student workbook with scaffolded practice unit 4a...trigonometric functions lesson 1: inverses of...

Student Workbookwith Scaffolded Practice

Unit 4A

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1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7456-8 U4A

Copyright © 2014

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

EDUCATIONWALCH

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

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Program pages

Workbook pages

Introduction 5

Unit 4A: Mathematical Modeling of Inverse, Logarithmic, and Trigonometric Functions

Lesson 1: Inverses of FunctionsLesson 4A.1.1: Determining Inverses of Quadratic Functions . . . . . . . . . . . . . .U4A-5–U4A-34 7–16

Lesson 4A.1.2: Determining Inverses of Other Functions . . . . . . . . . . . . . . . . U4A-35–U4A-63 17–26

Lesson 2: Modeling Logarithmic FunctionsLesson 4A.2.1: Logarithmic Functions as Inverses . . . . . . . . . . . . . . . . . . . . . . U4A-73–U4A-97 27–36

Lesson 4A.2.2: Common Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4A-98–U4A-115 37–46

Lesson 4A.2.3: Natural Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4A-116–U4A-137 47–56

Lesson 4A.2.4: Graphing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . U4A-138–U4A-163 57–68

Lesson 4A.2.5: Interpreting Logarithmic Models . . . . . . . . . . . . . . . . . . . . . U4A-164–U4A-182 69–80

Lesson 3: Modeling Trigonometric FunctionsLesson 4A.3.1: Graphing the Sine Function . . . . . . . . . . . . . . . . . . . . . . . . . . U4A-191–U4A-233 81–92

Lesson 4A.3.2: Graphing the Cosine Function. . . . . . . . . . . . . . . . . . . . . . . . U4A-234–U4A-272 93–104

Station ActivitiesSet 1: Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4A-303–U4A-311 105–114

Set 2: Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . U4A-318–U4A-326 115–124

Coordinate Planes 125–142

Table of Contents

CCSS IP Math III Teacher Resource© Walch Educationiii

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The CCSS Mathematics III Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

CCSS IP Math III Teacher Resource© Walch Educationv

Introduction

5

© Walch EducationU4A-5

UNIT 4A • MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONSLesson 1: Inverses of Functions

CCSS IP Math III Teacher Resource 4A.1.1

Name: Date:

Warm-Up 4A.1.1

A flooring company engineer was asked to install a rectangular dance floor with a total area given by the quadratic function A(x) = x(x – 8), where x represents the length of the floor in feet and x – 8 is the width. Pre-fabricated flooring panels can be ordered to accommodate different areas, so the engineer would like to know what the total areas could be for specific values of x.

1. Solve the quadratic function for x in terms of A, the total area of the dance floor. (Hint: Use the

quadratic formula, xb b ac

a

4

2

2

=− ± −

.)

2. What are the restrictions on the domain of x in the function A(x) = x(x – 8)? State the domain.

3. What are the restrictions on the domain of A in the function in problem 1?

4. What is a reasonable value for the length (x) of the dance floor, and what is the total area of a dance floor with that value of x?

5. What is a reasonable value for the area (A) of the dance floor, and what are the dimensions of that dance floor if the length = x and the width is x – 8?

Lesson 4A.1.1: Determining Inverses of Quadratic Functions

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U4A-14© Walch EducationCCSS IP Math III Teacher Resource

4A.1.1

Name: Date:

Scaffolded Practice 4A.1.1Example 1

Find the inverse, g(x), of the function f(x) = 4x2 and determine the domain value(s) over which the inverse exists.

1. Switch the domain and function variables, and then rename f(x) as g(x).

2. Solve the possible inverse for g(x).

3. Determine the domain of g(x).

4. Determine the range of g(x).

5. Determine the domain of f(x).

6. Determine whether the function f(x) exhibits one-to-one correspondence.

7. Determine the parts of its domain over which f(x) exhibits one-to-one correspondence.

8. Determine the range of f(x).

9. Match domains to ranges to find the inverse(s) of f(x).

continued

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Example 2

Find the inverse, f –1(x), of the function f(x) = x(x – 1) and determine the domain value(s) over which the inverse exists.

Example 3

Use a graphing calculator to graph the quadratic function f(x) = x2 – 3x – 1 and its inverse, g(x). Write an equation for the possible inverse, g(x), using algebraic methods. Then, verify your written equation using points from the graphs of f(x) and g(x).

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Problem-Based Task 4A.1.1: Sliding Rocks

A geologist for the Department of Natural Resources has been studying a rock formation popular with mountain climbers. The rocks form an overhang that is actually one large metamorphic rock block that is sliding past another rock along what is called a slip fault. The geologist has determined that the overhang is disappearing as the rock formation slides past the other rock at its base, at a rate of 10 centimeters per decade. The overhang’s movement is also accelerating at a rate of 2 centimeters per second squared. The overhang’s motion can be approximated by the quadratic function d(t) = 10t + t2, where d is the distance in centimeters the overhang slides and t is the time in decades. Write an inverse function for d(t) and determine the domain values over which the inverse exists in this real-world situation. How long will it take the overhang to move 1 meter?

How long will it take the overhang to move 1 meter?

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Name: Date:

For problems 1–3, write the inverse for each quadratic function and the restricted domain over which the quadratic function can have an inverse.

1. f(x) = 3x2 – 5

2. g(x) = 7x – x2

3. h(x) = (x + 1)(x – 2)

For problems 4–7, write the inverse for each quadratic function and the restricted domain over which the quadratic function and its inverse exist.

4. a(x) = 8x2 – 3

5. b(x) = x2 – 5x

6. c(x) = 3x2 – 2x – 1

7. d(x) = 2x2 + 1

For problems 8–9, use what you know about determining inverses of quadratic functions to answer the questions.

8. The area of a rectangular carpet is given by the quadratic function A(x) = x(x + 30), where A(x) is the area and x is the width of the carpet in feet. What are the restrictions on the domain of the function and its inverse?

9. The kinetic energy of a motorcycle is given by the function K v mv( )1

22= , where m is the mass

of the motorcycle and v is the velocity with which the motorcycle is moving. If the domain of

v is [–90, 120], over what domain(s) does the inverse exist?

Use the given information to complete all parts of problem 10.

10. A social media site has been operating for 8 years. The growth in the number of site users is approximated by the quadratic function N( y) = 25y2 – 85y + 30, where N( y) is the number of users in millions and y is the number of years since the site was launched.

a. Write an inverse for the function.

b. State the domain of the inverse.

c. State the domain of the inverse based on the real-world characteristics of the problem.

Practice 4A.1.1: Determining Inverses of Quadratic Functions

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Notes

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Notes

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Name: Date:

Warm-Up 4A.1.2

The courthouse clock lost power during an electrical storm, but the decorative pendulum bob that

normally swings once every 3 seconds continued to swing until friction brought it to a stop. At the

instant the clock lost power, the period of the pendulum T could be written as the radical function

( ) 2T LL

gπ= , where T is the amount of time in seconds needed for one complete swing of the

pendulum to and from its beginning location. The constant g is the magnitude of the acceleration

due to gravity (approximately 9.8 meters per second squared, or 9.8 m/s2), and L is the length of the

pendulum in meters. The period of the pendulum is proportional to the square root of its length.

Over what real-world domain does this function have an inverse?

1. What is the domain of the function?

2. Does the function have an inverse? Explain.

3. Write the inverse of the function.

4. What is the domain of the inverse?

5. What is the shared domain for the function and its inverse? Give a numerical answer using interval notation.

Lesson 4A.1.2: Determining Inverses of Other Functions

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4A.1.2

Name: Date:


Find the inverse of the function ( )1

f xx

x=

− if it exists, and determine the domain value(s) shared

by the inverse and the function.

1. Switch the domain and function variables, and then rewrite using inverse function notation.

2. Solve the possible inverse for f –1(x).

3. Determine the domain of f –1(x).

4. Determine the range of f –1(x).

5. Determine whether the function f(x) exhibits one-to-one correspondence.

6. Compare the domains of the function and its inverse to determine where f(x) has an inverse, f –1(x).

continued

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Example 2

Find the inverse of the radical function ( ) 23f x x= + if it exists, and determine the domain over which the function and its inverse exist.

Example 3

Find the inverse of the absolute-value function f x x( ) 2= + if it exists, and determine the domain and range of the inverse.

Example 4

Graph the cubic function f(x) = x3 – 4x2 + 3x and determine the restricted domain(s) over which its inverse exists.

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Name: Date:

Problem-Based Task 4A.1.2: The Rat Snake’s Many-Mouse Diet

In a multi-county region of southeast Georgia, the rat snake population is growing faster than the area can support. If the rat snakes’ usual prey populations become scarce, the snakes may resort to eating other small mammals, which could harm the balance of the existing ecosystem.

A herpetologist with the Georgia Department of Agriculture received a grant to introduce a bioactive chemical into the diet of rat snakes. This chemical would control the rat snake population so that beneficial small-mammal populations would not be endangered. The herpetologist plans to introduce the bioactive chemical into the diet of field mice, one of the rat snake’s prey species, by spraying the chemical on the field mice’s feeding habitats.

The amount of the bioactive chemical that can be tolerated by the field mice is given by the function A m m( ) 2 33= − , where m is the mass of an individual field mouse in grams, and A(m) is the amount (concentration) of bioactive chemical solution sprayed on the feeding habitats, with the concentration measured in grams of chemical per 100 liters of solution. Determine the domain of the function if the mice in the habitat each have a mass of 9 grams or less. How does the domain of the function compare to the domain of its inverse? What is the real-world significance of the domain of the inverse in terms of field-mouse mass and the amount of bioactive chemical used?

What is the real-world significance of the domain of

the inverse in terms of field-mouse mass and the

amount of bioactive chemical used?

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Name: Date:

For problems 1–4, write a function for each description, and then determine the inverse of the function.

1. a rational function that has a zero at x = 2 and is undefined at x = –3

2. a polynomial function that has solutions at x = 0 and x = 1

3. a radical function that is the square root of the sum of 1 more than 3 times a number

4. an absolute-value function that is half of the absolute value of the quantity 5 less 2 times a number

For problems 5–7, determine the shared restricted domain(s) over which each function and its inverse are defined.

5. ( )2

1f x

x=

−

6. ( ) 3 24g x x=

7. ( ) 4 2 2h x x= −

Practice 4A.1.2: Determining Inverses of Other Functions

continued

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Name: Date:

For problems 8–10, read the scenarios and use the information given in each to complete the problems.

8. At an automobile manufacturing plant, it takes one model of a robotic welder t seconds to weld

a car chassis frame. The time it takes a second model to weld the frame is t – 1 seconds. The

function that describes how long it takes the two welders working together to weld the frame

is 1

( )

1 1

1W t t t= +

−. Derive the inverse of the function and describe the restricted domain over

which it is defined.

9. A privately funded orbiting spacecraft has to decelerate in order to change the altitude of its orbital path around Earth. The velocity of the orbiter is given by the function v(x)2 = v0

2 + 2ax, where v0 is the orbiter’s velocity at the time its deceleration starts, a is the deceleration in kilometers per minute squared, and x is the distance in kilometers the orbiter travels during the deceleration burn. If v0 = 20 kilometers per minute and a = –5 kilometers per minute squared, write the inverse of the velocity function and explain any restrictions on the value of x in the inverse.

10. The amount of variation in the pressure that a natural gas pipeline can withstand without triggering an alarm at the closest monitoring station is given by the absolute function

( ) 25P x x= − . The alarm is triggered if the pressure variation function rises above 40. Write the inverse of this function and describe the restricted domain of the inverse and the function itself.

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Notes

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Notes

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U4A-73

UNIT 4A • MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONSLesson 2: Modeling Logarithmic Functions


© Walch Education

Name: Date:

Warm-Up 4A.2.1

In the metric system, sound intensity is measured in watts per square meter. For example, the average sound intensity at a rock concert is 0.1 watt per square meter. The threshold of human hearing is about 10–12 watt per square meter.

1. What is 0.1 watt per square meter written as an expression using a base of 10 and an exponent?

2. How much greater is the sound intensity at an average rock concert than the threshold of human hearing?

3. Write an exponential function with a base of 10 and a power of x to represent sound intensity, I(x).

Lesson 4A.2.1: Logarithmic Functions as Inverses

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Write the inverse logarithmic function of the exponential function f(x) = 0.1 • 20.3x.

1. Isolate the exponential term.

2. Rewrite the result as a logarithm.

3. Isolate the exponent variable, x.

4. Use the rules of logarithms to rewrite the result so that the simplest expression possible can be used to evaluate the function numerically.

5. Switch the domain and function variables to write the logarithmic inverse as a logarithmic function.

continued

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4A.2.1

Name: Date:

Example 2

Derive the exponential function on which the logarithmic function g(x) = log6 x2 – log

6 25 is based.

Example 3

Compare the domain and range of the inverse logarithmic functions of the exponential functions f(x) = 2 • 3–x and g(x) = 2 • 3–x + 4.

Example 4

Use a logarithmic function to solve the exponential equation x

x =−

4 5

3

.

Example 5

Write the domain and function value of the exponential function f(x) = 1.23 • 20.7x and its inverse at a domain value of x = 1.05. Use a graphing or scientific calculator and the rules of exponents and logarithms to verify your results.

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U4A-92CCSS IP Math III Teacher Resource 4A.2.1

© Walch Education


Name: Date:

Problem-Based Task 4A.2.1: Healing the Waters

The Chattamulgee River Keepers have embarked on a year-long river cleanup. They aim to work with businesses and towns along the river to reduce the acidity of the water so that a greater variety of plant and animal life can live in the river.

The river’s acidity can be measured in two ways:

• by determining the concentration of the chemical responsible for the river’s acidity. This

concentration is measured in milligrams per liter and is usually a number with a magnitude

of 10–6. This is a measure of the mass of a single hydronium ion in a solution measured

in liters by the pH of the solution, defined as C

C

= −pH = log

1log10 10 , where C is the

concentration of the chemical responsible for the level of acidity in the solution.

• by determining the pH level of the river. The pH level is the preferred measurement of river acidity because it is a number between 0 and 14 instead of a very small number represented by scientific notation. The pH of “neutral” water (non-acidic) is 7. Other common pH measures are bleach, which is very alkaline (the “opposite” of acidic) with a pH of 13, and gastric or stomach acid, which has a pH of 1.

At the beginning of the year, the pH of the river is 5.2. After one month of education and cleanup activities, the river pH is 5.4. By what amount did the concentration of the acid-producing chemical in the river change after one month? What will the pH of the river be at the end of the year? (Assume that the pH changes by the same amount each month during the year.)

What will the pH of the river be at the end of

the year?

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U4A-96



© Walch Education

Name: Date:

For problems 1–3, write the exponential function that matches each logarithmic function.

1. f(x) = 2 • log10 (x + 1) – 3

2. g xx

x( )

2•log 4

log 53

3

=+−

3. h(x) = 0.25 • log5 x–0.25

For problems 4–7, state the domain and range of the logarithmic function.

4. a(x) = 4 – log6 (x – 6)

5. b xx

=( )9

log 36

6. c(x) = log7 (2 – x) + 7

7. d x x( )1

4•log (2 1)3= −

Practice 4A.2.1: Logarithmic Functions as Inverses

continued

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U4A-97



© Walch Education

Name: Date:

Use the information given in each scenario to complete problems 8–10.

8. The intensity of a particular sound is 10–8 watt per meter squared. The intensity of a second

type of sound is 10–4 watt per meter squared. The decibel loudness of a sound is given by the

function D ii

i=

( ) log100

, in which i0 = 10–12 watt per meter squared, the threshold of human

hearing. How much greater is the loudness (in decibels) of the second sound than the first? How

much greater is the intensity of the second sound than the first?

9. In order to change their electrical properties, semiconductors are heated in a process called furnace annealing. The pH of an acid bath used to treat a semiconductor after it emerges from an annealing furnace is supposed to be 3.5. However, the pH can cover a range of 3.45 to 3.55 without the semiconductor being rejected. The pH is given by the formula pH = –log10 C, where C is the concentration of the acid bath. Find the range of concentrations of the acid bath within which the semiconductor treatment is acceptable.

10. The chance of rolling a 6 on a fair number cube with faces numbered 1–6 is 1 out of 6, and the chance of rolling a 6 twice in a row is 1 out of 36. What is the chance, C(n), of rolling a 6 ten times in a row? Write an exponential function for the chance C(n) of rolling a 6 n times in a row. Write the inverse C –1(n) using a logarithm.

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U4A-98



© Walch Education

Name: Date:

Warm-Up 4A.2.2

According to the special theory of relativity, the speed of light is the same throughout the universe.

It has a value of approximately 300,000,000 meters per second. One of the consequences of the

relativity theory is that the length L of an object will shrink the faster it moves. The function that

describes this is L v Lv

c( ) • 10

2

2= − , where v is the object’s velocity, c is the speed of light, L0 is the

starting length of the object, and L(v) is the object’s length at that velocity.

1. Write the function as a logarithm if L0 is 1 meter.

2. Calculate the length of the object at a velocity of 1 kilometer per second.

3. Explain how the length of the object at a velocity of 1 kilometer per second changes.

Lesson 4A.2.2: Common Logarithms

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4A.2.2

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Write the inverse of the exponential function f(x) = 3 • 102x.

1. Identify the base of the exponential function.

2. Switch the variables and use f –1(x) as the inverse function variable.


4. Write the inverse logarithmic function.

5. Simplify the result.

continued

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Example 2

Write the exponential function that has the inverse g(x) = 10 – 2 • log x.

Example 3

Rewrite the logarithmic function f(x) = 3 • log4 (x – 1) – 2 using common logarithms.

Example 4

Write the inverse of the exponential function g(x) = 2 • 3x – 4 using common logarithms. Evaluate any constants in the logarithmic function.

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U4A-110



© Walch Education

Name: Date:

Problem-Based Task 4A.2.2: Power Plotting and Semi-Logarithms

The graph of the common logarithmic function f(x) = log x is shown below on the left, beside a semi-logarithmic coordinate plane.

–10 –8 –6 –4 –2 2 4 6 8 10

x

10

8

6

4

2

–2

–4

–6

–8

–10

y

0

f x( ) = log x

3

2

1

0

–1

–2

–310–2 10–1 101 1021

A semi-logarithmic coordinate plane is a coordinate plane on which one axis is labeled in powers of 10 and the other axis is a standard linear axis. Either scale can be changed to produce a coordinate plane with four quadrants. Plot the points (log 0.01, 0.01), (log 0.1, 0.1), (log 1, 1), (log 10, 10), and (log 102, 102) on the provided semi-logarithmic coordinate plane. Describe the resulting graph.

Describe the resulting graph.

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U4A-114



© Walch Education

Name: Date:

For problems 1–3, write the inverse of the exponential function using common logarithms.

1. f(x) = 3 • 2x

2. g(x) = 4 • 32x

3. h(x) = 0.5 • 5–x

For problems 4–7, rewrite each logarithmic function as a logarithm with a base other than 10.

4. ( )2• log

log4=a x

x

5. ( )log5

5• log=b x

x

6. c xx

= −( ) log32

log

7. ( )log3• log

log 4 • log( 1)=

−d x

x

x

Practice 4A.2.2: Common Logarithms

continued

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U4A-115



© Walch Education

Name: Date:


8. The function R(y) = 3 • 109 • 2y describes the revenue R(y) in dollars from public computer “cloud” services from 2008 through 2013, where y = 1 for 2008. Write the inverse of this function as a common logarithmic function and determine the domain of the exponential function.

9. From 2001 to 2010, the production in megawatts of solar cells in China went from about 20 to about 11,000. The function describing this growth is P(n) = 4 • 2.15n, where P(n) is the produced capacity in megawatts and n = 1 for the year 2001. Write the inverse of this function using common logarithms.

10. A semi-logarithmic coordinate plane is a coordinate plane on which one axis is labeled in powers of 10 and the other axis is a standard linear axis. The following semi-logarithmic graph shows the cost of sequencing a human genome from 2007 to 2011. Use the graph to write a common logarithmic function for the cost C(y) as a function of the year y. Let y = 1 for 2007.

$10 million

$1 million

$100,000

$10,000

$1,000

2007 2008 2009 2010 2011 2012

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U4A-116



© Walch Education

Name: Date:

Warm-Up 4A.2.3

The growth of algae on a retention pond during a summer drought can be modeled by the exponential function r(t) = 2 • at – 1, where a is a constant, t is the number of months, and r(t) is the rate of growth in quarter acres of pond surface area per month. After one month, the growth rate r is approximately 4.437.

1. What is the value of the base of the exponential function, a?

2. The value of e is approximately equal to 2.71828. How does the calculated value for a compare to the value of the base of a natural logarithm e?

3. Why do you think a and e are not equal?

Lesson 4A.2.3: Natural Logarithms

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4A.2.3

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Write the simplified logarithmic function that is the inverse of the exponential function f(x) = 2 • e3x – 1.


2. Rewrite the exponential term to eliminate any other constants that can be removed.

3. Write the inverse logarithmic function.

4. Switch x and f(x) and replace f(x) with f –1(x).

5. If possible, simplify the function using any applicable rules of logarithms.

continued

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Example 2

Write an exponential function that is equivalent to the natural logarithm function g(x) = ln (4 – 5x)0.3.

Example 3

Solve the logarithmic equation 10 + 2 • ln (x – 4) = 5 for x. Express the result to three decimal places.

Example 4

For the following statement, write the natural logarithm function and its equivalent exponential function:

The function that is the opposite of four tenths of the natural log of the quantity eight reduced by two times a number.

Example 5

Write the logarithmic function f(x) = 5 + 2 • log5 (x – 2) as a natural logarithm function.

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© Walch Education


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Problem-Based Task 4A.2.3: Watch IT Grow!

According to a survey by a manufacturer of Internet networking hardware, the growth of global consumer Internet traffic grew from 2,280 petabytes (2.28 • 1018 bytes) per month in 2006 to 12,684 petabytes per month in 2010. Usage is projected to grow to 42,070 petabytes per month in 2014. It is estimated that the largest portion of Internet traffic in 2014 (about 46 percent) will result from users streaming videos. Use a graphing calculator to generate a natural logarithm function in order to describe the exponential growth. Then, write the corresponding exponential function as a check for the natural logarithm function. Let the year 2006 have a value of x = 1, with the following years numbered sequentially; e.g., 1, 2, 3. Round the constants in the functions to two significant figures.

Use a graphing calculator to

generate a natural logarithm function in order to describe

the exponential growth.

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© Walch Education

Name: Date:

For problems 1–4, use the rule bb

aa =logln

ln to write each function as a natural logarithm function.

1. f(x) = 2 • log3 (4x)

2. g(x) = 1 – log (x – 2)

3. f(x) = 5 • log6 (4 – x) – 3 • log6 (x – 2)

4. g(x) = [log2 (x + 1)]3

For problems 5–7, compare the values of the functions f(x) and g(x) at x = 2 using >, <, or = signs. Do not use a calculator.

5. f(x) = 2 • log x g(x) = log x

6. f(x) = 4 • ln (x – 1) g(x) = 2 • ln x

7. f(x) = log3 (x – 1) g(x) = ln (x – 1)

Practice 4A.2.3: Natural Logarithms

continued

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© Walch Education

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8. The exponential function N(x) = N0(x) • e–(ln 2)x gives the number of fair coins that land heads-up each time a group of coins is flipped. After each flip, the coins that land heads-up are removed. Calculate the number of coins remaining after 3 flips if there were 30 coins to start.

9. A researcher sprays a concentrated solution of bleach on a mildewed surface. The exponential function N(t) = N0(t) • e–t describes the amount of mildew that remains on the surface t seconds after the solution is sprayed. After how many seconds will half the original amount of mildew remain?

10. Carbon-14 is a radioactive isotope that decays at a steady rate over time. Because of this predictable decay rate, archaeologists can measure the amount of carbon-14 that remains in a fossil in order to determine the fossil’s age. The exponential function R(t) = R0(t) • e–ct models the amount of radioactive carbon-14 that remains in a fossil after the time t has elapsed. The constant c is 3.83 • 10–12 s–1 since t is in seconds (s) and the exponential term must not have any units. Write the exponential function as a natural logarithm function and find t in years if R(t) = 0.5 • R0(t).

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Warm-Up 4A.2.4

In preparation for the upcoming Mathlete Tournament, Mrs. Blake graphed two functions as shown. She asked her students to use this graph to compare the domain and range of each function.

–0.4 –0.2 0.20 0.4 0.6 0.8 1 1.2 1.4 1.6

x

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

–0.2

–0.4

–0.6

–0.8

–1

–1.2

–1.4

–1.6

y

f x( ) = ln x( )

g x( ) = log10

x( )

1. Over which restricted domain are the values of f(x) < g(x)?

2. Over which restricted domain are the values of f(x) > g(x)?

3. At what domain value is f(x) = g(x)?

4. What are the ranges of f(x) and g(x)?

Lesson 4A.2.4: Graphing Logarithmic Functions

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Sketch the graphs of the functions f(x) = 2 • log2 x and g(x) = 2 – log

2 x on a coordinate plane. Describe

the end behavior of each function.

1. Determine function values for f(1) and g(1) and write the results as ordered pairs.

2. Find the value(s) of x at which f(x) = g(x).

3. Find the function value at which the functions are equal.

4. Write the approximate point at which the functions intersect.

5. Determine additional points for sketching the graph of each function.

6. Write the additional points.

7. Plot and sketch a curve to connect the points for each function.

8. Describe what happens to the function values as x becomes very large and as x approaches 0.

continued

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Example 2

Sketch the graphs of f(x) = x + log x and g(x) = log x on a coordinate plane, using a graphing calculator if needed. Describe the end behavior of the function. Then, write f(x) in terms of g(x).

Example 3

Use the graph of the function f(x) and the function g(x) = 3 • log4 x to write the algebraic form of f(x).

Assume that f(x) is of the form a • log4 (x + b) and includes the point (15, 4). Then, describe how to

find the common solution of the two functions without finding x or the function value at x.

– 4 – 2 20 4 6 8 10 12 14

x

8

10

6

4

2

– 2

– 4

– 6

– 8

– 10

y

f(x)

g(x) = 3 • log4 x

continued

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Example 4

The graph shows a logarithmic function of the form f(x) = a • logb (x + c), where (–1, 2) is a point on

the function. The exact x-intercept can be determined directly from the graph. Write the x-intercept and estimate the value of the y-intercept from the graph. Then, calculate the actual y-intercept and compare this value with the estimate.

-10 -5 0 5 10 15

-10

-5

5

10

x

y

f(x)

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Problem-Based Task 4A.2.4: To the Rescue!

An emergency management agency wants to provide boaters in the state’s coastal waters with updated information on the average length of rescue attempts. The natural logarithm function N(t) = 12 – 6 • ln t is based on 28 recent rescue attempts. The time t, in hours, has a domain of [1, 6]. The range of the number of rescue attempts is [1, 12] and is restricted to whole numbers. For example, the greatest number of rescue attempts (12) occurred within the first hour after a distress call (t = 1). Use a graphing calculator to help sketch a graph of N(t). Interpret the domain and range for the function in terms of the context of this problem. Also, interpret the values of N(t) as its graph approaches the horizontal and vertical axes and at any intercepts on the graph.

Interpret the domain and range for the function in terms of the context of this

problem.

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For problems 1–4, sketch f(x) and g(x). Then, calculate the solution to the system of the two functions.

1. f(x) = 1 + 2 • log3 x g(x) = 2 – log3 x

2. f(x) = log (x + 1) g(x) = log (2 – x)

3. f(x) = 3 • ln (x + 1) g(x) = 2 • ln (x – 1)

4. f(x) = log4 x2 g(x) = log4 (3x + 4)

For problems 5–7, compare the domains and ranges of the three functions in each problem. Then, state the domain(s) over which all three functions are defined.

5. f(x) = 1 – ln x g(x) = 1 – 2 • ln x h(x) = 2 – ln x

6. f(x) = 2 • log (x – 1) g(x) = log (x – 2) h(x) = log (x + 1)

7. f(x) = 1 + 2 • log3 (x + 4) g(x) = 2 + 4 • log3 (x + 1) h(x) = 4 + log3 (x + 2)

Practice 4A.2.4: Graphing Logarithmic Functions

continued

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For problems 8–10, use the information in each problem to sketch a graph of the given function on a coordinate plane. Be sure to label the axes so that all the real-world parts of the domain and range are evident. Then, use your graph to solve the problem.

8. The longevity of soy-based ink-jet cartridges at a public transportation ticket kiosk varies according to the logarithmic function N(t) = 19 – 6 • ln t, where t is the time in weeks and N(t) is the number of cartridges used up during the time t. If the domain of the function is [2, 20], what is the range of the function? What kind of number is N(t)—a fraction, an integer, negative, positive?

9. The number of mechanical failures of a NASCAR stock car over the first 40 minutes of a race is modeled by the logarithmic function F(t) = –1.5 + 0.9 • ln t, where t is time in minutes. What is the minimum time elapsed after which a failure is predicted to occur according to this model?

10. In a youth hockey league, the team score keeper found that the time T(s) it takes a tied game to end with a winning score within 17 minutes of the overtime period is modeled by the function T(s) = 17 – 11 • ln s, in which s is the number of shots on goal taken by the winning team. What are the domain and range of the function?

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Warm-Up 4A.2.5

The number of people using social media, smartphones, and tablet computers has grown exponentially over the past 5 years. The following exponential functions model the growth in the number of users, in millions, for three examples of these technologies, where t represents time in years:

• a particular social media website: f (t) = e t – 1

• a popular smartphone device: g(t) = e 0.5t – 1

• a certain model of tablet computer: h(t) = e 0.1t – 1

Use the following graph of the exponential functions to complete the problems. Assume that each technology was launched at the same time and thus the numbers of users are equal when t = 0.

– 2 2 4 6 8 10

x

10

8

6

4

2

0

– 2

y

g t( ) = e0.5t – 1

f t( ) = et – 1

h t( ) = e0.1t – 1

Time (years)

Use

rs (m

illio

ns)

1. How does the growth in the number of users for each technology correlate with the coefficient of the power in the exponential term of each function?

2. Write the inverse logarithmic function of each given exponential function.

3. How do the terms in the logarithmic functions compare in relation to the exponential growth of each technology?

4. How can the logarithmic functions describe the growth of each technology?

Lesson 4A.2.5: Interpreting Logarithmic Models

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The number of electric vehicles (E) sold in the United States within 3 years of their introduction to the market can be modeled by the logarithmic function E(y) = 1.67 + 5.74 • ln y, where E(y) is the number of vehicles sold in thousands and y is the number of years after introduction to the market. The number of hybrid vehicles (H) sold within 3 years of their introduction to the market can be modeled by the logarithmic function H(y) = 0.78 + 2.4 • ln y. At what value of y was an equal number of each vehicle sold, and which type of vehicle had greater sales over the 3 introductory years? Explain your reasoning with references to the terms in the function and how they compare.

1. Set E(y) = H(y).

2. Solve the resulting equation for ln y.

3. Rewrite the resulting equation using an inverse function.

4. Interpret the resulting value of y in terms of the conditions of the original problem.

5. Check the value of y found in step 3 by substituting it back into the functions E(y) and H(y).

6. Show which vehicle type had greater sales over the 3-year period by comparing the characteristics of each function.

continued

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Example 2

A pyramid-shaped token for a board game consists of 4 congruent equilateral faces. Each face is a different color: blue, green, red, or white. Each face is equally likely to end up on the bottom if the token is rolled on the game board. What is the probability of the green face landing on the bottom? The probability of the same event a occurring n times is given by the function f(n) = an. Write a function for the green face landing on the bottom n times in a row. Then, write the inverse of the function, and explain what the inverse function describes in the context of this problem.

Example 3

The owners of a West Virginia pine forest and grasslands preserve introduced a breeding pair of quail to a specific part of the property. The population of quail can be modeled by an exponential function, but a statistics teacher at the local college came up with the logarithmic function M(n) = 8.33 – log (50 – n)4, which models how the number of quail offspring n relates to the number of months M since the introduction of the first pair to the preserve. What is the maximum number of quail (n

max) that can be

estimated using this model? Explain your answer, and state the domain for the function. What does the constant 8.33 mean in terms of this function model? Write the inverse of the logarithmic function.

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Problem-Based Task 4A.2.5: When Will It Beep?

Many models of household smoke detectors use a radioactive element as a component in the smoke-detection system. In one model, the radioactive element has a half-life of 450 years. This model will stop working (even if you change the battery annually) if the radioactivity drops to 0.2% of its original radioactivity level. The amount of radioactivity R(t) left after t years of radioactive decay is related to the original amount of radioactivity R0 by the exponential function R(t) = R0 • e –ct, where c is a constant that is unique to the radioactive element. How long can this smoke detector be expected to function properly? Based on this result, explain why it is recommended that the battery in a smoke detector be changed annually.

How long can this smoke detector be expected to

function properly?

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Use the information below to complete problems 1–3.

An aerial photograph of a section of undeveloped land documents the growth of a rapidly growing ground plant called kudzu over a 10-year period. A statistician estimates that the spread of the plant over the 10-year period can be modeled by the logarithmic function A(t) = –18.5 + 30.5 • ln t, where A is the acreage covered by kudzu and t is the number of years.

1. What is the value of A(1), and what meaning does it have in the context of this problem?

2. What is the domain of the function?

3. What is the range of the function?

Use the given information to complete problem 4.

4. The manufacturer of a brand of orange juice would like to decrease the acidity of the original product and market the low-acid version in special health food markets. If the original juice has a pH of 3.25, and the manufacturer wants to cut the acid-producing substances in the juice in half, what will the pH of the resulting product be? (Note: pH = –log c, where c is the concentration of hydronium or “acid” ions.)

Practice 4A.2.5: Interpreting Logarithmic Models

continued

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The graph shows the work done by a mechanic’s pneumatic car lift at two different temperatures. Use the graph to complete problems 5 and 6.

–1 1 2 3 4 5 6 7 8 9 10

x

10

9

8

7

6

5

4

3

2

1

0

–1

y

U v( ) = 5• ln v

Q v( ) = 8• ln v

5. Draw a vertical line on the graph that represents the difference between the work function at the lower temperature, U(v) = 5 • ln v, and the work function at the higher temperature, Q(v) = 8 • ln v, at a domain value of v = 2 where v is the volume of the liquid-compression chamber of the car lift.

6. Without using a calculator, write and simplify a mathematical expression that represents this difference.

continued

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Use the information below to complete problems 7–9.

A beagle can detect a biological trace element in the air in average concentrations of one part per ten million. Humans can detect similar substances in the air in average concentrations of one part per thousand.

7. Write fractions representing the sense of smell in beagles and humans.

8. Write common logarithms for each fraction determined in problem 7.

9. Compare the answers to problems 7 and 8, and describe a way of comparing the sense of smell of beagles and humans using powers of 10 and common logarithms.

Use the given information to complete problem 10.

10. The trumpet section of the school band has 6 trumpets that can produce a sound with an

intensity level of 90 decibels (dB). Three more trumpets will be added to the trumpet section

for the next school term. The band director says that this will make the band “50 percent

louder.” The math club president says the band will not be that loud. Who is correct, the

band director or the math club president? Explain your answer using the sound-intensity

function D II

I( ) 10 log

0

= •

, in which I0 is the threshold of human hearing of 10–12 watt per

square meter.

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UNIT 4A • MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONSLesson 3: Modeling Trigonometric Functions

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© Walch Education

Warm-Up 4A.3.1

Sofia is preparing a presentation to the city council on the need to lengthen several ramps at the skate park. Her presentation must include mathematical justifications for the new ramp lengths in order to prove her point to the council. To make her case, Sofia decided to create a table and a diagram to help the council members understand how she used sines to find the ramp lengths based on given angles. The diagram is finished, but the table is not.

In Sofia’s diagram, the sine ratio for angle A in right triangle ABC is defined as the ratio of the length of side a to the length of the hypotenuse, c.

B

CA

ca

b

Mathematically, the sine ratio of angle A can be written as sin =Aa

c. Similarly, the sine ratio of

angle B can be written as sin =Bb

c. Use the diagram and the following questions to complete the

table for Sofia’s presentation.

m A∠ m B∠ m C∠ a b c sin A sin B

30° 1

45° 1

60° 1

1. What is the relationship between the sum of ∠m A and ∠m B in each row? Use this information to fill in the missing angle measures for angles B and C.

2. What is the relationship between the lengths of the three sides of right triangle ABC? Use this information to fill in the missing lengths for b and c. Simplify the resulting values for the rows for 30°, 45°, and 60° as much as possible without using a calculator.

3. Use the definition of the sine ratio to compute the values of sin A and sin B in the table. What do you notice about the sine ratios for complementary angles? (Hint: Complementary angles have measures that add up to the measure of a right angle.)

Lesson 4A.3.1: Graphing the Sine Function

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Sketch the graph of f(x) = 2 sin x over the restricted domain [–2π, 2π].

1. Identify the amplitude and period of the function.

2. Identify any other coefficients or terms that would affect the shape of the graph.

3. Draw and label the axes for your graph based on steps 1 and 2.

4. Determine the values of the restricted domain for which the function value(s) equal 0. List the points corresponding to these zeros, and plot them on the graph.

5. Determine what values of the restricted domain are the maximum and minimum of the function value(s). List the points corresponding to these extremes, and plot them on the graph.

6. Plot points for the sines of 6

π,

4

π, and

3

π radians.

7. Plot additional points.

8. Compare the ordered pairs for the graphed points on either side of the x-axis.

9. Determine the function values for each value of x over the restricted domain of [π, 2π]. Plot points on the graph for this part of the domain.

10. Compare the points over the two restricted domains [0, π] and [π, 2π].

11. Predict what the shape of the function graph will be over the remainder of the domain, (–2π, 0).

12. Plot additional points on the graph to confirm your prediction. Then, draw a curve connecting the points across the domain [–2π, 2π].

13. Verify the resulting graph using a graphing calculator.

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Example 2

How many complete cycles of the sine function g(x) = sin 3x are found in the restricted domain [–180°, 180°]?

Example 3

Determine the coordinates of the point(s) that represent a maximum positive function value of the

function h(x) = 3 sin 2x over the restricted domain 3

4,5

3

π π−

.

Example 4

Sketch the graph of the function a(x) = 1 + 2 sin 3x over the restricted domain [0, 2π].

Example 5

Determine the coordinates of the points at which the first maximum and minimum function values

occur for the function ( ) sin3

π= +

c x x for values of x > 0.

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Problem-Based Task 4A.3.1: Going Through a Phase

Electricity generated in the United States is often delivered by alternating current from generators. In most homes and small businesses, the voltage of this alternating current has a magnitude from 110 to 120 volts. However, such alternating current actually oscillates (or moves back and forth) rapidly each second between a maximum and minimum voltage value that is given by the function V(t) = Vmaximum • sin (2πft), in which f is the frequency with which each cycle of the voltage V progresses from 0 to a maximum, down to a minimum, and back up to 0.

Since most household electrical current in the United States has a frequency, f, from 50 to 60 cycles per second, the function can be written as V(t) = Vmaximum • sin (100πt) or V(t) = Vmaximum • sin (120πt), depending on the frequency value.

The most common form of alternating current that comes from the generators is described as

three-phase current, because the current produced by each of three coils in the generator is out of

phase with the one that precedes it by 1

3 of a complete rotation of the generator shaft.

Write functions for the three alternating-current voltages that make up the three-phase electrical

energy that comes from such a generator with f = 60 cycles per second and Vmaximum = 170 volts. Then,

sketch the graph of each function on a coordinate

plane over the domain 0,1

10π

with an x-axis

interval of 1

100π. Finally, determine the domain

over which any of the three functions goes

through one complete cycle.

Determine the domain over which any of the three functions goes through one

complete cycle.

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For problems 1–4, refer to the provided graph to complete each problem. (Note: Some graphs show only part of a complete cycle. The x-axis of each graph is expressed in radians.)

1. Which function has the greater amplitude, f(x) or g(x)?

–10 –8 –6 –4 –2 2 4 6 8 10

x

10

8

6

4

2

0

–2

–4

–6

–8

–10

y

f(x)

g(x)

2. Which function has the greater period, f(x) or g(x)?

–10 –8 –6 –4 –2 2 4 6 8 10

x

10

8

6

4

2

–2

–4

–6

–8

–10

y

0

f(x)

g(x)

Practice 4A.3.1: Graphing the Sine Function

continued

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3. Determine the amount by which the functions f(x) and g(x) are out of phase.

–6 –4 –2 2 4 6

x

6

4

2

–2

–4

–6

y

0f(x)g(x)

4. Write the simplest form of the sine function shown.

–10 –8 –6 –4 –2 2 4 6 8 10

x

10

8

6

4

2

–2

–4

–6

–8

–10

y

0

f(x)

continued

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For problems 5–7, use the given information to find the requested values and coordinates.

5. At what value of x > 0 will the first maximum occur for the function f(x) = 4 sin 3x? Determine the coordinates of the point for this value of x.

6. At what value of x > 0 will the first zero occur for the function g(x) = 2 sin 0.5x? Determine the coordinates of the point for this value of x.

7. At what value of x > 0 will the first minimum occur for the function h(x) = sin (0.4x + 45°)? Determine the coordinates of the point for this value of x.

Use your knowledge of sine functions to complete problems 8–10.

8. The frequency of a sound wave is 750 cycles per second. If the sound intensity can be modeled by the sine function S(t) = 0.05 sin 750t, what is the period of the sound wave?

9. The voltage in an alternating current circuit can be modeled by the function V(t) = 175 sin (110πt). How often does the voltage reach a peak positive or negative value in 1 second?

10. The angle of a beam of light relative to a vertical line that passes through one material into

a second material is related to the angle of the beam of light in the second material by the

relationship na sin a = nb sin b. The constants na and nb differ for various materials. How are the

angles a and b related over the domain 0,2

π

for the following situations?

a. na > nb

b. na = nb

c. na = 2nb, and angles a and b are complementary

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Warm-Up 4A.3.2

Sofia is going to great lengths to convince the city council to build new ramps at the skate park. She knows her presentation is strong, but she wants to strengthen it with more mathematical justification for the new ramp lengths in order to prove her point to the council. To add to her presentation, Sofia wants to include a table and diagram for cosine values of the existing ramp angles to support her longer suggested ramp lengths. The diagram is finished, but the table is not.

In Sofia’s diagram, the cosine ratio for angle A in right triangle ABC is defined as the ratio of the length of side a to the length of the hypotenuse, c.

B

CA

ca

b

Mathematically, the cosine ratio of angle A can be written as Ab

ccos = . Similarly, the cosine ratio

of angle B can be written as Ba

ccos = . Use the diagram and the following questions to complete the

table for Sofia’s presentation.

m A∠ m B∠ m C∠ a b c cos A cos B

30° 1

45° 1

60° 1

1. What is the relationship between the sum of m A∠ and m B∠ in each row? Use this information to fill in the missing angle measures for angles B and C.

2. What is the relationship between the lengths of the three sides of right triangle ABC? Use this information to fill in the missing lengths for b and c. Simplify the resulting values for the rows for 30°, 45°, and 60° as much as possible without using a calculator.

3. Use the definition of the cosine ratio to compute the values of cos A and cos B in the table. What do you notice about the cosine ratios for complementary angles? (Hint: Complementary angles have measures that add up to the measure of a right angle.)

Lesson 4A.3.2: Graphing the Cosine Function

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Sketch the graph of f(x) = 3 cos x over the restricted domain [–2π, 2π].

1. Identify the amplitude and period of the function.

2. Identify any other coefficients or terms that would affect the shape of the graph.

3. Draw and label the axes for your graph based on steps 1 and 2.

4. Determine the values of the restricted domain for which the function value(s) equal 0. List the points corresponding to these zeros, and plot them on the graph.

5. Determine what values of the restricted domain are the maximum and minimum of the function value(s). List the points corresponding to these extremes, and plot them on the graph.

6. Plot points for the cosines of π6

, π4

, and π3

radians.

7. Plot additional points.

8. Compare the ordered pairs for the graphed points on either side of the x-axis.

9. Determine the function values for each value of x over the restricted domain of [π, 2π]. Plot points on the graph for this part of the domain.

10. Compare the points over the two restricted domains [0, π] and [π, 2π].

11. Predict what the shape of the function graph will be over the remainder of the domain, (–2π, 0).

12. Plot additional points on the graph to confirm your prediction. Then, draw a curve connecting the points across the domain [–2π, 2π].

13. Verify the resulting graph using a graphing calculator.

continued

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Example 2

How many complete cycles of the cosine function g(x) = cos 4x are found in the restricted domain [–270°, 270°]?

Example 3

Determine the coordinates of the point(s) that represent a maximum positive function value of the

function h(x) = 2 cos 3x over the restricted domain π π

−

3

2,2

3.

Example 4

Sketch the graph of the function a(x) = –2 + 4 cos 2x over the restricted domain [–π, π].

Example 5

Determine the coordinates of the points at which the first maximum and minimum function values

occur for the function π

= −

c x x( ) cos

4 for values of x > 0.

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Problem-Based Task 4A.3.2: Picking Up the Beat(s)

Cruz and Seth both play the six-string guitar. Seth’s guitar is already in tune, so Cruz starts tuning his guitar to it. Seth plucks the low E string, the thickest string; Cruz plucks the low E string on his guitar and changes its tuning to match the frequency of Seth’s E string. When the two strings sound like they are in tune, the two guitarists pluck the strings at the same time. Cruz listens for beats to determine the changing sound intensity of the tuned string. The number of beats per second indicates by what frequency the strings are out of tune. Therefore, though the strings may seem to be in tune, by counting beats, Cruz can confirm whether the two strings are exactly in tune.

The sound intensity of the first guitar’s string (the reference string on Seth’s guitar) varies with time like a cosine function, and can be given by I1(t) = A cos 2πf1t, in which A is the amplitude of the sound wave in millimeters at the instant the string is plucked, f1 is the frequency of the wave, and t is the time in seconds. If the second guitar’s string is plucked with exactly the same force, its sound intensity can be given by the function I2(t) = A cos 2πf2t.

When the two strings are plucked at the same time to tune one to the other, the two sound intensities cancel or reinforce each other according to the function Itotal = I1 + I2 = A cos 2πf1t + A cos 2πf2t. According to the rules by which the sums of cosine functions can be written as products, the changing sound intensity Itotal(t) of the guitar string being tuned can also be given by the product-of-

cosines function π π=+

•

−

I t A tf f

tf f

( ) 2 cos 22

cos 22total

1 2 1 2 .

In this function, the cosine terms refer to two different components of the sound intensity:

• The term π+

A tf f

2 cos 22

1 2 determines how the sound intensity varies according to

time. The amplitude of this sound wave varies between 2A and –2A over each period of the

cosine term.

• The term π−

tf f

cos 22

1 2 determines the number of beats per second due to the difference

in frequency of the strings, −f f1 2 , which is called the beat frequency. The beat frequency causes

the combined amplitude to vary in intensity by that number of times per second.

continued

97

U4A-266


Name: Date:


© Walch Education

If the amplitude of each plucked string is 1 millimeter, determine the amplitude of the combined wave of the two strings at the instant the strings are plucked. Then, find the number of beats produced in 1 second if the reference string has a frequency of 438 cycles per second and the string being tuned has a frequency of 442 cycles per second.

Determine the amplitude of the combined wave of the two strings

at the instant the strings are plucked.

98

U4A-270


Name: Date:


© Walch Education

For problems 1–4, refer to the provided graph to complete each problem. (Note: Some graphs show only part of a complete cycle. The x-axis of each graph is expressed in radians.)

1. Which function has the greater amplitude, f(x) or g(x)?

–6 –4 –2 2 4 6

x

5

4

3

2

1

0

–1

–2

–3

–4

–5

y

f(x)

g(x)

2. Which function has the greater period, f(x) or g(x)?

–3 –2 –1 1 2 3

x

3

2.5

2

1.5

1

0.5

0

–0.5

–1

–1.5

–2

–2.5

–3

y

f(x) g(x)

Practice 4A.3.2: Graphing the Cosine Function

continued

99

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© Walch Education

3. Determine the amount by which the two functions f(x) and g(x) are out of phase.

–3 –2 –1 1 2 3

x

4

3.5

3

2.5

2

1.5

1

0.50

–0.5

–1

–1.5

–2

–2.5

–3

–3.5

–4

y

f(x)

g(x)

4. Write the simplest form of the cosine function shown.

–3 –2 –1 1 2 3

x

8

7

6

5

4

3

2

1

0

–1

–2

y

f(x)

continued

100

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© Walch Education

For problems 5–7, use the given information to find the requested values and coordinates.

5. At what value of x > 0 will the first maximum occur for the function f(x) = 3 cos 4x? Determine the coordinates of the point for this value of x.

6. At what value of x > 0 will the first zero occur for the function g(x) = 0.2 cos 0.5x? Determine the coordinates of the point for this value of x.

7. At what value of x > 0 will the first minimum occur for the function h(x) = cos (0.6x + 30°)? Determine the coordinates of the point for this value of x.

Use your knowledge of cosine functions to complete problems 8–10.

8. The alarm in a smoke detector produces a high-pitched sound when smoke is detected. The intensity of the sound can be modeled by the function I(t) = A cos (3 • 104 • π • t). What are the period and frequency of the sound intensity? The frequency is measured in cycles per second.

9. The horizontal distance a football travels, unaided by gravity, is given by D(t) = v0 • t cos A0, where v0 is the horizontal velocity of the football as it leaves the kicker’s foot, t is the football’s “hang” time, and A0 is the measure of the angle at which the football is kicked.

a. If D(t) = 150 feet (50 yards on a football field) and the hang time is 5 seconds, what is the product v0 cos A0 in feet per second?

b. What is the range of the values of cos A0?

c. What is v0 if A0 = 60°?

d. Why can A0 not equal 0° or 90°?

10. In an electric guitar, the pickup is the device that changes the vibrations of the strings into electrical current. The magnetic flux created by the permanent magnet in the pickup is given by the function M(A, B, C) = AB cos C, in which A is the area enclosed by a loop of wire wrapped around the permanent magnet, B is the strength of the magnetic field, and C is the angle between the plane made by the loop of wire and the direction of the magnetic field relative to the loop of wire.

a. At what value(s) of C is M = 0?

b. At what value(s) of C is M a maximum?

c. What is that maximum over the domain for C of [–180°, 180°]?

d. State three possible combinations of values of A, B, and C that would make M what it is when it has a maximum value. (Note: A > 0, but B can be positive or negative.)

101

Notes

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103

Notes

Name: Date:

104

UNIT 4A • MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONSStation Activities Set 1: Inverse Functions

U4A-303

Name: Date:

CCSS IP Math III Teacher Resource© Walch Education

Station 1

Work with your group to complete each problem.

For problems 1–4, find the inverse of each given function. Circle the letter of the correct answer.

1. f x x( )= +3 2

a. f –1(x) = 1

3 2x −

b. f –1(x) = 3 2x +

c. f –1(x) = x − 2

3

2. f x x( )= +2 3

a. f –1(x) = 1

32x +

b. f –1(x) = 3 2x +

c. f –1(x) = x −3

3. f xx

( ) =4

2

a. f –1(x) = 214x

b. f –1(x) = x2

4

c. f –1(x) = ( )214x

4. f xx

( ) = 3

a. f –1(x) = x3

b. f –1(x) = 3x

c. f –1(x) = x3 continued

105


U4A-304

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CCSS IP Math III Teacher Resource © Walch Education

Use the given information to complete problems 5–7.

5. If f xx

( )= −32

, prove that f x x− = − +1 2 6( ) .

6. If f x x( )= −3 2 , prove that f x x− = +( )1132( ) .

7. If f xx

( )= +12

3 , prove that f xx

− =−

1 12 6

( ) .

106


U4A-305

Name: Date:


Station 2

Work with your group to find the inverse of the given function. Graph both f(x) and f –1(x).

1. f x x( )= +52

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

2. f x x( )= −3 1

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

continued

107


U4A-306

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3. f x x( )= +2 5

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

4. f x x( )= +312

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

continued

108


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5. f x x( )= −4 2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

6. What is the relationship between the graph of f(x) and the graph of f –1(x)?

109


U4A-308

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Station 3

Work with your group. For each function, a) provide the inverse; b) graph the function and its inverse; and c) state whether the original function is one-to-one.

1. f x x( ) ( )= +3 2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

2. f x x( )= −3 4

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

continued

110


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3. f xx

( ) =5

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

4. f x x( ) = 2 2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

111


U4A-310

Name: Date:


Station 4

Work with your group. For each function, a) provide the inverse; b) graph the function and its inverse; and c) state whether the original function is one-to-one.

1. f xx

( ) = 2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

2. f xx

( )=−3

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

continued

112


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3. f xx

( )= +1

41

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

113

UNIT 4A • MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONSStation Activities Set 2: Graphs of Trigonometric Functions

U4A-318

Name: Date:


Station 1

Work with your group to complete the following problems.

1. Construct a graph of the sine function by using a compass to transfer vertical distances from the unit circle to the coordinate axes below it. Begin by labeling the marked points on the unit circle in radians. The first four points are labeled for you.

y

x1 unit

3

6

4

1

0

–1

0

y

x

2 2

3 2

2. The terminal side of an angle rotates counterclockwise about the origin, starting at 0. What happens to the sine value of this angle on the unit circle?

continued

115


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3. Construct a graph of the cosine function by using a compass to transfer horizontal distances from the unit circle to the coordinate axes below it. Begin by labeling the marked points on the unit circle in radians. The first four points are labeled for you.

y

x1 unit

3

6

4

1

0

–1

0

y

x

2 2

3 2

4. The terminal side of an angle rotates counterclockwise about the origin, starting at 0. What happens to the cosine value of this angle on the unit circle?

116


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Station 2


For problems 1–4, refer to the graph.

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

–4

2

3

4

1. Which two functions are shown with dashed lines? What are their periods? Which function has asymptotes and where are they?

2. Describe the intersection points of the two dashed-line functions.

3. Which two functions are shown with solid lines? What are their periods? Which function has asymptotes and where are they?

4. Describe the intersection points of the two solid-line functions.

continued

117


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5. Identify the two trigonometric functions in the graph. Draw the asymptotes for each function.

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

–4

2

3

4

6. Identify the two trigonometric functions in the graph. Draw the asymptotes for each function.

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

–4

2

3

4

118


U4A-322

Name: Date:


Station 3

Work with your group to complete the following problems. Use the provided coordinate planes and colored pencils to graph the functions described.

1. Graph y = sin x. Then use a different color to graph the sine function that has three times the amplitude and one-half the period. What is the equation of the transformed function?

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

2

3

2. Graph y = cos x. Then use a different color to graph the cosine function that has one-half the amplitude and twice the period. What is the equation of the transformed function?

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

2

continued

119


U4A-323

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3. Graph y = tan 0.5x. Then use a different color to show the tangent graph that has a phase shift

of π2

to the right. What is the equation of the transformed function?

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

–4

2

3

4

4. Write the equations of the two sine functions shown in the graph.

continued

120


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5. Three graphs are shown: one trigonometric function and two transformations of it. The graph of the original function is solid, one transformation is bold, and the other transformation is dashed. Determine the equations for all three graphs.

5π

2

3π

2

π

2

4π3π2ππ 7π

2

0

y

x

1

–1

–2

–3

–4

2

3

4

121


U4A-325

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Station 4


1. Show that the graphs of y = sin x and y = sin–1 x are reflections across the line y = x. Use a different color for each graph.

π

2

π

2

π

π

0

y

x

π

2–

π

2–

π–

π–

3π

2

2. What does the graph in problem 1 tell you about the relationship between the graphs of y = sin x and y = sin–1 x?

3. Show that the graphs of y = cos x and y = cos–1 x are reflections across the line y = x. Use a different color for each graph.

π

2

π

2

π

π

0

y

x

π

2–

π

2–

π–

π–

3π

2

continued

122


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4. What does the graph in problem 3 tell you about the relationship between the graphs of y = cos x and y = cos–1 x?

5. Graph y = sin–1 x and y = cos–1 x. Remember that these functions have restricted ranges.

π

2

π

0–1–2 1 2

y

x

π

2–

π

2

π

0–1–2 1 2

y

x

π

2–

123

student workbook with scaffolded practice unit 4a...trigonometric functions lesson 1: inverses of...

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