Pre Calculus Workbook

Study Notebook

Cover: Jason Reed/Photodisc/Getty Images

Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 ISBN: 978-0-07-893814-6 MHID: 0-07-893814-7 Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 079 18 17 16 15 14 13 12 11 10 09

ContentsChapter 1Before You Read ............................................. 1 Key Points ........................................................ 2 1-1 Functions .................................................... 3 1-2 Analyzing Graphs of Functions and Relations ............................................. 5 1-3 Continuity, End Behavior, and Limits ......... 7 1-4 Extrema and Average Rates of Change ....................................................... 9 1-5 Parent Functions and Transformations........................................ 11 1-6 Function Operations and Composition of Functions ......................... 13 1-7 Inverse Relations and Functions .................................................. 15 Tie It Together................................................ 17 Before the Test .............................................. 18

Chapter 4Before You Read ........................................... 49 Key Points ...................................................... 50 4-1 Right Triangle Trigonometry ..................... 51 4-2 Degrees and Radians............................... 53 4-3 Trigonometric Functions on the Unit Circle ........................................... 55 4-4 Graphing Sine and Cosine Functions .................................................. 57 4-5 Graphing Other Trigonometric Functions .................................................. 59 4-6 Inverse Trigonometric Functions .................................................. 61 4-7 The Law of Sines and the Law of Cosines ................................................ 63 Tie It Together................................................ 65 Before the Test .............................................. 66

Chapter 2Before You Read ........................................... 19 Key Points ...................................................... 20 2-1 Power and Radical Functions .................. 21 2-2 Polynomial Functions ............................... 23 2-3 The Remainder and Factor Theorems ................................................. 25 2-4 Zeros of Polynomial Functions ................. 27 2-5 Rational Functions .................................... 29 2-6 Nonlinear Inequalities ............................... 31 Tie It Together................................................ 33 Before the Test .............................................. 34

Chapter 5Before You Read ........................................... 67 Key Points ...................................................... 68 5-1 Trigonometric Identities ............................ 69 5-2 Verifying Trigonometric Identities ............. 71 5-3 Solving Trigonometric Equations.............. 73 5-4 Sum and Difference Identities .................. 75 5-5 Multiple-Angle and Product-to-Sum Identities ......................... 77 Tie It Together................................................ 79 Before the Test .............................................. 80

Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 3Before You Read ........................................... 35 Key Points ...................................................... 36 3-1 Exponential Functions .............................. 37 3-2 Logarithmic Functions .............................. 39 3-3 Properties of Logarithms .......................... 41 3-4 Exponential and Logarithmic Equations.................................................. 43 3-5 Modeling with Nonlinear Regression ............................................... 45 Tie It Together................................................ 47 Before the Test .............................................. 48

Chapter 6Before You Read ........................................... 81 Key Points ...................................................... 82 6-1 Multivariable Linear Systems and Row Operations................................. 83 6-2 Matrix Multiplication, Inverses, and Determinants ..................................... 85 6-3 Solving Linear Systems Using Inverses and Cramers Rule ..................... 87 6-4 Partial Fractions ....................................... 89 6-5 Linear Optimization .................................. 91 Tie It Together................................................ 93 Before the Test .............................................. 94

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Chapter 7Before You Read ........................................... 95 Key Points ...................................................... 96 7-1 Parabolas ................................................. 97 7-2 Ellipses and Circles .................................. 99 7-3 Hyperbolas ............................................. 101 7-4 Rotations of Conic Sections ................... 103 7-5 Parametric Equations ............................. 105 Tie It Together.............................................. 107 Before the Test ............................................ 108

Chapter 10Before You Read ......................................... 137 Key Points .................................................... 138 10-1 Sequences, Series, and Sigma Notation ..................................... 139 10-2 Arithmetic Sequences and Series ........ 141 10-3 Geometric Sequences and Series ....... 143 10-4 Mathematical Induction ........................ 145 10-5 The Binomial Theorem ......................... 147 10-6 Functions as Infinite Series .................. 149 Tie It Together.............................................. 151 Before the Test ............................................ 152

Chapter 8Before You Read ......................................... 109 Key Points .................................................... 110 8-1 Introduction to Vectors ........................... 111 8-2 Vectors in the Coordinate Plane ...................................................... 113 8-3 Dot Products and Vector Projections .............................................. 115 8-4 Vectors in Three-Dimensional Space ..................................................... 117 8-5 Dot and Cross Products of Vectors in Space .................................... 119 Tie It Together.............................................. 121 Before the Test ............................................ 122

Chapter 11Before You Read ......................................... 153 Key Points .................................................... 154 11-1 Descriptive Statistics ............................ 155 11-2 Probability Distributions ........................ 157 11-3 The Normal Distribution........................ 159 11-4 The Central Limit Theorem .................. 161 11-5 Confidence Intervals ............................. 163 11-6 Hypothesis Testing ............................... 165 11-7 Correlation and Linear Regression ...... 167 Tie It Together.............................................. 169 Before the Test ............................................ 170

Chapter 9Before You Read ......................................... 123 Key Points .................................................... 124 9-1 Polar Coordinates ................................... 125 9-2 Graphs of Polar Equations ..................... 127 9-3 Polar and Rectangular Forms of Equations ................................ 129 9-4 Polar Forms of Conic Sections............... 131 9-5 Complex Numbers and De Moivres Theorem ............................. 133 Tie It Together.............................................. 135 Before the Test ............................................ 136

Chapter 12Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Before You Read ......................................... 171 Key Points .................................................... 172 12-1 Estimating Limits Graphically ............... 173 12-2 Evaluating Limits Algebraically ............. 175 12-3 Tangent Lines and Velocity .................. 177 12-4 Derivatives ........................................... 179 12-5 Area Under a Curve and Integration .... 181 12-6 The Fundamental Theorem of Calculus ............................................ 183 Tie It Together.............................................. 185 Before the Test ............................................ 186

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Note-Taking TipsYour notes are a reminder of what you learned in class. Taking good notes can help you succeed in mathematics. The following tips will help you take better classroom notes. Before class, ask what your teacher will be discussing in class. Review mentally what you already know about the concept. Be an active listener. Focus on what your teacher is saying. Listen for important concepts. Pay attention to words, examples, and/or diagrams your teacher emphasizes. Write your notes as clear and concise as possible. The following symbols and abbreviations may be helpful in your note-taking. Word or Phrase for example such as with without and Symbol or Abbreviation e.g. i.e. w/ w/o + Word or Phrase not equal approximately therefore versus angle Symbol or Abbreviation vs

Use a symbol such as a star ( ) or an asterisk () to emphasis important concepts. Place a question mark (?) next to anything that you do not understand. Ask questions and participate in class discussion. Draw and label pictures or diagrams to help clarify a concept.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

When working out an example, write what you are doing to solve the problem next to each step. Be sure to use your own words. Review your notes as soon as possible after class. During this time, organize and summarize new concepts and clarify misunderstandings.

Note-Taking Donts Dont write every word. Concentrate on the main ideas and concepts. Dont use someone elses notes as they may not make sense. Dont doodle. It distracts you from listening actively. Dont lose focus or you will become lost in your note-taking.

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Functions from a Calculus PerspectiveBefore You ReadBefore you read the chapter, respond to the following statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read Functions from a Calculus Perspective For a given function f, a value in the domain is represented by the dependent variable x and a value in the range of f is represented by the independent variable y. The graph of a continuous function has no breaks, holes, or gaps.

A function f is increasing on an interval if and only if for any two points in the interval, a positive change in x results in a negative change in f (x). A translation is a rigid transformation that has the effect of shifting the graph of a function up, down, left, or right.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

The inverse relation of a function is not necessarily a function.

Note-Taking Tips In addition to writing important definitions in your notes, be sure to include your own examples of the concepts presented. For example, when studying functions, be sure to include relations that are functions and relations that are not functions in your notes. When you take notes, make sure that someone who did not understand the topic would understand after reading what you have written.

Chapter 1

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Functions from a Calculus PerspectiveKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on functions, one fact might be that the rational and irrational number subsets form the real number set. After completing the chapter, you can use this table to review for your chapter test. Lesson 1-1 Functions Fact

1-2

Analyzing Graphs of Functions and Relations

1-3

Continuity, End Behavior, and Limits


1-4

Extrema and Average Rates of Change

1-5

Parent Functions and Transformations

1-6

Function Operations and Composition of Functions

1-7

Inverse Relations and Functions

Chapter 1

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1-1

Functions

What Youll Learn

Scan Lesson 1-1. Write two things that you already know about functions. 1.

2.

Active Vocabularydomain

New Vocabulary Write the definition next to each term.

dependent variableCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

function

independent variable

range

relation

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Lesson 1-1

(continued)

Main IdeaDescribe Subsets of Real Numbers pp. 45

DetailsComplete the table. Write each set of numbers in set-builder and interval notation, if possible. Set-Builder Notation

Set {-2, -1, 0, 1, } x4 -5 x < 22 x < -4 or x > 6 all multiples of 7

Interval Notation


Identify Functions pp. 58

Determine whether each relation represents y as a function of x. Write yes or no. 1. 2. {(2, -4), (-3, 7), (23, -5), (-3, 10)} The input value x is a Social Security number and the output value y is the owners first name.y

1.

2. 3.

3.

4.

x 2 1 0 1 2

y 1 0 0 1 3

x

4.

5.Chapter 1

y = x 2 + 3x - 4

5.Glencoe Precalculus

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1-2

Analyzing Graphs of Functions and Relations

What Youll Learn

Scan the Examples for Lesson 1-2. Predict two things that you think you will learn about functions and their graphs. 1.

2.

Active Vocabularyeven functionCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Match the term with its definition by drawing a line to connect the two. the x-intercept(s) of the graph of a function

line of symmetry

functions that are symmetric with respect to the origin

odd function

graphs that have this property can be rotated 180 with respect to a point and appear unchanged

point symmetry

the solution(s) of a given equation

roots

functions that are symmetric with respect to the y-axis

zeros

graphs that have this property can be folded along a line so that the two halves of the graph match exactly

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Lesson 1-2 (continued)

Main IdeaAnalyzing Function Graphs pp. 1316

DetailsUse the graph of each function to find its y-intercept and zeros.y y

x

x

y-intercept: zeros: Symmetry of Graphs pp. 1618

y-intercept: zeros:

Identify which function is even, which is odd, and which is neither. 1. 2. 3. f(x) = x 3 - x g(x) = 2x 4 + x - 1 h(x) = -3x 2 + 1 1. 2. 3.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Helping You Remember

Think about the different types of line symmetry: about the x-axis, about the y-axis, and about the origin. Provide examples of graphs illustrating each of the symmetries. x-axis symmetryy

y-axis symmetryy

origin symmetryy

x

x

x

Chapter 1

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1-3

Continuity, End Behavior, and Limits

What Youll Learn

Scan the text in Lesson 1-3. Write two facts that you learned about continuity. 1.

2.

Active Vocabulary

New Vocabulary Write the correct term next to each definition. a function that has no breaks, holes, or gaps in its graph


a concept describing how a function behaves at either end of its graph a function is said to have this form of a discontinuity at x = c if the absolute value of the function increases or decreases indefinitely as the x-values approach c from the left and the right the concept of approaching a value without necessarily ever reaching it a function is said to have this form of discontinuity at x = c if the function is continuous everywhere except for a hole at x=c a function that is not continuous a function is said to have this form of discontinuity at x = c if the limits of the function as x approaches c from the left and the right exist but have two distinct valuesChapter 1

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Main IdeaContinuity pp. 2426

DetailsComplete the table by providing your own verbal description of each type of discontinuity. Then provide an example to illustrate your verbal description.

Discontinuity Infinite

Verbal Description

Example

Jump Discontinuity


Removable or Point

End Behavior pp. 2829

Use the graph of f (x) = x 3 - 2x 2 - 5x + 6 to describe its end behavior.y

x

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1-4

Extrema and Average Rates of Change

What Youll Learn

Scan the text under the Now heading. List two things that you will learn in this lesson. 1.

2.


Active Vocabularyconstant

New Vocabulary Label the diagram with the terms listed at the left.

decreasing

y

increasingx

maximum

2

minimum

secant line

Chapter 1

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Main IdeaIncreasing and Decreasing Behavior pp. 3438

DetailsDraw the graph of a function modeling the indicated behavior throughout its domain. Increasing Function Decreasing Function Constant Function

Average Rate of Change pp. 3839

Find the average rate of change of f(x) = x 4 - 3x 2 + 6x on the interval [-1, 2].


Slope formula Substitute -1 for x 1 and 2 for x 2.

Evaluate f(3) and f(-2). Simplify.

Chapter 1

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Parent Functions and Transformations

What Youll Learn

Scan Lesson 1-5. Predict two things that you expect to learn based on the headings and Key Concept boxes. 1.

2.

Active Vocabulary

New Vocabulary Fill in each blank with the correct term.

absolute value functionCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

A(n) is a rigid transformation that has the affect of shifting the graph of a function up, down, left, or right.

identity function

A(n) functions in a family.

is the simplest of the

parent function

A(n) is a rigid transformation which produces a mirror image of the graph of a function with respect to a specific line.

reflection

The points with coordinates (a, a).

f (x) = x passes through all

transformations

A(n) of a parent function affects the appearance of the parent graph.

translationChapter 1

The V-shaped function.

, denoted as f (x) = |x|, is aGlencoe Precalculus

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Main IdeaParent Functions pp. 4546

DetailsEach graph is the parent function for a family of functions. Identify the parent function.y y y

x

x

x

Transformations pp. 4651

Define the three different transformations introduced in this lesson. TranslationCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Reflection

Dilation

Chapter 1

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1-6

Function Operations and Composition of FunctionsScan Lesson 1-6. List two headings that you would use to make an outline of this lesson. 1.

What Youll Learn

2.

Active Vocabulary

Review Vocabulary Define function in your own words. (Lesson 1-1)

function


Define relation in your own words. (Lesson 1-1) relation

Define roots in your own words. (Lesson 1-2) roots

New Vocabulary Write the definition next to the term. composition

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Main IdeaOperations with Functions pp. 5758

DetailsGiven f (x) = 2x + 1 and g (x) = 4x 2 - 1, find each function and its domain. 1. (f + g)(x) = Domain: 2. (f g)(x) = Domain: 3. (f g)(x) = Domain: 4.f ()(x) = g

Domain:

Composition of Functions pp. 5860

Given f (x) = x + 1, g(x) = x 2 2, and h(x) = x 2 + 3x 4, find each composition. 1. 2. 3.


[ f g ](x) =g[ f (x) ] =

[ h f ](x) =


In the composition f g, which is read as f composition g or f of g, the function g is applied first then f. Think of a mnemonic device for remembering how to find the composition of two functions f and g so that you are not confused when you see f g or g f.

Chapter 1

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1-7

Inverse Relations and Functions

What Youll Learn

Scan the Examples for Lesson 1-7. Predict two things that you think you will learn about inverse relations. 1.

2.

Active Vocabulary

Review Vocabulary Define domain in your own words. (Lesson 1-1)

domain

Define range in your own words. (Lesson 1-1)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

range

New Vocabulary Fill in each blank with the correct term or phrase. inverse relations If a function passes the horizontal line test, then it is said to be , because no x-value is matched with more than one y-value and no y-value is matched with more than one x-value.

inverse function

Two relations are if and only if one relation contains the element (b, a) whenever the other relation contains the element (a, b).

one-to-one

If the inverse relation of a function f is also a function, then it is called the of f.

Chapter 1

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Main IdeaInverse Functions pp. 6566

DetailsGraph each function using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. 1. 2. 3. 4. 5. f (x) = x 3 + 13 g (x) = 2-x

h(x) = -2|x - 4| + 1 g (x) = f (x) = x 3 + x 2 - 3xx+3 x-2 x+2 x-4

Find Inverse Functions pp. 6669

Find the inverse of f (x) = .

Original function

Replace f (x) with y.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exchange x and y.

Solve for y. Replace y with f 1(x).


In Lesson 1-6, you learned how to find the composition of two functions. Explain what role the composition of functions plays in determining whether two functions are inverses of one another.

Chapter 1

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Functions from a Calculus PerspectiveTie It TogetherUse the graph of f (x) to complete the graphic organizer.y

x

domain: range:

y-intercept(s): zeros:

Intervals on which f (x) is:Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

End Behavior constant left end right end

increasing

decreasing

Determine whether f(x) has each of the following characteristics. Write yes or no. If yes, state the x-values where they occur.

Point(s) of Discontinuity infinite jump point

Extrema relative maximum absolute maximum

relative minimum

absolute minimum

Chapter 1

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Functions from a Calculus PerspectiveBefore the TestNow that you have read and worked through the chapter, think about what you have learned and complete the table below. Compare your previous answers with these. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Functions from a Calculus Perspective For a given function f, a value in the domain is represented by the dependent variable x and a value in the range of f is represented by the independent variable y. The graph of a continuous function has no breaks, holes, or gaps. A function f is increasing on an interval if and only if for any two points in the interval, a positive change in x results in a negative change in f (x). A translation is a rigid transformation that has the effect of shifting the graph of a function up, down, left, or right.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

After You Read

The inverse relation of a function is not necessarily a function.

Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes, personal tutors, and practice tests to help you study for concepts in Chapter 1.

Are You Ready for the Chapter Test?Use this checklist to help you study. I completed the Chapter 1 Study Guide and Review in the textbook. I took the Chapter 1 Practice Test in the textbook. I used the online resources for additional review options. I reviewed my homework assignments and made corrections to incorrect answers. I reviewed all vocabulary terms from the chapter.

Study Tip When studying for tests, create and use graphic organizers to show relationships between concepts.Chapter 1

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Power, Polynomial, and Rational FunctionsBefore You ReadBefore you read the chapter, respond to these statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read Power, Polynomial, and Rational Functions When solving a radical equation, you should check for extraneous solutions.

The graph of a polynomial function may contain breaks, holes, gaps, or sharp corners.

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form x - c.


A polynomial function of degree n, where n > 0 has at least one zero in the real number system.

If the degree of the numerator of a rational function is the same as the degree of the denominator, then the graph will have no horizontal asymptote.

Note-Taking Tips In addition to writing definitions in your notes, be sure to include examples and graphs to display the concepts presented. For example, when studying rational functions be sure to include notes on how to find the vertical asymptote(s), horizontal asymptote, intercepts, and holes. Take notes that are specific enough that you will understand what they mean when you read them a few days later.

Chapter 2

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Power, Polynomial, and Rational FunctionsKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on power and radical functions, one fact might be to describe the shape of the graph of an even-degree and an odd-degree radical function. After completing the chapter, you can use this table to review for your chapter test. Lesson 2-1 Power and Radical Functions Fact

2-2 Polynomial Functions

2-3 The Remainder and Factor TheoremsCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

2-4 Zeros of Polynomial Functions

2-5 Rational Functions

2-6 Nonlinear Inequalities

Chapter 2

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2-1

Power and Radical Functions

What Youll Learn

Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1.

2.

Active Vocabularyextraneous solution

New Vocabulary Fill in the blank with the correct term. A is any function of the form f(x) = axn where a and n are nonzero constant real numbers.


monomial function

A positive integer.

is a power function in which n is a

power function

A is a function that has at least one radical expression containing the independent variable.

radical function

Solutions that do not satisfy the original equation are called .

Review Vocabulary Define function in your own words. (Lesson 1-1) function

Chapter 2

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Lesson 2-1

(continued)

Main IdeaPower Functions pp. 8689

DetailsIdentify the power functions in this list by placing each function in the correct box. f(x) = 2x 31 f(x) = x

f(x) = 3 x f(x) = -5x 32

f(x) = 4x - 1

1 f(x) = 5-x

Power Functions

Not Power Functions

Radical Functions pp. 8991

Find the _______ range, and intercepts of domain, 3 8 - 3x . f(x) = 2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

domain: range:

x-intercept: y-intercept:

Helping You Remember np

How can you remember the domain of f(x) = x where n is even or n is odd? Sketch a graph of each.

n is even.y

n is odd.y

x

x

Chapter 2

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2-2

Polynomial Functions

What Youll Learn

Scan the Examples for Lesson 2-2. Predict two things that you think you will learn about polynomial functions. 1.

2.

Active Vocabularypolynomial functionCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Match the term with its definition by drawing a line to connect the two. the coefficient of the variable with the greatest exponent

leading coefficient

the zero of a factor (x - c) that occurs more than once in the completely factored form of f(x)

turning point

the number of times a zero is repeated

repeated zero

functions formed from the sums and differences of monomial functions and constants

multiplicity

points that indicate where the graph of the function changes from increasing to decreasing, and vice versa

Chapter 2

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Lesson 2-2

(continued)

Main IdeaGraph Polynomial Functions pp. 97102

DetailsComplete the chart for the Leading Term Test for Polynomial End Behavior.

n is odd, an is positive. lim f(x) =x -

n is odd, an is negative. lim f(x) =x -

lim f(x) =x

lim f(x) =x

n is even, an is positive. lim f(x) =x -

n is even, an is negative. lim f(x) =x -

lim f(x) =x

lim f(x) =x

Model Data p. 103

Write a polynomial function to model the number of DVD players sold during February in the United States as a function of the number of years t since 2000 as shown in the chart.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Years Since 2000 Number of DVD Players

4 919,295

5 590,128

6 770,132

7 1,117,899


Look up the words quartet, quarter, and quarto. How can the meaning of these words help you remember the meaning of quartic function?

Chapter 2

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2-3

The Remainder and Factor Theorems

What Youll Learn

Scan the lesson. Write two things that you already know about solving quadratic equations. 1.

2.

Active Vocabulary

Review Vocabulary Define domain in your own words. (Lesson 1-1)

domainCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Define range in your own words. (Lesson 1-1) range

New Vocabulary Write the correct term next to each definition. a shortcut for dividing a polynomial by a linear factor of the form (x - c) the process of using synthetic division to evaluate a function the quotient when a polynomial is divided by one of its binomial factors (x - c)

Chapter 2

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Lesson 2-3

(continued)

Main IdeaDivide Polynomials pp. 109112

DetailsFill in each box with the division indicated.

Long Division x + 3 3x3 + 6x2 + x 2

Synthetic Division 3 3 6 1 -2

The Remainder and Factor Theorems pp. 112114

Determine if each expression is a factor of f(x) = x3 - 2x2 - 13x - 10. Write yes or no. 1. 2. 3. 4. 5. 6. x-2 x+1 x+3 x-5 x+2 x-1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.


Suppose the remainder when f(x) is divided by (x - c) is 0. Explain what this tells you about factors, zeros of the function, and x-intercepts of the graph.

Chapter 2

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2-4

Zeros of Polynomial Functions

What Youll Learn

Scan the text in Lesson 2-4. Write two facts that you learned about zeros of polynomial functions. 1.

2.

Active VocabularyDescartes Rule of Signs



Rational Zero Theorem

Chapter 2

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Lesson 2-4

Fundamental Theorem of Algebra

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Main IdeaReal Zeros pp. 119123

DetailsUse the Rational Zero Theorem, the Upper and Lower Bound Tests, or Descartes Rule of Signs to find the zeros of f(x) = x3 - 7x - 6. Find all possible rational zeros. Determine an interval within which all of the real zeros are located. Determine the number of positive and negative real zeros.

Find the real zeros.


Complex Zeros pp. 123126

Find the minimum degree of the polynomial function with real coefficients that have the given zeros. 1. 2. 3. 2, -71 6, -6, 3

3 + 7 , 2 - 8i

Chapter 2

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Rational Functions

What Youll Learn

Scan the text under the Now heading. List two things that you will learn in the lesson. 1.

2.

Active Vocabularyhole

New Vocabulary Label the diagram with the correct terms.y


horizontal asymptotex

rational function

vertical asymptote

Vocabulary Link A synonym for extraneous is irrelevant. Explain how this relates to extraneous solutions. extraneous

Chapter 2

29

Glencoe Precalculus

Lesson 2-5

NAME

DATE

PERIOD

Lesson 2-5

(continued)

Main IdeaRational Functions pp. 130136

DetailsExplain how to find each characteristic for a rational function. Characteristic hole(s) Method

vertical asymptote(s) horizontal asymptote


oblique asymptote

Rational Equations pp. 136137

x 2 2 Solve = + . Write the reason for 2 x - 3x - 4 x-4 x+1

each step in the solution.2 (x 4)(x + 1) 2 x 2 ( x - 3x - 4 ) = (x 4)(x + 1)( + ) x-4 x+1

2 = x(x + 1) + 2(x 4) 2 = x2 + 3x 8 0 = x2 + 3x 10 0 = (x + 5)(x 2) x = 5 or x = 2

Chapter 2

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Glencoe Precalculus

NAME

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2-6

Nonlinear Inequalities

What Youll Learn

Scan Lesson 2-6. List two headings that you would use to make an outline of this lesson. 1.

2.

Active Vocabularycritical number

New Vocabulary Fill in each blank with the correct term. A number line showing the test intervals for solving an inequality is called a .

polynomial inequality

The numbers used to make a sign chart are called .


rational inequality

has the general form f(x) 0, A f(x) < 0, f(x) 0, f(x) > 0, or f(x) 0.

sign chart

A a rational function.

is an inequality formed using

Review Vocabulary Define polynomial function in your own words. (Lesson 2-2) polynomial function

Chapter 2

31

Glencoe Precalculus

Lesson 2-6

NAME

DATE

PERIOD

Lesson 2-6

(continued)

Main IdeaPolynomial Inequalities pp. 141143

DetailsComplete the chart to solve x3 + 3x2 - 6x - 8 > 0. Let f(x) = x3 + 3x2 - 6x - 8. Factor f(x) and find all of the real zeros. Determine the end behavior of f(x).

Complete the sign chart.

Write the solutions.

Solve Rational Inequalities pp. 143144

4 1 Solve < . Complete the steps shown. x+5 2x + 3 1 Subtract from each (2x + 3) side.


Use the LCD to rewrite each fraction. Then add.

Find the critical numbers.

Complete the sign chart using these numbers.

Solution

Chapter 2

32

Glencoe Precalculus

NAME

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Power, Polynomial, and Rational FunctionsTie It TogetherComplete the graphic organizer by providing examples of each type of function. Types of Functions

Polynomial Function

Rational Function

Power Function

Radical Function

Complete the organizer by providing an example of each inequality and its solution. Nonlinear Inequalities

Polynomial Inequality

Rational Inequality


Explain how each theorem or rule is used. Theorem / Rule Rational Zero Theorem How it is used

Descartes Rule of Signs

Factor Theorem

Remainder Theorem

Chapter 2

33

Glancoe Precalculus

NAME

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Power, Polynomial, and Rational FunctionsBefore the TestNow that you have read and worked through the chapter, think about what you have learned. Complete the table below. Compare your previous answers with these. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Power, Polynomial, and Rational Functions When solving a radical equation, you should check for extraneous solutions. The graph of a polynomial function may contain breaks, holes, gaps, or sharp corners. Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form x - c. A polynomial function of degree n, where n > 0 has at least one zero in the real number system. If the degree of the numerator of a rational function is the same as the degree of the denominator, then the graph will have no horizontal asymptote. Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes, personal tutors, and practice tests to help you study for concepts in Chapter 2.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

After You Read


Study Tip Review your homework assignments. Redo any problems that you missed when they were first assigned.Chapter 2

34

Glancoe Precalculus

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Exponential and Logarithmic FunctionsBefore You ReadBefore you read the chapter, respond to these statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read Exponential and Logarithmic Functions The function f(x) = x 3 is an exponential function.

A natural logarithm is a logarithm with base e.

log b x + log b y = log b (x + y)

36 x + 1 = 6 (x + 1)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

2

Data exhibiting rapid growth or decay can be modeled using a logarithmic function.

Note-Taking Tips Before each lesson, scan through the lesson. Write any questions that you think of in your notes. As you study the lesson, record the answers to your questions. Always write clear, concise notes so they can be easily read when studying for a quiz or test.

Chapter 3

35

Glencoe Precalculus

NAME

DATE

PERIOD

Exponential and Logarithmic FunctionsKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on logarithmic functions, one fact might be that a logarithmic function is the inverse of an exponential function. After completing the chapter, you can use this table to review for your chapter test. Lesson 3-1 Exponential Functions Fact

3-2

Logarithmic Functions

3-3

Properties of Logarithms


3-4

Exponential and Logarithmic Equations

3-5

Modeling with Nonlinear Regression

Chapter 3

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Glencoe Precalculus

NAME

DATE

PERIOD

3-1

Exponential Functions

What Youll Learn

Scan Lesson 3-1. Predict two things that you expect to learn based on the headings and figures in the lesson. 1.

2.

Active Vocabularyalgebraic function



continuous compound interest

exponential function

natural base

transcendental function

Chapter 3

37

Glencoe Precalculus

Lesson 3-1

NAME

DATE

PERIOD

Lesson 3-1

(continued)

Main IdeaExponential Functions pp. 158161 Complete the chart.

Details

Exponential Function domain range x-intercept y-intercept asymptote

Growth

Decay

end behavior


Exponential Growth and Decay pp. 161165

Complete the steps in the solution of the problem below. Beth invests $500 at a 4% interest rate that is compounded quarterly. If she makes no other deposits or withdrawals, what will her balance be after 10 years? Compound interest formula Substitute. Simplify. After 10 years, her balance will be .

Chapter 3

38

Glencoe Precalculus

NAME

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3-2

Logarithmic Functions

What Youll Learn

Scan the Examples for Lesson 3-2. Predict two things that you think you will learn about logarithmic functions. 1.

2.

Active Vocabularybase

New Vocabulary Label the diagram with the correct terms.

f(x) = log b x }

common logarithmCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

logarithm

log 10 x }

logarithmic function with base b

ln x}

natural logarithm

log 2 8 = 3}

exponent

Chapter 3

39

Glencoe Precalculus

Lesson 3-2

NAME

DATE

PERIOD

Lesson 3-2

(continued)

Main IdeaLogarithmic Functions and Expressions pp. 172174

DetailsEvaluate each expression. 1. log 4 161 3. log 2 8

2. ln e 5 4. log 8 81 6. ln 4

5. log 6 6

3

__

(e )

Graphs of Logarithmic Functions pp. 175177

Use the graph of f(x) = log x to describe the transformation that results in each function. Then sketch the graphs of the functions. 1. g(x) = log (x - 2) 2. h(x) = 2 log x + 1



Write an example that can help you remember how to change an equation from logarithmic form to exponential form.

Chapter 3

40

Glencoe Precalculus

NAME

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3-3


What Youll Learn

Scan the text in Lesson 3-3. Write two facts that you learned about properties of logarithms as you scanned the text. 1.

2.

Active Vocabulary

Review Vocabulary Use each term in a sentence that shows its mathematical meaning.

exponential functionCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

natural logarithm

transcendental function

common logarithm

Chapter 3

41

Glencoe Precalculus

NAME

DATE

PERIOD

Lesson 3-3

(continued)

Main IdeaProperties of Logarithms pp. 181-183

DetailsFill in the table with an example of each property. Property Product Property Example

Quotient Property

Power Property

Change of Base Formula pp. 183184

Apply the change of base formula to evaluate each logarithm. Round answers to the nearest hundredth. 1. 2. 3. 4. log4 6Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

log3 96 log 2 14

log51 26


To help you remember that the log of a product is equal to the sum of the logs, relate this fact to a property of exponents. Give an example of each relationship.

Chapter 3

42

Glencoe Precalculus

NAME

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3-4

Exponential and Logarithmic Equations

What Youll Learn

Scan Lesson 3-4. List two that headings that you would use to make an outline of this lesson. 1.

2.

Active Vocabulary

Review Vocabulary List the three properties of exponents that you learned in Lesson 3-3. Provide an example of each property.



Vocabulary Link When you solved radical equations, you needed to watch for extraneous solutions that would cause the expression under the radical to be negative. What extraneous solutions might occur when solving a logarithmic equation?

Chapter 3

43

Glencoe Precalculus

Lesson 3-4

NAME

DATE

PERIOD

Lesson 3-4

(continued)

Main IdeaOne-to-One Property of Exponential Functions pp. 190191 Solve each equation. 1. 2. 3. 4. One-to-One Property of Logarithmic Functions pp. 191195 4 x = 16 x - 21 () 3a

Details

= 27 43

___

log 5 25 = x1 1 ln = x 2

( )

Complete the steps to solve log2 10 = log2 x + log2 (x - 3). Original equation Product Property Distributive Property One-to-One PropertyCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Subtract. Factor. Solve. Check solutions.


Many students have trouble recognizing when they can solve an exponential equation using the One-to-One Property and when they must use logarithms. Explain how they could remember this.

Chapter 3

44

Glencoe Precalculus

NAME

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3-5

Modeling with Nonlinear Regression

What Youll Learn

Scan the lesson. Write two things that you already know about modeling data. 1.

2.

Active Vocabulary

Review Vocabulary Exponential Function Define exponential function in your own words. (Lesson 3-1)

exponential functionCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Define logarithmic function in your own words. (Lesson 3-2) logarithmic function

New Vocabulary Fill in the blank with the correct term. linearize models growth that was A initially exponential but that slows down and levels out, approaching a horizontal asymptote. To linearize data would be to transform the data so that it appears to cluster about a line.

logistic growth function

Chapter 3

45

Glencoe Precalculus

Lesson 3-5

NAME

DATE

PERIOD


Main IdeaExponential, Logarithmic, and Logistic Modeling pp. 200204

DetailsSketch a graph for each model. Quadratic Regression

Linear Regression

Power Regression

Exponential Regression

Logarithmic Regression

Logistic Regression


Linearizing Data pp. 204206

Linearize the data according to the power model and write an equation of a line of best fit. Then use this linear model for the transformed data to find a model for the original data. x y 1 2 2 16 3 54 4 128 5 250 6 432

Chapter 3

46

Glencoe Precalculus

NAME

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PERIOD

Exponential and Logarithmic FunctionsTie It TogetherComplete the first and third columns with one or more details. Complete the middle column with a description of the relationship between the rectangular boxes. Exponential Function Logarithmic Function

Shape of Graph

Shape of Graph


Solving Exponential Equations

Solving Logarithmic Equations

f(x) = e x

f(x) = ln x

Chapter 3

47

Glencoe Precalculus

NAME

DATE

PERIOD

Exponential and Logarithmic FunctionsBefore the TestNow that you have read and worked through the chapter, think about what you have learned and complete the table below. Compare your previous answers with these. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Exponential and Logarithmic Functions The function f(x) = x 3 is an exponential function. After You Read

A natural logarithm is a logarithm with base e.

log b x + log b y = log b (x + y)

36 x + 1 = 6 (x + 1)

2

Data exhibiting rapid growth or decay can be modeled using a logarithmic function.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.



Study Tips Review information every day to keep it fresh in your mind and to help reduce the amount of studying before test day. Look over your notes, and review your corrected homework. If you have any questions about any of the concepts, ask your teacher before the day of the test.Chapter 3

48

Glencoe Precalculus

NAME

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Trigonometric FunctionsBefore You ReadBefore you read the chapter, think about what you know about trigonometric functions. List three things that you already know about them in the first column. Then list three things that you would like to learn about them in the second column. K What I know W What I want to find out


Note-Taking Tips Do not write every word. Concentrate on the main ideas and concepts. Draw and label diagrams that correspond to the main ideas and concepts in your notes. Always write clear and concise notes so that they can be easily read when studying for a quiz or exam. Read your notes later in the day and rewrite any parts that are confusing.Chapter 4

49

Glencoe Precalculus

NAME

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Trigonometric FunctionsKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on right triangle trigonometry, one fact might be that the cosecant, secant, and cotangent functions are reciprocal functions of the sine, cosine, and tangent functions, respectively. After completing the chapter, you can use this table to review for your chapter test. Lesson 4-1 Right Triangle Trigonometry Fact

4-2

Degrees and Radians

4-3

Trigonometric Functions on the Unit Circle


4-4

Graphing Sine and Cosine Functions

4-5

Graphing Other Trigonometric Functions

4-6

Inverse Trigonometric Functions

4-7

The Law of Sines and the Law of Cosines

Chapter 4

50

Glencoe Precalculus

NAME

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4-1

Right Triangle Trigonometry

What Youll Learn

Scan the examples for Lesson 4-1. Predict two things that you think you will learn about right triangle trigonometry. 1.

2.

Active Vocabularycosine

New Vocabulary Fill in each blank with the correct term. If is an acute angle and the sine of is x, then the of x is the measure of angle . Let be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer to the lengths of the side opposite , the side adjacent to , and the hypotenuse, respectively. Then () = .adj hyp


inverse cosine

inverse sine

If is an acute angle and the tangent of is x, then the of x is the measure of angle . Let be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer to the lengths of the side opposite , the side adjacent to , and the hypotenuse, respectively. Then opp () = .adj

inverse tangent

sine

If is an acute angle and the cosine of is x, then the of x is the measure of angle . Let be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer to the lengths of the side opposite , the side adjacent to , and the hypotenuse, opp respectively. Then () = .hyp

tangent

Chapter 4

51

Glencoe Precalculus

Lesson 4-1

NAME

DATE

PERIOD

Lesson 4-1

(continued)

Main IdeaValues of Trigonometric Ratios pp. 220222

DetailsFind the exact values of the six trigonometric functions of .5 13

12

sin = csc = Solving Right Triangles pp. 222226

cos = sec =

tan = cot =

Solve ABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.b a 30 20


B=

a

b


In your own words, describe the relationship that exists between the sine and inverse sine functions. Be specific.

Chapter 4

52

Glencoe Precalculus

NAME

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4-2

Degrees and Radians

What Youll Learn


2.

Active Vocabularycoterminal angles initial side standard position terminal sideCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Label the diagram with the terms listed at the left. Positive iny

x

vertex

Vocabulary Link Sector is a word that is used in everyday English. Find the definition of sector using a dictionary. Explain how the English definition can help you remember how sector is used in mathematics.

Chapter 4

53

Glencoe Precalculus

Lesson 4-2

NAME

DATE

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Main IdeaAngles and Their Measure pp. 231234

DetailsIdentify all angles that are coterminal with 30. Then find and draw one positive and one negative angle coterminal with 30.

Application with Angle Measure pp. 235237

LOGOS The logo shown is painted on the center of a high school football field. Find the approximate area of the shaded region. Area

10


10 ft

10 ft

10


Suppose is a central angle in a circle of radius r. Explain how to find the length of the intercepted arc s. Include a drawing with your explanation.

Chapter 4

54

Glencoe Precalculus

NAME

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4-3

Trigonometric Functions on the Unit Circle

What Youll Learn

Scan the text in Lesson 4-3. Write two facts that you learned about reference angles. 1.

2.

Active Vocabularycircular function



period

periodic function

quadrantal angle

reference angle

unit circleChapter 4

55

Glencoe Precalculus

Lesson 4-3

NAME

DATE

PERIOD


Main IdeaTrigonometric Functions pp. 242247 Complete the table. sin cos tan 30 or 6

Details

45 or 4

60 or 3

Trigonometric Functions on the Unit Circle pp. 247250

Fill in the blanks to complete the unit circle. Then, use the unit circle to find the values.y (0, 1) 2

3

(, ) (, )1 23 2

90

60 45 30

4 6

2 2

2 2


(, )3 2

1 2

0

0

(1, 0) x

7 sin = 6

cos 300 =5 sec = 4

2 tan = 3

cot 60 =Chapter 4

csc 150 =Glencoe Precalculus

56

NAME

DATE

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4-4

Graph Sine and Cosine Functions

What Youll Learn

Scan Lesson 4-4. Write two things that you already know about the sine and cosine functions. 1.

2.

Active Vocabularyamplitude

New Vocabulary Match the term with its definition by drawing a line to connect the two. any transformation of a sine function

frequency

half the distance between the maximum and minimum values of a sinusoidal function


midline

the distance between any two sets of repeating points on the graph of a sinusoidal function

period

phase shift

the difference between the horizontal position of a sinusoidal function and that of an otherwise similar sinusoidal function

sinusoid

a vertical translation of a sinusoidal function

vertical shiftChapter 4

the reference line about which a sinusoidal function oscillates

57

Glencoe Precalculus

Lesson 4-4

the number of cycles a sinusoidal function completes in a one unit interval

NAME

DATE

PERIOD


Main IdeaTransformations of Sine and Cosine Functions pp. 256262

DetailsState the amplitude, period, frequency, phase shift, and vertical shift of y = 2 cos (x - 3) + 4. Amplitude: Frequency: Vertical shift: Period: Phase shift:

Applications of Sinusoidal Functions p. 263

METEOROLOGY The average monthly temperatures for Chicago, Illinois, are shown. Month Temp. (F) Month Temp. (F) Jan 22 July 73.3 Feb 27 Aug 71.7 Mar 37.3 Sep 63.8 Apr 47.8 Oct 52.1 May 58.7 Nov 39.3 June 68.2 Dec 27.4

Source: U.S. National Oceanic and Atmospheric Administration a. Write a function that models the monthly temperatures, using x = 1 to represent January. Step 1 Make a scatter plot of the data and choose a model.Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Step 2 Use the maximum and minimum values of the data to find a, b, c, and d. a = c = b = d =

Step 3 Write the function using the values for a, b, c, and d.

b.

According to your model, what is Chicagos average monthly temperature in May?

Chapter 4

58

Glencoe Precalculus

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4-5

Graphing Other Trigonometric Functions

What Youll Learn


2.

Active Vocabulary__________________________Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Write the correct term next to each definition. an object is in this type of motion when the amplitude is determined by the function y = kect the reduction in the amplitude of a sinusoidal wave

__________________________

__________________________

a function of the form y = f(x) sin bx or y = f(x) cos bx, where f(x) is the damping factor the resulting wave when the amplitude of a sinusoidal function is reduced the function f(x) in a damped trigonometric function of the form y = f(x) sin bx or y = f(x) cos bx

__________________________

__________________________

sinusoid

Chapter 4

59

Glencoe Precalculus

Lesson 4-5

Review Vocabulary Define sinusoid in your own words. (Lesson 4-4)

NAME

DATE

PERIOD


Main IdeaTangent and Reciprocal Functions pp. 269274

DetailsLocate the vertical asymptotes, and sketch the graph of y = tan 2x - 1. Vertical asymptotes: Graph:y

1- 3 4 - 4 4 3 4

x

Damped Trigonometric Functions pp. 275276

Identify the damping factor f(x) of each function. Then describe the behavior of the graph.2 1. y = x sin x 3


2. y = (x + 3)2 cos x

3. y = 4x sin x


Explain the relationship between the cosine and secant, the sine and cosecant, and the tangent and cotangent functions.

Chapter 4

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Glencoe Precalculus

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4-6

Inverse Trigonometric Functions

What Youll Learn


2.

Active Vocabulary

Review Vocabulary Define function in your own words. (Lesson 1-1)

function

Define inverse function in your own words. (Lesson 1-7)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

inverse function

New Vocabulary Write the definition next to each term. arcsine function

arccosine function

arctangent function

Chapter 4

61

Glencoe Precalculus

Lesson 4-6

NAME

DATE

PERIOD


Main IdeaInverse Trigonometric Functions pp. 280285 Complete each table. Domain

Details

y = tan x

y = tan1 x

Range

Compositions of Trigonometric Functions pp. 286287

Find the exact value of each expression, if it exists.1 1. cos cos-1 = 2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

(

)

2. sin-1 cos =

(

4

)

3. cos tan-1 = 4. tan sin-1 - = 2

(

- 3 33

)

(

)


Explain the relationship that exists between the sine function and the inverse sine function. Provide two examples.

Chapter 4

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Glencoe Precalculus

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4-7

The Law of Sines and the Law of CosinesScan Lesson 4-7. Predict two things that you expect to learn based on the headings and Key Concept boxes. 1.

What Youll Learn

2.

Active Vocabularyambiguous case

New Vocabulary Match the term with its definition by drawing a line to connect the two. a formula used to solve an oblique triangle when given the measures of three sides or the measures of two sides and the included angle


Herons Formula

triangles that are not right triangles

Law of Cosines

a formula used to solve an oblique triangle when given the measures of two angles and a nonincluded side, two angles and the included side, or two sides and a nonincluded angle

Law of Sines

a formula of finding the area of a triangle when the measures of all three sides are known

oblique triangles

while solving an oblique triangle when given the measures of two sides and a nonincluded angle there may be no solution, one solution, or two solutions

Chapter 4

63

Glencoe Precalculus

Lesson 4-7

NAME

DATE

PERIOD


Main IdeaSolve Oblique Triangles pp. 291296

DetailsIdentify the law that you would use to solve an oblique triangle with the given measures. Oblique Triangles Known Measures two sides and a nonincluded angle two angles and the included side two sides and the included angle two angles and a nonincluded side three sides Law of Sines or Law of Cosines


Find Areas of Oblique Triangles pp. 296297

Use Herons Formula to find the area of each triangle. Round to the nearest tenth, if necessary. 1. a = 5 in., b = 12 in., c = 13 in. 2. a = 17 cm, b = 21 cm, c = 30 cm 3. a = 25 yd, b = 32 yd, c = 43 yd


In your own words, explain how to find the area of any triangle when you know the measure of two sides and the included angle. Use a diagram to illustrate.

Chapter 4

64

Glencoe Precalculus

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Trigonometric FunctionsTie It TogetherComplete the table. Function Domain Range Graphy

Period

1

y = sin x

- -1

x

2

y

y = sin1 x-1- 2

1

x

1

y

y = cos x-2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

2 -1

x

y

y = cos1 x-1 -

1

x

y 4 2 - -2 -4 x

y = tan x

2

y

y = tan1 x

-1- 2

1

x

Chapter 4

65

Glencoe Precalculus

NAME

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Before the TestReview the ideas that you listed in the table at the beginning of the chapter. Cross out any incorrect information in the first column. Then complete the table by filling in the third column. K What I know W What I want to find out L What I Learned




Study Tip Review your notes before the test. Ask questions if you need any topics clarified.Chapter 4

66

Glencoe Precalculus

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DATE

PERIOD

Trigonometric Identities and EquationsBefore You ReadBefore you read the chapter, think about what you know about trigonometric identities and equations. List three things that you already know about them in the first column. Then list three things that you would like to learn about them in the second column. K What I know ... W What I want to find out ...


Note-Taking Tips Write questions that you have about the lesson in the margin of your notes. Record the answers to your questions as you work through the lesson. Include graphs and charts that present the information introduced in a format that is easy to read and study.Chapter 5

67

Glencoe Precalculus

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Trigonometric Identities and EquationsKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on trigonometric identities, one fact might be that a function f is a cofunction of a function g if f () = g () when and are complementary angles. After completing the chapter, you can use this table to review for your chapter test.

Lesson 5-1 Trigonometric Identities

Fact

5-2

Verifying Trigonometric Identities

5-3

Solving Trigonometric Equations


5-4

Sum and Difference Identities

5-5

Multiple-Angle and Product-to-Sum Identities

Chapter 5

68

Glencoe Precalculus

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5-1

Trigonometric Identities

What Youll Learn


2.

Active Vocabulary

Review Vocabulary Complete each identity. (Lesson 4-1)1 = csc 1 = sec


1 = tan

sin = cos

cos = sin

1 = cos

New Vocabulary Fill in each blank with the correct term. identity A function f is a(n) of a function g if f() = g() when and are complementary angles.

trigonometric identity

A(n) is an equation in which the left side is equal to the right side for all values of the variable for which both sides are defined.

cofunction

A(n) trigonometric functions.

is an identity involving

Chapter 5

69

Glencoe Precalculus

Lesson 5-1

NAME

DATE

PERIOD

Lesson 5-1

(continued)

Main IdeaBasic Trigonometric Identities pp. 312314

DetailsComplete each of the following charts. List the Pythagorean identities.

List the cofunction identities.

List the odd-even identities.

Simplify and Rewrite Trigonometric Expressions pp. 315316

Simplify sin x - cos - x cos2 x.

(2

)


Original equation = = = = Cofunction Identity Factor sin x from each term. Pythagorean Identity Simplify.


Explain how the unit circle could be used to remember the odd-even identities. Include a diagram illustrating your explanation.y

x

Chapter 5

70

Glencoe Precalculus

NAME

DATE

PERIOD

5-2

Verifying Trigonometric Identities

What Youll Learn

Scan the Examples for Lesson 5-2. Predict two things that you think you will learn about verifying trigonometric identities. 1.

2.

Active Vocabularyverify an identityCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Write the definition next to the term.

Vocabulary Link Verify is a word that is used in everyday English. Use a dictionary to find a definition of the word verify. Explain how the English definition can help you remember what it means to verify an identity in mathematics.

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Lesson 5-2

(continued)

Main IdeaVerify Trigonometric Identities pp. 320323

DetailsComplete each of the following steps to verifysin x sin x + = 2 csc x . 1 + cos x 1 - cos x

Original identity Start with the left-hand side. Common denominator Combine like terms. Pythagorean IdentityCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Reciprocal Identity Identify Identities and Nonidentities p. 323 Use your graphing calculator to test whether sec2 x tan2 x - 1 = tan2 x is an identity. If it appears to be an identity, verify it. If it is not, find an x-value for which both sides are defined but not equal. Graph left side: Graph right side:

Is this an identity? x-value:

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5-3

Solving Trigonometric Equations

What Youll Learn

Scan Lesson 5-3. Predict two things that you expect to learn based on the headings and examples. 1.

2.

Active Vocabularyidentity

Review Vocabulary Define each term in your own words.


trigonometric identity

verify an identity

cofunction

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Lesson 5-3

(continued)

Main IdeaUse Algebraic Techniques to Solve pp. 327329

DetailsComplete each step to solve 6 tan x + 3 = 3 tan x. Original equation Subtract 3 tan x from each side. Subtract 3 from each side. Divide each side by 3. Find solution(s).

Use Trigonometric Identities to Solve p. 330

Complete each of the following steps to solve 2 sin2 x + cos x = 1 in the interval [0, 2]. Original equation Pythagorean Identity Distributive Property Subtract 1 from each side. Multiply by 1. Factor. Solve. Find solution(s).Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.


To help you remember the algebraic techniques that are used in solving trigonometric equations, write examples that involve factoring a quadratic trinomial and factoring the difference of two squares.

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5-4


What Youll Learn

Scan the lesson. Write two things that you already know about trigonometric functions. 1.

2.

Active Vocabulary

Review Vocabulary Find each value. (Lesson 4-3) sin = 3 5 tan = 6 cot = 2


cos = csc 7 =4

cos 4 =3

sin 3 =4

tan = 6

sec = 6

csc 2 =3

New Vocabulary Write the definition of reduction identity. reduction identity

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Main IdeaEvaluate Trigonometric Functions pp. 336340

Details Find the exact value of cos 5 .12

cos 5 = cos12

+ = 5 6 4 12

=

Cosine sum identity

=

Substitute. Multiply and combine like terms.

= Solve Trigonometric Equations p. 340

Find the solution to each expression in the interval [0, 2). Show your work.2 1. sin + x + sin - x =

(6

)

(6

)

2


2. tan ( + x) + tan ( + x) = 2 3

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5-5

Multiple-Angle and Product-to-Sum Identities

What Youll Learn


2.

Active Vocabulary

Review Vocabulary Complete each identity. (Lesson 5-1) sin (-) = tan (-) = sec (-) = cos (-) = csc (-) = cot (-) =


New Vocabulary Complete each identity. sine double angle identity cosine double angle identity tangent double angle identity sin (2) = cos (2) =

tan (2) =

sine half-angle identity

1 sin =

(2 )

cosine half-angle identity

1 cos =

(2 )

tangent half-angle identityChapter 5

1 tan =

(2 )

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Main IdeaUse Multiple-Angle Identities pp. 346349

Details5 Complete each of the following steps. If sin = in

the interval , , find sin 2, cos 2, and tan 2.

(2 )

13

sin 2 = = = cos 2 = = = tan 2 = =

Sine Double-Angle Identity5 12 sin = , cos = 13 13

Simplify. Cosine Double-Angle Identity5 12 cos = , sin = 13 13

Simplify. Tangent Double-Angle IdentityCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

5 tan = 12

= Use Product-to-Sum Identities pp. 350351

Simplify.

Rewrite each product as a sum or difference. 1. cos 2x sin 4x = 2. sin 5 sin 3 = Find each exact value. 3. sin 50 + sin 40 = 4. cos - cos 3 =8 8

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Trigonometric Identities and EquationsTie It TogetherComplete the graphic organizer. Trigonometric Identities

Pythagorean Identities



Double-Angle Identities

Half-Angle Identities

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Trigonometric Identities and EquationsBefore the TestReview the ideas that you listed in the table at the beginning of the chapter. Cross out any incorrect information in the first column. Then complete the third column of the table. K What I know W What I want to find out L What I Learned




Study Tip Make flashcards to help you memorize each of the trigonometric identities. Write an identity on one side of the card and its name on the other. Review the flashcards daily.

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Systems of Equations and MatricesBefore You ReadBefore you read the chapter, respond to these statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read Systems of Equations and Matrices A multivariable linear system is a system of linear equations in more than two variables.

Matrix multiplication is a commutative operation.

Cramers Rule uses inverse matrices to solve square systems of linear equations.


When finding the partial fraction decomposition of an improper rational expression, you should first divide the numerator by the denominator using polynomial division. If a linear programming problem can be optimized, it occurs at an interior point of the region representing the set of feasible solutions.

Note-Taking Tips Before each lesson, scan through the lesson. Write any questions that come to mind in your notes. As you work through the lesson, record the answers to your questions. If you are unable to answer all of your questions, ask your teacher for help. You may wish to use a highlighting marker to emphasize important concepts in your notes.

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Systems of Equations and MatricesKey PointsScan the pages in the chapter. Write at least one specific fact concerning each lesson. For example, in the lesson on linear optimization, one fact might be that there may be multiple solutions to a linear programming problem. After completing the chapter, you can use this table to review for your chapter test. Lesson 6-1 Multivariable Linear Systems and Row Operations Fact

6-2

Matrix Multiplication, Inverses, and Determinants

6-3

Solving Linear Systems Using Inverses and Cramers Rule


6-4

Partial Fractions

6-5

Linear Optimization

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6-1

Multivariable Linear Systems and Row Operations

What Youll Learn

Scan the examples for Lesson 6-1. Predict two things that you think you will learn about solving systems of linear equations. 1.

2.

Active Vocabularyaugmented matrixCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Label each diagram with a term listed at the left. x - 2y + 3z = 7 2x + y + z = 4 -3x + 2y - 2z = -10

coefficient matrix

1 -2 elementary row operations 2 -3

3

7

1 -2 2 -31 -1 2 3 2

3

1 1 4 2 -2 -101 R1 2

1 1 2 -27 2

multivariable linear system

-2R 1 + R 2

0 -3

5 -5 -10 2 -2 -10

row-echelon form

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Lesson 6-1

(continued)

Main IdeaGaussian Elimination pp. 364368

DetailsPerform the indicated elementary row operations to produce the row-echelon form of the given system of linear equations. x - y + 2z = 6 2x + y + 5z = -9 y - 2z = -5

Augmented matrix

1 -1 2 6 2 1 -5 -9 0 1 -2 -5

1 R2 3

1 -1 0

2

6

1 -2 -5

1 -1 R 2 - 2R 1 0 1

2

6 R3 - R2

1 -1 2 6 0 1 -3 -7

-2 -5

Gauss-Jordan Elimination pp. 369371

Solve the system of equations using Gauss-Jordan elimination. x-y+z=1 2x + y - z = -4 -x + 2y + 3z = 7


Write the augmented matrix.

Apply elementary row operations to obtain a row-echelon form.

Apply elementary row operations to obtain zeros above the leading 1s in each row.

Write the solution to the system.Chapter 6

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6-2

Matrix Multiplication, Inverses, and DeterminantsScan the text under the Now heading. List two things that you will learn about in the lesson. 1.

What Youll Learn

2.

Active VocabularydeterminantCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

New Vocabulary Fill in each blank with the correct term. The multiplicative identity for a set of square matrices is called the .

identity matrix

The multiplicative inverse of a square matrix is called its matrix.

inverse

A matrix that has an inverse is said to be

.

invertible

A(n) inverse. a c

is a matrix that does not have an

singular matrix

Let A be the matrix

b . The number ad - bc is called the d

of the matrix A.

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Lesson 6-2

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Main IdeaMultiply Matrices pp. 375378

DetailsUse the following matrices to illustrate the Associative Property of Matrix Multiplication. A= -1 3 2 -1 -5 1 ,C= ,B= 0 -4 1 -3 -3 0 (AB)C = A(BC)

)

=

)

-1 3 -5 1 = 0 -4 -3 0 =

Inverses and Determinants pp. 379382

Find the determinant and inverse of Determinant: Inverse:

-7 4 . 5 -3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.


Identify which of the properties listed below apply to matrices. If a property does not apply, provide a counterexample. Associative Property of Multiplication Commutative Property of Multiplication Distributive Property

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6-3

Solving Linear Systems Using Inverses and Cramers RuleScan the lesson. Write two things that you already know about solving systems of linear equations. 1.

What Youll Learn

2.

Active Vocabulary

Review Vocabulary Define augmented matrix in your own words. (Lesson 6-1)


Define inverse matrix in your own words. (Lesson 6-2)

New Vocabulary Write the definition next to each term. Cramers Rule

square system

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Lesson 6-3

(continued)

Main IdeaUse Inverse Matrices pp. 388389

DetailsUse an inverse matrix to solve the system of equations. -2x + 5y = 17 3x - 7y = -24 Write the system in matrix form A X = B.

Find A-1. A-1 =

Multiply A-1 by B. X=

Use Cramers Rule pp. 390391

Use Cramers Rule to find the solution of the system of linear equations, if a unique solution exists. 2x - 3y = -7 x + 4y = 2 Calculate the determinant of the coefficient matrix.


Solve for x.

Solve for y.

Write the solution to the system.

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6-4

Partial Fractions

What Youll Learn


2.

Active Vocabulary

Review Vocabulary Define polynomial function in your own words. (Lesson 2-2)


Define rational function in your own words. (Lesson 2-5)

New Vocabulary Write the correct term next to each definition. one of the rational expressions in the sum of two or more rational expressions that represents a rational function the sum of the rational expressions that represent a rational functionChapter 6

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Lesson 6-4

(continued)

Main IdeaLinear Factors pp. 398400

Details8x - 7 Find the partial fraction decomposition of . 2 x -x -2

Form of partial fraction decomposition Multiply each side by the LCD. Distributive Property Group like terms. Equate the coefficients to obtain a system of two equations. Solve the system. Write the partial decomposition. Irreducible Quadratic Factors pp. 401 Write the appropriate form of the partial fraction decomposition for each rational expression. Do not solve for the variables. 1. 2. 32x3 + 5x2 + 2x + 9 (x + 2)2 (x2 - 7) 4x2 - x + 8 x + 4x



A classmate told you that the appropriate form

2x2 + 7x + 3 A of the partial fraction decomposition of is . Explain your classmates (x + 3)2 (x + 3)2

error. Then give the appropriate form of the decomposition.

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6-5

Linear Optimization

What Youll Learn


2.

Active Vocabulary

New Vocabulary Match the term with its definition by drawing a line to connect the two. the process of finding a minimum or maximum value for a specific quantity

constraintsCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

feasibl

Pre Calculus Workbook

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