st box purchased
b. the 2nd box purchased
c. the 3rd box purchased
Solution: Since P(ticket) 5 .2, P(no ticket) 5 1 2 .2 5 .8.
a. P(ticket in 1st box) 5 .2
b. P(first ticket in 2nd box) 5 P(no ticket, ticket)
5 P(no ticket) 3 P(ticket)
5 .8 3 .2
5 .16
c. P(first ticket in 3rd box) 5 P(no ticket, no ticket, ticket)
5 P(no ticket) 3 P(no ticket) 3 P(ticket)
5 .8 3 .8 3 .2
5 .128
Copyright © 2009 by Amsco School Publications, Inc.
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It is interesting to note that the greatest probability that a ticket will be found
occurs for the first box! The results of the example can be generalized.
•
If p is the probability of success, then
P(first success occurs on the rth trial) 5 (1 2 p)r21 3 p
In 1–3, find all probabilities to three decimal places.
1. A battery manufacturer determines that 5 out of every 100 batteries are
defective. Find the probability that a quality control engineer finds the first
defective battery in the
a. 4th battery tested
b. 7th battery tested
2. Let a sum of 8 when two dice are rolled be considered a success.
a. Find P(8) in fractional form.
b. Find the probability that the first sum of 8 occurs on the 5th roll.
3. At Rosedale High School, the probability that a student studies a foreign
language is .6. If students at Rosedale High are randomly chosen and asked
if they study a foreign language, find the probability that
a. the first student who studies a foreign language is the 6th student asked
b. the first student who does not study a foreign language is the 10th student
asked
A simulation can be used with regard to a geometric distribution. This type of simulation is sometimes called a wait-time simulation since we continue the simulation
until a condition occurs.
Consider the example that involved buying cereal boxes until a free movie ticket
was found. Since the tickets are placed in 20% of the boxes, assign the digits 0 and 1
to a free ticket. Using a random-number table or a calculator, begin each simulation
by noting the number of digits necessary until a 0 or 1 occurs. Record the number of
digits including the first success. Repeat the simulation for a total of 20 times.
For example, using the T1-83+/84+, execute the command
randInt(0, 9)
repeatedly until a 0 or 1 appears. (randInt is found in the MATH PROB menu.)
Count the number of executions including the success and record the number. Then
begin the next simulation. The results of 20 simulations are recorded in the table
below.
Simulation
First success
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
15
4
1
1
1
3
5
1
2
18
2
3
3
2
2
3
6
5
1
1
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In the table, the ordered pair (1, 15) tells us that in the 1st simulation, the first success occurred in the 15th try. The ordered pair (2, 4) tells us that in the 2nd simulation,
the first success occurred in the 4th try, and so on.
The empirical probability that the first ticket is in the third box is then
number of simulations where
success occurs in 3rd execution
total number of simulations
4
5 20
5 .2
Note that these particular simulations gave a result greater than the theoretical
value. Also, since the sum of the digits of all the trials is 79, the average number of
boxes needed to obtain a ticket is estimated as
79
20
5 3.95
so we expect to purchase a minimum of 4 boxes before finding a prize.
4. The probability that a construction company will be awarded a project is .4.
a. Conduct 10 simulations using a calculator to estimate the probability that the
company will land its first project on its 4th bid.
(1) Describe the method of assignment of digits to the situation.
(2) Describe how each simulation is carried out.
(3) Show your work to find the empirical probability.
b. Compare the calculated probability with the theoretical value.
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Algebra 2 and Trigonometry:
Chapter Sixteen Test
Write your answers legibly in the space provided below. Show any work on scratch
paper. An incorrect answer with sufficient work may receive partial credit. A correct
answer with insufficient work may receive only partial credit. All scratch paper must
be turned in at the conclusion of this test.
In 1–8, evaluate each expression.
1. 6P6
2. 7P3
3.
20C18
4.
13C0
5. Q 10 R
3
6.
1099 P1
7. (12C4)(8C5)
8.
11P5
1 9 P4
9. A fast-food store offers the following choices:
Hamburger: Single / Double / Triple
Roll: White / Whole Wheat / Sesame
Topping: Ketchup / Cheese / Onion / Chili / Relish / Mayonnaise
How many different orders are possible for a hamburger on a roll with
one topping?
10. In how many different ways can the letters of the word CIVILIZATION be
arranged? (Answer may be left in factorial notation.)
11. What is the probability that a random arrangement of the letters of the
word CIVILIZATION begins and ends with I?
12. Every week, Sharon works 4 days and does not work the other 3 days. In how
many different ways can her work schedule be arranged?
13. The weather channel gives the probability of rain on Saturday as 30% and the
probability of rain on Sunday as 40%. What is the probability that it will rain
on one or both days?
14. An equilateral triangle is drawn on a square target such that the base of the
triangle is one side of the square. If a dart hits the target, what is the probability
that it lands in the triangle?
15. A summer reading list includes 8 novels, 10 nonfiction books, and 5 plays. How
many different selections of 2 novels, 3 nonfiction books, and 1 play can be
chosen from the list?
16. A 20-member club has an equal number of freshmen, sophomores, juniors, and
seniors. If 5 members are selected at random, what is the probability that at
least 4 will be seniors?
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17. The owner of Katonah Krafts estimates that 1 out of every 3 persons who come
into the store makes a purchase. Today, 4 people came into the store the first
hour it was open. What is the probability that at least one person made
a purchase?
18. Of the students at Eastchester High School, 2 out of 3 have jobs for the summer.
If 10 students are selected at random, what is the probability that exactly 4 will
have jobs? Round your answer to four decimal places.
19. On a multiple-choice test, the probability of selecting the correct answer by
guessing is 14. What is the probability that a student who guesses every answer
will have exactly 13 correct answers on a test with 20 questions? Round your
answer to four decimal places.
20. The buyer for a shoe store estimates that 10% of the shoes sold are size 6. If the
store sold 25 pairs of shoes yesterday, to four decimal places, what is the
probability that at most 2 pairs of size 6 shoes were sold?
In 21 and 22, use the normal approximation to estimate each probability. Round
answers to four decimal places.
21. A fair coin is tossed 100 times. What is the probability of getting at least
45 heads?
22. The probability that the Cougars will win any game is .4. What is the
probability that the team will win between 10 and 20 of the 50 games it plays
this season?
23. Expand the binomial (x 1 2)6.
24. Expand the binomial (3x 1 2y)4.
25. What is the fourth term of (a 2 2b)7?
26. What is the fifth term of A 3 1 3t B ?
27. Use the binomial theorem to write an expression for the volume of a cube with
sides measuring (2n 2 3) inches.
5
28. What is the sum of the numerical coefficients of the expansion of (x 1 y)20?
(Hint: Use (1 1 1)20.)
Bonus: In a small town, a study showed that 10% of registered voters had lived at
the same address all their lives. A sample group of registered voters is to be
selected at random from the town. How large of a group is needed to have
the probability be greater than 40% that at least one person has always lived
at the same address? (Use a calculator and trial and error.)
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SAT Preparation Exercises (Chapter 16)
I. MULTIPLE-CHOICE QUESTIONS
In 1–15, select the letter of the correct
answer.
6.
4
3
1. How many distinct arrangements of the
letters of the word TENNESSEE can be
made?
(A) 3,780
(B) 7,560
(C) 10,480
(D) 15,120
(E) 30,240
2
1
5
2. How many ways can a row of four girls
followed by four boys be arranged if the
girls must sit together and the boys must
sit together?
(A) 16
(B) 256
(C) 576
(D) 10,080
(E) 40,320
The probability of getting each number
on the spinner is proportional to the number. For example, P(4) 5 2 ? P(2). What is
the probability of getting a sum of 5 from
two spins?
2
(A) 45
4
(B) 45
3. A building has two small elevators. Each
elevator can safely carry 3 people. There
are 5 people waiting to take the elevators
to the second floor. If both elevators
arrive at the first floor at the same time,
how many ways can the people group
themselves in the elevators?
(A) 4
(B) 6
(C) 10
(D) 12
(E) 20
8
(D) 225
2
(E) 75
4. Mrs. Grasso’s eleventh-grade class has 12
boys and 14 girls. In how many ways can
at least 2 boys be selected for a committee
of 3 students?
(A) 66
(B) 286
(C) 924
(D) 1,144
(E) 4,004
5. A spinner is divided into red, white, and
blue sectors. The area of the white sector
is twice the area of the red, and the area
of the blue sector is one and a half times
the area of the white. The spinner is spun
twice. What is the probability that one
spin lands on red and one spin lands on
blue?
1
(A) 12
(B) 16
(D) 21
(E) 23
3
(C) 12
Copyright © 2009 by Amsco School Publications, Inc.
4
(C) 75
7.
a
b
A target consists of one square inside
another. If a dart has an equally likely
chance of striking any point on the target,
what is the probability that it will strike
the shaded region?
2
(A) ba2 2 1
2
(D) ba2 1 1
2
(B) 1 2 ba2
(E) ba
2
(C) ba2
8. The math department has 8 teachers.
What size subcommittee should be
selected to provide the largest number of
combinations of teachers on the committee?
(A) 8
(B) 7
(C) 6
(D) 5
(E) 4
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9. If 10 coins are flipped simultaneously and
P(k) is the probability of getting k heads,
10
what is the value of a P(k) ?
k50
1
10
(A) 0
(B)
(D) 1
(E) 10
(C)
1
210
10. If 10 coins are flipped simultaneously and
P(k) is the probability of not getting
10
k heads, what is a P(k) ?
k50
1
10
(A) 0
(B)
(D) 1
(E) 10
11. If xCx–2 5 15, then x is
(A) 3
(B) 4
(D) 6
(E) 7
(C)
1
210
(C) 5
12. Q 25 R 2 Q 25 R 1 Q 25 R 2 c 1 Q 25 R
0
2
24
1
25
2 Q R equals
25
(A) 0
(B) 1
(C) 2
(D) 4
(E) 25
x
x
13. If Q 1 R 2 Q 0 R 5 24, then x is
(A) 4
(D) 24
(B) 6
(E) 25
Date
II. STUDENT-PRODUCED RESPONSE
QUESTIONS
In 16–24, you are to solve the problem.
16. The probability of the Jaguars winning
any one game is 30%. To four decimal
places, what is the probability that the
Jaguars will win exactly 2 games in a
5-game series?
17. How many different ways can a group of
3 dogs and 2 cats be chosen if a pet store
has 10 dogs and 8 cats?
18. A committee of 7 students is to be chosen
from a group of 20 seniors and 14 juniors.
Find, to four decimal places, the probability that the committee will include 4 seniors and 3 juniors.
19. The probability of Slugger Sam getting a
hit in any one at bat is .75. To four decimal places, what is the probability that
Sam will get at most 5 hits in his next 6 atbats?
20. A gardener will plant 3 yellow tulips, 2
red tulips, and 4 purple tulips along a
straight path. How many different
arrangements can she make?
21. How many diagonals are there in a regular hexagon?
(C) 12
14. What is the sum of the numerical coefficients of (x 2 y)10?
(A) 2252
(B) 25
(C) 0
(D) 5
(E) 252
15. The expression (2 2 i)5 where i 5 !21 is
equivalent to
(A) 238 2 41i
(B) 38 2 39i
(C) 42 1 40i
(D) 258 1 56i
(E) 142 1 120i
Copyright © 2009 by Amsco School Publications, Inc.
22. What is the coefficient of the x5 term of
the product of (x3 1 3x2 1 3x 1 1) and
(x5 1 5x4 1 10x3 1 10x2 1 5x 1 1)?
23. If n is an integer between 0 and 100, then
100 Cn ? (100 2 n)! is equal to 100 P100 2 n for
what values of n?
24. A fair die is tossed three times. What is
the probability that, on at least two of the
throws, the face that appears on top is
greater than 4?
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Summary of Formulas and Trigonometric Relationships
Functions and Relations
• To determine if a relation is a function, use the vertical line test.
• To determine if a function is one-to-one, use the horizontal line test.
• If f(x) is a function, then:
The graph of f(x) 1 a is the graph of f(x) moved a units up or down.
The graph of f(x 1 a) is the graph of f(x) moved a units to the left when a is positive and |a|
units to the right when a is negative.
The graph of 2f(x) is the graph of f(x) reflected in the x-axis.
The graph of f(2x) is the graph of f(x) reflected in the y-axis.
When a . 1, the graph of af(x) is the graph of f(x) stretched vertically by a factor of a (or,
alternatively, af(x) is the graph of f(x) compressed horizontally by a factor of a).
When 0 , a , 1, the graph of af(x) is the graph of f(x) compressed vertically by a factor of a
(or, alternatively, af(x) is the graph of f(x) stretched horizontally by a factor of a).
Absolute Value Equations and Inequalities
• If x 5 k for positive k, then x 5 2k or x 5 k.
• If x , k for positive k, then 2k , x , k.
• If x . k for positive k, then x . k or x , 2k.
• If x 5 k for negative k, then the solution set is { }.
• If x , k for negative k, then the solution set is { }.
• If x . k for negative k, then the solution set is the real numbers.
Quadratic Inequalities in One Variable
• To solve a quadratic inequality:
1. Find the roots of the corresponding equality.
2. The roots of the equality separate the number line into two or more intervals.
3. Test a number from each interval. An interval is part of the solution if the test number makes
the inequality true.
• Alternatively, to solve 0 . ax2 1 bx 1 c or 0 , ax2 1 bx 1 c, graph y . ax2 1 bx 1 c or
y , ax2 1 bx 1 c, respectively. The solution is the set of x-coordinates of the points common to
the graph of y . ax2 1 bx + c or y , ax2 1 bx 1 c and the x-axis.
232
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Summary of Formulas and Trigonometric Relationships 233
Quadratic Inequalities in Two Variables
• The solution set of y . ax2 1 bx 1 c is the set of coordinates of the points above the graph of
y 5 ax2 1 bx 1 c.
• The solution set of y , ax2 1 bx 1 c is the set of coordinates of the points below the graph of
y 5 ax2 1 bx 1 c.
Quadratic Equations
b
b
, c 2 4a
• The vertex of the parabola y 5 ax2 1 bx 1 c is A 22a
B and the axis of symmetry is
b
x 5 22a.
2
• To complete the square in ax2 1 bx 1 c 5 0:
1. Divide by a if a 1.
2. Isolate the terms in x on one side of the equation.
2
3. Add the square of one-half the coefficient of x or A 12b B to both sides of the equation.
4. Write the square root of both sides of the resulting equation and solve for x.
• The quadratic formula gives the roots of ax2 1 bx 1 c 5 0 as
2
x 5 2b 6 "2ab 2 4ac
• In ax2 1 bx 1 c 5 0, the value of b2 2 4ac is the discriminant.
The number
of x-intercepts
of the function is
When the discriminant
b2 2 4ac is
The roots of the
equation are
. 0 and a perfect square
real, rational, and unequal
2
. 0 and not a perfect square
real, irrational, and unequal
2
50
real, rational, and equal
1
,0
imaginary numbers
0
• If A and B are two roots of a quadratic equation, then a quadratic equation with these roots is
x2 2 (A 1 B)x 1 A ? B 5 0
• In ax2 1 bx 1 c 5 0:
2ba is equal to the sum of the roots.
ac is equal to the product of the roots.
Complex Numbers
• For any integer n:
i 4n 5 1
i 4n 1 2 5 21
i 4n 1 1 5 i
i 4n 1 3 5 2i
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234 Summary of Formulas and Trigonometric Relationships
• The complex conjugate of a 1 bi is a 2 bi. Also, (a 1 bi)(a 2 bi) 5 a2 1 b2.
• The multiplicative inverse of any non-zero complex number a 1 bi is
1
a 1 bi
a
b
5 a2 1
b2 2 a2 1 b2 i
Sequences and Series
• For an arithmetic sequence:
an 5 a1 1 (n 2 1)d 5 an–1 1 d
Sn 5 n2 (a1 1 an) 5 n2 f2a1 1 (n 2 1)dg
• For a geometric sequence:
an 5 a1rn–1 5 an21r
a1 (1 2 rn)
1 2 r
Sn 5
• For a geometric series, if r , 1:
`
a
n21
5 1 21 r
a a1r
n51
Exponential Growth and Decay
• If a quantity A0 is increased or decreased by a rate r per interval of time, compounded n times
per interval, its value A after t intervals of time is
A 5 A0 A 1 1 nr B
nt
• If the increase or decrease is continuous for t units of time, the formula becomes
A 5 A0ert
Exponents and Logarithms
Powers
Logarithms
Exponent of Zero
b0 5 1
logb 1 5 0
Exponent of One
b1 5 b
logb b 5 1
Products
bx1y 5 bxby
bx
y
Quotients
bx2y 5 b
Powers
(bx) y 5 bxy
Power of a Product
(bc) x 5 bxcx
Power of a Quotient
A bc B
x
x
5 bcx
logb cd 5 logb c 1 logb d
logb dc 5 logb c 2 logb d
logb cn 5 n logb c
logb (cd) n 5 n logb (cd)
5 n(log b c 1 log b d)
n
logb A dc B 5 n log b A dc B
5 n(log b c 2 log b d)
• To change the base of a logarithm to any base b:
log y
logb y 5 log b
ln y
logb y 5 ln b
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Summary of Formulas and Trigonometric Relationships 235
Trigonometry of the Right Triangle
In right ABC:
hyp
sin A 5 hyp
opp
csc A 5 sin1 A 5 opp
adj
sec A 5 cos1 A 5 adj
hyp
cos A 5 hyp
opp
adj
sin A
tan A 5 cos
A 5 adj
cot A 5 tan1 A 5 opp
Trigonometry of the Unit Circle
• If P(x, y) is a point on the unit circle and ROP is an angle in standard position with measure
u, then
x 5 cos u
y 5 sin u
• If P(x, y) is any point in the coordinate plane, then the formulas become
where r 5 "x2 1 y2.
x 5 r cos u
y 5 r sin u
Trigonometric Function Values of Special Angles
u (degrees)
0°
30°
45°
60°
90°
u (radians)
0
p
6
p
4
p
3
p
2
sin u
0
1
2
!2
2
!3
2
1
cos u
1
!3
2
!2
2
1
2
0
tan u
0
!3
3
1
!3
undefined
Reference Angles
If u is a second-, third-, or fourth-quadrant angle:
Second Quadrant
Reference Angle
Third Quadrant
180 2 u
u 2 180
Fourth Quadrant
360 2 u
sin u
sin (180 2 u)
2sin (u 2 180)
2sin (360 2 u)
cos u
–cos (180 2 u)
2cos (u 2 180)
cos (360 2 u)
tan u
–tan (180 2 u)
tan (u 2 180)
2tan (360 2 u)
Radian Measure
• To convert x in radians to degrees, multiply x by 180
p .
p
• To convert x in degrees to radians, multiply x by 180
.
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236 Summary of Formulas and Trigonometric Relationships
Trigonometric Functions
For y 5 a sin b(x 1 c) or y 5 a cos b(x 1 c):
• The amplitude, a, is the maximum value of the function and 2a is the minimum value.
• b is the number of cycles in the 2p interval (also called the angular frequency), the period 2p
is
b
|b|
the length of the interval for one cycle, and the frequency 2p
is the reciprocal of the period.
• The phase shift is 2c. If c is positive, the graph is shifted c units to the left. If c is negative, the
graph is shifted c units to the right.
Function
Domain (n is an integer)
Range
Sine
All real numbers
[21, 1]
Cosine
All real numbers
[21, 1]
p
2
1 np
Tangent
All real numbers except
Cotangent
All real numbers except np
p
2
All real numbers
1 np
Secant
All real numbers except
Cosecant
All real numbers except np
All real numbers
(2`, 21] < [1, `)
(2`, 21] < [1, `)
Sine Function with
a Restricted Domain
Inverse Sine Function
y 5 sin x
y 5 arcsin x or y 5 sin21 x
Domain 5 U x : 2p2 # x # p2 V
Range 5 {y : 21 # y # 1}
Domain 5 {x : 21 # x # 1}
Cosine Function with
a Restricted Domain
Inverse Cosine Function
y 5 cos x
y 5 arccos x or y 5 cos21 x
Domain 5 {x : 0 # x # p}
Domain 5 {x : 21 # x # 1}
Range 5 {y : 21 # y # 1}
Range 5 {y : 0 # y # p}
Tangent Function with
a Restricted Domain
Inverse Tangent Function
y 5 tan x
y 5 arctan x or y 5 tan21 x
Domain 5 Ux : 2p2 , x , p2 V
Range 5 {y : y is a real number}
Domain 5 {x : x is a real number}
Range 5 U y : 2p2 # y # p2 V
Range 5 Uy : 2p2 , y , p2 V
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Summary of Formulas and Trigonometric Relationships 237
Trigonometric Identities
Cofunction Identities
cos u 5 sin (90° 2 u)
sin u 5 cos (90° 2 u)
tan u 5 cot (90° 2 u)
cot u 5 tan (90° 2 u)
sec u 5 csc (90° 2 u)
csc u 5 sec (90° 2 u)
Reciprocal Identities
Quotient Identities
Pythagorean Identities
sec u 5 cos1 u
sin u
tan u 5 cos
u
cos2 u 1 sin2 u 5 1
csc u 5 sin1 u
u
cot u 5 cos
sin u
1 1 tan2 u 5 sec2 u
cot u 5 tan1 u
Sums of Angle Measures
cos (A 1 B) 5 cos A cos B 2 sin A sin B
cot2 u 1 1 5 csc2 u
Differences of Angle Measures
cos (A – B) 5 cos A cos B 1 sin A sin B
sin (A 1 B) 5 sin A cos B 1 cos A sin B
sin (A – B) 5 sin A cos B 2 cos A sin B
A 1 tan B
tan (A + B) 5 1tan
2 tan A tan B
A 2 tan B
tan (A – B) 5 1tan
1 tan A tan B
Double-Angle Identities
sin (2A) 5 2 sin A cos A
cos (2A) 5 cos2 A 2 sin2 A
cos (2A) 5 2 cos2 A 2 1
cos (2A) 5 1 2 2 sin2 A
Half-Angle Identities
sin 12A 5 6#1 2 2cos A
cos 12A 5 6#1 1 2cos A
2 cos A
tan 12A 5 6#11 1
cos A
A
tan (2A) 5 1 22tan
tan2 A
Trigonometric Relationships
• The Law of Cosines gives a relationship between an angle, the opposite side, and the two adjacent sides in ABC:
a2 5 b2 1 c2 2 2bc cos A
2
c2 2 a2
cos A 5 b 1 2bc
b2 5 a2 1 c2 2 2ac cos B
2
c2 2 b2
cos B 5 a 1 2ac
c2 5 a2 1 b2 2 2ab cos C
2
b2 2 c2
cos C 5 a 1 2ab
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238 Summary of Formulas and Trigonometric Relationships
• The Law of Sines gives a relationship between all the sides and angles of ABC:
a
sin A
5 sinb B 5 sinc C
• Given an angle and two adjacent sides of a triangle, the area of ABC can be calculated using
the following formula:
Area of ABC 5 12bc sin A 5 12ac sin B 5 12ab sin C
• In ABC, to determine the number of possible triangles given a, b, and mA (ambiguous
case), let h 5 b sin A:
C
C
C
C
b
h
b
A
A
A is:
and:
Possible
triangles:
b
ha
B
b
a h a
B
A
B9
C
C
a
B
h
a
A
h
B
a
b
B
a
h b
A
A
B
Acute
a,h
Acute
h5a
Acute
h,a,b
Acute
a.b
Obtuse
a#b
Obtuse
a.b
None
One,
right Two
One
None
One
Statistics Formulas
1
x , of a set of N numbers is equal to the sum of the numbers divided by N or N
• The mean, 2
a x.
• The median of a set of N numbers arranged in numerical order is the middle number.
a. If N is odd, the median is the number in the N 21 1 position.
N
b. If N is even, the median is the average of the numbers in the N
2 and 2 1 1 positions.
• The mode is the number that occurs most often in a set of data.
a. If two or more numbers occur more often than all other data values and these two or more
numbers have the same frequency, then each of these numbers is a mode.
b. If each data value occurs with the same frequency, then the data set has no mode.
• Quartiles are determined when a set of data is arranged in ascending numerical order, usually
from left to right.
a. The second quartile, Q2, is the median of the data, separating the data into two halves.
b. The first quartile, Q1, is the median of the lower half of the data, not counting the median.
c. The third quartile, Q3, is the median of the upper half of the data, not counting the median.
• The range is the difference between the highest and lowest numbers in a data set.
• The interquartile range is the difference between the upper and lower quartiles.
• The five statistical summary is the minimum, first quartile, median, third quartile, and maximum
of a data set.
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Summary of Formulas and Trigonometric Relationships 239
• An outlier is any data value that is more than third quartile 1 1.5 3 (interquartile range) or
less than first quartile 2 1.5 3 (interquartile range).
x) 2.
• The population variance, s2, of a set of data is n1 a (x 2 2
1
22
• The sample variance, s2, of a set of data is n 2
1 a (x 2 x) .
• The population standard deviation, s, is the square root of the population variance or
1
x) 2.
#n a (x 2 2
• The sample standard deviation, s, is the square root of the sample variance or
1
x) 2.
#n 2 1 a (x 2 2
Statistics Formulas for Grouped Data
If a set of data is grouped in terms of the frequency f of a given value x:
xf
x , is a f .
• The mean, 2
a
f(x 2 2
x )2
• The population variance, s2, is a
.
• The sample variance, s2, is
af
2
2
a f(x 2 x )
.
A afB 2 1
• The population standard deviation, s, is
• The sample standard deviation, s, is
É
2
a f(x 2 x )
.
f
a
É
22
a f(x 2 x ) .
A afB 2 1
Probability Formulas
• If A and B are independent events, then the probability of both occurring is P(A) 3 P(B) .
• In permutations (or arrangements), order is important.
The permutation of n things, taken n at a time is
nPn
5 n! 5 n(n 2 1)(n 2 2) c3 ? 2 ? 1
The permutation of n things, taken r at a time, where r # n is
n!
n(n 2 1)(n 2 2) c or nPr 5 (n 2
nPr 5 8
r)!
r factors
The permutation of n things where there are a1, a2, c, ar repetitions is
n!
a1! ? a2! ? c ? ar!
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240 Summary of Formulas and Trigonometric Relationships
• In combinations (or sets), order is not important.
The combination of n things, taken n at a time, where r # n is
nCr
nCn
P
5 nr!r or nCr 5 (n 2n!r)!r!
5 1 and nC0 5 1.
For whole numbers n and r where r # n: nCr 5 nCn – r .
• In a Bernoulli experiment or binomial experiment, if the probability of success is p and the
probability of failure is q 5 1 2 p, then the probability of r successes in n trials is
r n2r
nCrp q
r
The probability of at most r successes in n trials is a nCipiqn2i.
i50
n
The probability of at least r successes in n trials is a nCipiqn2i.
i5r
Normal and Binomial Distributions
• For a normal distribution, the following relationships exist.
The mean and the median of the data values lie on
the line of symmetry of the curve.
Approximately 68% of the data values lie within
one standard deviation from the mean.
Approximately 95% of the data values lie within
two standard deviations from the mean.
Approximately 99.7% of the data values lie within
three standard deviations from the mean.
99.7% of data values
95% of data values
68% of
data values
13.5% 34% 34% 13.5%
x–23s x–22s x–2s x– x–1s x–12s x–13s
Normal distribution
• If x is a data value of a normal distribution, its z-score is
x 2 2
x
x 2 2
x
z-score 5 standard
s
deviation 5
• A binomial distribution of n trials with probability of success of p can be approximated by a
normal distribution with mean np and standard deviation !np(1 2 p) .
• When n is a positive integer, (x 1 y)n can be expressed as a polynomial called the binomial
expansion:
n
(x 1 y)n 5 a nCixn 2 iyi
i50
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Summary of Formulas and Trigonometric Relationships 241
Linear Regression
• The regression line is appropriate only for data that appears to be linearly related.
• A high correlation coefficient, r, does not necessarily mean that one variable causes the other.
• When r 5 1 or 21, there is a perfect linear relationship between the data values.
• When r 5 0, no linear relationship exists between the data values.
• When r is close to 1, the data has a strong linear relationship. Values between 0 and 1 indicate
various degrees of moderate correlation.
• The sign of r matches the sign of the slope of the regression line.
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Table of Measures
Length
English System
1 foot (ft) 12 inches (in.)
1 yard (yd) 3 feet
1 mile (mi) 1,760 yards
1 mile (mi) 5,280 feet
Metric System
1 centimeter (cm) 10 millimeters (mm)
1 decimeter (dm) 10 centimeters
1 meter (m) 10 decimeters
1 meter 100 centimeters
1 kilometer (km) 1,000 meters
Weight (Mass)
English System
1 pound (lb) 16 ounces (oz)
1 ton (T) 2,000 pounds
Metric System
1 gram (g) 1,000 milligrams (mg)
1 kilogram (kg) 1,000 grams
1 metric ton (MT) 1,000 kilograms
Liquid Measure
English System
1 pint (pt) 16 fluid ounces (fl oz)
1 quart (qt) 2 pints
1 gallon (gal) 4 quarts
Metric System
1 liter (L) 1,000 milliliters (mL)
1 kiloliter (kL) 1,000 liters
1 cubic centimeter (cm3) 1 milliliter
1 liter 1,000 cubic centimeters
Time
English System
1 minute (min) 60 seconds (sec)
1 hour (hr) 60 minutes
1 day 24 hours
1 week 7 days
1 year 52 weeks
1 year 12 months
Metric System
Same as the English system
Conversion factors from the English system to the metric system:
Length
1 inch 2.54 centimeters
1 foot 0.305 meter
1 yard 0.914 meter
1 mile 1.61 kilometers
Holistic Scoring Rubric
242
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Holistic Scoring Rubric
To be used with Cumulative Reviews (Chapters 2–16)
Part II (2 credits)
Score
General Description
2 Credits
Complete and correct
•
•
•
•
•
Demonstrates thorough understanding of the concepts involved.
Employs appropriate strategies.
Performs all calculations correctly.
Graphs and diagrams are accurate and appropriate.
Deductive arguments are used to justify conclusions.
Response Criteria
1 Credit
Partially correct
•
•
•
•
•
Demonstrates limited understanding of the concepts involved.
Employs appropriate strategies.
Performs most calculations correctly.
Graphs and diagrams are accurate and appropriate.
Arguments are constructed with some mathematical basis.
0 Credits
Completely incorrect
•
•
•
•
•
Demonstrates insufficient understanding of the concepts involved.
Employs inappropriate strategies.
Performs most calculations incorrectly.
Graphs and diagrams are neither accurate nor appropriate.
Arguments are constructed with no mathematical basis.
Part III (4 credits)
Score
General Description
Response Criteria
4 Credits
Complete and correct
•
•
•
•
•
3 Credits
Fundamentally correct
• Demonstrates satisfactory understanding of the concepts involved.
• Employs appropriate strategies.
• Performs most calculations correctly; flaws reflect inattentive
execution of mathematical procedures.
• Graphs and diagrams are accurate and appropriate.
• Arguments are constructed with some mathematical basis; flaws
reflect inattentive execution of written arguments.
2 Credits
Partially correct
•
•
•
•
•
Demonstrates adequate understanding of the concepts involved.
Employs appropriate strategies.
Performs most calculations correctly.
Graphs and diagrams are accurate and appropriate.
Arguments are constructed with some mathematical basis.
1 Credit
Incomplete and flawed,
but not completely
incorrect
•
•
•
•
•
Demonstrates limited understanding of the concepts involved.
Employs inappropriate strategies.
Performs most calculations correctly.
Some graphs and diagrams are accurate or appropriate.
Some correct reasoning or justification is used.
0 Credits
Completely incorrect
•
•
•
•
•
Demonstrates insufficient understanding of the concepts involved.
Employs inappropriate strategies.
Performs calculations with major errors.
Graphs and diagrams are neither accurate nor appropriate.
Arguments are constructed with no mathematical basis.
Demonstrates thorough understanding of the concepts involved.
Employs appropriate strategies.
Performs all calculations correctly.
Graphs and diagrams are accurate and appropriate.
Deductive arguments are used to justify conclusions.
243
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244 Summary of Formulas and Trigonometric Relationships
Part IV (6 credits)
Score
General Description
6 Credits
Complete and correct
•
•
•
•
•
Response Criteria
5 Credits
Essentially correct
• Demonstrates satisfactory understanding of the concepts involved.
• Employs appropriate strategies.
• Performs most calculations correctly; flaws reflect inattentive
execution of mathematical procedures.
• Graphs and diagrams are accurate and appropriate.
• Deductive arguments are used to justify conclusions.
4 Credits
Fundamentally correct
• Demonstrates satisfactory understanding of the concepts involved.
• Employs appropriate strategies.
• Performs most calculations correctly; flaws reflect inattentive
execution of mathematical procedures.
• Graphs and diagrams are accurate and appropriate.
• Arguments are constructed with some mathematical basis; flaws
reflect inattentive execution of written arguments.
3 Credits
Partially correct
•
•
•
•
•
Demonstrates adequate understanding of the concepts involved.
Employs appropriate strategies.
Performs most calculations correctly.
Graphs and diagrams are accurate and appropriate.
Arguments are constructed with some mathematical basis.
2 Credits
Partially incorrect
•
•
•
•
•
Demonstrates limited understanding of the concepts involved.
Employs appropriate strategies.
Performs most calculations correctly.
Graphs and diagrams are accurate and appropriate.
Arguments are constructed with some mathematical basis.
1 Credit
Incomplete and flawed,
but not completely
incorrect
•
•
•
•
•
Demonstrates limited understanding of the concepts involved.
Employs inappropriate strategies.
Performs most calculations correctly.
Some graphs and diagrams are accurate or appropriate.
Some correct reasoning or justification is used.
0 Credits
Completely incorrect
•
•
•
•
•
Demonstrates insufficient understanding of the concepts involved.
Employs inappropriate strategies.
Performs calculations with major errors.
Graphs and diagrams are neither accurate nor appropriate.
Arguments are constructed with no mathematical basis.
Demonstrates thorough understanding of the concepts involved.
Employs appropriate strategies.
Performs all calculations correctly.
Graphs and diagrams are accurate and appropriate.
Deductive arguments are used to justify conclusions.
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Conversion Chart
To be used with Cumulative Reviews (Chapters 2–16)
Chart for Converting Total Cumulative Review Raw Scores to
Scaled Cumulative Review Scores
Raw
Score
Scaled
Score
Raw
Score
Scaled
Score
Raw
Score
Scaled
Score
44
100
29
75
14
45
43
98
28
73
13
42
42
96
27
71
12
40
41
95
26
70
11
37
40
93
25
68
10
34
39
91
24
66
9
31
38
90
23
64
8
28
37
88
22
62
7
25
36
87
21
60
6
22
35
85
20
58
5
19
34
84
19
56
4
15
33
82
18
54
3
11
32
80
17
51
2
8
31
79
16
49
1
4
30
77
15
47
0
0
To determine the student’s cumulative review score, find the student’s total raw score in the column labeled “Raw Score” and then locate the scaled score that corresponds to that raw score. This
scaled score is analogous to the scaled score given for comparable work on the Regents examination.
A passing score is any scaled score of 65 or higher. Cumulative reviews that earn a scaled score
between 60 and 64 may be scored a second time using the Holistic Scoring Rubric to ensure that
responses to questions in parts II, III, and IV received all merited credits.
245