student workbook with scaffolded practice unit 4b...speed. then, determine how far she had traveled...

Student Workbookwith Scaffolded Practice

Unit 4B

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1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7456-8 U4B

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 4B: Mathematical Modeling and Choosing a ModelLesson 1: Creating Equations

Lesson 4B.1.1: Creating Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . U4B-7–U4B-35 7–16

Lesson 4B.1.2: Representing and Interpreting Constraints . . . . . . . . . . . . . . . . U4B-36–U4B-53 17–26

Lesson 4B.1.3: Rearranging Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4B-54–U4B-81 27–36

Lesson 2: Transforming a Model and Combining FunctionsLesson 4B.2.1: Transformations of Parent Graphs . . . . . . . . . . . . . . . . . . . . . .U4B-90–U4B-115 37–48

Lesson 4B.2.2: Recognizing Odd and Even Functions . . . . . . . . . . . . . . . . . .U4B-116–U4B-130 49–58

Lesson 4B.2.3: Combining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-131–U4B-144 59–68

Lesson 3: Comparing Properties Within and Between FunctionsLesson 4B.3.1: Reading and Identifying Key Features of

Real-World Situation Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-155–U4B-189 69–82

Lesson 4B.3.2: Calculating Average Rates of Change . . . . . . . . . . . . . . . . . . .U4B-190–U4B-211 83–92

Lesson 4B.3.3: Comparing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-212–U4B-234 93–102

Lesson 4: Choosing a ModelLesson 4B.4.1: Linear, Exponential, and Quadratic Functions . . . . . . . . . . .U4B-248–U4B-272 103–114

Lesson 4B.4.2: Piecewise, Step, and Absolute Value Functions . . . . . . . . . .U4B-273–U4B-296 115–128

Lesson 4B.4.3: Square Root and Cube Root Functions . . . . . . . . . . . . . . . . . .U4B-297–U4B-319 129–140

Lesson 5: Geometric ModelingLesson 4B.5.1: Two-Dimensional Cross Sections of

Three-Dimensional Objects . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-331–U4B-349 141–152

Lesson 4B.5.2: Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-350–U4B-369 153–164

Lesson 4B.5.3: Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-370–U4B-392 165–178

Station ActivitiesSet 1: Choosing a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-429–U4B-436 179–186

Set 2: Geometric Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U4B-441–U4B-444 187–190

Coordinate Planes 191–220

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

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UNIT 4B • MATHEMATICAL MODELING AND CHOOSING A MODELLesson 1: Creating Equations

U4B-7

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CCSS IP Math III Teacher Resource 4B.1.1

© Walch Education

Warm-Up 4B.1.1

An aquarium-supply manufacturer conducts yearly follow-up customer surveys in order to track the performance of its products. The graph shows the percentage of aquarium heating elements that customers reported had to be replaced over the course of 7 years.

1 2 3 4 5 6 7 8

x

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

y

Time in years

Perc

enta

ge

of r

epla

cem

ents

1. How does the independent variable (time) relate to the dependent variable (percentage of replacements)?

2. What type of equation(s)—linear, exponential, quadratic, and/or logarithmic—could this graph represent? Explain each in terms of the graphed relationship between the time and the percentage of replacements.

3. What are some of the limitations on the values of the percentage and time variables in this example?

Lesson 4B.1.1: Creating Equations in One Variable

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U4B-17CCSS IP Math III Teacher Resource

4B.1.1© Walch Education

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Scaffolded Practice 4B.1.1Example 1

Aaron wants to have his company’s logo printed on tablet computer cases, so that he can give away the cases as part of a marketing campaign. A specialty printing company will charge Aaron a $750 fee to design and print the personalized cases, plus the cost of the actual cases. The price of each case is $3. How many personalized cases can Aaron purchase if his budget is $1,200?

1. Write an equation in words for determining the total cost to produce the personalized cases.

2. Write an equation for the cost for n cases.

3. Determine how much money Aaron will have left to spend on cases after paying the fee.

4. Determine the number of cases Aaron can buy.

5. Use the equation to check the result.

continued

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U4B-18CCSS IP Math III Teacher Resource 4B.1.1

© Walch Education

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Example 2

Chloe is driving back home from college for summer vacation. She fuels up her gas tank and then drives for a certain amount of time before passing a roadside attraction. She drives on without stopping, and 3 hours after leaving her college, she has driven 120 miles past the attraction. Seven hours after leaving her college, she has driven 400 miles past the attraction. Write a linear equation in one variable for the distance Chloe covers in t hours, and describe the domain of the linear equation. Assuming that Chloe travels at a constant speed without stopping, use the equation to determine her speed. Then, determine how far she had traveled before she passed the roadside attraction.

Example 3

Write a quadratic equation in one variable that is true for the three data points (0, 0), (1, 2), and (2, 8) by solving a system of three equations based on the standard form of a quadratic equation, y = ax2 + bx + c. Then, use your equation to find the y-value for a fourth point on the same graph that has an x-value of 3.

Example 4

The data shows the current i in milliamps (mA) in a cell phone circuit in fractions of a second after a cell-tower signal is received.

Time, t (s) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Current, i (mA) 2 3 5 8 13 21 34 55 89

Graph these points on a graphing calculator. Use the graph to write an exponential equation of the general form y = abx that approximately fits the data. Then, rewrite the resulting equation so that it includes a power of 10.

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Problem-Based Task 4B.1.1: How Long Will the Fertilizer Last?

Two teams from the state agriculture department are conducting a long-term study of how long nitrogen-based fertilizer stays in the soil. Over several years beginning in 1982 (Year 1), members of each team took soil samples from a field to which the fertilizer was applied, and measured the amount of remaining nitrogen-15 isotope (N-15) from the fertilizer. Both teams’ samples contained identical amounts of N-15 each year. However, when the teams used their existing data to predict the amount of N-15 that will be present in the soil in the year 2056, their predictions were different: Team A predicted that 10–5 kg of N-15 will remain, but Team B predicted that all the fertilizer will be gone by 2056.

The data in the table shows the amount in kilograms of N-15 remaining in a soil sample for given years. Each amount is represented as a decimal fraction of the original amount applied to the field in 1982. The original amount of the sample is the same for each team, so it is assigned a value of 1; the amounts in subsequent years are fractions of that amount.

Remaining Nitrogen-15 (kg)

Year 1 (1982) Year 2 (1983) Year 25 (2006) Year 75 prediction (2056)

Team A 11

2

1

1210–5

Team B 11

2

1

120

The two teams would like to come up with an equation in one variable to describe how the amount of nitrogen-15 decreases as a function of the number of years since the application of the fertilizer. Team A thinks an exponential equation would be the most accurate model, but Team B thinks a quadratic equation would be better. The teams agree to derive both forms of equations and decide which one is a better predictor of the amount of N-15 that will be present in 2056. Each team plans to use its own prediction for 2056 to write its equations.

Use a graphing calculator to derive an exponential equation and a quadratic equation for each team’s data set and prediction. Discuss the advantages and the limitations of the equations for the data sets, including comments about the domain (years) and the range (amount of N-15) of the equations for the years after 2056.

Use a graphing calculator to derive

an exponential equation and a

quadratic equation for each team’s

data set and prediction.

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© Walch Education

For problems 1–3, write an equation in one variable that fits the data points exactly without using a calculator.

1. (–2, –3) and (3, 2)

2. (4, 1), (3, –2), and (–2, 3)

3. (–1, 5), (0, 1), and (1, 0.2)

For problems 4–7, write an equation in one variable in the simplest form of the equation type listed, using all three of the graphed data points.

0.5 1 1.5 2 2.5 3 3.5 4

x

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

y

4. an exponential equation

5. a linear equation

6. a logarithmic equation

7. a quadratic equation

Practice 4B.1.1: Creating Equations in One Variable

continued

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U4B-35

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© Walch Education

Read the scenario that follows, and use the information in it to complete problems 8–10.

Dane and his friend Gale rode their in-line skates to a friend’s barbecue. They traveled the same route. Dane took off from a starting point (0, 0) and skated at a constant speed of 20 kilometers per hour for 30 minutes. Then, he skated at a constant speed of 10 kilometers per hour for another 15 minutes. Gale skated from the same starting point (0, 0) to the same destination. The distance Gale skated is modeled by a quadratic equation for the route’s entire duration.

8. Write a linear equation for the first segment of Dane’s trip.

9. Write a linear equation for the second segment of Dane’s trip.

10. Write a quadratic equation for Gale’s distance traveled as a function of time.

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U4B-36

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© Walch Education

Warm-Up 4B.1.2

A chemist finds that the logarithmic function P(t) = –1 + ln (2t + 1) models the probability that a certain compound will be produced in the presence of a catalyst. In the equation of the function, P(t) represents the probability that the compound will be produced, and t represents the elapsed time in seconds after the catalyst is introduced to the reactor. The graph represents the equation. Use the equation and its graph to solve the problems that follow.

–2 –1 1 2 3 4

x

2

1.5

1

0.5

0

–0.5

–1

–1.5

–2

–2.5

–3

–3.5

–4

y

P(t) = –1 + ln (2t + 1)

1. What constraints exist for the argument of the natural logarithm?

2. What constraints exist for the dependent variable, P(t)?

3. What real-world constraints exist for the independent variable, t?

4. What is the minimum value of t for which P(t) ≥ 0?

5. How does this value of t compare to the answer to problem 1, and what does this value mean in the context of the problem?

6. What are the equations of the lines that represent the constraints found in problems 1–5?

Lesson 4B.1.2: Representing and Interpreting Constraints

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Meredith runs the souvenir store for a minor-league baseball team. She checked into how much it would cost to have smartphone cases made that feature an image of the team’s mascot. One manufacturer charges a $250 fee to design and print the personalized cases, plus the cost of the actual cases. The first 50 cases cost $5 each, and the next 100 cases cost $3 each. Write an equation to determine how many cases Meredith can purchase with a budget of $750. Determine the constraints on the terms of the equation based on the situation, then apply the constraints to solve for the number of cases that can be purchased.

1. Write an equation in words for determining the total cost to produce the personalized cases.

2. Write an equation for the cost of n cases if n is 50 or less.

3. Write an equation for the cost C(n, m) if n is the number of cases that are $5 each and m is the number of cases that are $3 each.

4. List any constraints on the value of the expression m.

5. Write an equation for the amount of money Meredith has available to spend as a function of the cost equation identified in step 3.

6. List any constraints on the terms on the right side of the resulting equation.

7. Substitute this constraint into the equation from step 5 and simplify.

8. Apply any remaining constraints to the value of m and determine the total number of cases Meredith can purchase for $750.

continued

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© Walch Education

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Example 2

Certain medical tests require that patients be injected with liquids containing trace amounts

of radioactive elements in order to track the movement of blood in the circulatory system. The

concentration of the radioactive tracer substance diminishes in the human body over time according

to the function =−

+ −

( )

2

1 4 2C t a

t

t t, in which a is a constant unique to the tracer (a > 0), t is time

in hours, and C(t) is the concentration of the tracer in milligrams per liter. Identify real-world and

mathematical constraints on t, the time that the tracer is in the body, which allow C(t) to be defined in

the context of the situation.

Example 3

A sports shop is holding a sale on a particular brand of tennis racket. The total retail value of the initial stock of rackets was $7,500. The sale price is $120 per racket. Write an equation for the value of the rackets remaining after n rackets are sold. Then, find the number of rackets sold when the total value, V(n), reaches 0, and explain what this reveals about how much “below retail” the sale price is.

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Problem-Based Task 4B.1.2: Home-Team Hoops

The home basketball team, the Panthers, scored 85 points, which consisted of x two-point goals, y three-point goals, and z one-point free throws. The data point describing the two- and three-point goals and the one-point free throws is (x, y, z). Describe any restrictions on the type of numbers represented by x, y, and z. Write a linear equation in one variable for the total score in terms of x, y, and z. Determine three data points that would result in the Panthers’ score of 85 points. Then, determine the values of x, y, and z for three teams that lost to the Panthers (assume that the Panthers scored 85 in each of the three games).

Suppose that the Panthers scored 3 fewer two-point goals than a fourth team, the Vikings, but then the Panthers also score 2 more three-point goals and twice as many one-point free throws. Do the Panthers win the game, and if so, by how many points? Assume that no game ends in a tied score.

Do the Panthers win the game, and

if so, by how many points?

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U4B-53

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© Walch Education

For problems 1–3, determine the restricted domain that corresponds to the range constraint on the dependent variable in the equation.

1. y = –3x + 5 if y > –1

2. y = 2x2 – 3 if y ≤ 11

3. y = 2 + 3x if y < 4

For problems 4–7, use the given constraints on the variable(s) to determine the solution set(s) of the system.

4. x + y > 3, x – y < 2 if x > 1

5. 2x – y ≤ 3, x – 2y ≥ 3 if x ≤ –2

6. x + 2y < 3, 3x – 2y > 1 if x < 0 and y > 1

7. x ≤ 2y, y ≥ 2x if x < 2 and y > 2

Use the given information to solve problems 8–10.

8. The distance an object falls under the influence of gravity is given by =( )1

22h t gt , where h(t) is

distance in meters and t is time in seconds. If g = –9.8 meters per second squared, what are the

constraints on the values of the variables h(t) and t?

9. A school club is selling canvas tote bags to raise money for a trip to an amusement park. The total cost of the bags is $750. How many bags do the club members have to sell at $12 per bag before they start making money for their trip?

10. A state environmental protection agency considers the presence of a toxic substance in soil “negligible” when the amount of the substance reaches 0.1% of the amount originally measured. The equation that measures the reduction in the toxic substance is given by A(t) = A0 • e–t, in which A0 is the original amount of the toxic substance and t is the time in years. What is the domain of the time variable t for the amount of the toxic substance to be declared “negligible”?

Practice 4B.1.2: Representing and Interpreting Constraints

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Notes

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Notes

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U4B-54

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© Walch Education

Warm-Up 4B.1.3

Mahomet is an artist who creates cast-iron geometric sculptures. The developers of an upscale office building want him to make a sculpture for the building’s front lobby. Mahomet has determined the sculpture will be a right circular cone with a volume of 12π cubic meters, a slant height s of 5 meters, and a radius r of 3 meters. He needs to determine the cone’s actual height to make sure the sculpture will fit in the lobby, which has a height of 5 meters.

The volume of a right circular cone is given by the formula V r h1

32π= , in which h is the height of

the cone and r is the radius of the cone’s base. The height h is perpendicular to the base, and the slant

height s is the distance from the vertex of the cone to a point on the circumference of its circular base.

Use the formula and the given information to complete the problems that follow.

hs

r

1. Rewrite the given formula in terms of the height, h.

2. Use the known values and the rewritten formula to solve for the cone’s height, h.

3. Using the variables s, h, and r, how could the Pythagorean Theorem be used to determine the height, h? Write this formula in terms of h.

4. Using this alternate formula, what is the cone’s height, h?

5. Compare your results from problems 2 and 4. Are they the same? Explain.

6. Based on your results, will Mahomet’s cone fit in the lobby?

Lesson 4B.1.3: Rearranging Formulas

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© Walch Education

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The pressure and temperature inside an insulated hot-beverage bottle is related to the volume of the bottle and the amount of beverage in it by a real-world form of the ideal gas law. The ideal gas law is given by the formula PV = nRT, in which n is the number of moles (a unit of counting) of the gas in a container, P is the pressure the gas exerts on the container, V is the volume of the container, and T is the temperature in degrees Kelvin. The only constant in the formula is R. Rearrange the formula to show how the temperature T is affected by doubling each variable n, P, and V. (Note: All of the quantities in the formula are nonzero.)

1. Isolate temperature, T, in the given formula, PV = nRT.

2. Determine how T is affected in the rearranged formula if n is doubled and P and V stay the same.

3. Determine how T is affected in the rearranged formula if P is doubled and n and V stay the same.

4. Determine how T is affected in the rearranged formula if V is doubled and n and P stay the same.

continued

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Example 2

The distance, d, an object moves in one direction can be described by the formula d vt at1

22= + ,

in which v is the velocity of the object when timing starts, a is the acceleration of the object when

timing starts, t is the duration of the motion, and d is the initial distance of the object relative to some

arbitrary reference point (e.g., the origin on a coordinate plane) when the timing starts. Rearrange

the formula for d so that it can be solved for t using the quadratic formula, xb b ac

a

4

2

2

=− ± −

. In

this problem, the quantities a, d, and v are constants that can be positive, negative, or 0. Apply the

condition t > 0 to the resulting values of t and interpret the result(s) for real-world motion if the

following values are given: a = –2 meters per second squared, d = 10 meters from a reference point,

and v = 5 meters per second. Describe the motion of the object using these values.

Example 3

The formula for a standard earthquake-body wave scale, mb, is given by m

A

TQb log=

+ , in which A

is the amplitude of the ground motion in microns (10–6 meter), T is the period of the wave, and Q is a

correction constant. Determine a formula for the frequency of the earthquake wave if the frequency F

is defined as the reciprocal of the wave period. Rearrange the formula to find the range of F values for

when the range of T values is [4, 5] seconds. Then, rearrange the formula to find the range of values of

A for when the range of values of mb is [6, 9], the range of T values is [4, 5] seconds, and Q = 2.

Example 4

The diffusion rate of a gas in the combustion chamber of a diesel engine is directly proportional to the square root of the molecular mass of the gas. This relationship is given by the formula r k m•= , in which k is a constant. Find the diffusion rate for two gases A and B if the molecular mass of gas A is three more than two times the molecular mass of gas B. Describe how the resulting two rates are related.

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Problem-Based Task 4B.1.3: A Math Melody Mash-Up

The school band director also teaches physics. He has asked some of the school orchestra to learn more about the science behind their instruments. In one class, he explained how these three formulas apply to stringed musical instruments such as guitars, violins, and pianos:

m

Lµ = v

F

µ= v = f • λ

In the formulas, the quantity µ (the lowercase Greek letter mu) is a constant. It is the mass m per unit length of a string of length L. The quantity v is the velocity of a sound wave produced by the string, and is equal to the product of the frequency f of the sound and its wavelength λ (the lowercase Greek letter lambda). The quantity F is the tension in the string.

Derive a formula for the tension in a string in terms of all of the other variables. Then, explain how the tension can stay the same for a guitar string if the frequency is changed from 275 cycles per second to 400 cycles per second. What is the wavelength of the waves corresponding to these frequencies if the length of the string is 1 meter and the string has a mass of 100 grams?

Derive a formula for the tension

in a string in terms of all of the other

variables.

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© Walch Education

For problems 1–4, rearrange the given formulas to complete the problems.

1. If V = I • R and P = V • I, write P in terms of R.

2. If k AeE

RTa

=−

, write a formula for Ea.

3. Given that [H+] and [OH–] are ions, find [H+] and [OH–] if pH+ + pOH– = 14, pH+ = 2 • pOH–, pH+ = –log [H+], and pOH– = –log [OH–].

4. Which is greater, the surface tension factor for a ring of diameter l and a circumference C, or the

surface tension factor for a wire of length l? Use the formulas sF

C2•ring ring= and s

F

l2•wire wire=

to justify your answer.

Practice 4B.1.3: Rearranging Formulas

continued

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U4B-81

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© Walch Education

Use the given information to complete problems 5–10. Rearrange the formulas if necessary.

5. How is Li related to Lv if c > v and Li > 0 in the formula L Lv

cv i• 1

2

2= − ?

6. The magnetic field, B, near a wire carrying a current, I, is given by the formula BI

r

•

40µπ

= , in

which r is the distance from the wire to a point in space at which the field is measured. What

happens to the values of B if I varies between –1 and 1 ampere and r decreases by 5 meters?

7. What is the range of values for fc if DRT

N fA c•= and 103 < D < 105?

8. The formula for the ideal gas law, PV = nRT, relates the number of moles of a gas, n, to its

pressure P, temperature T, and the volume V of the container in which it is held. The quantity

R is a constant. The quantity Vm is the molar volume and is defined by the formula VV

nm= . All

of the quantities are positive. Use the formula for the ideal gas law to write the molar volume in

terms of P, R, and T.

9. Find a if KC D

A B

c d

a b

•

•eq= and Keq > 1.

10. The surface area of a cylinder is given by the formula A(r, h) = 2πr2 + 2πrh. Find h if A(r, h) = 9π.

34

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Notes

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UNIT 4B • MATHEMATICAL MODELING AND CHOOSING A MODELLesson 2: Transforming a Model and Combining Functions

U4B-90

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© Walch Education

Warm-Up 4B.2.1

Five bicyclists—Anh, Brayden, Caroline, Damian, and Ganesh—are training for a race. The following graph shows the distance traveled in kilometers by each bicyclist relative to a landmark (0, 0) over the course of x hours. The graphed functions for the five bicyclists are a(x), b(x), c(x), d(x), and g(x), respectively. Anh, Brayden, and Ganesh took off from the landmark at (0, 0). Caroline and Damian started their rides from their respective houses, represented by the y-intercept for their functions, c(x) and d(x). Let the function a(x) of the first bicyclist, Anh, be the parent function a(x) = x, and the four other functions, b(x), c(x), d(x), and g(x), be transformations of the parent function. Use this information and the graph to answer the questions that follow.

–10 –8 –6 –4 –2 2 4 6 8 10

x

10

8

6

4

2

0

–2

–4

–6

–8

–10

yg(x) = 3x

a(x) = x

b(x) = 2x

c(x) = x + 3

d(x) =

x + 1

Hours

Dis

tanc

e tr

avel

ed (k

m)

1. Which graph or graphs represent the transformation k • a(x)?

2. Which graph or graphs represent the transformation a(kx)?

3. Which graph or graphs represent the transformation a(x + k)?

4. Which graph or graphs represent the transformation a(x) + k?

Lesson 4B.2.1: Transformations of Parent Graphs

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Compare the graphs of the exponential functions a(x) = ex + 2, b(x) = ex + 2, c(x) = e2x, and d(x) = 2 • ex. Compare each graph’s x- and y-intercepts, domain, and range to those of the parent function, f(x) = ex. Finally, determine the value of x at the intersection point of the functions a(x) and b(x).

1. Graph each of the given functions on a graphing calculator.

2. Determine and compare the domains and ranges of the functions.

3. Determine and compare the y-intercepts of the functions.

4. Determine and compare the x-intercepts of the functions.

5. Determine the value of x at the intersection point of the functions a(x) and b(x).

continued

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© Walch Education

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Example 2

The graph shows a parent quadratic function f(x) = x2 and two quadratic functions, g(x) and h(x), derived from it. Use the maximum or minimum point of the quadratic functions to derive their equations, and write the functions in a form that indicates the transformation(s) of the parent function. Then, describe what transformation(s) the parent function underwent to result in each transformed function.

Example 3

Identify the transformation(s) of the parent function f(x) = log x that result in the function g(x) = 1 + 2 • log (3x + 4). Describe the effect of the transformation(s) on the domain and range of the function g(x). Finally, determine the y-intercept(s) of g(x).

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Problem-Based Task 4B.2.1: An Absolute Gas?

Hydraulic fracturing, or “fracking,” is the process of injecting pressurized fluid into shale rocks in order to fracture them, which releases natural gas trapped by the rocks. A particular fracking well has three 400-liter gas tanks, each containing a different formulation of natural gas extracted from the ground. A technician checked the pressure and temperature gauges on the three tanks twice during his shift, and recorded the results in the table shown. The temperature is measured in degrees Celsius (°C) and the pressure is measured in kilopascals (kPa).

Pressure 1 (kPa) Temperature 1 (°C) Pressure 2 (kPa) Temperature 2 (°C)Gas A 80 0 109 100Gas B 60 0 82 100Gas C 100 0 137 100

Assume that the relationship between the pressure P (the dependent variable) and the temperature t (the independent variable) can be represented by a linear function. Derive the family of linear functions for this situation, and state the transformation(s) of the parent function f(x) = x that result in the form of the three linear functions in the family. At what temperature is the pressure of each gas equal to 0 kPa?

At what temperature is

the pressure of each gas equal

to 0 kPa?

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For problems 1–3, use the graph of functions f(x) and g(x) to determine the value of the constant for the given general form of both functions. Then, write the equations of f(x) and g(x) in the given form.

1. general form: (x + a)2

–6 –4 –2 2

x

10

8

6

4

2

0

y

g(x)f(x)

2. general form: ex + a

– – 6 – 4 – 2 2

x

6

4

2

0

– 2

y

g(x)

f(x)

3. general form: log (x + a)

–4 –2 2 4

x

4

2

0

–2

–4

y

g(x) f(x)

Practice 4B.2.1: Transformations of Parent Graphs

continued

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For problems 4–6, use the graph to determine the domain and range for each given function. Write your answers in interval notation.

– 10 – 8 – 6 – 4 – 2 2 4 6 8 10

x

10

8

6

4

2

0

– 2

– 4

– 6

– 8

– 10

y

h(x)

g(x)

f(x)

4. f(x)

5. g(x)

6. h(x)

continued

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Use the information given in each problem to complete problems 7–10.

7. The area of a rectangle is given by the function A(x) = x(x – 2). What is the value of x if the area is doubled?

8. A vehicle’s distance d in miles is given by the function d(r, t) = r • t, in which r is the rate in miles per hour and t is the time in hours. What happens to the value of d if the rate is tripled and the time increases by 3 hours?

9. The growth factor for one generation of a bacteria culture is given by the function =g tt

( )ln2

,

in which t is the time it takes for the generation to mature to its full size based on the energy

resources and environmental conditions available in the culture. What happens to t if g(t) is cut

in half?

10. The profit p from selling n units of a product is given by the function p(n) = 250n – 75,000. What is the domain of n if the range of p(n) is (0, 5000)? How does the domain of n change if the upper bound of the range of p(n) is decreased by 2,500?

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Warm-Up 4B.2.2

The functions f(x) = 3x3 – 2x, g(x) = 1 + log (10x), and ( )=h x e x( ) 2 2 are models for changes in the population of three groups of kangaroo rats at a time x in months in the presence of predators (x < 0) and in the absence of predators (x > 0). The population is the dependent variable. Use the graph to answer the questions that follow.

– 5 – 4 – 3 – 2 – 1 1 2 3 4 5

x

5

4

3

2

1

0

– 1

– 2

– 3

– 4

– 5

y

g(x) = 1 + log (10x)

f(x) = 3x3 – 2x

h(x) = 2(ex2

)

1. How do the function values of f(x) and f(–x) compare?

2. How do the function values of g(x) and g(–x) compare?

3. How do the function values of h(x) and h(–x) compare?

Lesson 4B.2.2: Recognizing Odd and Even Functions

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Show that the function f(x) = 2x2 – x is neither even nor odd by using two opposite values of x.

1. Evaluate f(x) at a value of x greater than 0.

2. Evaluate f(x) at a value of x that is the opposite of the value of x used in the previous step.

3. Summarize your findings.

continued

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Example 2

Describe how the graph of the function g(x) = x3 – 2x can be used to determine if the function is even or odd.

Example 3

Describe how the graph of the function h(x) = 6x6 – 2x2 – 1 can be used to determine if the function is even or odd.

Example 4

Compare the functions f(x) = log x2 and g(x) = 2 • log x using the definitions of even and odd functions.

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Problem-Based Task 4B.2.2: Odd and Even Velocity Functions

The velocity of a plane or ship has a direction and a magnitude. For example, a plane can be described as traveling in a northeast direction from a starting point at 350 miles per hour. On a coordinate plane, this velocity can be described as traveling at a specific velocity to the east and at a specific velocity to the north that result in the “total” velocity to the northeast, as shown in the following graph.

(v cos , v sin )θ θ

θ0 x

vy

vx

y

v

Notice that the velocity in the northeast direction is in the first quadrant and is labeled v. Its eastern part lies along the positive x-axis and is labeled vx. Its northern part lies along the positive y-axis and is labeled vy. The angle θ is measured counterclockwise from the positive x-axis. If the plane or ship were to travel in a different direction at the same speed from the starting point, its velocity would stay the same, but its direction would change and so would the values of vx and vy.

Notice that vx and vy form the sides of a right triangle with v as the hypotenuse. Recall the cosine and sine functions of an angle θ, where θ is a positive angle that measures less than 90°:

θ =coslength of adjacent side

length of hypotenuse

θ =sinlength of opposite side

length of hypotenuse

Thus, for this situation, the ratios can be

written as θ =v

vxcos and θ =

v

vysin . Are the

sine and cosine functions for this scenario even,

odd, or neither?

Are the sine and cosine

functions for this scenario even, odd, or

neither?

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For problems 1–4, change or remove terms to first rewrite each original function as an even function, then as an odd function.

1. a(x) = 5 – 2x + x2

2. = −b x x( ) –2• 3 4

3. c(x) = x2 • log (x2 + x)

4. d(x) = cos x + sin x

For problems 5–7, use the following graph to determine the equation of each given function, then identify the function as even, odd, or neither.

–10 –8 –6 –4 –2 2 4 6 8 10

x

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

y

g(x)

f(x)

h(x)

5. f(x)

6. g(x)

7. h(x)

Practice 4B.2.2: Recognizing Odd and Even Functions

continued

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Use the information and the graph that follows to complete problems 8–10.

In a microelectronics circuit, the current I and voltage V fluctuate with time t according to the functions I(t) = 2 • cos at and V(t) = 10 • sin at, as shown on the graph. The third curve on the graph, P(t), is the power produced by the circuit. The power function P(t) is the product of the current and voltage functions.

–10 –8 –6 –4 –2 2 4 6 8 10

x

25

20

15

10

5

0

–5

–10

–15

–20

–25

y

P(t)

V(t)

I(t)

8. Are the functions I(t) and V(t) even, odd, or neither?

9. What does the graph of P(t) indicate about its classification as even, odd, or neither? Is the graph of P(t) the graph of a cosine or a sine function? Explain your answers.

10. Recall that the period of a cosine or a sine function is the distance (which is time t in this problem) between successive maximum or minimum points on a graph. How does the period of P(t) compare to the period of the current and voltage functions?

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U4B-131

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Warm-Up 4B.2.3

Ella is an artist who uses a three-dimensional printer to produce custom-designed plastic toys for specialty retailers. Ella charges a $1,500 set-up fee to program and use the printer for a specific design, and $25 for the first toy produced. The cost for each additional toy is discounted by 0.5 percent of the original cost of the first toy.

1. Without including the set-up charge, what would be the cost of producing 50 toys?

2. Write a general function for the cost c(n) of producing n toys without the set-up charge.

3. Write a function for the total cost T(c) of producing n toys, including the set-up charge and the cost c(n) of n toys.

Lesson 4B.2.3: Combining Functions

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Find (f + g)(x) if =−

f xx

x( )

1 and =

−+

g xx

x( )

1

2. Determine the domains over which f(x), g(x), and

(f + g)(x) are defined.

1. To find (f + g)(x), find f(x) + g(x).

2. Determine the domain of f(x).

3. Determine the domain of g(x).

4. Determine the domain of the combined function (f + g)(x).

continued

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Example 2

Find (f • g)(x) if = −f x x( ) 3 and = −g x x( ) 4 . Determine the domains over which f(x), g(x), and (f • g)(x) are defined.

Example 3

Find the function that results from the composition ( )( )f g x if f(x) = log x and = −g x x( ) 1 . Determine the domains of f(x), g(x), and ( )( )f g x .

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Problem-Based Task 4B.2.3: Blender Compositions

Ice cream shops often use an industrial-strength high-speed blender to mix various flavorings or toppings with milk and ice cream to create milkshakes. Unlike traditional household blenders, these powerful machines feature a long blending apparatus that mixes the ingredients. Blender manufacturers use a function called the sedimentation constant to quantify how fast a blender can mix a customer’s request in order to market their blenders to restaurant owners. The sedimentation constant also gives a qualitative measure of how creamy or thick the blend is.

The sedimentation constant c is given by the function cv

s2ω

ω( )= , in which v is the speed (in cm

per second) at which a solid topping particle travels from the blending apparatus to the edge of the

mixing container, s is the distance in centimeters the solid topping particle travels from the blending

apparatus to the edge of the mixing container, and the Greek lowercase letter ω (omega) is the

angular or rotational speed of the mixture. The rotational speed ω is measured in radians per second

and is given by the function tt

2ω

π( )= , in which t is the rotational time in seconds.

Write a composition function for the sedimentation constant c that is a function of the rotational time t. Then, use the relationship s = v • m to write the sedimentation constant c in terms of the rotational time t and the “topping migration time” m (in seconds) from the blending apparatus to the edge of the mixing container. Finally, determine the units of the sedimentation constant. What factors might affect the sedimentation constant that are not directly represented mathematically in its function? What factors

might affect the sedimentation constant that are not directly

represented mathematically in

its function?

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For problems 1–3, identify h(x) as a combination, a composition, or both.

1. ( ) ( )= h x f x2

2. ( )( ) ( )= h x f f x

3. ( )( ) ( )= h x f f x2

For problems 4–7, f(x) = sin (x + 1) and g(x) = x + 1. Write the equation of the combination or composition given and determine its domain.

4. ( )

f

gx and ( )

g

fx

5. ( )( )−f g x and ( )( )−g f x

6. ( )( )f g x and ( )( )g f x

7. ( )( )f f x and ( )( )g g x

Practice 4B.2.3: Combining Functions

continued

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Use the given information to complete problems 8–10.

An animator using CGI software to create a 3-D animation starts out by drawing a wire frame, or grid, to represent the object being animated. The wire frame outlines the basic geometric shape that surrounds all the parts of the object. The animator’s latest project is to create an animation of a hobby rocket that is launched and follows an erratic path above the ground until it burns out after 10 seconds. The part of the wire frame that represents the space formed by the rocket’s path from launch to landing can be modeled by a rectangular solid. The solid’s base has a length of x and a width of x + 20; the solid’s height is given by 3x – 1. The rectangular solid’s wire frame is no larger than the flight path—that is, there is no extra margin of space around the path. After t seconds, the rocket is at the corner of the rectangular space that is diagonally across from the corner of the rectangular space from which it was launched. The variable x is related to the time of flight by the function x(t) = 50 • t. (Note: The time for the rocket to fall back to Earth is not a factor in this problem.)

8. Write a function V(x) for the volume of the wire space formed by the rocket.

9. Write a function V(t) for the volume of the wire space as a function of the flight time of the rocket.

10. What is the volume of the wire space in which the rocket could have flown at t = 5 seconds? What is the maximum volume of the space during the rocket’s flight?

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UNIT 4B • MATHEMATICAL MODELING AND CHOOSING A MODELLesson 3: Comparing Properties Within and Between Functions

U4B-155

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Warm-Up 4B.3.1

The exponential function N(t) = 5(1 – 2–t), graphed as follows, describes the number of breeding pairs of cougars that can be supported by a nature preserve. The independent variable t represents the time in years. Use the function equation and the graph to complete the problems.

–2 2 4 6 8 10

10

8

6

4

2

–2

t

N(t) = 5(1 – 2– t)

N(t)

1. What is the domain of t?

2. Based on the graph and the domain, describe what happens to the function values as time increases.

3. Rewrite the function with a positive exponent.

4. Use the result of problem 3 to describe what happens to the function values as t increases.

5. What are the maximum and minimum function values? Explain what these values mean within the context of the problem.

Lesson 4B.3.1: Reading and Identifying Key Features of Real-World Situation Graphs

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The function for finding the amount of time it takes two workers to complete a job is given by

= +t t t

1 1 1

both 1 2

. Compare this to the function = +1 1 1

2 1 2f d d, which relates the focal length f

2 of a lens

to the distances between the lens and an object (d1) and between the lens and the image of the

object created by the lens (d2). F

2 and F

1 represent the focal points of the two-sided lens. How are the

domains and ranges of these two functions alike and different? Solve each function for the function

variable on the left side of each equation. How do the variable relationships differ?

1. State the restrictions, if any, on the domain and range of the first function, tboth

.

2. State the restrictions, if any, on the domain and range of the second function, f2.

3. Solve the function = +t t t

1 1 1

both 1 2

for tboth

.

4. Solve the function = +1 1 1

2 1 2f d d for f

2.

5. Use the restrictions on the domain and range of each function to compare the relationships between the variables in the functions.

continued

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Example 2

Working together, carpenters Benjamin and Ava can build a table in 4 hours. Use the function

=+

tt t

t t

•both

1 2

1 2

to create a graph and to find the time required for each carpenter to build a table

separately if Benjamin takes t hours and Ava takes t + 2 hours.

Example 3

The diagram shows the distance (d1) between an object and a lens and the distance (d

2) between the

lens and the image it creates of the object. F1 and F

2 represent the focal points of the two-sided lens.

(Note that these points are symmetric about the lens at line l.) Use the diagram and the function

=+•

11 2

1 2

fd d

d d to find d

1 and d

2 if the focal length f

1 of the lens is –1 and d

2 = d

1 + 2.

f1

f2

F1

F2

d1

d2

0

Object

Image

Lensl

continued

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Example 4

The table shows the number of earthquakes of six different average magnitudes detected over the course of a year by the Earthquake Hazards Program of the U.S. Geological Survey.

Magnitude, m 6.0 6.5 7.0 7.5 8.0 8.5Number, N 210 56 15 3 1 0

Use a graphing calculator to find a natural logarithm function that fits this data. Then choose the scales for a graph of the data based on real-world factors. (Note: The earthquake magnitude is the power of an exponent, which can be positive, negative, or 0.) Describe the shape of the resulting graph.

Example 5

Modify the earthquake data from Example 4 (repeated below) by adding a second dependent variable in the graphing calculator table for the natural logarithm of the number of earthquakes—ln N—of each value of m. Graph this new variable and the earthquake magnitude m and describe the resulting graph. Suggest an explanation for the shape and domain of the new graph given that the earthquake magnitude m is defined as the power of an exponent. What restrictions exist on m?

Magnitude, m 6.0 6.5 7.0 7.5 8.0 8.5Number, N 210 56 15 3 1 0

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Problem-Based Task 4B.3.1: Tracking Carbon Dioxide

Scientists at the Mauna Loa Observatory in Hawaii have been tracking the levels of carbon dioxide in the atmosphere since the 1950s. The table shows the scientists’ measurements of the amount of carbon dioxide in the atmosphere in parts per million over a six-decade period.

Year, t 1960 1970 1980 1990 2000 2010Carbon dioxide, C 315 324 337 352 368 388

The concentration of carbon dioxide in the atmosphere can be modeled by the function C(t) = at + b + c • sin 20πt, in which a is the concentration in 1960 (t = 1), b is the concentration when measurements started, and c is a constant that indicates by how much the carbon dioxide concentration varies during each year of the decade. The argument of the sine function shows that the carbon dioxide fluctuates from a maximum to minimum value annually.

Use a graphing calculator to analyze the data in the table and fit an equation to it. Graph the table data and compare it to the given function, C(t) = at + b + c • sin 20πt. How are they alike and how are they different? Modify the function for C(t) by adding a sine term in which the annual fluctuation of the carbon dioxide concentration is 3 parts per million. Finally, use the function to predict the carbon dioxide concentration in April 2019.

Use the function to predict the carbon dioxide

concentration in April 2019.

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The following graph shows the rate at which different-sized groups of 2, 4, 8, and 16 fashion designers—a(t), b(t), c(t), and d(t), respectively—created 20 different but specific designs for a new line of clothing made from biodegradable natural fabrics and recyclable synthetic materials. The vertical axis represents the number of designs completed, and the horizontal axis represents the time in days. Use the graph to complete problems 1–3.

0.5 1 1.5 2 2.5 3

x

24

22

20

18

16

14

12

10

8

6

4

2

0

y

d(t)

b(t)c(t)

a(t)

1. Describe the change in the number of designs completed for all four groups from time t = 0 to t = 1.

2. Describe the change in the number of designs completed for all four groups from time t = 2 to t = 3.

3. Describe how the number of completed designs changes as the group size increases.

Practice 4B.3.1: Reading and Identifying Key Features of Real-World Situation Graphs

continued

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The following graph shows the annual consumption of coal by Asian nations and Oceania from 2001 through 2010 in comparison to coal consumption in North America over the same time span. Use the graph to complete problems 4–6.

20 4 6 8 10 12

x

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

y

Asia and Oceania

North America

4. What kind of function would best represent both sets of data?

5. Describe the change in function values across the time interval.

6. At what time was the coal consumption of Asian nations and Oceania 3 times that of North America?

continued

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Scientists at the Jet Propulsion Laboratory at the California Institute of Technology track the number of asteroids passing close to Earth as detected by an unmanned satellite called the Wide-Field Infrared Survey Explorer (WISE). Such near-Earth objects (NEOs) are grouped into categories according to their diameters. The following graph shows a plot of four categories of NEOs. The table shows the NEO categories c, the diameter ranges in meters, and the number n of NEOs in hundreds detected over a given time period. Use the graph and table to complete problems 7–10.

10 2 3 4 5

x

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

y

Category, c 1 2 3 4NEO diameter range (meters) 100–300 300–500 500–1,000 > 1,000Number, n (hundreds) 163 24 15 9

7. What does the graph imply about the number of detected NEOs that are less than 100 meters in diameter?

8. What does the graph imply about the number of detected NEOs that are greater than 1,000 meters in diameter?

9. Explain how and why both an exponential and a linear function can be used to describe this data.

10. Compare the exponential and the linear model for this data and draw a conclusion about which function would “best” describe the data.

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Warm-Up 4B.3.2

For a given week at a local gas station, the average price of gasoline was $3.50 per gallon, and the average price of diesel was $3.30 per gallon. The following table shows the price in dollars for various purchases of equal amounts of gasoline and diesel, where x represents the number of gallons purchased, g(x) is the function for gasoline purchases, and d(x) is the function for diesel purchases.

x 5 10 15 20g(x) 17.50 35 52.50 70d(x) 16.50 33 49.50 66

1. How does the value of d(5) – d(15) compare to the value of d(15) – d(5)?

2. How does the value of g(5) – g(15) compare to the value of g(15) – g(5)?

3. Compare g(x) – d(x) to the value of d(x) – g(x) for two of the given values of x.

4. How do the values of g(x) and d(x) change when x increases by 1?

Lesson 4B.3.2: Calculating Average Rates of Change

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Compare the average rate of change of the function f(x) = sin x over the restricted domain [0°, 45°] with the average rate of change of the same function over the restricted domain [45°, 90°].

1. Calculate the values of f(0°) and f(45°).

2. Convert degree measures to radian measures.

3. Calculate the rate of change over the restricted domain [0°, 45°].

4. Calculate the rate of change over the restricted domain [45°, 90°].

5. Summarize the results of steps 1–4 for the two average rates of change for f(x) = sin x.

continued

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Example 2

Two school clubs are selling plants to raise money for a trip. The juggling club started out with 250 plants and had 30 left after selling them for 7 days. The mock trial club started out with 450 plants and had 50 left after selling them for 8 days. Compare the rates at which the two clubs are selling their plants. If the two clubs started selling plants at the same time, which club will sell out of plants first? Explain.

Example 3

Find the average rate of change of the function f(x) = x3 + 2x2 – x – 2 over the restricted intervals [–2, –1] and [–1, 1]. Determine if the rate of change over both intervals is positive, negative, 0, undefined, and/or constant. Use the graph of f(x) to explain how the average rate of change relates to the behavior of the graph across its whole domain.

Example 4

Three linear functions—f(x), g(x), and h(x)—share a common intersection point of (1, 1) on a coordinate plane. The function f(x) also passes through the point (–2, –3), while the function g(x) passes through the point (–2, 3), and the function h(x) passes through the point (2, 3). Which function has the greatest rate of change? What are the equations of the functions?

Example 5

The table gives the function values for f(x) = log x over the restricted domain [0.001, 1,000]. Use the data to calculate and compare the rates of change of f(x) for the intervals [0.001, 0.01], [0.01, 0.1], [0.1, 1], and [1, 10]. Describe how the rates change as the values of x become larger and smaller.

x 0.001 0.01 0.1 1 10 100 1,000f(x) –3 –2 –1 0 1 2 3

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Problem-Based Task 4B.3.2: When Seahorses Attack

The dwarf seahorse is an efficient predator of one of its favorite foods—copepods, which are tiny relatives of crayfish, lobsters, and shrimp. The seahorse’s success rate when hunting copepods is about 94 percent. Copepods move with an average speed of 500 body lengths per second; they vary from 1 to 2 millimeters in length. The ergonomic shape of the dwarf seahorse’s head limits turbulence as the seahorse moves toward its prey. When the dwarf seahorse gets to within 1 millimeter of a stationary copepod, it strikes, with a strike time of 1 millisecond. Use this data to show that the speed of the seahorse’s strike is greater than the speed of a 1-mm copepod attempting to escape.

Use this data to show that

the speed of the seahorse’s strike is greater than the speed of a 1-mm

copepod attempting to escape.

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For problems 1–4, find the average rate of change between x = 1 and x = 2 for the function given.

1. a(x) = 0.1x – 0.2

2. b(x) = 0.2x – 0.1

3. c(x) = 10–x

4. d(x) = 1 – 10–x

For problems 5–7, determine the average rate of change over the domain intervals [0.01, 0.1] and [0.1, 1] for the function given.

5. f(x) = 1 – 0.01x

6. g(x) = 1 – 0.01x3

7. = +h x x( ) 0.042

Practice 4B.3.2: Calculating Average Rates of Change

continued

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A nonprofit organization commissioned a study of the number its members who are also members of two different social network websites over a 13-month period. The following graph shows the number of organization members in hundreds who used the two social networks each month, with the function f(t) representing users of one social network and g(t) representing users of the second social network. Use the graph to complete problems 8–10.

20 4 6 8 10 12 14

x

700

650

600

550

500

450

400

350

300

250

200

150

100

50

y

f(t)

g(t)

Time in months

Soci

al m

edia

use

rs (h

un

dre

ds)

8. Describe the average rate(s) of change of the function f(t) over a 10-month period of the study and the overall average rate of change in the function value from t = 1 to t = 10.

9. Describe the average rate(s) of change of the function g(t) over the 10-month period of the study and the overall average rate of change in the function value from t = 1 to t = 10.

10. Compare the average rates of change for both functions over the period of the study, and compare the number of members who use each social network.

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Warm-Up 4B.3.3

Oceanographers use software to forecast ocean wave height. Such wave forecasts can be used to warn vessels to stay in port, or let surfers know when to grab their boards. Ocean forecasting software relies on mathematical models.

Nyala is a graduate student studying oceanography. One of her assignments for her Introduction to Ocean Modeling class is to determine and compare the average rates of change in the heights of two particular ocean waves, represented by the functions f(x) = 3 sin 2x and g(x) = 2 sin 3x. Each wave’s height is measured in meters. Use this information to solve the problems that follow.

1. Calculate the rate of change of the function f(x) over the interval π

0,12

.

2. Calculate the rate of change of the function g(x) over the interval π

0,12

.

3. Compare the results for the two functions.

Lesson 4B.3.3: Comparing Functions

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Two functions f(x) = –2(x – 5)2 + 3 and g(x) = 3 + 8x – 4x2 each represent the volume of a rectangular solid. Which solid’s function, f(x) or g(x), has the greater maximum volume?

1. Determine the coordinates of the maximum point of f(x).

2. Determine the coordinates of the maximum point of g(x).

3. Determine which function models the greatest maximum volume of its associated rectangular solid.

continued

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Example 2

Sand dune “booming” or singing refers to the audible, tone-specific sounds made by shifting sand dunes under the influence of wind. A team of biophysicists compared the loudness of several sand dunes’ booms with the decibel ratings (or sound intensity) of the booms. The table shows some of the team’s data, with x representing the loudness x in joules (energy) for each sound and f(x) representing the decibel rating.

x –0.999 –0.99 –0.9 0 9 99f(x) –1 0 1 2 3 4

Describe how the domain and function values change over the interval [–0.999, 99]. Then compare the rate of change of the function values over the restricted intervals of [–0.999, –0.99] and [9, 99].

Example 3

Use the data from the table in Example 2 (shown in the table that follows) to find the function that fits the data if the general form of the function is f(x) = a + log (x + b).

x –0.999 –0.99 –0.9 0 9 99f(x) –1 0 1 2 3 4

Example 4

Members of a Florida high school’s nature club tracked the estimated number of tree frogs living in a nearby cypress tree for 9 weeks. The estimates are based on the recorded volume of the frogs’ singing over a 24-hour period on the same day of each week. The table shows the club’s estimates for the number of frogs n per week, with the independent variable t representing time in weeks.

Estimated number of frogs, n

10 14 25 27 29 32 35 36 38

Weeks, t 1 2 3 4 5 6 7 8 9

Use a graphing calculator to graph the data. Then, use the graph to find both an exponential function of the form f(x) = a • bx and a power function of the form g(x) = a • xb to fit the data. Compare how closely each function equation models the data on the graph.

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Problem-Based Task 4B.3.3: Comparing Energy Sources

The U.S. Energy Information Administration has collected data on global energy use of five types of energy sources since 1990: renewables (such as biomass, hydroelectric, solar, tidal, and wind energy), nuclear energy, coal, natural gas, and liquid hydrocarbons (i.e., liquid natural gas and petroleum). It has also projected the use of those types of energy to the year 2030 as an aid to those developing federal energy law and policy. The graph shows the actual and projected usage amounts for each type of energy source.

50 10 15 20 25 30

t

500

450

400

350

300

250

200

150

100

50

E(t)

R(t)N(t)

C(t)

G(t)

L(t)

Renewables R(t) = 13t + 337Nuclear energy N(t) = 11t + 309Coal C(t) = 10t + 290Natural gas G(t) = 6t + 209Liquid hydrocarbons L(t) = 3t + 137

The horizontal axis runs from 1990 (t = 1) to 2030 (t = 30). The vertical axis is the amount of energy E(t) used or projected in quadrillions of BTUs, a unit of energy measurement. The specific equation for each graphed function is listed in the table.

Which fuel type has the greatest average rate of change of usage and which has the least? At what time will any one of the fuel types overtake the usage of any one of the other fuel types? How does the combined use of coal and natural gas compare with the use of renewables? At what time is the use of renewables greater than the combined use of coal and natural gas, if ever?

Which fuel type has the greatest average rate of

change of usage and which has

the least?

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The following graph shows the number of boxes of popcorn n(t) sold during a basketball game. The time t covers from 0 to 2 hours in 0.1-hour intervals. Use the graph to complete problems 1–3.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

t

10

9

8

7

6

5

4

3

2

1

0

n(t)

1. How many different maximum function values occur during the game?

2. How many different minimum function values occur during the game?

3. At what time(s) during the game do the maximum and minimum values occur?

Practice 4B.3.3: Comparing Functions

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Use the given information to complete problems 4–7.

For a class project, a group of physics students dropped a bag of sand from the top of a parking garage while another member of the group shot a video of the falling bag from the ground. Afterward, the students reviewed the video to determine how fast the bag was moving 1 second after it was dropped. It took the bag 3 seconds to hit the ground. The function h(t) = –16t2 describes the height from which the bag fell in feet, with t representing time in seconds. Find the average rate of speed of the bag in feet per second for the intervals in problems 4–6, then use these rates to answer problem 7.

4. [1, 1.1]

5. [1, 1.01]

6. [1, 1.001]

7. What is the speed of the bag at a time of t = 1 second? What does the speed function at any time t appear to be, and how are its features related to those of the distance function for any time t?

For problems 8–10, compare the average rates of change over the interval [1, 1.1] for each function pair. Use a calculator to estimate the rates.

8. a(x) = –1 + ex; b(x) = sin x

9. c(x) = x2; d(x) = x

10. e(x) = 1 + log x; =f x x( )

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UNIT 4B • MATHEMATICAL MODELING AND CHOOSING A MODELLesson 4: Choosing a Model

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Warm-Up 4B.4.1

The graph shows three functions, f(x) = 5x, g(x) = 5x2, and h(x) = 5x, which are competing models used by a field zoologist in the Great Plains to describe the increase in a prairie dog population. The zoologist would like to know which model best describes the increasing population. Use the graph to complete the problems that follow.

–1 –0.5 0.5 1 1.5 2

x

6

5

4

3

2

1

0

–1

–2

–3

y

f(x) = 5x

g(x) = 5x2

h(x) = 5x

1. Compare the average rate of change of the three functions over the domain interval [0, 1], which corresponds to 1 year.

2. Compare the y-intercepts of the functions. What do they mean in the context of the problem?

3. What is the solution for the system of all three functions and what is its meaning in the context of the problem?

Lesson 4B.4.1: Linear, Exponential, and Quadratic Functions

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A function commonly used to represent bounded growth of a real-world function value is the

exponential-linear combination function of the general form f(x) = a(1 – b–x), in which a represents

the horizontal asymptote y = a toward which the values of f(x) converge. Use the following graph to

determine a and b using the point 1,8

3

, then write an exponential-linear combination function

that models the graph.

1 2 3 4 5

x

5.5

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

y

f(x)

1. Use the graph to identify the asymptote toward which the function values converge as the values of x increase.

2. Determine the value of a in the general form of the combination function.

3. Update the general form of the combination function with the value of a.

4. Find f(0) algebraically and compare it to the graph.

5. Use the point 1,8

3

to find the value of b.

6. Write the exponential-linear combination function.

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Example 2

The graph models the distance, s(t), traveled by a speeding car moving at a constant rate r of 80 miles

per hour, and the distance covered by a state patrol car, p(t), until the patrol car intercepts the

speeding car after 5 seconds (t). The general equations for the cars are s(t) = rt for the speeding car and

p t at( )1

22= for the state patrol car, in which a is the acceleration in miles per second squared. Write the

specific equations for the two distances at a time of 5 seconds, and determine appropriate horizontal

and vertical axis scales on the graph to reflect the given problem conditions.

x

y

s(t)p(t)

Time (seconds)

Dis

tanc

e (m

iles)

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Example 3

Cloud storage refers to keeping data on a remote database instead of your own computer or mobile device in order to save hard drive space. Cloud storage users pay for this service either as part of a data plan subscription or separately. Two market research teams used function models to describe the declining cost of cloud-based computer-storage services. Team A used an exponential function, represented in the graph by A(t); Team B used a quadratic function, represented by B(t). The graphs of the functions are shown over a 10-month period with “the present” represented by the region at which both models reach a minimum function value. The y-axis represents the price in dollars per gigabyte of storage. Domain values that are greater than those represented by the minimum points are projections based on the models. Describe how well the two models fit the data of the changing cost of cloud storage. Use function characteristics to compare the models. Describe the limitations of these models for predicting future changes in the cost of cloud services.

1 2 3 4 5 6 7 8 9 10

x

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

y

A(t)

B(t)

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Problem-Based Task 4B.4.1: Comparing Social Media Growth

An Internet market research company compared the growth in users of a popular social media website in the United States with the site’s user-base growth in the rest of the world. The graph and table show the number of active site users (in millions) for each quarter of a year from 2010 through the third quarter of 2013.

The market researchers want to compare and project the growth in the two markets using exponential, linear, and quadratic function models. Use the table data and a graphing calculator to generate exponential, linear, and quadratic function models for the United States users and the non-United States users. Then, graph the models and see how they compare to the data points shown on the graph. Which function model would be best for each set of data points?

Quarter and year

Users in millionsU.S. Non-U.S.

Q1 2010 10 30Q2 2010 11 40Q3 2010 12 49Q4 2010 13 54Q1 2011 18 68Q2 2011 24 85Q3 2011 27 101Q4 2011 30 117Q1 2012 35 138Q2 2012 38 151Q3 2012 41 167Q4 2012 45 185Q1 2013 49 204Q2 2013 50 218Q3 2013 51 232

Q2‘10

Q4‘10

Q2‘11

Q4‘11

Q2‘12

Q4‘12

Q2‘13

Q4‘13

Q2‘14

Q4‘14

Q1‘10

Q3‘10

Q1‘11

Q3‘11

Q1‘12

Q3‘12

Q1‘13

Q3‘13

Q1‘14

Q3‘14

x

240

220

200

180

160

140

120

100

80

60

40

20

0

y

Time (in quarters of a year)

Use

rs (i

n m

illio

ns)

U.S.

Non-U.S.

Which function model would be

best for each set of data points?

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A study of Wi-Fi data speeds and the distance at which the signals can be detected produced the average results shown in the table. (“Distance” refers to how far the Wi-Fi signal traveled from the device transmitting it to the device receiving the signal.) Each speed value s can be written as an ordered pair (0, s) and each distance value d can be written as an ordered pair (d, 0). Use the table to complete problems 1–3.

Wi-Fi speed (megabits per second)

Distance (meters)

450 18110 5550 91

1. If a line is drawn connecting the pairs of distances and speeds on a graph, what would the slope of that line represent? (Hint: It is an average rate of change.)

2. Calculate the slopes of the lines connecting each distance-speed pair of values, and explain what the different values mean in the context of the problem.

3. Which function model—exponential, linear, or quadratic—would best describe the relationship between distance and speed? Explain your answer.

For problems 4–10, determine the type of function—exponential, linear, or quadratic—that is most appropriate to model the situation described. Then, use the chosen function model to complete the problem.

4. A credit card company compounds annual interest daily. What is the domain for a calculation of the interest rate for 1 year that is not a leap year?

Practice 4B.4.1: Linear, Exponential, and Quadratic Functions

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5. The school physical education program requires students to be timed for a 50-yard run. What is the range of the average student speeds?

6. A bowler starts out with a league average of 160 points per game and then has an average of 164 and 166 points per game after the next two weeks, respectively. What is the asymptote of the bowler’s average if this trend continues and only whole numbers are used to express the average?

student workbook with scaffolded practice unit 4b...speed. then, determine how far she had traveled...

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