unit 8: exponential functions - tcann - home8... · · 2015-03-09use the exponential properties...

Intermediate Algebra Unit 8: Exponential

Functions

Intermediate Algebra Unit 8: Exponential Functions

1

Objectives: page

exponential rules 2

properties of exponents and scientific notation 3 – 4

simplifying expressions with exponents 5

multiplication properties of exponents 6

mixed practice 7

solving exponential equations 8 – 9

graphing exponential equations 10 – 12

rational exponents 13 – 16

review questions 17 – 18

equations with fractional or negative exponents 19 – 21

writing exponential functions 22 – 25

exponential growth and decay 26 – 31

doubling growth & half-life decay 32 – 34

compound interest 35 – 38

review questions 39 – 41


2

Exponential Rules:

Use the exponential properties to simplify and rewrite the following expressions for all real numbers x and y, where a > 0 and b > 0:

(1) ax • ay = (6) a-x =

(2) (ax)y = (7) a0 =

(3) (ab)x = (8) ax = ay iff

(4) =

x

ba

(9) when x ≠ 0, ax = bx iff

(5) =y

x

aa (10) =n

1

x


3

Properties of Exponents and Scientific Notation:

Perform the indicated operation using exponential properties and simplify your answer using positive exponents:

(1) x4 • x3 (2) 36 • 312 (3) x-5 • y4 • y7

(4) ( )32b (5) ( )542 (6) ( )( )1452 yx2yx7 −−

(7) 5

8

xx (8) 4

12

55 (9) 3

3

44

(10) ( )0986 zyx17 (11) 7

3

bb (12) 6x−

(13) 3

3

2

yx

−−

(14)

3

63

24

yxyx

−

(15) 5

54

y2yx16−

−


4

Standard Form: or decimal form is the way we usually represent numbers Scientific Notation: is a way to express very large or very small numbers:

10 a n× where 10a1 <≤ and n is an integer Express each number in scientific notation: (16) 6,380,000 (17) 0.000047

Use properties of exponents to multiply and divide numbers in scientific notation:

(18) ( )( )75 102104 ×× (19) ( )( )62 103107.2 ×× −

(20) ( )( )83 107105 ×× (21) ( )24101.4 −×

(22) 3

5

104.1103.6

×× (23) 3

7

106.1102.19

−××


5


6


7


8

Solving Exponential Equations: Exponential Definitions: Using Exponential Properties to Solve for x:

(1) ( )33 2x5 += (2) 1x2x5 93 −=

(3) 2564 2x9 =− (4) 3212 4n =+

(5) 4mm

8191 +=

(6) 27x + 2 = 92x – 1


9

Solve each equation and check your solution:

(7) 2 3x + 5 = 128 (8) 5 2x + 3 = 125

(9) 1612 3y =− (10) 22x = 85 – x

(11) 16n = 8 n + 1 (12) 10 x – 1 = 100 2x – 3


10

Graphing Exponential Equations: Exponential Functions:

Examine the following equations by graphing all of them simultaneously on your graphing calculator. Then complete the chart below:

y = 0.5x y = 0.75x y = 2x y = 5x

Characteristics of graphs of y = nx n > 1 0 < n < 1

domain

range

y-intercept

behavior


11

On the given set of axes, graph each function and state the domain and range:

(1) y = 3(2)x (2) y = 2(0.5)x (3) y = 1.5(2)x

Domain: Range:

Domain: Range:

Domain: Range:

(4) y = 2x + 2 (5) y = 4x – 3 (6) y = 2(0.4)x – 1

Domain: Range:

Domain: Range:

Domain: Range:

(7) y = 5(0.2)x – 2 (8) y = 3(2)x – 1 – 2 (9) y = 2(0.5)x + 1 + 2

Domain: Range:

Domain: Range:

Domain: Range:

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y


12

Determine whether each function represents exponential growth or decay. (Use your calculator to verify your answers.)

(10) y = 3(6)x (11) x

1092y

= (12) y = 10 -x

(13) x

358y

= (14) y = 2(2.5)x (15) y = 5(0.6)x

(16) y = 0.1(2)x (17) y = 5 • 4 -x (18) ( )x7142

1y =


(19) 32x – 1 = 3x + 2 (20) 4x + 1 = 82x + 3

(21) 2x + 1 = 8 (22) 104x + 1 = 100x – 2


13

Rational Exponents:


14

Examples:

(1) Evaluate ( )2

32

0

218x3

++− (2) Evaluate

31f if f(x) = 8x

(3) Evaluate ( ) 32

1x −+ if x = 7 (4) Evaluate ( ) 1021

3xx4 −++ if x = 4

(5) Evaluate f(9) if ( )2

21

0 xxxf−

+= (6) Evaluate 2

121

0 xxx5 −+−

if x = 16

(7) Evaluate f(-2) if f(x) = 3x (8) Evaluate f(-4) if f(x) = (2x)2


15

(9) Evaluate f(64) if ( ) 32

0 xx16)x(f += (10) Evaluate f(8) if 32

32

0 xxx)x(f−

++=

Solve each equation:

(11) 2713x = (12) 53x + 2 = 25

(13) 43x + 5 = 16 (14) 4x = 8x – 1


16

Evaluate each expression:

(1) 23

9 (2) 32

1000 (3) 53

32 (4) 32

)8(−

(5) 32

64 (6) 23

64 (7) 23

4−

− (8) 32

27−

(9) 25

4− (10) 43

81 (11) ( )21

136− (12) ( )31

08

(13) 21

254

(14)

21

169 −

(15)

31

271

− (16)

31

271 −

(17) 23

21

22 • (18) 21

2 44 • (19) 21

23

99−

• (20) 32

31

88−

•

(21) 10 55 −+ (22) 2

10 44

−+− (23) ( ) 2

10 88 + (24) ( ) 2

10 33 −+

(25) Find the value of 31

a2 when 827a =

(26) Evaluate 21

0 aa−

+ when a = 9

(27) If k = 4, find the value of ( )23

0k9

(28) Find the value of ( )43

1m− when m = 16

(29) If 32

x)x(f = , find the value of f(-216)

(30) If 23

x)x(f−

= , find the value of f(100)

(31) If the function 21

x4)x(g−

= , find the value of g(25)


17

Review Questions: For each function given below:

(a) State whether the function represents growth, decay or neither (b) Graph the function and list at least 5 points in the given chart, including

the y-intercept. Circle the y-intercept in the chart. (c) State the domain and range

(1) ( )x42y −= (a) (c) Domain: Range:

(2) x

214y

=

(a) (c) Domain: Range: (3) x35y −•= (a) (c) Domain: Range:

x y

x

y

x y

x

y

x y

x

y


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(4) The functions graphed in questions (1) – (3) are asymptotic to the x-axis. Explain what that means.

(5) Find the value of 21

0 aa3 + when a = 25 (6) Given 23

x)x(f−

= find f(9)

Solve the following equations showing all work:

(7) 1xx 749 += (8) 2xx 464 +=

(9) 7x3x2 322 −+ = (10) x2

x

821 −=


19

Solving Equations with Fractional or Negative Exponents:


20

Equations with Fractional or Negative Exponents: Solve each equation and check your solution:

(1) 7x 21

= (2) 9y 2 =−

(3) 151x 34

=− (4) 3x2 21

=−

(5) 5x 31

= (6) 2a 41

=−


21


(7) 102y 23

=+−

(8) 43x2 41

=+−

(9) 4x 32

= (10) 31b 2

1

=−

(11) 162x2 34

= (12) 205x4 32

=−


22

Writing Exponential Functions: Exponential Functions: y = a • bx Write an exponential function whose graph passes through the given points:

(1) (0, -2) and (-2, -32) (2) (0, 3) and (1, 15)

(3) (0, 7) and (2, 63) (4) (0, -5) and (-3, -135)

(5) (0, 0.2) and (4, 51.2) (6) (0, -0.3) and (5, -9.6)


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For the following questions, round all multipliers (b values) to the nearest hundredth: (7) In 1983, there were 102,000 farms in Minnesota, but by 1998, this number had

dropped to 80,000. (a) Write an exponential function in the form y = abx that could be used to model the farm

population, y, is Minnesota. Write the function in terms of x, the number of years since 1983.

(b) Suppose the number of farms continues to decline at the same rate. Estimate the number of farms in 2010.

(8) The number of bacteria in a colony is growing exponentially, as there are 100 present

at 2 p.m. and 4000 present at 4 p.m. (a) Write an exponential function to model the population y of bacteria x hours after 2 p.m.

(b) How many bacteria were there at 7 p.m. that day?


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(9) During the 19th century, rabbits were brought to Australia. Since the rabbits had no natural enemies on that continent, their population increased rapidly. Suppose there were 65,000 rabbits in Australia in 1865 and 2,500,000 in 1867. (a) Write an exponential function that could be used to model the rabbit population y in

Australia. Write the function in terms of x, the number of years since 1865.

(b) Assume that the rabbit population continued to grow at that rate. Estimate the Australian rabbit population in 1872.

(10) The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.

(a) Write an exponential function to model the population y bacteria after x days.

(b) How many bacteria are there after 6 days? (11) A college with a graduating class of 4000 students in the year 2012 predicts that it

will have a graduating class of 4862 in 4 years. Write an exponential function to model the number of students y in the graduating class t years after 2012.


25

Write an exponential function whose graph passes through the given points: (12) (0, 3) and (-1, 6)

(13) (0, -18) and (-2, -2)


26

Exponential Growth and Decay: Finding the multiplier for growth or decay: Find the multiplier for each situation:

(1) 12% growth (2) 25% decay (3) 7.5% decay

(4) 8.2% decay (5) 1% growth (6) 0.5% decay

Use a growth or decay model to solve each question: (7) A new school district is experiencing an annual growth rate of 9.5%. The school population

is now 5600 students. What is the approximate predicted population:

(a) 3 years from now? (b) 5 years from now? (c) 10 years from now?

(8) The rate in the number of reported cases of robbery is dropping at about 7% per year in a given region of the country. The number of cases reported this year was approximately 156,000. If the number continues to drop at this rate, what is the approximate predicted number of cases:

(a) 1 year from now? (b) 3 years from now? (c) 5 years from now?


27

Using a table (and a calculator) to find a specific value for an exponential function:

Example: Your doctor prescribes a medication for your allergies. After each 1 hour interval, only 90% of the medication present 1 hour ago remains in your system. If you take a 100-milligram tablet, in approximately how many hours will only 50% of the medication remain in your system?

Solution: The multiplier is 100% – 10% = 90% or 0.9 Complete a table using 100(0.9)x, where x is a positive integer:

x 1 2 3 4 5 6 7 100(0.9)x 90 81 72.9 65.61 59.05 53.14 47.83

Answer: 50% of the medication will be left in your system between 6 and 7 hours after the

initial dose.

For each, write an exponential function and complete a table to answer the question: (9) You invest $5000 in an account that earns interest at an effective rate of 8.4%

per year. In how many years will you have over $6800 in the account?

x

(10) If you invest $50,000 in a high interest account that earns interest at an effective rate of 13.8% per year, how many years will it take to double your money?

x

Challenge: (11) After 2 hours, only 75% of a new medication remains in your body. If you take an

80-milligram tablet, and this rate of decay is constant, in approximately how many hours will less than 15 milligrams remain in your system?

x


28

Using Exponential Functions for Real World Applications: Exponential growth: Exponential decay: Exponential Growth or Decay: N = N0 (1 + r)t

(1) In 2000, the chicken population on Farmer Fred’s farm was 10,000. The number of chickens increased at a rate of 9% per year. Predict the population in 2005.

(2) Suppose the value of a computer depreciates at a rate of 25% a year. Determine the value of a laptop computer two years after it has been purchased for $3,750.

(3) Sally bought a printer for $250 in 2009, and by 2011 the value had become $150. To the nearest hundredth of a percent, find the rate of depreciation for the printer.


29

(4) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a rate of 14% per week. What is the expected population in 11 weeks?

(5) In 1990, Exponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%. Find the projected population for 2010.

(6) A cup of coffee contains 130 mg of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how much caffeine will remain in the body after 6 hours?

(7) Danny bought a collector’s edition football jersey for $1000 that appreciated in value, and in 5 years was worth $2150. Find, to the nearest hundredth of a percent, the rate of appreciation for the jersey.

(8) Find the projected population of each location in 2015: (a) In Honolulu, Hawaii, the population was 836,231 in 1990. The average

yearly rate of growth is 0.7%. (b) The population in Kings County, New York has demonstrated an average

decrease of 0.45% over several years. The population in 1997 was 2,240,384.


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(9) The first stage of the Saturn 5 rocket that propelled astronauts to the Moon burned about 8% of its available fuel every 15 seconds, and carried about 600,000 gallons at liftoff. (a) What is the multiplier for the fuel loss of the first stage over 15 seconds? (b) How much first-stage fuel remains after 2 minutes? (c) Approximately how long after liftoff was half of the first-stage fuel used?

(10) The population in Sacramento, California, was 369,365 in 1990 and was growing at a rate of about 2.5% per year. (a) What was the multiplier for Sacramento’s population growth? (b) Write the function that models Sacramento’s projected population growth. (c) What was the projected population of Sacramento for the year 2000? (d) During what year will Sacramento’s projected population double?

(11) Suppose the annual rate of inflation averages 3%. Use this information to answer the following questions: (a) If a tune-up for your car presently costs $40, estimate how much it will cost 3

years from now. And also estimate the cost 10 years from now. (b) If you live in an apartment for which the monthly rent is $300, about how much will the

rent be 3 years from now? And also estimate the rent 10 years from now.


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(12) In 2012, the pig population on Farmer Fred’s farm was 8,000. The number of pigs increased at a rate of 6% per year. Predict the population in 2020.

(13) Suppose the value of a car depreciates at a rate of 18% a year. Determine the value of a car three years after it has been purchased for $22,700.

(14) Tony bought a collectible baseball card for $125 in 2006, and by 2010 the value had become $150. To the nearest hundredth of a percent, find the rate of appreciation for the baseball card.


32

Do Now: (1) The population of Los Angeles County was 9,145,219 in 1997. If the average

growth rate is 0.45%, predict the population in 2010.

Doubling Time Growth:

( )dt

0 2NN =

Half-Life Decay:

ht

0 21NN

=

(2) E. coli has a doubling time of 25 minutes. In a lab, if an experiment starts with a population of 1000 E. coli, how many bacteria will be present in 3 hours?

(3) The radioactive isotope gallium 67, used in the diagnosis of tumors, has a biological half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams (rounded to the nearest integer) will be left after 1 week?


33

(4) Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a half-life of 2.67 days. If the initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be left over after: (a) ½ day (b) 1 week

(5) If Farmer Fred uses 25 pounds of insecticide, assuming its half-life is 12 years, how many pounds (rounded to the nearest tenth) will still be active after: (a) 5 years (b) 20 years

(6) Mexico has a population of about 100 million people, and it is estimated that the population will double in 21 years. If population growth continues at the same rate, what will be the population in: (a) 15 years (b) 30 years

(7) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth continues at the same rate, find the population (rounded to the nearest million) in: (a) 10 years (b) 30 years


34

Exponential Half-Life Decay can be modeled by the equation ht

0 21NN

=

Solve each question: (15) There are 10 grams of Curium-245 which has a half-life of 9,300 years. How many grams

will remain after 37,200 years?

(16) There are 80 grams of Cobalt-58 which have a half-life of 71 days. How many grams will remain after 213 days?

(17) The half-life of Rhodium-105 is 1.5 days. If there are initially 7500 grams of this isotope, how many grams, to the nearest thousandth of a gram, would remain after 30 days?

(18) Two hundred ten years ago there were 132,000 grams of Cesium-137, which has a half-life of 30 years. How many grams are there today?

(19) In a nuclear reaction, 150 grams of Plutonium-239 are produced, which has a half-life of 24,400 years. How many grams would remain after one million years? (Express your final answer in scientific notation.)


35

Do Now:

(1) A laser printer was purchased for $300 in 2001. If its value depreciates at a rate of 30% a year, determine how much it will be worth in 2007.

(2) Suppose you want to buy a car that costs $11,800. The expected depreciation of the car is 20% per year. You take out a four-year loan to pay for the car. How much will the car be worth after you have paid off the loan in four year?

The general equation for exponential growth is modified for finding the balance in an account that earns compound interest.

Compound Interest: tn

nr1PA

+=


36

Compound Interest:

(1) Using the formula tn

nr1PA

+= , complete the following table if $1000 is invested at a rate

of 10% for one year. The value of n varies as indicated: Number of Times Compounded per year n value

tn

nr1P

+ = A

Annually

Bi-Annually

Quarterly

Monthly

Weekly

Daily

Hourly

Minutely

(2) Describe what happens as the value of n increases? Explain your answer. (3) What would happen if the value of P increased? decreased? Explain your answer. (4) What would happen if the value of t increased? decreased? Explain your answer.


37

(5) If Annette invested $1,000 in an account paying 10% compounded monthly, how much will be in the account at the end of 10 years?

(6) Steve would like to have $20,000 cash for a new car 5 years from now. How much should be placed in an account now if the account pays 9.75% compounded weekly?

(7) Suppose $2,500 is invested at 7% compounded quarterly. How much money will be in the account in: (a) ¾ year (b) 15 years


38

(8) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in: (a) ½ year (b) 10 years

(9) How much money must Sally invest for a new yacht if she wants to have $50,000 in her account that earns 5% compounded quarterly after 15 years?

(10) Tanya won $5,000 in a raffle. She would like to invest her winnings in a money market account that provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in the account at the end of 10 years? Show your work and explain your answer.


39

Review Questions: (1) Write an exponential function whose graph passes through the points (0, -2) and (3, -6.75).

(Show all work as done in class for full credit.)

(2) A high school with a graduating class of 6500 students in the year 2011 predicts that it will have a graduating class of 8860 in 2015.

(a) Write an exponential function to model the number of students y in the graduating class x years after 2011. Round the multiplier (b value) to the thousandth place.

(b) Use the equation from part (a) to predict the number of students in the graduating class of 2020.


40

(3) Find the multiplier for each situation: [plug in to (1 + r) or (1 – r) and solve]

(a) 2.4% decay (b) 72% growth

(4) A certain car depreciates in value 18% each year.

(a) Write an exponential function to model the depreciation of a car that costs $40,000 when purchased new.

(b) Suppose the car was purchased in 2010. What is the first year the car will be worth less than half its original value?


41

(5) You purchase a collectable baseball card for $90.00, and the value of the card appreciates at a rate of 2.5% per year. How much will the card be worth in 20 years, assuming that rate of growth stays the same?

(6) Alex invested $2500 at a rate of 2.3% at Citibank. The interest is compounded daily. How much money will Alex have in his account after 6 years? Calculate the balance for compounded monthly and quarterly as well.

(7) The Shaffer family bought a house 12 years ago for $95,000. The house is now worth $167,000. Assuming a steady rate of growth, what was the yearly rate of appreciation, rounded to the nearest tenth of a percent?


42

unit 8: exponential functions - tcann - home8... · · 2015-03-09use the exponential properties...

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