Lesson 8-8 Exponential Growth and Decay 437

Exponential Growth and DecayLesson Preview

Part 1 Exponential Growth

In 1990, Florida’s population was about 13 million. Since 1990, the state’spopulation has grown about 1.7% each year. This means that Florida’s populationis growing exponentially.

To find Florida’s population in 1991, multiply the 1990 population by 1.7% and addthis to the 1990 population. So the population in 1991 is (1.7% + 100%) of the1990 population, or 101.7% of the 1990 population. Here is a function that modelsFlorida’s population since 1990.

population in millions

y = 13.0(1.017)x number of years since 1990

101.7% as a decimal

The following is a general rule for modeling exponential growth.

8-88-8

Check Skills You’ll Need (For help, go to Lesson 4-3.)

Use the formula I ≠ prt to find the interest for principal p, interest rate r, andtime t in years.

1. principal: $1000; interest rate: 5%; time: 2 years $100

2. principal: $360; interest rate: 6%; time: 3 years $64.80

3. principal: $2500; interest rate: 4.5%; time: 2 years $225

4. principal: $1680; interest rate: 5.25%; time: 4 years $352.80

5. principal: $1350; interest rate: 4.8%; time: 5 years $324

New Vocabulary • exponential growth • growth factor • compound interest• interest period • exponential decay • decay factor

What You’ll LearnTo model exponentialgrowth

To model exponentialdecay

. . . And WhyTo find the balance of a bank account, as in Examples 2 and 3

OBJECTIVE

2

OBJECTIVE

1

1OBJECTIVE

1Interactive lesson includes instant self-check, tutorials, and activities.

Exponential Growth

Key Concepts Rule Exponential Growth

can be modeled with the function

y = a ? bx for a . 0 and b . 1.

starting amount (when x = 0)

y = a ? bx exponent

The base, which is greater than 1, is the growth factor.

Exponential growth

ConnectionReal-World

In 2000, Florida’s populationwas about 16 million. Roughly23% of the population wasunder the age of 18.

1. Plan

Lesson Preview

Check Skills You’ll Need

Proportions and Percent EquationsLesson 4-3Exercise 53Extra Practice, p. 705

Lesson Resources

Teaching ResourcesPractice, Reteaching, Enrichment

Reaching All StudentsPractice Workbook 8-8Spanish Practice Workbook 8-8Technology Activities 8Hands-On Activities 19Basic Algebra Planning Guide 8-8

Presentation Assistant Plus!Transparencies• Check Skills You’ll Need 8-8• Additional Examples 8-8• Student Edition Answers 8-8• Lesson Quiz 8-8PH Presentation Pro CD 8-8

Computer Test Generator CD

TechnologyResource Pro® CD-ROM Computer Test Generator CDPrentice Hall Presentation Pro CD

www.PHSchool.comStudent Site• Teacher Web Code: aek-5500• Self-grading Lesson QuizTeacher Center• Lesson Planner• Resources

Plus

✓

8-88-8

437

Before the LessonDiagnose prerequisite skills using:• Check Skills You’ll Need

During the LessonMonitor progress using:• Check Understanding• Additional Examples• Standardized Test Prep

After the LessonAssess knowledge using:• Lesson Quiz• Computer Test Generator CD

Ongoing Assessment and Intervention✓✓

Modeling Exponential Growth

Medical Care Since 1985, the daily cost of patient care in community hospitals inthe United States has increased about 8.1% per year. In 1985, such hospital costswere an average of $460 per day.

a. Write an equation to model the cost of hospital care.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of years since 1985.Let y = the cost of community hospital care at various times.Let a = the initial cost in 1985, $460.Let b = the growth factor, which is 100% + 8.1% = 108.1% = 1.081.

Write y = 460 ? 1.081x

b. Use your equation to find the approximate cost per day in 2000.

y = 460 ? 1.081x

y = 460 ? 1.08115 2000 is 15 years after 1985, so substitute 15 for x.

< 1480 Use a calculator. Round to the nearest dollar.

The average cost per day in 2000 was about $1480.

a. Suppose your community has 4512 students this year. The student population isgrowing 2.5% each year. Write an equation to model the student population.

b. What will the student population be in 3 years?

When a bank pays interest on both the principal and the interest an account hasalready earned, the bank is paying An is thelength of time over which interest is calculated.

Compound Interest

Savings Suppose your parents deposited $1500 in an account paying 6.5% interestcompounded annually (once a year) when you were born. Find the account balanceafter 18 years.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of interest periods.Let y = the balance.Let a = the initial deposit, $1500.Let b = 100% + 6.5% = 106.5% = 1.065.

Write y = 1500 ? 1.065x

= 1500 ? 1.06518 Once a year for 18 years is 18 interest periods.Substitute 18 for x.

< 4659.98 Use a calculator. Round to the nearest cent.

The balance after 18 years will be $4659.98.

a. Suppose the interest rate on the account in Example 2 was 8%. How muchwould be in the account after 18 years? $5994.03

b. Another formula for compound interest is B = p(1 + r)x, where B is thebalance, p is the principal, and r is the interest rate in decimal form. Use thisformula to find the balance in the account in part (a). $5994.03

c. Critical Thinking Explain why the two formulas for finding compound interestare actually the same.

Check Understanding 22

EXAMPLEEXAMPLE22

interest periodcompound interest.

Check Understanding 11

EXAMPLEEXAMPLE11

Calculator Hint

To evaluate 460 ? 108115, press

460 1.081

15 .

438 Chapter 8 Exponents and Exponential Functions

���� �

Deposit $1500

Interest compounded annually 6.5%

Balance after 18 years $4659.98

a. y ≠ 4512 ?1.025x b. about 4859 students

(1 ± r) is the same as 100% ± 100r% written as a decimal.

438

2. Teach

Math Background

Exponential functions are widelyused to model many types ofgrowth and decay. The graph ofan exponential growth functionrises from left to right at an ever-increasing rate while that of anexponential decay function fallsfrom left to right at an ever-decreasing rate.

Teaching Notes

Teaching Tip

Even though students mayunderstand the word exponent,they may not understand whatgrowing exponentially means.Have students extend this table.

Multiply by 2 Square2 24 48 16

64 256

Continue until the student sees that the geometric sequenceformed with the common ratio 2grows much more slowly than thesequence formed by squaring(using the exponent 2).

Alternative Method

Have students solve the problemusing the [TABLE] function on agraphing calculator. First put theequation into . Then press2nd [TABLE]. Use the arrows toscroll to x = 18. The amount inthe y-column is 4660. Theamounts in the y-column havebeen rounded to the nearesttenth. Ask students to find how long it took to double the amount deposited. Guidestudents to look in the y-column for the amount closest to 3000. a little over 11 years

Y=

EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

OBJECTIVE

1

Reaching All StudentsBelow Level Have students draw a treediagram illustrating the following: oneperson sends an e-mail to two friends;then each person forwards the e-mailto two friends, and so on.

Advanced Learners Ask students toexplain whether the consumption perperson of whole milk in the UnitedStates as modeled in Example 5 willever reach 0 gal/person.

English LearnersSee note on page 440.Error PreventionSee note on page 441.

When interest is compounded quarterly (four times per year), you divide theinterest rate by 4, the number of interest periods per year. To find the number ofpayment periods, you multiply the number of years by the number of interestperiods per year.

Compound Interest

Savings Suppose the account in Example 2 paid interest compounded quarterlyinstead of annually. Find the account balance after 18 years.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of interest periods.Let y = the balance.Let a = the initial deposit, $1500.

Let b = 100% +There are 4 interest periods in 1 year,so divide the interest into 4 parts.

= 1 + 0.01625 = 1.01625

Write y = 1500 ? 1.01625x

= 1500 ? 1.0162572 Four interest periods a year for 18 years is 72 interest periods. Substitute 72 for x.

< 4787.75 Use a calculator. Round to the nearest cent.

The balance after 18 years will be $4787.75.

a. Suppose the account in Example 3 paid interest compounded monthly. Howmuch money would be in the account after 18 years? $4817.75

b. You deposit $200 into an account earning 5%, compounded monthly. How muchwill be in the account after 1 year? After 2 years? After 5 years?

Part 2 Exponential Decay

The graphs at the right show exponentialgrowth and exponential decay. For exponential growth, as x increases,y increases exponentially. For exponential decay, as x increases, y decreases exponentially.

O

y

4

x2 4

2

exponentialgrowth

y � 2(1.5)x

6

6

8

exponentialdecay

y � 2(0.5)x

Check Understanding 33

6.5%4

EXAMPLEEXAMPLE33

Annual Interest Rate of 8%

Compounded

annually

Periods per Year

1

Interest Rate per Period

8% every year

semi-annually 2 8%2 � 4% every 6 months

quarterly 4 8%4 � 2% every 3 months

monthly 12 8%12 � 0.6% every month

1OBJECTIVE

2 Exponential Decay

Lesson 8-8 Exponential Growth and Decay 439

���� �

Deposit $1500

Interest compounded quarterly 6.5%

Balance after 18 years

$4787.75

$210.23; $220.99; $256.67

439

Teaching Tip

Ask: How much more interest didthe account in Example 3 earnthan the account in Example 2?$127.77

Additional Examples

In 1998, a certain town had apopulation of about 13,000 people.Since 1998, the population hasincreased about 1.4% a year.a. Write an equation to modelthe population increase. y ≠ 13,000 ? 1.014x

b. Use your equation to find theapproximate population in 2006.about 14,529 people

Suppose you deposit $1000 in a college fund that pays 7.2%interest compounded annually.Find the account balance after 5 years. about $1415.71

Suppose the account in theabove problem paid interestquarterly instead of annually. Find the account balance after 5 years. about $1428.75

33

22

11

EXAMPLEEXAMPLE33

440 Chapter 8 Exponents and Exponential Functions

A real-world example of exponential decay is radioactive decay, in whichradioactive elements break down by releasing particles and energy.

Medicine The half-life of a radioactive substance is the length of time it takes forone half of the substance to decay into another substance. To treat some forms ofcancer, doctors use radioactive iodine. The half-life of iodine-131 is 8 days. Apatient receives a 12-mCi (millicuries, a measure of radiation) treatment. Howmuch iodine-131 is left in the patient 16 days later?

In 16 days, there are two 8-day half-lives.

After one half-life, there are 6 mCi left in the patient.

After two half-lives, there are 3 mCi left in the patient.

a. How many half-lives of iodine-131 occur in 32 days? 4 half-livesb. Suppose you start with a 50-mCi sample of iodine-131. How much iodine-131 is

left after one half-life? After two half-lives? 25 mCi; 12.5 mCic. Chemistry Cesium-137 has a half-life of 30 years. Suppose a lab stored a 30-mCi

sample in 1973. How much of the sample will be left in 2003? In 2063?

The function y = a ? bx can model exponential decay as well as exponential growth.

When a number is decreased by 5%, the result is 95% of the original number.So when you find the decay factor, think 100% minus the percent a number is decreasing.

Modeling Exponential Decay

Milk Consumption Since 1980, the number of gallons of whole milk each person inthe United States drinks each year has decreased 4.1% each year. In 1980, eachperson drank an average of 16.5 gallons of whole milk per year.

a. Write an equation to model the gallons of whole milk drunk per person.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of years since 1980.Let y = the consumption of whole milk, in gallons.Let a = 16.5, the initial number of gallons in 1980.Let b = the decay factor, which is 100% - 4.1% = 95.9% = 0.959.

Write y = 16.5 ? 0.959x

EXAMPLEEXAMPLE55

Check Understanding 44

EXAMPLEEXAMPLE Real-World Problem Solving44

Key Concepts Rule Exponential Decay

The function y = a ? bx models exponential decay for a . 0 and 0 , b , 1.

starting amount (when x = 0)

y = a ? bx exponent

The base, which is between 0 and 1, is the decay factor.

ConnectionReal-World

One cup of milk contains 300 mg of calcium. The bodyabsorbs about 32% of thecalcium in milk.

15 mCi; 3.75 mCi

Reading Math

Marie Curie(1867–1934) receivedNobel prizes in physicsand chemistry for herpioneering work withradioactive elements.The curie (a unit ofradioactivity) is namedfor Marie Curie.

440

Teaching Notes

English Learners

Help student understand the termdecay. Explain that decay is theopposite of growth; it’s a breakingdown or decrease. Elicit examplessuch as tooth decay or an orangeturning moldy and decaying. Pointout that exponential decay is theopposite of exponential growth.The same model can be used bysubstituting a decay factor for thegrowth factor.

Additional Examples

Technetium-99 has a half-lifeof 6 hours. Suppose a lab has 80 mg of technetium-99. Howmuch technetium-99 is left after24 hours? 5 mg

Suppose the population of a certain endangered species has decreased 2.4% each year.Suppose there were 60 of theseanimals in a given area in 1999.a. Write an equation to modelthe number of animals in thisspecies that remain alive in thatarea. y ≠ 60 ? 0.976x

b. Use your equation to find theapproximate number of animalsremaining in 2005. about 52 animals

Closure

Ask students to explain thedifference between modelingusing an exponential growthfunction and modeling using anexponential decay function. In an exponential growthfunction, you add the percent ofincreased growth to 100% to findthe base. In an exponential decayfunction, you subtract thepercent of decreased growthfrom 100% to find the base.

55

44

EXAMPLEEXAMPLE55

OBJECTIVE

2

b. Use your equation to find the approximate consumption per person of wholemilk in 2000.

y = 16.5 ? 0.959x

y = 16.5 ? 0.95920 2000 is 20 years after 1980, so substitute 20 for x.

< 7.1 Use a calculator. Round to the nearest tenth of a gallon.

The average annual consumption of whole milk in 2000 was about 7 gal/person.

Statistics In 1990, the population of Washington, D.C., was about 604,000 people.Since then the population has decreased about 1.8% per year.a. What is the initial number of people? 604,000b. What is the decay factor? 0.982c. Write an equation to model the population of Washington, D.C., since 1990.d. Suppose the current trend in population change continues. Predict the

population of Washington, D.C., in 2010. about 420,017 people

Practice and Problem Solving

Identify the initial amount a and the growth factor b in each exponential function.

1. g(x) = 20 ? 2x 2. y = 200 ? 1.0875x 3. y = 10,000 ? 1.01x 4. f(t) = 1.5 t

5. Suppose the population of a city is 50,000 and is growing 3% each year.a. The initial amount a isj. 50,000b. The growth factor b is 100% + 3%, which is 1 + j = j. 0.03; 1.03c. To find the population after one year, you multiply 50,000 ? j. 1.03d. Complete the equation y = j ? jj to find the population after x years.e. Use your equation to predict the population after 25 years.

Each percent is an annual interest rate. In the formula y ≠ a ? bx, what value wouldyou use for b?

6. 4%1.04 7. 5%1.05 8. 3.7%1.037 9. 8.75% 10. 0.5%

Assume each interest rate below is an annual interest rate. Find the interest ratefor an account that is compounded quarterly and monthly. 11–15. See left.

11. 3% 12. 4% 13. 4.5% 14. 7.6% 15. 6.25%

Find the balance in each account.

16. $4000 principal earning 6% compounded annually, after 5 years $5352.90

17. $12,000 principal earning 4.8% compounded annually, after 7 years

18. $500 principal earning 4% compounded quarterly, after 6 years $634.87

19. $20,000 deposit earning 3.5% compounded quarterly, after 10 years $28,338.18

20. Chemistry The half-life of iodine-124 is 4 days. A technician measures a 40-mCi sample of iodine-124.a. How many half-lives of iodine-124 occur in 16 days? 4 half-livesb. How much iodine-124 is in the sample 16 days after the technician measures

the original sample? 2.5 mCi

Example 4(page 440)

Examples 2, 3(pages 438, 439)

Example 1(page 438)

Check Understanding 55

Practice and Problem SolvingFor more practice, see Extra Practice.EXERCISES

Practice by ExampleAA

Lesson 8-8 Exponential Growth and Decay 441

5c. y ≠ 604,000 ? (0.982)x

20; 2 200; 1.0875 10,000; 1.01 1; 1.5

5d. 50,000; 1.03; x

about 104,689 people

1.0051.0875

11. 0.75%, 0.25%

12. 1%; 0. %

13. 1.125%; 0.375%

14. 1.9%; 0.6 %

15. 1.5625%; 0.5208 %3

3

3

$16,661.35

441

3. Practice

Error PreventionExercises 6–10 Remind studentsthat money invested at interest‘grows.’ The base is the growthfactor, the percent of growth, as a decimal, added to 1.00.

Assignment Guide

ObjectiveCore 1–19, 31–43

Extension 54

ObjectiveCore 20–30, 44–52

Extension 53, 55

Standardized Test Prep 56–59

Mixed Review 60–63

CC

BBAA

CC

BBAA

1

2

Enrichment 8-8

Reteaching 8-8

Algebra 1 Chapter 8 Lesson 8-8 Practice 9

Name Class Date

Practice 8-8 Exponential Growth and Decay

Write an exponential function to model each situation. Find each amountafter the specified time.

1. Suppose one of your ancestors invested $500 in 1800 in an accountpaying 4% interest compounded annually. Find the account balance ineach of the following years.

a. 1850 b. 1900 c. 2000 d. 2100

2. Suppose you invest $1500 in an account paying 4.75% annual interest.Find the account balance after 25 yr with the interest compounded thefollowing ways.

a. annually b. semiannually c. quarterly d. monthly

3. The starting salary for a new employee is $25,000. The salary for thisemployee increases by 8% per year. What is the salary after each of thefollowing?

a. 1 yr b. 3 yr c. 5 yr d. 15 yr

4. Carbon-14 has a half-life of 5,700 years. Scientists use this fact todetermine the age of things made of organic material. Suppose theaverage page of a book containing approximately 0.5 mg of carbon-14 isput into a time capsule. How much carbon-14 will each page containafter each of the following numbers of years?

a. 5700 b. 11,400 c. 22,800 d. 34,200

5. The tax revenue that a small city receives increases by 3.5% per year. In1990, the city received $250,000 in tax revenue. Determine the taxrevenue in each of the following years.

a. 1995 b. 1998 c. 2000 d. 2006

6. Suppose the acreage of forest is decreasing by 2% per year because ofdevelopment. If there are currently 4,500,000 acres of forest, determinethe amount of forest land after each of the following.

a. 3 yr b. 5 yr c. 10 yr d. 20 yr

7. A $10,500 investment has a 15% loss each year. Determine the value ofthe investment after each of the following.

a. 1 yr b. 2 yr c. 4 yr d. 10 yr

8. A city of 2,950,000 people has a 2.5% annual decrease in population.Determine the city’s population after each of the following.

a. 1 yr b. 5 yr c. 15 yr d. 25 yr

9. A $25,000 purchase decreases 12% in value per year. Determine thevalue of the purchase after each of the following.

a. 1 yr b. 3 yr c. 5 yr d. 7 yr

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Practice 8-8

442 Chapter 8 Exponents and Exponential Functions

21. Chemistry The half-life of carbon-11 is 20 min.A sample of carbon-11 has 25 mCi.a. How many half-lives of carbon-11 occur in 1 hour? 3 half-livesb. How much carbon-11 is in the sample 1 hour after the original

sample is measured? 3.125 mCi

Identify the decay factor in each function.

22. y = 5 ? 0.5x 0.5 23. f(x) = 10 ? 0.1x 0.1

24. g(x) = 100 ? 25. y = 0.1 ? 0.9x 0.9

Identify each function as exponential growth or exponential decay.

26. y = 0.68 ? 2x 27. y = 2 ? 0.68x 28. y = 68 ? 2x 29. y = 68 ? 0.2x

30. Cars The value of a new car decreases exponentially. Suppose your motherbuys a new car for $22,000. The value of the car decreases by 20% each year.a. What is the initial price of the car? The decay factor? $22,000; 0.8b. Write an equation to model the value of the car x years after she buys it.c. Find the value of the car after 6 years. $5767.17

Write an exponential function to model each situation. Find each amount after thespecified time. 31. y ≠ 130,000 ? (1.01)x; about 142,179 people

31. A population of 130,000 grows 1% per year for 9 years.

32. A population of 3,000,000 decreases 1.5% annually for 10 years.

33. A $2400 principal earns 7% compounded annually for 10 years.

34. A $2400 principal earns 7% compounded monthly for 10 years.

35. Education Since 1985, the average annual cost y (in dollars) for tuition andfees at public two-year colleges in the United States has increased about 6.5%per year. In 1985, tuition and fees were an average of $584 per year.a. Write an equation to model the cost of two-year colleges. Predict the

average annual cost for 2005. y ≠ 584 ? (1.065)x; $2057.81b. Open-Ended Predict the average annual cost for the year you plan to

graduate from high school. Check students’ work.

Tell whether each graph shows a linear function, an exponential function,or neither. Justify your reasoning. 36–39. See margin.

36. 37.

38. 39.

Apply Your SkillsBB

23Q23R

x

Example 5(page 440)

exp. growth exp. decay exp. growth exp. decay

30b. y ≠ 22,000 ? (0.8)x

32. y ≠ 3,000,000 ? (0.985)x;about 2,579,191people

33. y ≠ 2400 ? (1.07)x;$4721.16

y ≠ 2400 ? (1.00583333)x; $4823.19

442

Exercises 26–29 Remind studentsthat a growth factor is greaterthan 1 and a decay factor is lessthan 1.

Exercises 36–39 Suggest studentsreview Lessons 5-3 and 8-7.

38. Exponential function; it is a curve with y-values that increase asx-values increase.

39. Neither; it decreases,then increases, unlike anexponential function.

pages 441–444 Exercises

36. Linear function; it is astraight line.

37. Neither; it is not just onestraight line.

40. linear function

Ox

60

20

2 4

y41. exponential function

Ox

60

20

2 4

y

Lesson 8-8 Exponential Growth and Decay 443

Graph the function represented in each table. Then tell whether the tablerepresents a linear function or an exponential function.

40. 41. 42.

43. Writing Would you rather have $500 in an account paying 6% interestcompounded quarterly or $600 in an account paying 5% compoundedannually? Summarize your reasoning. See margin.

How many half-lives occur in each period of time?

44. 2 days (1 half-life = 8 h) 45. 300 years (1 half-life = 75 yr)

46. Medicine The function y = 15 ? 0.84x models the amount y of a 15-mg dose of antibiotic remaining in the bloodstream after x hours.a. Estimation Use the graphing calculator screen

to estimate the half-life of this antibiotic in the bloodstream. about 4 h

b. Use your estimate to predict the fraction of the dosethat will remain in the bloodstream after 8 hours.

c. Verify your prediction by using the function to findthe amount of antibiotic remaining after 8 hours. See margin.

47. Population Growth Since 1990, the population of Virginia has grown at anaverage annual rate of about 1%. In 1990, the population was about 6,284,000.a. Write an equation to model the population growth in Virginia since 1990.b. Suppose this rate of growth continues. Predict Virginia’s population in 2010.

By which percent would you multiply a number to decrease it by the given amount?

48. 6% 94% 49. 12% 88% 50. 3.5% 96.5% 51. 53.9% 46.1%

52. a. Estimation Use the graph at the right.Estimate the half-life of cesium-134. 2 years

b. Suppose a scientist had 800 mCi ofcesium-134 in a sample. After how manyyears would the sample have 200 mCi of cesium-134? 4 years

53. Credit Card Balances Suppose you charge $250 for a new suit. If you do not pay the whole amount the first month, you are charged 1.8% monthly interest on youraccount balance. Suppose you can make a $30 payment each month.a. What is your balance after your first payment? $220.00b. How much interest are you charged after your first payment? $3.96c. What is your balance just before you make your second payment? $223.96d. What is your balance after your second payment? $193.96e. How many months will it take for you to pay off the entire bill? 9 monthsf. How much interest will you have paid in all? $18.07

ChallengeCC 420

500

400

300

200

100

6 8 10 12 14Time (years)

Lev

el (

mC

i)

Cesium-134 Decay

14

X=4.0106383 Y=7.4542313

Antibiotic Decay in theBloodstream

Xmin=0Xmax=13

Ymin=0Ymax=15

x

1

2

3

4

y

3

9

15

21

x

1

2

3

4

y

3

9

27

81

x

1

2

3

4

y

20

40

60

80

Reading Math

For help with Exercise46 go to page 445.

40–42. See margin p. 442.

4 half-lives6 half-lives

a. y ≠ 6,284,000 ? (1.01)x b. 7,667,674 people

443

4. Assess

Alternative Assessment

Group students in pairs. Instructone student to write a questionthat can be modeled by anexponential growth function.Have the other student write aquestion that can be modeled byan exponential decay function.Direct students to exchangeproblems, write the appropriatefunction, and solve.

Lesson Quiz 8-8

1. Identify the originalamount a and the growthfactor b in the exponentialfunction y = 10 ? 1.036x. a ≠ 10, b ≠ 1.036

2. A population of 24,500people has been increasingat a rate of 1.8% a year.What will be the populationin 15 years if it continues at that rate? about 32,017 people

3. Write an exponentialfunction to represent $2000 principal earning5.6% interest compoundedannually. y ≠ 2000 ? 1.056x

4. Find the account balance on $3000 principal earning6.4% interest compoundedquarterly for 7 years. about$4678.91

5. The half-life of a certainsubstance is 4 days. If youhave 100 mg of the sub-stance, how much of it will remain after 12 days?12.5 mg

6. The value of a $1200computer decreases 27%annually. What will be thevalue of the computer after3 years? about $466.82

42. linear function

Ox

20

10

2 4

y43. Answers may vary.

Sample: $600; even after10 years, there is moremoney in the accountwith an initial deposit of$600 ($977.34) than thereis in the account with an

initial deposit of $500($907.01).

46c. about 3.7 mg usingthe function and 15 mg 3 ≠ 3.75 mgusing the prediction

14

444 Chapter 8 Exponents and Exponential Functions

54. Data Collection Complete the table at the right using any ball. The height 0 is the starting height.Record the maximum height after the first, second,and third bounce. a-b. Check student’s work.a. Graph your data.b. Write an exponential decay function that models

your data.

55. On January 1, 2000, Chessville had a population of 40,000 people. Itspopulation increases 7% each year. On the same day, Checkersville had apopulation of 60,000 people. Its population decreases 4% each year. Duringwhat year will the population of Chessville exceed that of Checkersville?

56. For which function will values of y decrease as values of x increase? CA. y = 12.5(1.325)x B. y = 300(1.06)x

C. y = 5000(0.98)x D. y = 1.02x

57. Suppose you deposit $1000 in an account earning 6% interest. You makeno further deposits to the account and interest is compounded semi-annually. What is the balance after 5 years? HF. $538.62 G. $1006.00 H. $1343.92 I. $1790.85

58. Read the passage below and answer the following problem.

Suppose $24 had been invested in 1626 in an account paying 4.5% interestcompounded annually. Which amount is closest to the balance in 2000? AA. $339 million B. $89 million C. $9400 D. $8900

59. Which is greater, the amount in an account that pays 5% interestcompounded quarterly for 5 years or the amount in an account that pays5.5% compounded annually for 5 years? Assume the accounts start withthe same amount. Show your work. See left.

Graph each function. 60–62. See margin.

60. y = 2 ? 10x 61. f(x) = 100 ? 0.9x 62. g(x) = ? 0.1x

63. Geography In 2000, about 1.4 3 104 ships passed through the Panama Canal.About 5.2 3 107 gallons of water flow out of the canal with each ship. Abouthow many gallons of water flowed out of the canal with ships in 2000? Writeyour answer in scientific notation. 7.28 3 1011 gal

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Lesson 8-7

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Reading Comprehension

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Standardized Test Prep

Mixed ReviewMixed Review

2003

59. [2] Using $1000 fordeposit, quarterly:1000(1 ± )20

N 1282.04; annually: 1000(1 ± 0.055)5 N1306.96. The accountpaying 5.5% will begreater (OR equivalentexplanation).[1] correct approach

with minorcomputational error

0.054

Take It to the NETOnline lesson quiz atwww.PHSchool.com

Web Code: aea-0808

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Standardized Test Prep

ResourcesFor additional practice with avariety of test item formats:• Standardized Test Prep, p. 451• Test-Taking Strategies, p. 446• Test-Taking Strategies with

Transparencies

Exercise 56 Remind students thatthe y-values decrease as x-valuesincrease in exponential decayfunctions, but they increase inexponential growth functions.

pages 441-444 Exercises

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Reading Math Reading a Graph 445

Read the exercise below, and then learn how to use a graphing calculator tosolve it. Check your understanding by solving the exercise at the bottom of the page.

Medicine The function y = 15 ? 0.84x models the amount y of a 15-mg dose ofantibiotic remaining in the bloodstream after x hours.a. Estimation Use the graphing calculator screen to estimate the half-life of this

antibiotic in the bloodstream.b. Use your estimate to predict the fraction of the dose that will remain in the

bloodstream after 8 hours.c. Verify your prediction by using the function to find the amount of antibiotic

remaining after 8 hours.

To solve this problem, use the graphing calculator screen shown at the left.Parts of this screen are explained below.

A The point shown on the screen is a point (x, y) on the graph.B and C These are the x-coordinates and y-coordinates of the

highlighted point. The coordinates of the point are (4.0106383, 7.4542313).

D and E These give the domain and range for the x- and y-axes. Thescreen displays the graph for x-values from 0 to 13 and y-values from 0 to 15.

a. Half-life is the time required for the body to eliminate half of the initial dose.How do the coordinates (4.0106383, 7.4542313) relate to the half-life of theantibiotic? At time x = 0, the amount in the bloodstream is y = 15. When y = = 7.5, the corresponding value of x represents the half-life. When y < 7.5, x < 4, so the half-life is about 4 hours.

b. If 7.5 mg remain after 4 hours, then = 3.75 mg will remain after 8 hours.

c. y = 15 ? 0.84x

y = 15 ? 0.848 Substitute 8 for x.y < 3.72 Use a calculator.

The amount of antibiotic remaining after 8 hours will be about 3.72 mg.

EXERCISEMemory Suppose the function y = 40 ? 0.75x models the number y of foreign-language words recalled from a list of 40 words after x weeks (without additionalpractice or study).a. Estimation Use the graphing calculator to estimate the number of vocabulary

words recalled after 5 weeks. about 9 or 10 wordsb. Verify your prediction by using the function to find the number of vocabulary

words recalled after 5 weeks. about 9.49 or 9 words

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Antibiotic Decay in theBloodstream

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Reading a Graph

Reading GraphsStudents must be able to make ameaningful connection betweenthe graph, the function, and thesituation in the problem. They willuse the graph and the coordinatesof points to answer questions.They will use the graphingcalculator to find coordinates ofpoints on the graph. This featurehelps students see that a graph isa useful tool to make estimatesand answer questions.

Teaching Notes

Remind students to look at thefunction and try to make sense ofit by asking themselves questions.In the example they could ask:What does this function tell meabout the amount of medicine inthe bloodstream after x numberof hours? Do I expect the amountto decrease or increase over time?Does y decrease as x increases inthis example? How do I know?What is the amount of medicinewhen x = 0? Does that answermake sense? These questions helpstudents make sense of thefunction and should also helpthem read the graph.

Error PreventionSome students may think thatraising a number to a poweralways gives a number larger thanthe starting number. If they don’tunderstand how the function inthe first example can model anamount that is decreasing, havethem try some simple examples such as A B2 and A B3. They should see that the values decrease as theexponents increase.

ExerciseHave students work independentlyon a similar problem. Then havethem verify their predictions bysubstituting into the function, afterwhich they should pair up to sharetheir answers with each other. Youcan take note of the kinds ofdiscussions students have with eachother to assess their understanding.

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