8.1 Apply Exponent Properties Involving Products · Apply Exponent Properties Involving Products Goal p Use properties of exponents involving products. 8.1. VOCABULARY Order of magnitude

VOCABULARY

Order of magnitude

Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Algebra 1 Notetaking Guide 193

Your Notes

PRODUCT OF POWERS PROPERTY

Let a be a real number, and let m and n be positive integers.

Words: To multiply powers having the same base, .

Algebra: am p an 5 a

Example: 56 p 53 5 5 5 5

Simplify the expression.

a. 22 p 23 5 2

5 2

b. w9 p w2 p w7 5 w

5 w

c. 44 p 4 5 44 p 4

5 4

5 4

d. (26)(26)6 5 (26) p (26)6

5 (26)

5 (26)

Example 1 Use the product of powers property

When simplifying powers with numerical bases only, write your answers using exponents.

Apply Exponent PropertiesInvolving ProductsGoal p Use properties of exponents involving products.

8.1

VOCABULARY

Order of magnitude The order of magnitude of a quantity is the power of 10 nearest the quantity.


Your Notes

PRODUCT OF POWERS PROPERTY


Words: To multiply powers having the same base, add the exponents .

Algebra: am p an 5 a m 1 n

Example: 56 p 53 5 5 6 1 3 5 5 9


a. 22 p 23 5 2 2 1 3

5 2 5

b. w9 p w2 p w7 5 w 9 1 2 1 7

5 w 18

c. 44 p 4 5 44 p 4 1

5 4 4 1 1

5 4 5

d. (26)(26)6 5 (26) 1 p (26)6

5 (26) 1 1 6

5 (26) 7

Example 1 Use the product of powers property


Apply Exponent PropertiesInvolving ProductsGoal p Use properties of exponents involving products.

8.1

Your NotesPOWER OF A POWER PROPERTY


Words: To find a power of a power, .

Algebra: (am)n 5 a

Example: (34)2 5 3 5 3


a. (52)3 5 5 5 5

b. (n7)2 5 n 5 n

c. [(23)5]3 5 (23)

5 (23)

d. [(z 2 4)2]5 5 (z 2 4)

5 (z 2 4)

Example 2 Use the power of a power property

POWER OF A PRODUCT PROPERTY

Let a and b be real numbers, and let m be a positive integer.

Words: To find a power of a product, find the .

Algebra: (ab)m 5

Example: (23 p 17)5 5


a. (4 p 16)7 5

b. (23rs)2 5 ( )2 5 ( )2 p 2 p 2

5

c. 2(3rs)2 5 2( )2 5 2( 2 p 2 p 2) 5

Example 3 Use the power of a product property

When simplifying powers with numerical and variable bases, evaluate the numerical power.

194 Lesson 8.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your NotesPOWER OF A POWER PROPERTY


Words: To find a power of a power, multiply exponents .

Algebra: (am)n 5 a mn

Example: (34)2 5 3 4 p 2 5 3 8


a. (52)3 5 5 2 p 3 5 5 6

b. (n7)2 5 n 7 p 2 5 n 14

c. [(23)5]3 5 (23) 5 p 3

5 (23) 15

d. [(z 2 4)2]5 5 (z 2 4) 2 p 5

5 (z 2 4) 10

Example 2 Use the power of a power property

POWER OF A PRODUCT PROPERTY

Let a and b be real numbers, and let m be a positive integer.

Words: To find a power of a product, find the power of each factor and multiply .

Algebra: (ab)m 5 ambm

Example: (23 p 17)5 5 235 p 175


a. (4 p 16)7 5 47 p 167

b. (23rs)2 5 ( 23 p r p s )2 5 ( 23 )2 p r 2 p s 2

5 9r2s2

c. 2(3rs)2 5 2( 3 p r p s )2 5 2( 3 2 p r 2 p s 2) 5 29r2s2

Example 3 Use the power of a product property




Your Notes

1. (27)8(27)5 2. k3 p k p k2 3. (p3)4

4. [(q 1 8)2]6 5. (8cd)2 6. 2(5z)3

Checkpoint Simplify the expression.

Simplify x2 p (3x3y)3.

Solution

x2 p (3x3y)3 5 property

5 property

5 property

Example 4 Use all three properties

7. (2x5)4 8. (3y3)4 p y5


Homework


Your Notes

1. (27)8(27)5 2. k3 p k p k2 3. (p3)4

(27)13 k6 p12

4. [(q 1 8)2]6 5. (8cd)2 6. 2(5z)3

(q 1 8)12 64c2d2 2125z3


Simplify x2 p (3x3y)3.

Solution

x2 p (3x3y)3 5 x2 p 33 p (x3)3 p y3 Power of a product property

5 x2 p 27 p x9 p y3 Power of a power property

5 27x11y3 Product of powers property

Example 4 Use all three properties

7. (2x5)4 8. (3y3)4 p y5

16x20 81y17


Homework

8.2 Apply Exponent PropertiesInvolving QuotientsGoal p Use properties of exponents involving quotients.

QUOTIENT OF POWERS PROPERTY

Let a be a nonzero real number, and let m and n be positive integers such that m > n.

Words: To divide powers having the same base, the exponents.

Algebra: am }

an 5 a , a Þ 0

Example: 47 }

42 5 4 5 4


a. 612 }

65 5 6 5 6

b. (22)7

} (22)4

5 (22) 5 (22)

c. 42 p 48

} 44 5 4}

44

5 4

5

d. 1 } y9 p y12 5

y12 }

y9

5 y

5

Example 1 Use the quotient of powers property


Your Notes


8.2 Apply Exponent PropertiesInvolving QuotientsGoal p Use properties of exponents involving quotients.

QUOTIENT OF POWERS PROPERTY

Let a be a nonzero real number, and let m and n be positive integers such that m > n.

Words: To divide powers having the same base, subtract the exponents.

Algebra: am }

an 5 a m 2 n , a Þ 0

Example: 47 }

42 5 4 7 2 2 5 4 5


a. 612 }

65 5 6 12 2 5 5 6 7

b. (22)7

} (22)4

5 (22) 7 2 4 5 (22) 3

c. 42 p 48

} 44 5 4

10 }

44

5 4 10 2 4

5 46

d. 1 } y9 p y12 5

y12 }

y9

5 y 12 2 9

5 y3

Example 1 Use the quotient of powers property


Your Notes


Your NotesPOWER OF A QUOTIENT PROPERTY

Let a and b be real numbers with b Þ 0, and let m be a positive integer.

Words: To find a power of a quotient, find the power of the and the power of the and divide.

Algebra: 1 a } b 2 m 5 , b Þ 0 Example: 1 4 } 7 2 3 5 ____ ____

Example 2 Use the power of a quotient property


a. 1 r } s 2 5 5

____

b. 1 2 4 } w 2 3 5 1 2 3 5 5 5

____ ______ ______ ______


1. (28)8

} (28)5

2. 35 p 34

} 33

3. 1 2 r } 3 2 2 4. 1 5 } t 2

4



Your NotesPOWER OF A QUOTIENT PROPERTY

Let a and b be real numbers with b Þ 0, and let m be a positive integer.

Words: To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.

Algebra: 1 a } b 2 m 5 am }

bm , b Þ 0 Example: 1 4 } 7 2 3 5 43 }

73 ____ ____

Example 2 Use the power of a quotient property


a. 1 r } s 2 5 5 r

5 }

s5

____

b. 1 2 4 } w 2 3 5 1 24 } w 2 3 5 (24)3 }

w3 5 264 }

w3 5 2 64 } w3

____ ______ ______ ______


1. (28)8

} (28)5

2. 35 p 34

} 33

(28)3 36

3. 1 2 r } 3 2 2 4. 1 5 } t 2

4

r2 } 9 625 }

t4



Your Notes

Homework

Example 3 Use properties of exponents

Simplify 1 2y7 }

y5 2 3.

Solution

1 2y7 }

y5 2 3 5 property

5 property

5 property

5 property

5. 1 7y3z } y 2 2 6. 2s4

} t p 1 2t } s 2 3

7. 1 6m3n2 } 3mn 2 3 8. 4a }

b2 p 1 2a2b3 } a 2 4



Your Notes

Homework


Simplify 1 2y7 }

y5 2 3.

Solution

1 2y7 }

y5 2 3 5 (2y7)3

} (y5)3

Power of a quotient property

5 23 p (y7)3

} (y5)3

Power of a product property

5 8y21

} y15

Power of a power property

5 8y6 Quotient of powers property

5. 1 7y3z } y 2 2 6. 2s4

} t p 1 2t } s 2 3

49y5z2 16st2

7. 1 6m3n2 } 3mn 2 3 8. 4a }

b2 p 1 2a2b3 } a 2 4

8m6n3 64a5b10



8.3 Define and Use Zero andNegative ExponentsGoal p Use zero and negative exponents.

DEFINITION OF ZERO AND NEGATIVE EXPONENTS

Words Algebra Example

a to the zero power a0 5 , a Þ 0 50 5 is 1.

a2n is the reciprocal a2n 5 , a Þ 0 221 5 of an. ___ ____

an is the reciprocal an 5 , a Þ 0 2 5 of a2n. ____ ____

Example 1 Use definition of zero and negative exponents

Evaluate the expression.

a. 223 5 Definition of ____

5 Evaluate exponent. ___

b. (210)0 5 Definition of

c. 1 1 } 4 2 23

5 Definition of

____

5 Evaluate exponent.

___

5 Simplify.

d. 027 5 a2n is defined only for a number a.


Your Notes

8.3 Define and Use Zero andNegative ExponentsGoal p Use zero and negative exponents.

DEFINITION OF ZERO AND NEGATIVE EXPONENTS

Words Algebra Example

a to the zero power a0 5 1 , a Þ 0 50 5 1 is 1.

a2n is the reciprocal a2n 5 1 } an , a Þ 0 221 5 1 }

2

of an. ___ ____

an is the reciprocal an 5 1 } a2n , a Þ 0 2 5 1 }

221 of a2n. ____ ____

Example 1 Use definition of zero and negative exponents


a. 223 5 1 } 23

Definition of negative ____ exponents

5 1 } 8 Evaluate exponent.

___

b. (210)0 5 1 Definition of zero exponent

c. 1 1 } 4 2 23

5 1 } 1 1 } 4 2 3

Definition of negative

____ exponents

5 1 } 1 } 64

Evaluate exponent.

___

5 64 Simplify.

d. 027 5 undefined a2n is defined only for a nonzero number a.


Your Notes

Your NotesPROPERTIES OF EXPONENTS

Let a and b be real numbers, and let m and n be integers.

am p an 5 a property

(am)n 5 a property

(ab)m 5 property

am }

an 5 a , a Þ 0 property

1 a } b 2 m 5 , b Þ 0 property ____


a. (25)4 p (25)24 5 Product of powersproperty

5 exponents.

5 Definition of

b. (522)22 5 property

5 exponents.

5 Evaluate power.

c. 1 } 422 5 Definition of

5 Evaluate power.

d. 32 }

321 5 property

5 exponents.

5 Evaluate power.

Example 2 Evaluate exponential expressions


Your NotesPROPERTIES OF EXPONENTS

Let a and b be real numbers, and let m and n be integers.

am p an 5 a m 1 n Product of powers property

(am)n 5 a mn Power of a power property

(ab)m 5 ambm Power of a product property

am }

an 5 a m 2 n , a Þ 0 Quotient of powers property

1 a } b 2 m 5 am }

bm , b Þ 0 Power of a quotient property ____


a. (25)4 p (25)24 5 (25)4 1 (24) Product of powersproperty

5 50 Add exponents.

5 1 Definition of zero exponent

b. (522)22 5 522 p (22) Power of a power property

5 54 Multiply exponents.

5 625 Evaluate power.

c. 1 } 422 5 42 Definition of

negative exponents


d. 32 }

321 5 32 2 (21) Quotient of powers property

5 33 Subtract exponents.


Example 2 Evaluate exponential expressions


Your Notes

Homework

1. 1 1 } 8 2 21 2. 1 } 322

3. 621 } 6 4. (521)2

Checkpoint Evaluate the expression.


Simplify the expression 2w23x } (2wx)2

. Write your answer using

only positive exponents.

Solution

2w23x } (2wx)2

5 Definition of negative exponents ___________

5 property ___________

5 property ___________

5 property ___________

5. 6fg24

} 2f2g

6. (3yz2)22



Your Notes

Homework

1. 1 1 } 8 2 21 2. 1 } 322

8 9

3. 621 } 6 4. (521)2

1 } 36 1 } 25



Simplify the expression 2w23x } (2wx)2

. Write your answer using

only positive exponents.

Solution

2w23x } (2wx)2

5 2x } w3(2wx)2

Definition of negative exponents ___________

5 2x } w3(4w2x2)

Power of a product property ___________

5 2x } 4w5x2

Product of powers property ___________

5 1 } 2w5x

Quotient of powers property ___________

5. 6fg24

} 2f2g

6. (3yz2)22

3 } fg5 1 }

9y2z4



Define and Use Fractional Exponents

Goal p Use fractional exponents.

a. 25 1 } 2

5 b. 362 1 } 2

5

5 5

5

c. 4 5 } 2

5 d. 492 3 } 2

5

5 5

5 5

5 5

5 5

5

Example 1 Evaluate expressions involving square roots

1. 16 3 } 2 2. 64

2 1 } 2

3. 1442 3 } 2

4. 25 5 } 2


202 8.3 Focus On Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes

Focus On OperationsUse after Lesson 8.3

Define and Use Fractional Exponents

Goal p Use fractional exponents.

a. 25 1 } 2

5 Ï}

25 b. 362 1 } 2

5 1 }

36 1 } 2

5 5 5 1 }

Ï}

36

5 1 } 6

c. 4 5 } 2

5 4 1 1 } 2 2 p 5

d. 492 3 } 2

5 49 1 1 } 2 2 p 1 23 2

5 1 4 1 } 2 2 5 5 1 49

1 } 2 2 23

5 1 Ï}

4 2 5 5 1 Ï}

49 2 23

5 25 5 723

5 32 5 1 } 73

5 1 } 343

Example 1 Evaluate expressions involving square roots

1. 16 3 } 2 64 2. 64

2 1 } 2 1 }

8

3. 1442 3 } 2

1 } 1728

4. 25 5 } 2 3125


202 8.3 Focus On Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes

Focus On OperationsUse after Lesson 8.3

Homework

Your Notes

5. 64 2 } 3 6. 125

2 4 } 3

7. 1 1 } 4 2 2 3 } 2

1 1 } 4 2

5 } 2 8. 1 27 2

5 } 3 1 27 2

2 4 } 3



a. 152 1 } 2 p 15 5 } 2

5 b. 4 4 } 3 p 4 }

4 1 } 3 5

5 5

5 5

5 5

5

Copyright © Holt McDougal. All rights reserved. 8.3 Focus On Operations • Algebra 1 Notetaking Guide 203

a. 64 1 } 3 5 b. 216

2 1 } 3 5

5 5

5 5

c. 8 4 } 3 5 d. 27

2 2 } 3 5

5 5

5 5

5 5

5 5

5

Example 2 Evaluate expressions involving cube roots

Homework

Your Notes

5. 64 2 } 3 16 6. 125

2 4 } 3 1 }

625

7. 1 1 } 4 2 2 3 } 2

1 1 } 4 2

5 } 2 1 }

4 8. 1 27 2

5 } 3 1 27 2

2 4 } 3 3



a. 152 1 } 2 p 15 5 } 2

515 1 2

1 } 2 2 1

1

5 } 2 2

b. 4 4 } 3 p 4 }

4 1 } 3 5 4

1 4 } 3 2 11

}

4 1 } 3

5 15 4 } 2 5 4

1 7 } 3 2 }

4 1 } 3

5 152 5 4 1 7 } 3 2 2 1 1 }

3 2

5 225 5 42

5 16

Copyright © Holt McDougal. All rights reserved. 8.3 Focus On Operations • Algebra 1 Notetaking Guide 203

a. 64 1 } 3 5

3 Ï}

64 b. 2162 1 } 3

5 1 }

216 1 } 3

5 3 Ï}

43 5 1 }

3 Ï}

216

5 4 5 1 } 6

c. 8 4 } 3 5 8 1

1 } 2 2 p 4 d. 27

2 2 } 3 5 27 1 1 } 3 2 p 1 22 2

5 1 8 1 } 3 2 4 5 1 27

1 } 3 2 22

5 1 3 Ï}

8 2 4 5 1 3 Ï}

27 2 22

5 24 5 322

516 5 1 }

32

5 1 } 9

Example 2 Evaluate expressions involving cube roots

8.4 Use Scientific NotationGoal p Read and write numbers in scientific notation.

VOCABULARY

Scientific notation Your Notes

SCIENTIFIC NOTATION

A number is written in scientific notation when it is of the form where 1 ≤ c < 10 and n is an integer.

Number Standard form Scientific notation

Sixteen million

Two hundredths

a. 7,820,000 5 3 10 Move decimal point places to the . Exponent is .

b. 0.00401 5 3 10 Move decimal point places to the . Exponent is .

Example 1 Write numbers in scientific notation

a. 3.89 3 109 5 Exponent is . Move decimal point

places to the .

b. 9.097 3 1025 5 Exponent is . Move decimal point places to the

.

Example 2 Write numbers in standard form


8.4 Use Scientific NotationGoal p Read and write numbers in scientific notation.

VOCABULARY

Scientific notation A number is written in scientific notation when it is of the form c 3 10n where 1 ≤ c < 10 and n is an integer.

Your Notes

SCIENTIFIC NOTATION

A number is written in scientific notation when it is of the form c 3 10n where 1 ≤ c < 10 and n is an integer.

Number Standard form Scientific notation

Sixteen million 16,000,000 1.6 3 107

Two hundredths 0.02 2 3 1022

a. 7,820,000 5 7.82 3 10 6 Move decimal point 6 places to the left . Exponent is 6 .

b. 0.00401 5 4.01 3 10 23 Move decimal point 3 places to the right . Exponent is 23 .

Example 1 Write numbers in scientific notation

a. 3.89 3 109 5 3,899,000,000 Exponent is 9 . Move decimal point 9 places to the right .

b. 9.097 3 1025 5 0.00009097 Exponent is 25 . Move decimal point 5 places to the left .

Example 2 Write numbers in standard form


Your Notes


1. Write the number 0.0899 in scientific notation. Then write the number 6.0001 3 107 in standard form.

Checkpoint Complete the following exercise.

Order 3.2 3 1024, 0.0004, and 2.8 3 1025 from least to greatest.

SolutionStep 1 Write each number in scientific notation, if

necessary.

0.0004 5

Step 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10.

Because 1025 1024, you know that is less than both and

. Because 3.2 4, you know that is less than .

So, < < .

Step 3 Write the original numbers in order from least to greatest.

Example 3 Order numbers in scientific notation

2. Order 225,000, 1,740,000, and 1.75 3 105 from least to greatest.


Your Notes


1. Write the number 0.0899 in scientific notation. Then write the number 6.0001 3 107 in standard form.

8.99 3 1022; 60,001,000


Order 3.2 3 1024, 0.0004, and 2.8 3 1025 from least to greatest.

SolutionStep 1 Write each number in scientific notation, if

necessary.

0.0004 5 4 3 1024

Step 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10.

Because 1025 < 1024, you know that 2.8 3 1025 is less than both 3.2 3 1024 and 4 3 1024 . Because 3.2 < 4, you know that 3.2 3 1024 is less than 4 3 1024 .

So, 2.8 3 1025 < 3.2 3 1024 < 4 3 1024 .

Step 3 Write the original numbers in order from least to greatest.

2.8 3 1025; 3.2 3 1024; 0.0004

Example 3 Order numbers in scientific notation

2. Order 225,000, 1,740,000, and 1.75 3 105 from least to greatest.

1.75 3 105; 225,000; 1,740,000


Your Notes

Evaluate the expression. Write your answer in scientific notation.

a. (5.6 3 1024)(1.4 3 1025) 5 (5.6 p 1.4) 3 (1024 p 1025) Commutative property

and associative property

5 3 Product of powers property

b. (3.2 3 102)3

5 3 Power of a product property

5 3 Power of a power property

5 ( ) 3 Write in scientific notation.

5 3 ( ) Associative property

5 Product of powers property

c. 3.5 3 1023 }}

1.75 3 1025

5 3.5 } 1.75 3 1023 }

1025 Product rule for fractions

5 3 Quotient of powers property

Example 4 Compute with numbers in scientific notation

Homework

3. (2.01 3 1027)2 4. 4.8 3 1024 }

6 3 1024

Checkpoint


Your Notes

Evaluate the expression. Write your answer in scientific notation.

a. (5.6 3 1024)(1.4 3 1025) 5 (5.6 p 1.4) 3 (1024 p 1025) Commutative property

and associative property

5 7.84 3 1029 Product of powers property

b. (3.2 3 102)3

5 3.23 3 (102)3 Power of a product property

5 32.768 3 106 Power of a power property

5 ( 3.2768 3 101 ) 3 106 Write 32.768 in scientific notation.

5 3.2768 3 ( 101 3 106 ) Associative property

5 3.2768 3 107 Product of powers property

c. 3.5 3 1023 }}

1.75 3 1025

5 3.5 } 1.75 3 1023 }

1025 Product rule for fractions

5 2 3 102 Quotient of powers property

Example 4 Compute with numbers in scientific notation

Homework

3. (2.01 3 1027)2 4. 4.8 3 1024 }

6 3 1024

4.0401 3 10214 8 3 1021

Checkpoint


Your Notes

8.5 Write and Graph ExponentialGrowth FunctionsGoal p Write and graph exponential growth models.

VOCABULARY

Exponential function

Exponential growth

Compound interest

Write a rule for the function.

x 22 21 0 1 2

y 2 } 9 2 } 3 2 6 8

SolutionStep 1 Tell whether the function is exponential. Here the

y-values are multiplied by for each increase of 1 in x, so the table represents an exponential function of the form where .

Step 2 Find the value of a by finding the value of y when x 5 0. When x 5 0, y 5 5 5 . The value of y when x 5 0 is , so .

Step 3 Write the function rule. A rule for the function is y 5 .

Example 1 Write a function rule


Your Notes

8.5 Write and Graph ExponentialGrowth FunctionsGoal p Write and graph exponential growth models.

VOCABULARY

Exponential function A function of the form y 5 abx where a Þ 0, b > 0, and b Þ 1

Exponential growth A quantity that increases by the same percent over equal time periods

Compound interest Interest earned on both an initial investment and on previously earned interest

Write a rule for the function.

x 22 21 0 1 2

y 2 } 9 2 } 3 2 6 8

SolutionStep 1 Tell whether the function is exponential. Here the

y-values are multiplied by 3 for each increase of 1 in x, so the table represents an exponential function of the form y 5 abx where b 5 3 .

Step 2 Find the value of a by finding the value of y when x 5 0. When x 5 0, y 5 ab0 5 a p 1 5 a . The value of y when x 5 0 is 2 , so a 5 2.

Step 3 Write the function rule. A rule for the function is y 5 2 p 3x .

Example 1 Write a function rule


Your Notes

Graph the function y 5 3x. Identify its domain and range.

SolutionStep 1 Make a table by choosing a

x

y

1

3

5

7

9

12123 3

few values for x and finding the values of y. The domain is .

x 22 21 0 1 2

y

Step 2 Plot the points.

Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is .

Example 2 Graph an exponential function

Graph y 5 2 p 3x. Compare the graph with the graph of y 5 3x.

SolutionTo graph each function, make x y 5 3x y 5 2 p 3x

22

21

0

1

2

a table of values, plot the points, and draw a smooth curve through the points.

x

y

2

6

10

14

18

12123 3

Because the y-values for y 5 2 p 3x are the corresponding y-values for y 5 3x, the graph of y 5 2 p 3x is a of the graph of y 5 3x.

Example 3 Compare graphs of exponential functions


Your Notes

Graph the function y 5 3x. Identify its domain and range.

SolutionStep 1 Make a table by choosing a

x

y

1

3

5

7

9

12123 3

y 5 3x

(2, 9)

(1, 3)

(0, 1)( )1922,

( )1321,

few values for x and finding the values of y. The domain is all real numbers .

x 22 21 0 1 2

y 1 } 9 1 }

3

1 3 9


Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers .


Graph y 5 2 p 3x. Compare the graph with the graph of y 5 3x.

SolutionTo graph each function, make x y 5 3x y 5 2 p 3x

22 1 } 9 2 }

9

21 1 } 3 2 }

3

0 1 2

1 3 6

2 9 18

a table of values, plot the points, and draw a smooth curve through the points.

x

y

2

6

10

14

18

12123 3

y 5 3xy 5 2 ? 3x

Because the y-values for y 5 2 p 3x are 2 times the corresponding y-values for y 5 3x, the graph of y 5 2 p 3x is a vertical stretch of the graph of y 5 3x.



Your Notes

1. Write a rule for the function.

x 22 21 0 1 2

y 2 1 } 16 2 1 } 4 21 24 216

2. Graph y 5 4x. Identify its

x

y

2

6

10

14

12123 322

domain and range.

3. Graph y 5 22 p 3x. Compare

x

y

3

9

12123 323

29

215

the graph with the graph of y 5 3x.

Checkpoint Complete the following exercises.


Your Notes

1. Write a rule for the function.

x 22 21 0 1 2

y 2 1 } 16 2 1 } 4 21 24 216

y 5 21 p 4x

2. Graph y 5 4x. Identify its

x

y

2

6

10

14

12123 3

y 5 4x

22

(0, 1)

(1, 4)

(2, 16)

( )11622,

( )1421,

domain and range.

The domain is all real numbers. The range is all positive real numbers.

3. Graph y 5 22 p 3x. Compare

x

y

3

9

12123 3

y 5 3x

y 5 22 ? 3x23

29

215

the graph with the graph of y 5 3x.

The graph of y 5 22 p 3x is a vertical stretch and a reflection in the x-axis of the graph of y 5 3x.



Your Notes

Homework

EXPONENTIAL GROWTH MODEL

y 5 a(1 1 r)t

a is the . r is the .

1 1 r is the . t is the .

Investment You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years?

Solution

The initial amount is , the interest rate is , or , and the time period is .

y 5 a(1 1 r)t Write exponential growth model.

5 (1 1 ) Substitute for a, for r, and for t.

5 250( )10 Simplify.

ø Use a calculator.

You will have in 10 years.

Example 4 Solve a compound interest problem

4. In Example 4, suppose the annual interest rate is 5%. How much will your investment be worth in 10 years?



Your Notes

Homework

EXPONENTIAL GROWTH MODEL

y 5 a(1 1 r)t

a is the initial amount . r is the growth rate .

1 1 r is the growth factor . t is the time period .

Investment You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years?

Solution

The initial amount is $250 , the interest rate is 4% , or 0.04 , and the time period is 10 years .

y 5 a(1 1 r)t Write exponential growth model.

5 250 (1 1 0.04 ) 10 Substitute 250 for a, 0.04 for r, and 10 for t.

5 250( 1.04 )10 Simplify.

ø 370.06 Use a calculator.

You will have $370.06 in 10 years.

Example 4 Solve a compound interest problem

4. In Example 4, suppose the annual interest rate is 5%. How much will your investment be worth in 10 years?

about $407.22



8.6 Write and Graph ExponentialDecay FunctionsGoal p Write and graph exponential decay functions.

VOCABULARY

Exponential decay

Graph the function y 5 1 1 } 3 2 x and identify its domain and

range.

SolutionStep 1 Make a table of values.

The domain is .

x 22 21 0 1 2

y


x

y

1

3

5

7

9

12123 3

Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is .



Your Notes

8.6 Write and Graph ExponentialDecay FunctionsGoal p Write and graph exponential decay functions.

VOCABULARY

Exponential decay A quantity that decreases by the same percent over equal time periods

Graph the function y 5 1 1 } 3 2 x and identify its domain and

range.

SolutionStep 1 Make a table of values.

The domain is all real numbers .

x 22 21 0 1 2

y 9 3 1 1 } 3 1 }

9


x

y

1

3

5

7

9

12123 3

(22, 9)

(21, 3)

(0, 1) ( )192,

( )131,

y 5x( )1

3

Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers .



Your Notes

Your Notes

Graph y 5 2 p 1 1 } 3 2 x. Compare the graph with the graph

of y 5 1 1 } 3 2 x.

Solution

x y 5 1 1 } 3 2 x y 5 2 p 1 1 } 3 2 x

22

21

0

1

2

x

y

1

3

5

7

9

12123 3

Because the y-values for y 5 2 p 1 1 } 3 2 x are

the corresponding y-values for y 5 1 1 } 3 2 x, the graph of

y 5 2 p 1 1 } 3 2 x is a of the graph of

y 5 1 1 } 3 2 x.


1. Graph y 5 22 p 1 1 } 3 2 x. Compare the graph with the

graph of 1 1 } 3 2 x.

x

y

3

9

12123 323

29

215



Your Notes

Graph y 5 2 p 1 1 } 3 2 x. Compare the graph with the graph

of y 5 1 1 } 3 2 x.

Solution

x y 5 1 1 } 3 2 x y 5 2 p 1 1 } 3 2 x

22 9 18

21 3 6

0 1 2

1 1 } 3 2 }

3

2 1 } 9 2 }

9

x

y

1

3

5

7

9

12123 3

y 5x( )1

3

y 5 2 ? x( )1

3

Because the y-values for y 5 2 p 1 1 } 3 2 x are 2 times

the corresponding y-values for y 5 1 1 } 3 2 x, the graph of

y 5 2 p 1 1 } 3 2 x is a vertical stretch of the graph of

y 5 1 1 } 3 2 x.


1. Graph y 5 22 p 1 1 } 3 2 x. Compare the graph with the

graph of 1 1 } 3 2 x.

x

y

3

9

12123 323

29

215

y 5x( )1

3

y 5 22 ? x( )1

3

The graph of y 5 22 p 1 1 } 3 2 x

is a vertical stretch and a

reflection in the x-axis of

the graph of y 5 1 1 } 3 2 x.



Your Notes

Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

SolutionThe graph represents

x

y

1

3

5

12121

3 5

(1, 1)

(0, 5) (y 5 abx where 0 < b < 1). The y-intercept is , so a 5 . Find the value of b by using the point (1, 1) and a 5 .

y 5 abx Write function.

5 p b Substitute.

5 b Solve.

A function rule is .

Example 3 Classify and write rules for functions

EXPONENTIAL GROWTH AND DECAY

Exponential Growth Exponential Decay

y 5 abx, a > 0 y 5 abx, a > 0 and b > 1 and 0 < b < 1

x

y

(0, a)

x

y

(0, a)


EXPONENTIAL DECAY MODEL

y 5 a(1 1 r)t

a is the . r is the .

1 2 r is the . t is the .

Your Notes

Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

SolutionThe graph represents

x

y

1

3

5

12121

3 5

(1, 1)

(0, 5) exponential decay (y 5 abx where 0 < b < 1). The y-intercept is 5 , so a 5 5 . Find the value of b by using the point (1, 1) and a 5 5 .

y 5 abx Write function.

1 5 5 p b 1 Substitute.

0.2 5 b Solve.

A function rule is y 5 5(0.2)x .

Example 3 Classify and write rules for functions

EXPONENTIAL GROWTH AND DECAY

Exponential Growth Exponential Decay

y 5 abx, a > 0 y 5 abx, a > 0 and b > 1 and 0 < b < 1

x

y

(0, a)

x

y

(0, a)


EXPONENTIAL DECAY MODEL

y 5 a(1 1 r)t

a is the initial amount . r is the decay rate .

1 2 r is the decay factor . t is the time period .

Your Notes

Homework

Population The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city's population since 1995. Then find the population in 2003.

SolutionLet P be the population of the city (in thousands), and let t be the time (in years) since 1995. The initial value is , and the decay rate is .

P 5 a(1 2 r)t Write exponential decay model.

5 (1 2 )t Substitute for a, and for r.

5 Simplify.

To find the population in 2003, years after 1995, substitute for t.

P 5 Substitute for t.

ø Use a calculator.

The city's population was about in 2003.

Example 4 Use the exponential decay model

2. The graph of an exponential function passes through the points (0, 4) and (1, 10).

x

y

2

6

10

14

1212322

3

Graph the function. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

3. In Example 4, suppose that the decay rate of the city's population remains the same beyond 2003. What will be the population in 2020?



Your Notes

Homework

Population The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city's population since 1995. Then find the population in 2003.

SolutionLet P be the population of the city (in thousands), and let t be the time (in years) since 1995. The initial value is 357 , and the decay rate is 0.015 .

P 5 a(1 2 r)t Write exponential decay model.

5 357 (1 2 0.015 )t Substitute 357 for a, and 0.015 for r.

5 357(0.985)t Simplify.

To find the population in 2003, 8 years after 1995, substitute 8 for t.

P 5 357(0.985)8 Substitute 8 for t.

ø 316.3 Use a calculator.

The city's population was about 316,300 in 2003.

Example 4 Use the exponential decay model

2. The graph of an exponential function passes through the points (0, 4) and (1, 10).

x

y

2

6

10

14

1212322

3

(1, 10)

(0, 4)

Graph the function. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

Exponential growth;y 5 4(2.5)x

3. In Example 4, suppose that the decay rate of the city's population remains the same beyond 2003. What will be the population in 2020?

about 244,700



Graph the sequence 243, 81, 27, 9, 3, …. Let each term’s position number be the x-value. The term is the corresponding y-value.

Position, x 1 2

Term, y 243 27

Example 2 Graph a geometric sequence

Copyright © Holt McDougal. All rights reserved. 8.6 Focus On Functions • Algebra 1 Notetaking Guide 215

Relate Geometric Sequences to Exponential FunctionsGoal p Identify, graph, and write geometric sequences.

VOCABULARY

Geometric sequence

Common ratio

Tell whether the sequence 243, 81, 27, 9, 3, ... is arithmetic or geometric. Then write the next term of the sequence.

Solution

The first term is a1 5 . Find the of consecutive terms:

a2

} a1 5 5

a3 } a2 5 5 5 } 5 5 5

The ratios are . The sequence is . The

common ratio is .

The next term of the sequence is a6 5 p 5 .

Example 1 Identify a geometric sequence

The sequence is a function. The domain is the set of positive numbers (all positive whole numbers). The range is the set of terms.

1 2 3 4 5 6

y

x

200

50

100

150

Your Notes

Focus On FunctionsUse after Lesson 8.6

Graph the sequence 243, 81, 27, 9, 3, …. Let each term’s position number be the x-value. The term is the corresponding y-value.

Position, x 1 2 3 4 5 6Term, y 243 81 27 9 3 1

Example 2 Graph a geometric sequence

Copyright © Holt McDougal. All rights reserved. 8.6 Focus On Functions • Algebra 1 Notetaking Guide 215

Relate Geometric Sequences to Exponential FunctionsGoal p Identify, graph, and write geometric sequences.

VOCABULARY

Geometric sequence Sequence where the ratio of any term to the previous term is constant

Common ratio The constant ratio in a geometric sequence

Tell whether the sequence 243, 81, 27, 9, 3, ... is arithmetic or geometric. Then write the next term of the sequence.

Solution

The first term is a1 5 243. Find the ratios of consecutive terms:

a2

} a1 5 81

} 243 5 1 } 3 a3

} a2 5 27

} 81 5 1 } 3 a4

} a3 5

9 }

27 5 1 } 3

a5 } a4 5

3 } 9 5

1 } 3

The ratios are constant. The sequence is geometric. The

common ratio is 1 } 3 .

The next term of the sequence is a6 5 a5 p 1 } 3 5 1.

Example 1 Identify a geometric sequence

The sequence is a function. The domain is the set of positive numbers (all positive whole numbers). The range is the set of terms.

1 2 3 4 5 6

y

x

200

50

100

150

Your Notes

Focus On FunctionsUse after Lesson 8.6

Homework

Your NotesGENERAL RULE OF A GEOMETRIC SEQUENCE

The nth term of a with first a1 and common r is given by: an 5 .

Write a rule for the nth term of the geometric sequence 243, 81, 27, 9, 3, …. Then find a10 .

Solution

Write the general rule an 5 . Then substitute

values for and to give an 5 p . For the

tenth term, n 5 , so

a10 5 p 5 .

Example 3 Write a rule for a geometric sequence

1. Tell whether the sequence is arithmetic or geometric. Write the next term in the sequence.

7, 221, 63, 2189, …

2. Graph the sequence 7, 221, 63, 2189, …

3. Write a rule for the nth term of the geometric sequence and find a8. 3, 33, 363, 3993, …


1 2 3 4 5

400

600

200

�200

y

x

216 8.6 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Homework

Your NotesGENERAL RULE OF A GEOMETRIC SEQUENCE

The nth term of a geometric sequence with first term a1 and common ratio r is given by: an 5 a1r n21.

Write a rule for the nth term of the geometric sequence 243, 81, 27, 9, 3, …. Then find a10 .

Solution

Write the general rule an 5 a1r n21 . Then substitute

values for a1 and r to give an 5 243 p 1 1 } 3 2 n 21 . For the

tenth term, n 5 10, so

a10 5 243 p 1 1 } 3 2 10 21 5 1 } 81 .

Example 3 Write a rule for a geometric sequence

1. Tell whether the sequence is arithmetic or geometric. Write the next term in the sequence.

7, 221, 63, 2189, …

Geometric; 567

2. Graph the sequence 7, 221, 63, 2189, …

3. Write a rule for the nth term of the geometric sequence and find a8. 3, 33, 363, 3993, …

an 5 3 p 11n21; 58,461,513


1 2 3 4 5

400

600

200

�200

y

x

216 8.6 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Words to ReviewGive an example of the vocabulary word.

Review your notes and Chapter 8 by using the Chapter Review on pages 560–563 of your textbook.

Order of magnitude


Compound interest

Geometric sequence

Scientific notation

Exponential growth

Exponential decay

Common ratio

Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 217

Words to ReviewGive an example of the vocabulary word.

Review your notes and Chapter 8 by using the Chapter Review on pages 560–563 of your textbook.

Order of magnitude

The order of magnitude of 96,000,000 is 108.


y 5 3 p 2x

Compound interest

Interest earned on both an initial investment and previously earned interest

Geometric sequence

2, 10, 50, 250, 1250,…

Scientific notation

3.7 3 1025

Exponential growth

y 5 16(2.5)t

Exponential decay

y 5 2.85(0.05)t

Common ratio

3 for the sequence 4, 12, 36, 108,…

Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 217

Documents

8.1 Apply Exponent Properties Involving Products · Apply Exponent Properties Involving Products Goal p Use properties of exponents involving products. 8.1. VOCABULARY Order of magnitude