7-1 Think About a Plan · 2015. 3. 3. · b2 B. 81a4 b2c3 C. 3a4 b2c3 D. 3a4c3 b2 2. What is2a22 if a 525? F. 225 G. 25 H. 2 1 25 I. 1 25 3. Which of the following simplifi es to

Prentice Hall Algebra 1 • Practice and Problem Solving WorkbookCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

194

Name Class Date

7-1 Think About a PlanZero and Negative Exponents

Manufacturing A company is making metal rods with a target diameter of 1.5 mm. A rod is acceptable when its diameter is within 1023 mm of the target diameter. Write an inequality for the acceptable range of diameters.

Understanding the Problem

1. What is the target diameter for the metal rods?

2. How do you know if a rod is acceptable?

3. What is the problem asking you to determine?

Planning the Solution

4. What does the word “within” tell you about the acceptable range of

diameters?

5. What does this tell you about the type of inequality that you should use?

Getting an Answer

6. Write 1023 as a fraction and as a decimal.

7. Find the acceptable range for the diameters of the rods.

8. Write your answer as a single inequality.


195

Name Class Date

7-1 Practice Form G

Zero and Negative Exponents

Simplify each expression.

1. 130 2. 523

3. 3

324 4. 2

421

5. 2(7)22 6. 4621

7. 260 8. 2(12x)22

9. 1

80 10. 6bc0

11. 2(11x)0 12. a2

9b22

13. 3m28 p0 14. 5a24

2c

15. 23k23(mn)3

p28 16. a2m

3nb23

17. 822 q3 r25 18. 2(10a)24 b0

19. 11xy21z0

v23 20. 5m21

9(ab)24 c7


196

Name Class Date

Evaluate each expression for a 5 24, b 5 3, and c 5 2.

21. 3a21 22. b23

23. 4a2 b22 c3 24. 9a0 c4

25. 2a22 26. (2c)22

Write each number as a power of 10 using negative exponents.

27. 1

1000 28.

1

10

Write each expression as a decimal.

29. 1023 30. 8 ? 1024

31. Th e number of people who vote early doubles every week leading up to an election. Th is week 1200 people voted early. Th e expression 1200 ? 2w models the number of people who will vote early w weeks after this week. Evaluate the expression for w 5 23. Describe what the value of the expression represents in the situation.

32. A pizza shop makes large pizzas with a target diameter of 16 inches. A pizza

is acceptable if its diameter is within 3 ? 222 in. of the target diameter. Let d represent the diameter of a pizza. Write an inequality for the range of acceptable large pizza diameters in inches.

33. Open-Ended Choose a fraction to use as a value for the variable c. Find the

values of c21, c23, and c3.

7-1 Practice (continued) Form G

Zero and Negative Exponents


197

Name Class Date

7-1 Standardized Test PrepZero and Negative Exponents

Multiple Choice

For Exercises 1–6, choose the correct letter.

1. What is the simplifi ed form of 3a4b22c3?

A. 81a4c3

b2 B. 81a4

b2c3 C. 3a4

b2c3 D. 3a4c3

b2

2. What is2a22 if a 5 25? F. 225 G. 25 H. 2

1

25 I.

1

25

3. Which of the following simplifi es to a negative number?

A. 2424 B. (24)24 C. 424 D. 1

424

4. What is the simplifi ed form of2(14x)0 y27 z ?

F. 214z

y7 G. 14z

y7 H. z

y7 I. 2

z

y7

5. What is (2m)23n if m 5 2 and n 5 224?

A. 3 B. 23 C. 4 D. 24

6. What is the simplifi ed form of a2

5a3b23

?

F. 27

125a3 G. 227

125a3 H. 125a3

27 I. 2

125a3

27

Short Response

7. Th e number of bacteria in a culture quadruples every hour. Th ere were

65,536 bacteria in the culture at 8:00 a.m. Th e expression 65,536 ? 4h models the number of bacteria in the culture h hours after 8:00 a.m.

a. What is the value of the expression for h 5 24?

b. What does the value of the expression in part a represent?


198

Name Class Date

7-2 Think About a PlanScientifi c Notation

Physics Th e radius of a water molecule is about 1.4 angstroms. One angstrom is 0.00000001 cm. What is the diameter of a water molecule in centimeters? Use scientifi c notation.

1. What geometric facts and arithmetic operation(s) you are going to need to use to solve the problem? What is the relationship between diameter and radius? How do you convert between the given units?

2. How many angstroms are there in the diameter of a water molecule?

How will you turn angstroms into centimeters?

3. Rewrite 0.00000001 in scientifi c notation. Recall that a number written in scientifi c notation is in the form a 3 10n.

a = n = 0.00000001 =

4. Multiply the number of angstroms in the diameter of a water molecule by the length of an angstrom in centimeters to fi nd the number of centimeters in the diameter of a water molecule.


199

Name Class Date

7-2 Practice Form G

Scientifi c Notation

Is the number written in scientifi c notation? If not, explain.

1. 32.1 3 105 2. 5.6 3 1012

3. 4.6 3 1025 4. 0.7 3 1034

Write each number in scientifi c notation.

5. 3,200,000,000,000 6. 0.00000802

7. 70,030,000 8. 8.7 billion

Write each number in standard notation.

9. 3.37 3 1012 10. 3.060 3 107

11. 4.2 3 1026 12. 4.56 3 100

Simplify. Write each answer using scientifi c notation.

13. 5(3.2 3 1024) 14. 0.7(8.54 3 104)

15. 87(6.4 3 105) 16. 0.03(6 3 1027)

17. Writing Scientifi c notation is often used for working with very small or very large numbers. Describe two situations where using scientifi c notation might be appropriate.

18. Reasoning How does a number in scientifi c notation change when you multiply it by 100?

19. Country A has a population of 8.7 3 109. You hear that country B has twice as many people as country A and country C has twice as many people as country B.How many people live in country C?


200

Name Class Date


Scientifi c Notation

Write a number in scientifi c notation that is between the two given numbers.

20. 6.2 3 105, 9.6 3 104 21. 3.7 3 1023, 9.4 3 1022

22. 7.94 3 106, 7.93 3 107 23. 9 3 1026, 6 3 1027

Write a number in standard notation that is between the two given numbers.

24. 3.42 3 108, 3.421 3 108 25. 1.3 3 1024, 1 3 1023

26. 5.708 3 1026, 5.7008 3 1026 27. 1.2 3 100, 1.3 3 100

Write a number in words that is between the two given numbers.

28. 6.52 3 107, 1.2 3 108 29. 3.9 3 1025, 2.8 3 1024

30. Open-Ended Write two factors that, when multiplied together, produce a product of 3.6 3 108. One of the factors should be written in scientifi c notation.

31. Light travels at 1.86 3 105 miles per second. If a particle is traveling at half the speed of light, how fast is it moving?

32. An atom of carbon has a mass of 1.99 3 10223 grams.

a. What is the mass of two atoms of carbon? b. What is the mass of fi ve atoms of carbon?


201

Name Class Date

7-2 Standardized Test PrepScientifi c Notation

Multiple Choice


1. Which of the following expressions is written in scientifi c notation?

A. 73.4 3 105 B. 0.09 3 107 C. 80 3 103 D. 4.22 3 1025

2. Which of the following is 0.0000000708 written in scientifi c notation?

F. 7.08 3 1028 G. 7.8 3 1028 H. 708 3 10210 I. 70.8 3 1029

3. Which expression represents the largest number?

A. 40.1 3 1026 B. 4.1 3 1027 C. 0.411 3 1026 D. 0.04001 3 1025

4. Which expression is equal to 1

8000 written in scientifi c notation?

F. 8.0 3 103 G. 1.25 3 1024 H. 125 3 1025 I. 1.25 3 104

5. Which of the following statements is not true regarding an expression written in scientifi c notation in the form a 3 10n?

A. Th e value of a must be greater than or equal to 1 and smaller than 10. B. Th e value of n must be an integer. C. Doubling n results in a doubling of the value of the expression. D. Doubling a results in a doubling of the value of the expression.

6. Which number is equal to 7(3.5 3 104) written in scientifi c notation?

F. 24.5 3 704 G. 2.45 3 105 H. 24.5 3 104 I. 1.05 3 105

Short Response

7. A state government has 5.7 3 107 dollars invested in a pension fund for retired employees. It expects this investment to double in value every 8 years. What is the value of the investment after 8 years, 16 years and 24 years? Write your response in scientifi c notation.


202

Name Class Date

7-3 Think About a PlanMultiplying Powers With the Same Base

Chemistry In chemistry, a mole is a unit of measure equal to 6.02 3 1023 atoms

of a substance. Th e mass of a single neon atom is about 3.35 3 10223g. What is the mass of 2 moles of neon atoms? Write your answer in scientifi c notation.

Understanding the Problem

1. What is the mass of a single atom of neon?

2. How many atoms are in 1 mole?

Planning the Solution

3. Without solving, explain how you can determine the mass of 1 mole of neon atoms. How about 2 moles of neon atoms?

4. What exponent rules and guidelines will help you with your calculations?

Getting an Answer

5. Demonstrate how the numbers can be arranged to simplify the multiplication.

6. Complete your calculations.

7. Is your answer in scientifi c notation? Explain.


203

Name Class Date

7-3 Practice Form G

Multiplying Powers With the Same Base

Rewrite each expression using each base only once.

1. 45 ? 43 2. 24 ? 26 ? 22 3. 56 ? 522 ? 521

4. 1024 ? 104 ? 102 5. 79 ? 73 ? 7210 6. 92 ? 928 ? 96


7. z8z5 8. 24k23 ? 6k4 9. (25b3)(23b6)

10. (13x28)(3x10) 11. (22h5)(4h23) 12. 28n ? 11n9

13. (t3)(t6)(t9) 14. (2f 28)(4f 12) 15. (25d25)(6d2)

16. mn2 ? m2n24 ? mn21 17. (6a3b22)(24ab28) 18. (12mn)(2m3n22p5)(2m)

19. q4 ? r25 ? q3 ? r5 20. 23c7d22 ? 5c23d 21. fg21 ? f 3g5h2 ? 2h21

Simplify each expression. Write each answer in scientifi c notation.

22. (5 3 104)(1 3 107) 23. (3 3 1026)(2 3 1012) 24. (6 3 107)(4 3 1025)

25. (7 3 104)(7 3 1025) 26. (8 3 103) ? 1028 27. (9 3 1026)(2 3 1027)

Write each answer in scientifi c notation.

28. Th e population of a country in 1950 was 6.2 3 107. Th e population in 2030 is projected to be 3 3 102 times the 1950 population. If the projection is correct, what will the population of the country be in 2030?

29. Th e area of land that Rhode Island covers is approximately 1.5 3 103 square miles. Th e area of land that Alaska covers is a little more than 4.3 3 102 times the land area of Rhode Island. What is the approximate area of Alaska in square miles?

Complete each equation.

30. 922 ? 94 5 9u 31. 5u ? 53 5 525 32. 28 ? 2u 5 222

33. zu ? z25 5 z3 34. m23 ? m6 ? mu 5 m2 35. d 7 ? d213 ? d29 5 du


204

Name Class Date


Multiplying Powers With the Same Base

Find the area of each fi gure.

36. 3x2 2x

4x2

37.

6n 5

3n2

38.

2b2 4

5b

39. 3z4

Simplify each expression. Write each answer in scientifi c notation.

40. (7 3 1017)(8 3 10228) 41. (4 3 10211)(0.8 3 107) 42. (0.9 3 1015)(0.1 3 1026)

43. (0.8 3 105)(0.6 3 10217) 44. (0.5 3 103)(0.6 3 100) 45. (0.2 3 1011)(0.4 3 10214)

46. Th e diameter of the moon is approximately 3.5 3 103 kilometers. a. Th e diameter of Earth is approximately 3.7 times the diameter of the moon.

Determine the diameter of Earth. Write your answer in scientifi c notation. b. Th e distance from the center of Earth to the center of the moon is

approximately 30 times the diameter of Earth. Determine the distance from the center of Earth to the center of the moon. Write your answer in scientifi c notation.


47. 1

n28 ? n3 48. 1

x4 ? x29 49. 7k4(22k6 2 k)

50. 22x2(23x4 1 5) 51. 4x ? 4x11 ? 4 52. (n 1 2)5(n 1 2)23

53. Writing Explain what moving the decimal point 4 places to the right or to the left does to the value of a number. In scientifi c notation, what power of 10 would you multiply by to make up for the movement of the decimal point?


205

Name Class Date

7-3 Standardized Test PrepMultiplying Powers With the Same Base

Multiple Choice


1. What is the simplifi ed form of (23x3y2)(5xy21)?

A. 215x3

y2 B. 215x3y2 C. 215x4y D. 15x4y

2. What is the simplifi ed form of 29m22n5 ? 2m23n26?

F. 218m5n G. 218

m5n H.

218m6

n30 I. 218m6n30

3. A rectangular pasture has a fence around the perimeter. Th e length of the fence is 16x7 and the width is 48x4. What is the area of the pasture?

A. 3x3 B. 128x11 C. 768x11 D. 768x28

4. What is the simplifi ed form of (3.25 3 103)(7.8 3 106) written in scientifi c

notation?

F. 2.535 3 1010 G. 2.535 3 1018 H. 2.535 3 1019 I. 25.35 3 109

5. Th e size of a certain cell is 2.5 3 1029 m. Another cell is 1.5 3 103 times larger. How large is the larger cell in scientifi c notation?

A. 4 3 10227 m B. 4 3 10212 m C. 3.75 3 106 m D. 3.75 3 1026 m

Extended Response

6. Th e distance between two astronomical bodies is 6.75 3 106 miles. Th e

distance from the fi rst body to a third body is 3.56 3 104 times the distance between the fi rst two bodies.

a. Write an expression that represents the distance between the fi rst and third bodies.

b. Simplify the distance in part a. c. Write an expression that represents the distance light would travel from the

fi rst body to the third body and back to the fi rst. d. Simplify the distance the light travels in part c.


206

Name Class Date

7-4 Think About a PlanMore Multiplication Properties of Exponents

a. Geography Earth has a radius of about 6.4 3 106 m. What is the approximate surface area of Earth? Use the formula for

the surface area of a sphere, S 5 4π r 2. Write your answer in scientifi c notation.

b. Oceans cover about 70% of the surface of the Earth. About how many square meters of Earth’s surface are covered by ocean water?

c. Th e oceans have an average depth of 3790 m. Estimate the volume of water in Earth’s oceans.

1. What can you substitute into the formula?

2. What numbers need to be squared?

3. Calculate the surface area using 3.14 for π. Show your work. Write your answer in scientifi c notation.

4. For part b, set up a percent proportion with the appropriate units.

5. Solve your proportion. Show your work. Leave your answer in scientifi c notation.

6. Since you calculated the surface area of the ocean, how can you use your calculation to determine the volume of the water?

7. Determine the volume. Show your work. Write your answer in scientifi c notation.

6.4 3 106 m


207

Name Class Date

7-4 Practice Form G

More Multiplication Properties of Exponents


1. (z5)3 2. (m4)10 3. (v7)2 4. (k4)3

5. (x7)22 6. (r4)26 7. b(b28)23 8. h2(h7)0

9. (m2)7n5 10. (x6)2(y3)0 11. (g5)25(g6)22 12. (v2)3(w4)23

13. (6a)4 14. (5f )23 15. (9z)24 16. (10m3)22

17. (6j22)23 18. (9d10)22 19. (gh)0 20. (qr 6)4

21. (4a3)2a5 22. (m4n3)7(m4)3 23. (xy2)(xy2)21 24. z( y25z7)21y25

25. (7t23)3(s5t 4)2 26. m29(m21n)2n8 27. (3b24c22)6c3 28. 5x25y

2(2x214)2

Simplify. Write each answer in scientifi c notation.

29. (5 3 107)2 30. (2 3 104)6 31. (9 3 10212)2 32. (3 3 1028)3

33. (3.6 3 105)2 34. (9.3 3 1026)22 35. (1.7 3 1028)3 36. (6.24 3 1013)3

37. Th e radius of a cylinder is 5.4 3 106 cm. Th e height of the cylinder is

2.5 3 103 cm. What is the volume of the cylinder? (Hint: V 5 πr 2h)

38. Th e side length of a square is 9.6 3 105 in. What is the area of the square?

39. Th e side length of a cube is 3.78 3 103 ft. What is the volume of the cube?


208

Name Class Date


More Multiplication Properties of Exponents

Complete each equation.

40. (p4)u 5 p8 41. (zu)6 5 z224 42. (t12)u 5 1

43. (w3)u 5 w212 44. (n28)u 5 n8 45. 10(g 2)u 5 10g 6

46. (3au)3 5 27a29 47. (6q4ru)2 5 36q8 48. (x4y3)u 51

x 8y6

49. Writing Is (ym)n 5 (y n)m a true statement? Explain your reasoning.

50. Reasoning What is the diff erence between x4x3 and (x 4)3? Justify your

answer.


51. 23(2m)2 52. (68.68)8(68.68)28 53. (d2)25d3

54. (27p)3 1 7p3 55. 4a(08)b4(2b)27 56. (1025)3(9.9 3 10212)2

57. Th e volume of a circular cone can be determined by the formula

V 51

3 3.14 r

2h, where r is the radius of the base and h is the height of

the cone. Find the volume of the cone shown at the right in terms of x.

58. Th e volume of a sphere can be determined by the formula

V 54

3 3.14 r3, where r is the radius. Find the volume of the sphere

shown at the right in terms of t.

6t4

4x315


209

Name Class Date

7-4 Standardized Test PrepMore Multiplication Properties of Exponents

Gridded Response

Solve each exercise and enter your answer on the grid provided.

1. What is the simplifi ed form of (2 3 102)2 in standard notation?

2. What is the simplifi ed form of (0.00038 3 103)2 in standard notation?

3. Th e side of a square measures 2x2y3. What is the area of the square if x 5 22

and y 5 2?

4. Th e side of a square measures 3mn2. What is the area of the square if m 5 4 and n 5 22?

5. Th e radius of a circle is 0.00012 3 105 ft. What is the area of the circle? Use the

formula A 5 πr 2.

1. 2. 3. 4. 5. 1.

9876543

10

987654

210

9876543210

987

543210

987654321

9876543210

–

3

6

0

2

2.

9876543

10

987654

210

9876543210

987

543210

987654321

9876543210

–

3

6

0

2

3.

9876543

10

987654

210

9876543210

987

543210

987654321

9876543210

–

3

6

0

2

4.

9876543

10

987654

210

9876543210

987

543210

987654321

9876543210

–

3

6

0

2

5.

9876543

10

987654

210

9876543210

987

543210

987654321

9876543210

–

3

6

0

2


210

Name Class Date

7-5 Think About a PlanDivision Properties of Exponents

Physics Th e wavelength of a radio wave is defi ned as speed divided by frequency. An FM radio station has a frequency of 9 3 107 waves per second. Th e speed

of the waves is about 3 3 108 meters per second. What is the wavelength of the station?

1. Wavelength is speed divided by frequency. Which of the given values needs to be divided by the other?

divided by

2. What is the quotient of the fi rst part of the numbers?

3. What is the quotient of the powers of 10?

4. What is the quotient, written in scientifi c notation, of the speed divided by the frequency?

5. How many meters is the wavelength of the station?


211

Name Class Date

7-5 Practice Form G

Division Properties of Exponents


1. 56

52 2. 55

52

3. x7

x4 4. m23

m25

5. x6 y9

x2 y5 6. 21m8

3m2

7. a3

5b4

8. a3x

2yb3

9. a4

7b22

10. a23x4

2y5b23

11. a12p3

15pb4

12. aab3

a5bb22

13. a3x2y5z22

5xz5 b23 14.

(4m2) (3n5)

(2m23) (2mn)3

Explain why each expression is not in simplest form.

15. 24 r3 16. (3x)2

17. m3 n0 18. y5

y

Simplify each quotient. Write each answer in scientifi c notation.

19. 3.6 3 107

1.5 3 103 20. 4.5 3 1026

5 3 1022


212

Name Class Date


Division Properties of Exponents

21. Writing Explain how you divide expressions with numerators and denominators written in scientifi c notation. How do you handle the exponents? What do you do with the coeffi cients? Connect your response to the rules you have learned regarding the division properties of exponents.

22. A computer can do a computation in 6.8 3 1029 seconds. How many computations can the computer do in 5 minutes?

23. Error Analysis A student simplifi es the expression a64

32b3

as follows:

a64

32b35 f(6 4 3)422g3 5 (22)3 5 64. What mistake did the student make in

simplifying the expression? What is the correct simplifi cation?

24. Reasoning Th e division property of exponents says that to simplify powers with the same base you subtract the exponents. Use examples to show why powers need to have the same base in order for this technique to work.

25. Th e area of a triangle is 80x5 y3. Th e height of the triangle is x4 y. What is the

length of the base of the triangle?

26. Open-Ended First simplify the expression a12m5

15mb3

by raising each factor in

the parentheses to the third power and next reducing the result. Th en simplify by some other method. Explain your method. Are the results the same? Which method do you prefer?


213

Name Class Date

7-5 Standardized Test PrepDivision Properties of Exponents

Multiple Choice


1. Which of the following is equivalent to x5y2

xy2 when x 2 0 and y 2 0?

A. x6y5 B. x5y C. x4y D. x4

2. Which expression is not equal to 125?

F. 5°53

2

5¢

2

G. a53

54b23

H. 522

525 I. 5a55

53b

3. A fraction reduces to 36. If its numerator is (6x)5, what is its denominator?

A. 63x B. 63x5 C. 6x5 D. 67x5

4. Which is the correct simplifi cation of 5.4 3 1012

1.2 3 103 written in scientifi c notation?

F. 4.5 3 107 G. 4.5 3 109 H. 45 3 106 I. 6.48 3 107

5. Which of the following statements is not true regarding operations with exponents?

A. To divide powers with the same base, subtract the exponents. B. To subtract powers with the same base, divide the exponents. C. To multiply powers with the same base, add the exponents. D. To raise a power to a power, multiply the exponents.

6. Which of the following expressions is written in simplest form?

F. 35x2 G. (5y)3 H. a0b I. x2y3z

Short Response

7. A rectangle has an area of 12x3 y4 cm2. It has a width of 3xy cm. What ratio

would you set up to determine the length? What is the length of the rectangle?


214

Name Class Date

7-6 Think About a PlanExponential Functions

Computers A computer valued at $1500 loses 20% of its value each year. a. Write a function rule that models the value of the computer. b. Find the value of the computer after 3 yr. c. In how many years will the value of the computer be less than $500?

1. Make a table for which one column represents the number of years since the computer was purchased and the other represents the value of the computer at the end of each year.

2. Use the table to determine whether the function is exponential. Is there a common ratio? If so, what is it? What does the common ratio have to do with the situation represented in the problem? Where does it show up in an exponential function?

3. Remember the general form of an exponential function: f (x) 5 a ? bx.Given your table and situation, determine values for a and b. Write the appropriate exponential function.

4. Use the exponential function to answer part b.

5. Set your exponential function equal to $500 and solve or extend your table to determine the answer to part c.

0 $15001234

Years Since Purchase Value of Computer


215

Name Class Date

7-6 Practice Form G

Exponential Functions

Determine whether each table or rule represents an exponential function. Explain why or why not.

1. 2. x 1 2 3 4y 3 9 15 21

3. y 5 5 ? 2x 4. y 5 6 ? x3

5. y 5 3x 2 8 6. y 5 4 ? 0.3x

Evaluate each function for the given value.

7. f (x) 5 5x for x 5 4 8. h(t) 5 3 ? 4t for t 523 9. y 5 8 ? 0.7x for x 5 3

Graph each exponential function.

10. f (x) 5 3x 11. y 5 0.25x 12. y 5 8 ? 1.2x

13. An investment of $8000 in a certain Certifi cate of Deposit (CD) doubles in value every seven years. Th e function that models the growth of this investment is f (x) 5 8000 ? 2x, where x is the number of doubling periods. If the investor does not withdraw any money from this CD, how much money will be available for withdrawal after 28 years?

14. A population of amoebas in a petri dish will triple in size every 20 minutes. At the start of an experiment the population is 800. Th e function y 5 800 ? 3x, where x is the number of 20 minute periods, models the population growth. How many amoebas are in the petri dish after 3 hours?

15. A new car costs $15,000 to build in 2010. Th e company’s fi nancial analysts expect costs to rise by 6% per year for the 10 years they are planning to build the car. Th e cost to build the car can be modeled by the function f (t) 5 15,000 (1.06)t , where t is the number of years after 2010. How much will it cost the company to build the car in 2017?

x 1 2 3 4y 3 6 12 24


216

Name Class Date


Exponential Functions

Evaluate each function over the domain {22, 21, 0, 1, 2, 3 }. As the values of the domain increase, do the values of the range increase or decrease?

17. f(x) 5 3x 18. y 5 4.2x

19. m(x) 5 0.3x 20. g(t) 5 4 ? 3x

21. y 5 50 ? 0.1x 22. f(x) 5 2 ? 4x

Which function has the greater value for the given value of x?

23. y 5 5x or y 5 x5 for x 5 2 24. y 5 300 ? x3 or y 5 100 ? 3x for x 5 4

Solve each equation.

25. 3x 5 81 26. 5 ? 2x 5 40

27. 4x 1 4 5 68 28. 3 ? 2x 2 16 5 80

29. Reasoning Th e function that models the growth of a $1000 investment that earns 7% per year is f(x) 5 1000(1.07)x. How do you think you would write a function that models the growth of $1500 that earns 8% per year? Use that function to determine how much money a person would have after 5 years if she invested $1500 in an account earning 8% per year.

30. Writing Discuss the diff erences between exponential functions with a base

of 2 and 3, y 5 2x and y 5 3x, and quadratic and cubic functions y 5 x2 and

y 5 x3. Focus on the shapes of the diff erent graphs and rates of growth.

31. Open-Ended Find the value of each of the functions a) f(x) 5 2x2 and b) f(x) 5 2 ? 2x for x 5 5. Write another quadratic function and another exponential function with a base of two whose values at x 5 5 are between the values you found for functions a and b.


217

Name Class Date

7-6 Standardized Test PrepExponential Functions

Multiple Choice


1. Which of the following equations represents an exponential function?

A. y 5 3x2 B. y 5 3(2x) C. y 5 2x3 D. y 5 5x 1 4

2. What is the solution to 4 ? 3x 5 36? F. 2 G. 3 H. 4 I. 6

3. Which function has the greatest value for x 5 3?

A. f (x) 5 2 ? 4x B. f (x) 5 4 ? 2x C. f (x) 5 4 ? x2 D. f (x) 5 2 ? x4

4. When graphed, which function will not have an x-intercept?

F. y 5 2x 1 3 G. y 5 2x3 H. y 5 3x2 I. y 5 3 ? 2x

5. Which of the following statements is not true regarding the function y 5 2x?

A. Th e function is an exponential function. B. Th e function has a domain of all real numbers. C. As the value of x gets very large, the value of y gets close to zero. D. As the value of x increases by one, the value of y doubles.

6. Which of the following ordered pairs is on the graph of y 5 3 ? 4x?

F. (0, 0) G. (1, 81) H. (2, 24) I. (2, 48)

Short Response

7. You invest $1000 at 7% interest compounded annually. What is the exponential equation that models this situation? According to your equation, how much will the investment be worth 3 years later?


218

Name Class Date

7-7 Think About a PlanExponential Growth and Decay

Business You open a bank account to save for college and deposit $400 in the account. Each year, the balance in your account will increase 5%. a. Write a function that models your annual balance. b. What will be the total amount in your account after 7 yr?

1. Make a table that has one column representing the number of years the account has been open, a second column representing the balance in the given year, and a third column representing the total balance.

2. Th e fi rst row of the table represents year 0. Year 0 is the start up year, the year with a balance of $400. If you want to know how much you have after three years, you will read your balance from the fourth line of the table, which starts with the number 3. In order to answer the question in part b, what line of the table should you use? What number would start that line? Why?

3. Recall that the general form of an exponential function is f (x) 5 a ? b x. Given

the situation described, what is a? What is b? Write the function that models your annual balance. Check to make sure the function you created provides the same values you put into your table for years 1, 2 and 3.

a = b = f(x) =

4. Use the exponential function and extend the table to answer part b.

0123

Years Account Open Balance this Year Total Balance


219

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7-7 Practice Form G

Exponential Growth and Decay

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

1. f (x) 5 3 ? 5x 2. y 5 250 ? 1.065x

3. g (t) 5 3.5t 4. h(x) 5 5 ? 1.02x

Find the balance in each account after the given period.

5. $8000 principal earning 5% compounded annually, after 6 yr

6. $2000 principal earning 5.4% compounded annually, after 4 yr

7. $500 principal earning 4% compounded quarterly, after 10 yr

8. $6500 principal earning 2.8% compounded monthly, after 2 yr

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

9. y 5 8 ? 0.8x 10. f (x) 5 12 ? 0.1x

State whether the equation represents exponential growth, exponential decay, or neither.

11. y 5 0.82 ? 3x 12. f (x) 5 5 ? 0.3x

13. f (x) 5 18 ? x 2 14. y 5 0.9x

15. Th e town manager reports that revenue for a given year is $2.5 million. Th e budget director predicts that revenue will increase by 4% per yr. If the director’s prediction holds true, how much revenue will the town have available 10 years from the date of the town manager’s report?

16. A wildlife manager determines that there are approximately 200 deer in a certain state park.

a. Th e population is growing at a rate of 7% per year. How many deer will live in the park after 4 years?

b. If the carrying capacity of this park is 350 deer, how long will it take for the deer population to reach carrying capacity?


220

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Exponential Growth and Decay

17. Open-Ended Write an exponential function that begins its rapid increase when 2 # x # 3. Write another that begins its rapid increase when 3 # x # 4. Write a third that begins its rapid increase when 6 # x # 8.

18. A business purchases a computer system for $3000. If the value of the system decreases at a rate of 15% per year, how much is the computer worth after 4 years?

19. Writing Explain the diff erence in how you would model the following situations. Person A puts $1000 in a safe in his home, and puts in an additional $50 per year. Person B puts $1000 in an investment that earns 5% per year. Why is one exponential and the other linear? How would their graphs compare? How would their values compare over time?

State whether each graph shows an exponential growth function, an exponential decay function, or neither.

20.

xO

y

2

4

2

2

4

4 42

21.

xO

y

4

2

6

2

2

4 42

22.

xO

y

2

4

2

2

4

4 42

23.

x

y

2

4

2

2

4

4 42

24. Reasoning Can the graph of an exponential function ever have a y-intercept of 0? Why or why not?


221

Name Class Date

7-7 Standardized Test PrepExponential Growth and Decay

Multiple Choice


1. Which of the following functions models exponential decay? A. f (x) 5 12 ? 3x B. y 5 2 ? 0.8x C. y 5 23x

2 D. f (x) 5 1.8x

2. Suppose you deposit $500 into an account earning 7.8% interest compounded annually. How long will it take for the account to be worth $1000?

F. between 6 and 7 years H. between 8 and 9 years G. between 7 and 8 years I. between 9 and 10 years

3. What is always true regarding functions that model exponential growth? A. Th eir graphs are always symmetric about the y-axis. B. Th e range depends on the growth factor. C. Th e value of y is always greater than the value of x. D. Th e value of y decreases as x gets more negative.

4. What is 3 ? 2x if x 5 3? F. 18 G. 24 H. 125 I. 216

5. Where will the graphs of y 5 4x, y 5 1.2x and y 5 0.6x intersect?

A. (0, 0) B. (1, 0) C. (0, 1) D. (1, 1)

Short Response

6. Suppose you deposit $2000 into an account earning 5% interest compounded quarterly. To the nearest dollar, what is the balance after 4 years?

Documents

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