Algebra 1 Part 2 – Review of Exponents



Math 1: Notes on Exponent Rules

Rule 1: Multiplying Powers With the Same Base – __________________________________ __________

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic]

❖ Rule 2: Dividing Powers With the Same Base –

6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic]

❖ Rule 3: Zero Power Property –

11. [pic] 12. [pic] 13. [pic]

14. [pic] 15. [pic]

❖ Rule 4: Negative Exponents –

16. [pic] 17. [pic] 18. [pic]

19. [pic] 20. [pic]

21. [pic] 22. [pic] 23. [pic]

24. [pic] 25. [pic]

❖ Rule 5: Raising a Power to a Power –

26. (36)2 27. (n2)5 28. [pic]

29. (a-4)7 30. T2(T7)-2

❖ Rule 6: Raising a Product or a Quotient to a Power –

31. (5d2)3 32. (x2y)4 33. (4g5)-2

34. (-4pq2r3)2 35. (-2a2b)3(a2b6)4

36. [pic] 37. [pic] 38. [pic]

39. [pic] 40. [pic]

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Growing Sequences

Problem Situation: The Brown Tree Snake

The Brown Tree Snake is responsible for entirely wiping out over half of Guam’s native bird and lizard species as well as two out of three of Guam’s native bat species. The Brown Tree Snake was inadvertently introduced to Guam by the US military due to the fact that Guam is a hub for commercial and military shipments in the tropical western Pacific. It will eat frogs, lizards, small mammals, birds and birds' eggs, which is why Guam’s bird, lizard, and bat population has been affected. Listed in the table below is the data collected on the Brown Tree Snake’s invasion of Guam.

The number of snakes for the first few years is summarized by the following sequence:

1, 5, 25, 125, 625, . . .

• What are the next three terms of the sequence?

• How did you predict the number of snakes for the 6th, 7th, and 8th terms?

• What is the initial term of the sequence?

• What is the pattern of change?

• Do you think the sequence above is an arithmetic sequence? Why or why not?

Growing Sequences: Review of Arithmetic Sequences

Arithmetic sequence -

Common difference (d) -

Initial term (a1) -

Examples:

For each sequence find the initial term, common difference and next two terms.

2, 5, 8, 11, 14, . . .

7, 3, –1, –5, . . .

3, 11, 19, 27, 35, . . .

Geometric sequence –

Common ratio (r) -

Examples:

For each sequence find the initial term, common ratio and next two terms.

1, 2, 4, 8, 16, . . .

81, 27, 9, 3, 1, 1/3, . . .

1/2, 1, 2, 4, 8, . . .

2/9, 2/3, 2, 6, 18, . . .

Practice with Sequences

For a sequence, write arithmetic and the common difference or geometric and the common ratio. If a sequence is neither arithmetic nor geometric, write neither.

1) 2, 6, 18, 54, 162, ... _____________________ common __________ = ____

2) 14, 34, 54, 74, 94, ... _____________________ common __________ = ____

3) 4, 16, 36, 64, 100, ... _____________________ common __________ = ____

4) 9, 109, 209, 309, 409, ... _____________________ common __________ = ____

5) 1, 3, 9, 27, 81, ... _____________________ common __________ = ____

Given the initial term and either common difference or common ratio, write the first 6 terms of the sequence.

6) a1 = 7, r = 2 _________________________________________________

7) a1 = 7, d = 2 _________________________________________________

8) a1 = 3, r = 5 _________________________________________________

9) a1 = 4, d = 15 _________________________________________________

Guided Practice: Arithmetic and Geometric Sequences

1. You visit the Grand Canyon and drop a penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on.  What is the total distance the object will fall in 6 seconds?

a) Is this sequence arithmetic or geometric?

b) What is the initial term?

c) What is the rate of change (common ratio or common difference)?

2. The sum of the interior angles of a triangle is 180º, of a quadrilateral is 360º and of a pentagon is 540º.  Assuming this pattern continues, find the sum of the interior angles of a dodecagon (12 sides).

a) Is this sequence arithmetic or geometric?

b) What is the initial term?

c) What is the rate of change (common ratio or common difference)?

3. After knee surgery, your trainer tells you to return to your jogging program slowly.  He suggests jogging for 12 minutes each day for the first week.  Each week thereafter, he suggests that you increase that time by 6 minutes per day.  How many weeks will it be before you are up to jogging 60 minutes per day?

a) Is this sequence arithmetic or geometric?

b) What is the initial term?

c) What is the rate of change (common ratio or common difference)?

4. You complain that the hot tub in your hotel suite is not hot enough.  The hotel tells you that they will increase the temperature by 10% each hour.  If the current temperature of the hot tub is 75º F, what will be the temperature of the hot tub after 3 hours, to the nearest tenth of a degree?

a) Is this sequence arithmetic or geometric?

b) What is the initial term?

c) What is the rate of change (common ratio or common difference)?

5. A culture of bacteria doubles every 2 hours.  If there are 500 bacteria at the beginning, how many bacteria will there be after 24 hours?

a) Is this sequence arithmetic or geometric?

b) What is the initial term?

c) What is the rate of change (common ratio or common difference)?

Determine if the sequence is geometric. If it is, find the common ratio.

6. -1, 6, -36, 216, . . . 9. -1, 1, 4, 8, . . .

7. 4, 16, 36, 64, . . . 10. -3, -15, -75, -375, . . .

8. -2, -4, -8, -16, . . . 11. 1, -5, 25, -125, . . .

Classify each sequence as arithmetic or geometric. Then find the common ratio or the common difference and the next three terms.

5. -7, -3, 1, 5, ____, ____, ____

6. 2, 10, 50, 250, ____, ____, ____

7. 11, 23, 35, 47, ____, ____, ____

8. 8, 3, -2, -7, ____, ____, ____

9. 0.7, 1.5, 2.3, 3.1, ____, ____, ____

10. -3, -6, -12, -24, ____, ____, ____

11. 80, 20, 5, 1.25, ____, ____, ____

12. 4, 6, 9, 13.5, ____, ____, ____

Classify each sequence as arithmetic or geometric. Then find the 12th term.

13. 18, 7, -4, -15, …

14. 1.1, 2.2, 4.4, 8.8, …

15. 56, 28, 14, 7, …

16. 98, 101, 104, 107, …

17. 2, -6, 18, -54, …

18. 1, -1.5, -4, -6.5, …

19. 2, 5, 10, 17, 26, …

From Geometric Sequences to Tables and Graphs

The geometric sequence from the Brown Tree Snake problem (1, 5, 25, 125, 625 . . .) can be written in the form of a table, as shown below:

|Year |0 |1 |2 |3 |4 |

|# of Snakes |1 |5 |25 |125 |625 |

The Brown Tree Snake was first introduced to Guam in year 0. At the end of year 1, five snakes were found; at the end of year 2, twenty-five snakes were discovered, and so on. Since we now have a table of the information, a graph can be drawn, where the year is the independent variable (x) and the number of snakes is the dependent variable (y). See below:

Notice that the graph of the table is not a straight line. Therefore, the graph is not linear in nature, which we already from the fact that the sequence is not arithmetic. Rather, the graph is curved and moves in a growing fashion very rapidly due to the fact that the common ratio r of this sequence is 5. The curved graph of this problem situation is known as an exponential growth function. An exponential growth function occurs when the common ratio r is greater than one. Tables and graphs make viewing the data from the problem situation easier to see and we can easily see from either the table or graph that in year 3, the snake population is 125.

Let us look at a similar population growth for a certain kind of lizard in both a table and graph. Use either one or both to answer the questions below.

|Year |0 |1 |2 |3 |4 |5 |

|Population |10 |20 |40 |80 |160 |320 |

Notice from the shape of the graph that the information is exponential in nature.

1. What information does the point (2, 40) on the graph tell you?

2. What information does the point (1, 20) on the graph tell you?

3. When will the population exceed 100 lizards?

4. Explain how to find the common ratio, using either the table or graph.

5. If the information from the table were written as a sequence, what is the initial term?

6. How could we find the 10th term in the table, graph, or sequence?

The Mice Problem

A population of mice has a growth factor (otherwise known as the common ratio) of 3. After 1 month, there are 36 mice. After 2 months, there are 108 mice.

1. How many mice were in the population initially (at 0 months)? Explain how you found this number.

2. Write a sequence to show how the mice population is growing.

3. Is this sequence arithmetic or geometric? Explain how you know.

4. Now, put your sequence into the table below.

|Months |0 |1 |2 |3 |

|Number of Mice | | | | |

5. Is the graph of the table going to be a straight line or a curve? Explain your answer.

6. Graph the table to make sure of your answer on the graph. Make sure you label and title the graph.

a. What is your scale for the x-axis?

b. What is your scale for the y-axis?

Who Wants to Be Rich?

Students at a local school want to have a quiz show called Who Wants to Be Rich? Contestants will be asked a series of questions. A contestant will play until he or she misses a question. The total prize money will grow with each question answered correctly. Lucy and Pedro are on the prize winnings committee and have different view of how prize winning should be awarded. Their plans are outlined below for your consideration. Review them by answering the questions following the plans. Remember that the committee has a fixed amount of money to use for this quiz show.

1. Lucy proposes that a contestant receives $5 for answering the first question correctly. For each additional correct answer, the total prize would increase by $10.

a. For Lucy’s proposal, complete the table below.

|Number of questions |1 |2 |3 |4 |5 |6 |

|Number of Bacteria |1 |4 |16 | | | |

1. Graph the data in the table to the right. Be sure to label your graph and axes.

2. Is this graph linear or exponential?

3. Write the NOW-NEXT form to show the

pattern of growth.

NEXT = ______ ( NOW

4. What is the common ratio r?

5. Use the common ratio r to write a rule to showing how to calculate the number of bacteria y after x hours.

y = the number of bacteria produced in that hour

x = the number of hours

r = the common ratio or rate of change

a1 = the initial term of the sequence or the starting point

Use the above information to write the explicit form of the exponential function

y = a1 ( rx. Notice how similar it is to the NOW-NEXT recursive form.

NEXT = NOW ( r

y = a1 ( rx

y = 1 ( 4x

The NEXT and y components both represent the number of bacteria generated during the hour. The NOW and a1 both represent the starting point and r is the rate of change or the common ratio, which is 4 in this example.

6. Use the rule in step 6 to determine the number of bacteria in the colony after 7 hours. Verify the number of bacteria by either continuing the table in step 1 or continuing the graph in step 2.

7. After how many hours will there be at least 1,000,000 bacteria in the colony?

8. Suppose that instead of 1 bacterium, 50 bacteria land in your mouth. Write an explicit equation which describes the number of bacteria y in this colony after x hours.

9. What is different in this equation from the equation in step 6?

10. Using your new equation, determine the number of bacteria in the colony after 8 hours and after 10 hours.

11. Which method for determining the number of bacteria is easier for you? Using a table, graph, NOW-NEXT, or equation? Explain.

Guided Practice: More Bacteria: The bacteria E. coli often causes illness among people who eat the infected food. Suppose a single E. coli bacterium in a batch of ground beef begins doubling every 10 minutes.

1. Complete the table below to determine how many bacteria there will be after 10, 20, 30, 40, and 50 minutes have elapsed (assuming no bacteria die).

|10-min Period |1 |2 |3 |4 |5 |

|Number of Bacteria |2 | | | | |

2. Graph the data on the table. Be sure to title

your graph and label your axes.

3. Write two rules that can be used to calculate the number of bacteria in the food after any number of 10-minute periods.

4. What is the initial value?

5. What is the common ratio?

6. Use your rule(s) to determine the number of bacteria after 2 hours.

7. When will the number of bacteria reach 100,000?

Students at a high school conducted an experiment to examine the growth of mold. They set out a shallow pan containing a mixture of chicken broth, gelatin, and water. Each day, the students recorded the area of the mold in square millimeters. The students wrote the exponential equation m = 50(3d) to model the growth of the mold. In this equation, m is the area of the mold in square millimeters after d days.

8. What is the area of the mold at the start of the experiment?

9. What is the growth factor or common ratio?

10. What is the area of the mold after 5 days?

11. On which day will the area of the mold reach 6,400 mm2?

12. An exponential equation can be written in the form y = a(bx), where a and b are constant values.

a. What value does b have in the mold equation? What does this value represent?

b. What value does a have in the mold equation? What does this value represent?

Independent Practice: Charity Donations

Mari’s wealthy Great-aunt Sue wants to donate money to Mari’s school for new computers. She suggests three possible pans for her donations.

Plan 1: Great-aunt Sue’s first plan is give money in the following way: 1, 2, 4, 8, . . . . She will continue the pattern in this table until day 12. Complete the table to show how much money the school would receive each day.

|Day |1 |

|0 |100 |

|1 |180 |

|2 |325 |

|3 |583 |

|4 |1,050 |

A. The table shows the rabbit population growing exponentially.

1. What is the growth factor? Explain how you found your answer.

2. Assume this growth pattern continued. Write an equation for the rabbit population p for any year n after the rabbits are first counted. Explain what the numbers in your equation represent.

3. How many rabbits will there be after 10 years? How many will there be after 25 years? After 50

years?

4. In how many years will the rabbit population exceed one million?

B. Suppose that, during a different time period, the rabbit population could be predicted by the equation p = 15(1.2)n, where p is the population in millions, and n is the number of years.

1. What is the growth factor?

2. What was the initial population?

3. In how many years will the population double from the initial population?

4. What will the population be after 3 years? After how many more years will the population at 3 years double?

5. What will the population be after 10 years? After how many more years will the population at 10 years double?

6. How do the doubling time for parts (3) – (5) compare? Do you think the doubling time will be the

same for this relationship no matter where you start to count?

YOUR TURN

Let’s look at another population . . . the wolves of northern Michigan.

In parts of the United States, wolves are being reintroduced to wilderness areas where they had become extinct. Suppose 20 wolves are released in northern Michigan, and the yearly growth factor for this population is expected to be 1.2.

1. Make a table showing the projected number of wolves at the end of each of the first six years.

2. Write a NOW-NEXT equation that models the growth of the wolf population.

3. Write an explicit equation in function form that models the growth of the wolf population.

4. Using either equation, how long will it take for the new wolf population to exceed 100?

5. Using either equation, how many wolves will there be in 10 years? 15 years? 25 years?

Population Growth and Other Word Problems

The Elk Population

1) The table show that the elk population in a state forest is growing exponentially. What is the growth factor? Explain.

Growth of Elk Population

|Time (Year) |Population |

|0 |30 |

|1 |57 |

|2 |108 |

|3 |206 |

|4 |391 |

|5 |743 |

2) Suppose this growth pattern continues. How many elk will these be after 10 years? How many elk will there be after 15 years?

3) Write a NOW-NEXT equation you could use to predict the elk population p for any year n after the elk were first counted.

4) Use this equation to write an explicit equation in function notation to predict the elk population p for any year n after the elk were first counted.

5) In how many years will the elk population exceed one million?

For problems 6 and 7, write a NOW-NEXT equation and an explicit equation in function notation before find the solution(s) to the problems.

6) Suppose there are 100 trout in a lake and the yearly growth factor for the population is 1.5. How long will it take for the number of trout to double?

7) Suppose there are 500,000 squirrels in a forest and the growth factor for the population is 1.6 per year. Write an equation you could use to find the squirrel population p in n years.

8) Currently, 1,000 students attend East Garner IB Magnet Middle School. The school can accommodate 1,300 students. The school board estimates that the student population will grow by 5% per year for the next several years.

a) In how many years will the population outgrow the present building?

b) Suppose the school limits its growth to 50 students per year. How many years will it take for the population to outgrow the school?

9) Suppose that, for several years, the number of radios sold in the U.S. increased by 3% each year.

a) Suppose one million radios sold in the first year of this time period. About how many radios sold in each of the next 6 years?

b) Suppose only 100,000 radios sold in the first year. About how many radios sold in each of the next 6 years?

10) Suppose a movie ticket costs about $7, and inflation causes ticket prices to increase by 4.5% a year for the next several years.

a) At this rate, how much will tickets cost 5 years from now?

b) How much will a ticket cost 10 years from now?

c) How much will a ticket cost 30 years from now?

d) When will a ticket cost $25?

Declining Geometric Sequence Activity

Problem Situation:

The African Black Rhinoceros is the second largest of all land mammals and has been around for 40 million years. Prior to the 19th century, over 1,000,000 of the species roamed the plains of Africa; however, the number has been reduced by hunting and loss of natural habitat. The following sequence shows the population from the 1970s to early 1090s.

650,000; 195,000; 58,500; 17,550; 5,265

• What are the next three terms of the sequence?

• How did you predict the number of rhinoceros for the 6th, 7th, and 8th terms?

• What is the initial term of the sequence?

• What is the pattern of change?

• Do you think the sequence above is an arithmetic sequence? Why or why not?

• Do you think the sequence is a growing sequence? Why or why not?

Example 1: 4, 8/3, 16/9, 32/27, 64/81 . . .

Initial term:________ Common ratio:________

Example 2: 6, -3, 3/2, -3/4 . . .

Initial term:________ Common ratio:________

Guided Practice with Geometric Sequences

Determine if the sequence is geometric. If it is, find the common ratio.

1) 56, 28, 14, 7,... 2) 64, -48, 36, -27,... 3) 9, 6, 3, 0, -3, -6, …

4) 1000, 100, 10, . . . 5) 8, 2, ½, . . . 6) 18, 6, 2, . . .

Given the initial term and common ratio, write the first 6 terms of the sequence.

7) a1 = 7, r = 2/3 ___________________________________________________

8) a1 = 5, r = ½ ___________________________________________________

9) a1 = 3, r = 3/5 ___________________________________________________

10) a1 = 3/7, r = ¼ ___________________________________________________

Problem Situation:

A hot vanilla latte from McDonalds is poured into a cup and allowed to cool while you are riding to school. The difference between the latte temperature and room temperature is recorded every minute for 10 minutes. The sequence is found below:

80, 72, 65, 58, 52, 47, 43, 38, 34, 31, 28

11) Is this sequence geometric? If so, what is the approximate common ratio?

12) How is problem similar or different to the Black Rhinoceros problem in the lesson?

Independent Practice with Sequences

Are the following sequences arithmetic, geometric, or neither? If they are arithmetic, state the

value of d. If they are geometric, state r.

1. 6, 12, 18, 24, ... _______________________________________________________

2. 6, 11, 17, ... __________________________________________________________

3. 2, 14, 98, 686, ... ______________________________________________________

4. 160, 80, 40, 20, ... _____________________________________________________

5. -40, -25, -10, 5, .... _____________________________________________________

6. 7, -21, 63, -189, ... _____________________________________________________

7. 2/3, (2/3)2, (2/3)3, …___________________________________________________

8. 1/3, 4/3, 7/3, 10/3,…___________________________________________________

9. 10, 10/8, 10/64, …_____________________________________________________

10. 10, 80, 640, 5120, …____________________________________________________

11. 1/3, 8/3, 64/3, 512/3, …_________________________________________________

12. Which of the geometric sequences are growing?____________________________

13. Which of the geometric sequences are declining?____________________________

Real life application:

14. You throw a SuperBall on the cement as hard as you can and watch it bounce until it stops. You notice the first bounce reaches a height of 200ft, but the second bounce reaches only half of that height. How high will the 7th bounce reach?

a. What type of sequence is illustrated by this problem?

b. Is this sequence growing or declining?

Exponential Growth & Decay: Class Examples

1. The cost of a car is $10,000. If the price decreases 7% each year, what will the cost be after 10 years?

2. Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8% per year. In 1985, such hospital costs were an average of $460 per day. Find the cost per day in the year 2000.

3. Suppose your community has 4512 students this year. The student population is growing 2.5% each year. Write an equation to model the student population. What will the student population be in 3 years?

4. Since 1980, the number of gallons of whole milk each person drinks each year has decreased 4.1% each year. In 1980, each person drank an average of 16.5 gallons of whole milk per year. How many gallons will a person drink in the year 2000?

5. In 1990, the population of Washington D.C. was about 604,000 people. Since then the population has decreased about 1.8% each year. If the trend continues, what will the population be in 2010?

Half-Life

6. Cesium-137 has a half-life of 30 years. Suppose a lab stored a 30 mg sample in 1973, how much of the sample will be left in 2003? In 2063?

7. Technetium-99 has a half-life of 6 hours. Suppose a lab has 80 mg of technetium-99, how much is left after 24 hours?

Exponential Growth and Decay

1. Formula for Growth and Decay

2. A town with a population of 75,000 is increasing at a rate of 2% each year. What will the population be in 10 years?

3. Suppose your community has 4512 students this year. The student population is growing 2.5% each year.

a. Write an equation to model the student population.

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

4. Technetium-99 has a half-life of 6 hours. Suppose a lab has 80 mg of technetium-99. How much technetium-99 is left after 24 hours?

5. The half-life of ioding-131 is 8 days.

a. How many half-lives of iodine-131 occur in 32 days?

b. Suppose you start with a 50-millicuries sample of iodine-131. How much iodine-131 is left after one half-life? After two half-lives?

6. 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 3003? In 2063?

7. Suppose the population of a certain endangered species has decreased 2.4% each year. Suppose there were 60 of these animals in a given area in 1999.

a. Write an equation to model the number of animals in this species that remain alive in that area?

b. Use your equation to find the approximate number of animals remaining in 2005.

8. 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?

b. What is the decay factor?

c. 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.

9. At sea level the barometric pressure is 29.92 inches of mercury. For each male above sea level, the barometric pressure decreases by 18%. Find the barometric pressure 7 miles above sea level.

Compound Interest

Compound Interest formula:

y = a ( (1 + r )t

where a = the initial amount, r = the interest rate in decimal form, and t = time in years. Later we will look at investments where time is compounded more than once a year.

Practice

Write the formulas for the following compound interest problems.

1) You have an initial investment of $15,000 to be invested at a 6% interest rate compounded annually. What is the investment worth at the end of 5 years? What is the investment worth at the end of 15 years?

2) You have an initial investment of $7,000 to be invested at a 4.5% interest rate compounded annually. What is the investment worth at the end of 20 years? What is the investment worth at the end of 30 years?

3) Sam’s aunt Matilda gave him a stamp collection worth $2,500. Sam is considering selling the collection, but his aunt told him that, if he saved it, the stamps would increase in value. Sam decided to save the collection, and it’s value increased by 3.75% each year. Find the value of the collection 5 years from now. When will it be worth $5,000?

More Compound Interest . . . Making even more money by investing

Y = initial investment(1 + interest rate/number of time compounded)time ( times compounded in the year

Y = a ( (1 + r/n)nt, where a = initial amount, r = interest rate, t = time, and n = number of times the investment is compounded during one year. This is the formula that I want you to remember.

Practice

The scenario is that you have $5000 to invest and you want to know which of the following investment situations will give you the most money at the end of 5 years. The interest rate for all of the situations is 6%.

1. Calculate the investment if it is compounded annually.

2. Calculate the investment if it is compounded semi-annually (twice a year).

3. Calculate the investment if it is compounded quarterly (four times a year).

4. Calculate the investment if it is compounded monthly (12 times a year).

5. Calculate the investment if it is compounded daily (365 times a year).

6. What did you discover? Which situation will give you the most? Which situation is the most realistic for banks? Which situation is the most realistic for credit card companies? Explain your reasoning.

Independent Practice with Compound Interest

Write an equation for each problem situation in order to find the solution.

1) An investment of $75,000 increases at a rate of 12.5% per year. Find the value of the investment after 30 years. How much more would you have if the interest is compounded quarterly?

2) Suppose you invest $5000 at an annual interest of 7%, compounded semi-annually. How much will you have in the account after 10 years? Determine how much more you would have if the interest were compounded monthly.

3) Lisa invested $1000 into an account that pays 4% interest compounded monthly. If this account is for her newborn, how much will the account be worth on his 21st birthday, which is exactly 21 years from now?

4) Mr. Jackson wants to open up a savings account. He has looked at two different banks. Bank 1 is offering a rate of 5.5% compounded quarterly. Bank 2 is offering an account that has a rate of 8%, but is only compounded semi-annually. Mr. Jackson puts $6,000 in an account and wants to take it out for his retirement in 10 years. Which bank will give him the most money back?

5) Mason deposited $2,000 into a savings account that pay an annual interest rate of 9% compounded annually. Determine the amount of money in the savings account after 1 year, 5 years, 10 years and 20 years. Using the calculated values, construct a graph.

Find the balance in each account.

1. $4000 principal earning 6% compounded annually for 5 years

2. $12,000 principal earning 4.8% compounded annually for 7 years

3. $500 principal earning 4% compounded quarterly for 6 years

4. $20,000 deposit earning 3.5% compounded quarterly for 10 years

5. $2400 principal earns 7% compounded monthly for 10 years

Find each amount after the specified time.

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

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

8. A $22,000 car decreases by 20% each year for 6 years.

9. The half-life of iodine-124 is 4 days. A technician has a 40-mCi sample of iodine-124. How much iodine-124 is in the sample after 16 days?

Linear Functions versus Exponential Functions

The aim of this investigation is to develop students’ ability in recognizing data patterns likely to be modeled well by exponential growth functions. A further goal is to utilize graphing calculator experimentation to find a good regression model. Students should think analytically about the data being modeled as well as to use estimation and calculator-based tools.

Wolf Populations in the Midwest

Suppose that census counts of Midwest wolves began in 1980 and produced these estimates for several different years:

|Time Since 1980 (in years) |0 |2 |5 |7 |10 |13 |

|Estimated Wolf Population |100 |300 |500 |900 |1,500 |3,100 |

1) Plot the wolf population data on paper and decide whether a linear or exponential function seems likely to match the pattern of growth well. For the function type of your choice, experiment with different rules to see which rule provides a good model of the growth pattern. Provide your final rule here.

2) Use your graphing calculator to find both linear and exponential regression models for the given data pattern. Compare the fit of each function to the function you developed by experimentation in part 1.

3) What do the numbers in the linear and exponential function rules from part 2 suggest about the pattern of change in the wolf population?

4) Which model do you think best fits the data? Why?

5) Use the model for wolf population growth that you believe to be best to calculate population estimates for the missing years (1981, 1983, 1984, 1986, 1988, 1989, 1991, and 1992).

6) Use your model to give population estimates for the year 2000, 2005, and 2010. When will the population reach an estimated 500,000 wolves?

Alaskan Bowhead Whales

Suppose that census counts of Alaskan Bowhead Whales began in

1970 and produced these estimates for several different years:

|Time Since 1970 (in years) |0 |5 |15 |20 |26 |31 |

|Estimated Whale Population |5,040 |5,800 |7,900 |9,000 |11,000 |12,600 |

7. Plot the given whale population data on paper and decide which type of function seems likely to match the pattern of growth well. For the function type of your choice, experiment with different rules to see which provides a good model of the growth pattern. Provide your equation here.

8. Use your calculator to find both linear and exponential regression models for the data pattern. Compare the fit of each function to that of the function you developed by experimentation in problem 7.

9. Which model do you think best fits the data? Why?

10. What do the numbers in the linear and exponential function rules from problem 8 suggest about patterns of change in the whale population?

11. Use the model for whale population growth that you believe to be the best to calculate population estimates for the years 2002, 2005, and 2010.

12. When will the whale population reach 25,000?

Linear Versus Exponential Name:________________________________

Write a Now-Next Rule and an Input-Output Rule for each situation. Then, answer the question.

1. A population of rabbits doubles every 6 months. If we begin with 10 rabbits, how many rabbits will there be after 5 years? After 10 years?

|Number of |Number of |

|6 Month Periods |Rabbits |

| | |

| | |

| | |

| | |

| | |

2. A tournament starts with 128 participants. After each round of the tournament, half of the players are eliminated. How many players remain after 5 rounds?

|Round |Players |

| |Left |

| | |

| | |

| | |

| | |

| | |

3. Suppose you get two M&M’s for every question you answer correctly. How many M&M’s will you receive if you answer 10 questions correctly?

|Number Correct | | | | | |

|Number of M&M’s | | | | | |

Complete each table and classify it as linear or exponential growth. Then, write a Now-Next Rule and an Input-Output Rule for each table of data.

4.

|Years |0 |5 |10 |15 |20 |

|Value |$800 |$1200 |$1800 | | |

5.

|Months |3 |6 |9 |12 |15 |

|Number of |36 |72 |144 | | |

|Animals | | | | | |

6.

|Miles |0 |2 |4 |6 |8 |

|Driven | | | | | |

|Cost of |$19.50 |$20 |$20.50 | | |

|Rental Car | | | | | |

7.

|Hours |1 |2 |3 |4 |5 |

|Money |$135 |$170 |$205 | | |

|Earned | | | | | |

8.

|Years |1 |2 |3 |4 |5 |

|Number of |100 |170 |240 | | |

|MP3’s | | | | | |

9.

|Years |1 |2 |3 |4 |5 |

|Amount of |1600g |800g |400g | | |

|Matter | | | | | |

10. An investment of $10000 doubles in value every 13 years. How much is the investment worth after 52 years?

11. If you start with 50 bacteria and they triple ever 30 minutes, how many will there be after 5 hours?

Independent Practice Half-life Problems

Recall: The half-life of a radioactive substance is the time it takes for half of the material to decay. You are encouraged to make a table in order to generate some of the data for each problem situation below. Solve the following half-life problems by writing an equation and using the equation to find the solution. Make sure you find the initial value for each equation. The first problem has been partially worked in order to help you with the remaining problems.

1) A hospital prepared a 100-mg supply of technetium-99m, which has a half-life of 6 hours. Use the table below to help you understand how much of technetium-99m is left at the end of a 6-hour interval for 36 hours. Use this to help write an exponential function to find the amount of technetium-99m that remains after 75 hours.

The amount of technetium-99m is reduced by one half each 6 hours as shown in the table below. Fill in the missing information in the table.

|Number of 6-hour Intervals |0 |1 |2 |3 |4 |5 |6 |

|Number of Hours Elapsed |0 |6 | |18 |24 | |36 |

|Amount of Technetium-99m (mg) |100 |50 |25 | | |3.13 | |

2) Arsenic-74 is used to locate brain tumors. It has a half-life of 17.5 days. Write an exponential decay function for a 90-day sample. Use the function to find the amount remaining after 6 days.

3) Phospohoru-32 is used to study a plant’s use of fertilizer. It has a half-life of 14.3 days. Write the exponential decay function of a 50-mg sample. Find the amount of phosphorus-32 remaining after 84 days.

4) Iodine-131 is used to find leaks in water pipes. It has a half-life of 8.14 days. Write the exponential decay function for a 200-mg sample. Find the amount of iodine-131 remaining after 72 days.

5) Carbon-14 is used to determine the age of artifacts in carbon dating. It has a half-life of 5730 years. Write the exponential decay function for a 24-mg sample. Find the amount of carbon-14 remaining after 30 millennia (1 millennium – 1000 years).

Exponential Decay: Depreciation Problems

Most cars lose value each year by a process known as depreciation. You may have heard before that a new car loses a large part of its value in the first 2 or 3 years and continues to lose its value, but more gradually, over time. That is because the car does not lose the same amount of value each year, it loses approximately the same percentage of its value each year. What kind of model would be useful for calculating the value of a car over time?

Let us look at an example of depreciation: Suppose the value of car when new is $20,000 and it depreciates at a rate of 20% each year. What is the percentage rate of depreciation each year?

The percentage rate of depreciation is 20%, which means that 80% of the value of the car remains every year. We can calculate this percentage rate by subtracting 20% from 100% in order to calculate the value remaining of 80% each year.

What is the initial value of the car? $___________

What is the percentage rate? 100% - 20% = ___% each year

|Number of Years |0 |1 |2 |3 |4 |5 |

|Value of the Car |20,000 |16,000 | | |8,192 | |

Graph the table on the graph below.

Write an explicit equation for the data in order to calculate the value of deprecation for any year.

Y = initial value (1 – percentage rate of depreciation)time

Use this equation to find the depreciated value of the car for year 8.

When will the depreciated value of the car be worth $5000? _________________

YOUR TURN

Matt bought a new car at a cost of $25,000.  The car depreciates approximately 15% of its value each year. 

a.)  What is the percentage rate of depreciation for the value of this car? 

       (Remember that the percentage rate of depreciation is 0 ................
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