unit 6 – chapter 8

Unit 6 – Chapter 8

Unit 6• Chapter 7 Review and Chap. 8 Skills

• Section 8.1 – Exponent Properties with products

• Section 8.2 – Exponent Properties with quotients

• Section 8.3 – Define and Use Zero and Negative Exponents

• Section 8.4 – Use Scientific Notation

• Section 8.5 – Write and Graph Exponential Growth Functions

• Section 8.6 – Write and Graph Exponential Decay Functions

Warm-Up – Ch. 8

Daily Homework Quiz For use after Lesson 7.1

2. Solve the linear system by graphing.

2x + y = – 3 – 6x + 3y = 3

ANSWER (–1, –1)


ANSWER

A pet store sells angel fish for $6 each and clown loaches for $4 each . If the pet store sold 8 fish for $36, how many of each type of fish did it sell?

3.

2 angel fish and 6 clown loaches


Solve the linear system using substitution

1. –5x – y = 12

3x – 5y = 4

ANSWER (–2, –2)

2. 2x + 9y = –4

x – 2y = 11

ANSWER (7, –2 )


Solve the linear system using elimination.

1. –5x +y = 183x – y = –10

ANSWER (–4, –2)

2. 4x + 2y = 144x – 3y = –11

ANSWER (1, 5)

ANSWER (1, – 6)

3. – 7x – 3y = 11 4x – 2y = 16


A recreation center charges nonmembers $3 to use the pool and $5 to use the basketball courts. A person pays $42 to use the recreation facilities 12 times. How many times did the person use the pool.

4.

ANSWER 9 times


A group of 12 students and 3 teachers pays $57 for admission to a primate research center. Another group of 14 students and 4 teachers pays $69. Find the cost of one student ticket.

3.

ANSWER $3.50


Write a system of inequalities for the shaded region.

1.

ANSWER x < 2,

y > x +1

Prerequisite Skills VOCABULARY CHECK

1. Identify the exponent and the base in the expression 138.

2. Copy and complete: An expression that represents repeated multiplication of the same factor is called a(n) ? .

ANSWER

Exponent: 8, base: 13

ANSWER

Power

Prerequisite Skills SKILLS CHECK

Evaluate the expression.

3. x2 when x = 10 4. a3 when a = 3

5. r2 when r = 56

ANSWER

100

ANSWER

27

6. z3 when z = 12

2536

ANSWER ANSWER

18


Order the numbers from least to greatest.

7. 6.12, 6.2, 6.01 8. 0.073, 0.101, 0.0098

Write the percent as a decimal.

9. 4% 10. 0.5% 11. 13.8% 12. 145%

6.01, 6.12, 6.2

ANSWER

0.0098, 0.073, 0.101

ANSWER

0.04

ANSWER

0.005

ANSWER

0.138

ANSWER

1.45

ANSWER


13. Write a rule for the function.

f (x) = x + 2

ANSWER

Warm-Up – 8.1

Lesson 8.1, For use with pages 488-494


1. x4 when x = 3

2. a2 when a = –6

ANSWER 81

ANSWER 36

3. -a2 when a = –6

ANSWER -36


3. m3 when m = –5

4. A food storage container is in the shape of a cube.What is the volume of the container if one side is4 inches long? Use V = s3.

ANSWER 64 in.3


–125ANSWER

Vocabulary – 8.1• Power

• Repeated multiplication

• Exponent

• How many times to multiply a quantity

• Base

• Quantity multiplied

• Order of Magnitude

• The “power of 10” nearest the number

Activity• What is a2 * a3?

• Expand it out and combine like terms.

• Do you notice anything about the exponent?

• Try x3 * x4. What do you get? See any patterns?

• What is (x2)3?

• Expand it and combine like terms.

• What do you notice about the exponent?

• Try (x3)3

• What do you get? Notice any patterns?

Notes – 8.1 – Exponents with Products•Product of Powers Rule

• To Multiply Exponents w/ SAME BASE(!) •ADD THE EXPONENTS•am * an = a (m+n)

•Power of a Power Rule•To raise powers to a power w/SAME BASE(!)•MULTIPLY THE EXPONENTS•(am)n = a (m * n)

•Order of Magnitude•Round number to nearest power of 10•The exponent of 10 is the “order of magnitude”

Examples 8.1

GUIDED PRACTICE for Example 1

Simplify the expression.

= 391. 32 37 = 32 + 7

= 510

= (– 7)2+1

= (–7)3

3. (– 7)2(– 7) = (– 7)2 (– 7)1

4. x2 x6 x = x2 x6 x1

= x2 + 6 + 1

= x9

2. 5 59 = 51 + 951 59=

Use the power of a power propertyEXAMPLE 2

= 215 = (–6)10

= x8 = (y + 2)12

a. (25)3

= x2 4c. (x2)4

= (–6)2 5 b. [(–6)2]5

= (y+ 2)6 2 d. [(y + 2)6]2

= 25 3


= 414 = (–2)20

= n18 = (m + 1)20

= 42 75. (42)7 = (–2)4 5

= (m + 1)5 4 8. [(m + 1)5]4


= n3 6

6. [(–2)4]5

7. (n3)6

Use the power of a product propertyEXAMPLE 3

(9xy)2 = (9 x y)2 = b.

a. (24 13)8 = ?

a. 248 138

92 x2 y2 = 81x2y2b.

Use the power of a product propertyEXAMPLE 3

d. – (4z)2 = – (4 z)2 =

(–4z)2 = (–4 z)2c.

(–4)2 z2 = 16z2c.

d. – (42 z2) = –16z2

d. What is the order of magnitude of 90,000?

d. 90,000 is closest to 100,000 = 105, so Order of magnitude is 5.

d. What is the order of magnitude of 99?

d. 99 is closest to 100 = 102, so Order of magnitude is 2.

Use all three propertiesEXAMPLE 4

Simplify(2x3)2 x4

(2x3)2 x4 = 22 (x3)2 x4

= 4x10

Power of a product property

Power of a power property

Product of powers property

= 4 x6 x4

GUIDED PRACTICE for Examples 3, 4 and 5


(–3n)2 10.

9. (42 12)2

= 6561 m12n4



Product of powers property= 6561 m12 n4

= 94 (m3) 4 n 411. (9m3n)4

= 422 122

= (–3 n)2 = (–3)2 n2 = 9n2

GUIDED PRACTICE for Examples 3, 4 and 5




= 5 54 (x2)4

= 3125x8

= 5 625 x8

12. 5 (5x2)4

Solve a real-world problemEXAMPLE 5

Bees

In 2003 the U.S. Department of Agriculture (USDA) collected data on about 103 honeybee colonies. There are about 104 bees in an average colony during honey production season. About how many bees were in the USDA study?

Solve a real-world problemEXAMPLE 5

SOLUTION

To find the total number of bees, find the product of the number of colonies, 103, and the number of bees per colony, 104.

103•104 = 103+4 = 107

The USDA studied about 107, or 10,000,000, bees.

ANSWER

Warm-Up – 8.2



1. q3 when q = 14

2. c2 when c =35

ANSWER1

64

ANSWER9

25

3. A magazine had a circulation of 9364 in 2001. The circulation was about 125 times greater in 2006.Use order of magnitude to estimate the circulation in 2006.



ANSWER about 106 or 1,000,000

(–5n2)2 10. = (–5)2 (n2)2 = 25n4

Vocabulary – 8.2• No new vocab words!!

Activity• What is a3 / a2?

• Expand it out and cancel like terms.

• Do you notice anything about the exponents?

• Try x5/x2. What do you get? See any patterns?

• What is (1/2)3?

• Expand it and combine like terms.

• What do you notice about the exponent?

• Try (2/3)3

• What do you get? Notice any patterns?

Notes – 8.2 –Exponents with quotients•Quotient of Powers Rule

• To Divide Powers w/ SAME BASE(!) •SUBTRACT THE EXPONENTS•am / an = a (m-n)

•Power of a Quotient Rule•To raise fractions to a power:•Raise numerator and denominator to the power•(a/b)m = am/bm

Examples 8.2

EXAMPLE 1 Use the quotient of powers property

= 86

b. (– 3)9

(– 3)3

= (– 3)6

c. 54 58

57

= 512 – 7

= 55

810

84a. = 810 – 4

= (– 3)9 – 3

512

57=

EXAMPLE 1 Use the quotient of powers property

d. x61x4

= x6 – 4

= x2

x6

x4=



1. 611

65

= 66

2. (– 4)9

(– 4)2

= (– 4)7

3. 94 93

92

= 97 – 2

= 95

= 611 – 5

= (– 4)9 – 2

97

92=


4. y81y5

= y8 – 5

= y3

y8

y5=

EXAMPLE 2 Use the power of quotient property

x3

y3 = a. xy

3

(– 7)2

x2=b. 7

x –

2 – 7 x

2= 49

x2=

EXAMPLE 3 Use properties of exponents

Power of a quotient property


= 64x6

125y3Power of a power property


Power of a power propertya10

b5= 12a2

a10

2a2b5= Multiply fractions.

= 43 (x2)3

53y3

b. a2

b 12a2

5

Quotient of powers propertya8

2b5=

(4x2)3

(5y)3=a. 4x2

5y3

(a2)5

b512a2=

GUIDED PRACTICE for Examples 2 and 3





= x4

42y2

= x4

16y2

a2

b2=5. ab

2

(– 5)3

y3 = 125 y3–=– 5

y3

=6. = 5y

–3

( x2)2

(4y)2=7. x2

4y2




Multiply fractions.

8. 3t

2s 3 t5

1623 s3 33 t3

t5

16=

8 s3 27 t3

t5

16=

8 s3 27 16t3

t5

=

s3 t2

54=

EXAMPLE 4 Solve a multi-step problem

To construct what is known as a fractal tree, begin with a single segment (the trunk) that is 1 unit long, as in Step 0. Add three shorter segments that are unit long to form the first set of branches, as in Step 1. Then continue adding sets of successively shorter branches so that each new set of branches is half the length of the previous set, as in Steps 2 and 3.

12

Fractal Tree


a. Make a table showing the number of new branches at each step for Steps 1 - 4. Write the number of new branches as a power of 3.

b. How many times greater is the number of new branches added at Step 5 than the number of new branches added at Step 2?


a. Step Number of new branches

1 3 = 31

2 9 = 32

3 27 = 33

4 81 = 34

b. The number of new branches added at Step 5 is 35. The number of new branches added at Step 2 is 32. So, the number of new branches added at Step 5 is = 33 = 27 times the number of new

branches added at Step 2.

35

32

SOLUTION


FRACTAL TREE In Example 4, add a column to the table for the length of the new branches at each step. Write the length of the new branches as power of . What is the length of a new branchadded at Step 9?

12

9.

SOLUTION

( 19 )9 = 512

1units

ASTRONOMY

EXAMPLE 5Solve a real-world problem

The luminosity (in watts) of a star is the total amount of energy emitted from the star per unit of time. The order of magnitude of the luminosity of the sun is 1026 watts. The star Canopus is one of the brightest stars in the sky. The order of magnitude of the luminosity of Canopus is 1030 watts. How many times more luminous is Canopus than the sun?

EXAMPLE 5 Solve a real-world problem

Luminosity of Canopus (watts)Luminosity of the sun (watts) =

1030

10261030 - 26= = 104

ANSWER

Canopus is about 104 times as luminous as the sun.

SOLUTION


WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less luminous than Canopus, but Sirius appears to be brighter because it is much closer to the Earth. The order of magnitude of the luminosity of Sirius is 1028 watts. How many times more luminous is Canopus than Sirius?

12

9.

SOLUTION

Luminosity of Canopus (watts)Luminosity of the sun (watts) =

1030

10281030 - 28= = 102

ANSWER

Canopus is about 102 times as luminous as Sirius

Warm-Up – 8.3


1. Simplify (– 3x)2.

ANSWER 9x2

ANSWERa15

32b5

2. Simplify .a3

2b

5

3. The order of magnitude of Earth’s mass is about1027 grams. The order of magnitude of the sun’smass is about 1033 grams. About how many times as great is the sun’s mass as Earth’s mass?


ANSWER about 106

Vocabulary – 8.3• No New Vocabulary in this

section either!!

Activity• What was our rule for Quotients of Exponents?

• What is a2 / a3? Expand it and cancel like terms.

• Apply the quotient exponent rule. What do you get?

• Try x2/x5. What do you get when you apply the quotient exponent rule? See any patterns?

• Try x3/x3. What do you get? See any patterns?

• What is the square root of 64?

• What do you get when you raise 64^(1/2) on your calculator?

Notes – 8.3 – Zero and Negative Exponents• Zero Exponents

•Any number to the zero power is 1 EXCEPT 0!•Negative Exponents

•an is the reciprocal of a-n (an = 1 / a-n)•a-n is the reciprocal of an (a–n = 1 / an)•TO EVALUATE NEGATIVE EXPONENTS:

•Take the reciprocal and change the sign of the exponent.•Think “Change the line and flip the sign”

•Fractional Exponents•Raising a number to a fractional exponent looks like the following:

Notes – 8.3 – Zero and Negative Exponents

Summary of Exponent Rules – on page 504 of your textbook

Examples 8.3

Use definition of zero and negative exponents EXAMPLE 1

a. 3 – 2 Definition of negative exponents

1 9

= Evaluate exponent.

b. (–7 )0 Definition of zero exponent

13 2

=

= 1

Use definition of zero and negative exponents EXAMPLE 1

= 25 Simplify by multiplying numerator and denominator by 25.

d. 0 – 5 a – n is defined only for a nonzero number a.

10 5

(Undefined)=

=15

1 25

–21c.Definition of negative exponents

125

1 =

Evaluate exponent.


Definition of zero exponents


023

1. = 1


Definition of negative exponents

1 64

= Evaluate exponent.

2. (–8) – 21

(–8) 2=




18

1=

=12

13

12

3.–3

= 8 Simplify by multiplying numerator and denominator by 8.



4. (–1 )0 Definition of zero exponent= 1

EXAMPLE 2 Evaluate exponential expressions

a. 6– 4 64 Product of a power property

= 60 Add exponents.

= 1 Definition of zero exponent

= 6– 4 + 4


1256

= Evaluate power.

c. 1

3– 4Definition of zero exponent

Evaluate power.= 81

= 34


= 4– 4 Multiply exponents.


b. (4– 2)2

14

=4

2= 4– 2


1125

= Evaluate power.

d. 5– 1

52Quotient of powers property

= 5– 3 Subtract exponents.

153

= Definition of negative exponents

= 5– 1 – 2


5. 14– 3

Definition of negative exponent

Evaluate power.= 64




= 5 3 Multiply exponents.

6. (5– 3) – 1

= 125 Evaluate power.

= 43

– 15– 3=



Product of a power property

= (– 3)0 Add exponents.

= 1 Definition of zero exponent

= (– 3 ) 5 + ( – 5)7. (– 3 ) (– 3 ) – 55


Evaluate power.11296

=


8. 6– 2

62Quotient of powers property

= 6– 4 Subtract exponents.

164

= Definition of negative exponents

= 6– 2 – 2


Simplify the expression. Write your answer using only positive exponents.

a. (2xy – 5)3 = 23 x3 (y – 5)3

= 8 x3 y – 15

=y15

8x3





y5

(2x)2(– 4x2y2)=(2x)– 2y5

– 4x2y2b.

y5

(4x)2(– 4x2y2)=

y5

–16x4y2=

y3

16x4–=




Quotient of powers property

EXAMPLE 1 Fractional Exponents

Warm-Up – 8.4


1. Simplify using only positive exponents:x2y-3

2. Simplify using positive exponents 28x-2y-6

ANSWER x2/y3

ANSWER x2y6

4

1. Simplify using only positive exponents:(6x-2y3)-3

ANSWER x6 / (216y9)


1. Order the numbers 0.014, 0.1, 0.01 from least to greatest.

2. Find the ratio of the mass of the Milky Way galaxy, which is about 1044 grams, to the mass of the universe, which is about 1055 grams.

ANSWER 0.01, 0.014, 0.1

ANSWER 1

1011about

Vocabulary – 8.4• Scientific Notation

• Used to represent very large or very small numbers

Notes – 8.4 – Using Scientific Notation.1. To Write Scientific Notation

•Of the form C * 10n where 1<= C < 10 and n is an integer•n is the number of places to move the decimal

2. Comparing using Scientific Notation•I can only compare #’s in math that ??????•Convert all to standard form or Sci. Not.

•Order the exponents (powers of ten) first•Order the C’s second

Notes – 8.4 – Using Sci. Not. – cont1. To Add or Subtract using Scientific Notation

•Convert numbers so that n is the same•Line up decimals•Add the C’s and leave the n’s •Convert back to sci. not.•Example: Add 1.23*102 + 4.56*103

•Answer: 4.683*103

2. To Multiply or Divide using Scientific Not.•Multiply/Divide the C’s•Multiply/Divide the 10n

•Convert to proper sci. not.

Examples 8.4

EXAMPLE 1 Write numbers in scientific notation

4.259 107a. 42,590,000 =

b. 0.0000574 = 5.74 10-5

Move decimal point 7 places to the left.

Exponent is 7.

Move decimal point 5 places to the right.Exponent is – 5.

EXAMPLE 2 Write numbers in standard form

a. 2.0075 106 Exponent is 6.

Move decimal point 6 places to the right.

b. 1.685 10-4 Exponent is – 4.


= 2,007,500

= 0.0001685


Write the number 539,000 in scientific notation. Then write the number 4.5 3 10 – 4 in standard form.

1. 539,000 5.39 105= Move decimal point 5 places to the left.

Exponent is 5.

4.5 10 – 4 = 0.00045 Exponent is – 4.


Order numbers in scientific notation EXAMPLE 3

SOLUTION

STEP 1

Write each number in scientific notation, if necessary.

103,400,000 = 1.034 108 80,760,000 = 8.076 107

Order 103,400,000 , 7.8 10 , and 80,760,000 from least to greatest.

8

Order numbers in scientific notationEXAMPLE 3

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 107 < 108, you know that 8.076 107 is less than both 1.034 10 8 and 7.8 108. Because 1.034 < 7.8, you know that 1.034 108 is less than7.8 108.

So, 8.076 107 < 1.034 108 < 7.8 108.


STEP 3

Write the original numbers in order from least togreatest.

80,760,000; 103,400,000; 7.8 108

Compute with numbers in scientific notation EXAMPLE 4

Evaluate the expression. Write your answer in scientificnotation.a. (8.5 102)(1.7 106)

(8.5 • 1.7) (102•106)=

14.45 108=

Commutative property andassociative propertyProduct of powers property

1.445 109 = Product of powers property

Compute with numbers in scientific notation EXAMPLE 4

b. (1.5 10 3)– 2 (10 3)– 2= 1.52

(10 6)–= 2.25



(10 3)

c. (1.2 10 4)– 1.6

= 10 3 –

1.21.6

10 4Product rule for fractions

(10 7)= 0.75

(7.5 10 1)– = 10 7

7.5 (10 1 – = 10 7)

(10 6)= 7.5

Quotient of powers property

Write 0.75 in scientific notation.

Associative property



SOLUTION

STEP 1

Write each number in scientific notation, if necessary.

Order 2.7 × 10 5, 3.401 × 10 4, and 27,500 from least to greatest.

2.

27,500 = 2.75 × 104

Order numbers in scientific notation

STEP 2

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

So, 2.7 104 < 2.7 105 < 3.401 104


Because 104 < 105, you know that 3.401 104, 0.7 104 is less than both 2.7 105. Because 2.7 < 3.401, you know that 2.7 104 is less than 3.401 104


STEP 3

Write the original numbers in order from least togreatest.

27,500; 3.401 × 104, and 2.7 × 105


Evaluate the expression. Write your answer in scientific notation.

3. (1.3 10 5)– 2 2(10 5)–= 1.32

(10 10)–= 1.69



10 2

4. 4.5 10 5

– 1.5 =

10 2 –

4.51.5

10 5Product rule for fractions

10 7= 3 Quotient of powers property


5. (1.1 107) (1.7 102)

4.62 109=

Commutative property andassociative propertyProduct of powers property

Evaluate the expression. Write your answer in scientific notation.

(1.1 1.7) (102 107)=

Warm-Up – 8.5


1. Simplify – (– 4x2)3.

ANSWER 64x6

ANSWER4a6

9b2

2. Simplify2a3

3b

2

ANSWER 2b6

3a3

2. Simplify2a-3b5

3b-1

3. The table shows the cost of tickets for a matinee. Write a rule for the function.


ANSWER c = 2t + 1

Tickets, t 2 4 6 8

Cost, c 5 9 13 17

Vocabulary – 8.5• Exponential Function

• Function with a variable as an exponent y = abx

• Graph is NOT linear!!

• Exponential Growth

• Base must be b > 1

• Graph curves UP

• Exponential Decay

• Base must be 0 < b < 1

• Graph curves DOWN

• Compound Interest

• Interest earned on the initial investment and interest already earned

Notes – 8.5 – Exponential Growth Functions.• Exponential Functions are in the form of:

• Y = Abx

•A = Initial Value•b = Growth Factor

•If A > 0 and b > 1, it represents “Exponential Growth”

Notes – 8.5 – Exponential Growth Functions.• Exponential Growth is frequently used to calculate population growth and compound interest.•Uses the following model

Examples 8.5

EXAMPLE 1 Write a function rule

Write a rule for the function.

x –2 –1 0 1 2

y 2 4 8 16 32

SOLUTION

STEP 1

Tell whether the function is exponential.

EXAMPLE 1

Here, the y-values are multiplied by 2 for each increase of 1 in x, so the table represents an exponential function of the form y = a b x where b = 2.

y

X

3216842

210–1–2

+1 +1 +1 +1

2 2 2 2

Write a function rule

EXAMPLE 1

STEP 2Find the value of a by finding the value of y when x = 0. When x = 0, y = ab0 = a 1 = a. The value of y when x = 0 is 8, so a = 8.

STEP 3Write the function rule. A rule for the function is y = 8 2x.

Write a function rule


1. Write a rule for the function.

x –2 –1 0 1 2

y 3 9 27 81 243

SOLUTION

STEP 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 = a b x where b = 3.

y

X

343812793

210–1–2

+1 +1 +1 +1

3 3 3 3


STEP 2

STEP 3

Find the value of a by finding the value of y when x = 0. When x = 0, y = ab0 = a 1 = a. The value of y when x = 0 is 27, so a = 27.

Write the function rule. A rule for the function is y = 27 3x.

EXAMPLE 2 Graph an exponential function

Graph the function y = 2x. Identify its domain and range.

SOLUTION

STEP 1

Make a table by choosing a few values for x and finding the values of y. The domain is all real numbers.


STEP 2

Plot the points.

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.

STEP 3

EXAMPLE 3 Compare graphs of exponential functions

Graph the functions y = 3 2x and y = –3 2x. Compare each graph with the graph of y = 2x.

SOLUTION

To graph each function, make a table of values, plot the points, and draw a smooth curve through the points.


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

Because the y-values for y = –3 2x are – 3 times the corresponding y-values for y = 2x, the graph of y = – 3 2x is a vertical stretch with a reflection in the x-axis of the graph of y = 2x.


2. Graph y = 5x and identify its domain and range.

Domain: all real numbers, range: all positive real numbers

SOLUTION


3. Graph y = 2x. Compare the graph with the graph of y = 2x.

13

SOLUTION

The graph is a vertical shrink of the graph of y = 2x.


The graph is a vertical shrink with a reflection in the x–axis of the graph of y = 2x.

SOLUTION

4. Graph y = 2x. Compare the graph with the – 13

graph of y = 2x.


The owner of a 1953 Hudson Hornet convertible sold the car at an auction. The owner bought it in 1984 when its value was $11,000. The value of the car increased at a rate of 6.9% per year.

Collector Car

EXAMPLE 4

SOLUTION

a. Write a function that models the value of the car overtime.

b. The auction took place in 2004. What was the approximate value of the car at the time of the auction? Round your answer to the nearest dollar.

a. Let C be the value of the car (in dollars), and let t be the time (in years) since 1984. The initial value a is $11,000, and the growth rate r is 0.069.

Solve a multi-step problem

EXAMPLE 4

C = a(1 + r)t Write exponential growth model.

Substitute 11,000 for a and 0.069 for r. = 11,000(1 + 0.069)t

Simplify.= 11,000(1.069)t

b. To find the value of the car in 2004, 20 years after 1984, substitute 20 for t.

Substitute 20 for t.C = 11,000(1.069)20

Use a calculator.41,778≈


EXAMPLE 4

ANSWER

In 2004 the value of the car was about $41,778.


Standardized Test Practice EXAMPLE 5

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 5 years?

$300A $304.16B $781,250D$1344.56C

y = a(1 + r)t

= 250(1 + 0.04)5

= 250(1.04)5

304.16

Write exponential growth model.Substitute 250 for a, 0.04 for r, and 5 for t.

Simplify.Use a calculator.

You will have $304.16 in 5 years.

SOLUTION

Standardized Test PracticeEXAMPLE 5

ANSWER

A DCBThe correct answer is B.


5. What if ?In example 4, suppose the owner of the car Sold in 1994. Find the value of the car to the nearest dollar.

SOLUTION

Let C be the value of the car (in dollars), and let t be the time (in years) since 1984. The initial value a is $11,000, and the growth rate r is 0.069.


C = a(1 + r)t Write exponential growth model.

= 11,000(1 + 0.069)t

Simplify.= 11,000(1.069)t

To find the value of the car in 1994, 30 years after 1984, substitute 30 for t.

Substitute 30 for t.

Use a calculator.21,437≈

Substitute 11,000 for a and 0.069 for r.

C = 11,000(1.069)30

ANSWER In 2004 the value of the car was about $21,437.


6. What if ?In example 5, suppose the annual interest rate is 3.5%. How much will your investment be worth in 5 year.

SOLUTION

y = a(1 + r)t

= 250(1 + 0.035)

= 250(1.035)5

296.92

Write exponential growth model.

Substitute 250 for a, 0.035 for r, and 5 for t.

Simplify.

Use a calculator.

ANSWER You will have $296.92 in 5 years.

Warm-Up – 8.6


ANSWER 16

ANSWER 18

1. Evaluate .12

3

2. Evaluate .14

–2

ANSWER 4y2/x2

2. Evaluate .x2y

–2

3. The table shows how much money Tess owes afterw weeks. Write a rule for the function.


ANSWER m = 50 – 5w

Week, w 0 1 2 3

Owes, m 50 45 40 35

3. Given the following points, write the linear equation in Slope intercept form:(1,4) and (2,6)

ANSWER y = 2x + 2


You put $250 in a savings account that earns 5% annual interest compounded yearly. You do not make any deposits or withdrawals. What exponential function will model this growth? How much will your investment be worth in 5 years? In 20 years?

y = a(1 + r)t

= 250(1.05)5

319.07

You will have $319.07 in 5 years and 663.32 in 20 years.

SOLUTION

y = 250(1 + .05)t

= 250(1.05)20

663.32


If you buy a house for $100,000 and get a 4.5% loan for 30 years compounded annually, how much will you pay for the house at the end of the loan?

y = a(1 + r)t

= 100000(1.045)30

374,532

You will pay almost $375,000 for your $100,000 house!

SOLUTION

y = 100000(1 + .045)t


Some credit card companies charge 18% interest ANNUALLY compounded MONTHLY(!!). If you max out your $5000 credit limit and don’t make any payments for a year, how much will you owe? How about if you don’t make any payments for 2 years?

y = a(1 + r)t

= 5000(1.015)12

5978.09

You will pay $5978.09 after 1 year and $7,147.51 after two years.

SOLUTION

y = 5000(1 + 0.18/12)12t

Vocabulary – 8.6• Exponential Decay

• Y = abx

• a>0

• Base must be 0 < b < 1

• Graph curves DOWN from left to right

Notes – 8.6 – Exponential Decay• Exponential Functions are in the form of:

• Y = Abx

•A = Initial Value•b = Growth Factor

•If A > 0 and 0 < b < 1, it represents “Exponential Decay”•To Find the Exponential Rule

•Use points for x = 0 and x = 1•Plug in what you know and …..•Solve for both “a” and “b”

Notes – 8.6 – Exponential Decay Functions.• Exponential Decay is frequently used to calculate population decline and investments that lose money•Uses the following model

Examples 8.6


Graph the function y = x and identify its domain and range.

12

SOLUTION

STEP 1Make a table of values. The domain is all real numbers.

124y

210– 1– 2x

12

14


STEP 2

Plot the points.

STEP 3

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


Graph the functions y = 3 and y =

Compare each graph with the graph of y =.

12

13

– 12

12

x x

x


SOLUTION

= x12

y13

= xy 12

–x

0

2

1

–2

–1

y = 312 x

4

1

2

1214

12

3

6

3234

13

–

112

–

23

–

16

–

43

–


1212

12

12

– – 13

12

Because the y-values for y = 3 x are 3 times the

corresponding y-values for y = x, the graph of

y = 3 x is a vertical stretch of the graph of y = x.

Because the y-values for y = x are times the

corresponding y-values for y = x, the graph of

y = x is a vertical shrink with reflection in the

x-axis of the graph of y = x.

13

12

–

13

12

–12


SOLUTION

STEP 1Make a table of values. The domain is all real numbers.

y

210– 1– 2x

0.4 0.161–0.40.16

Graph the function y =(0.4) x and identify its domain and range.

2.


STEP 2

Plot the points.

STEP 3

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

EXAMPLE 4 Classify and write rules for functions

SOLUTION

The graph represents exponential growth (y = abx where b > 1). The y-intercept is 10, so a = 10. Find the value of b by using the point (1, 12) and a = 10.

y = abx Write function.

12 = 10 . b1 Substitute.

1.2 = b Solve.

A function rule is y = 10 (1.2)x.

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

EXAMPLE 4 Classify and write rules for functions

The graph represents exponential decay (y = abx where 0 < b < 1).The y-intercept is 8, so a = 8. Find the value of b by using the point (1, 4) and a = 8.



0.5 = b Solve.


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


4. The graph of an exponential function passes through the points (0, 10) and (1, 8). Graph the function. Tell whether the graph represents exponential growth or exponential decay. Write a rule for the function.


The graph represents exponential decay (y = abx where 0 < b < 1).The y-intercept is 10, so a = 10. Find the value of b by using the point (1, 8) and a = 10.

SOLUTION



0.8 = b Solve.


Solve a multi-step problemEXAMPLE 5

Forestry

The number of acres of Ponderosa pine forests decreased in the western United States from 1963 to 2002 by 0.5% annually. In 1963 there were about 41 million acres of Ponderosa pine forests.

a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time.

Solve a multi-step problem EXAMPLE 5

SOLUTION

P = a(1 – r)t Write exponential decay model.

= 41(1 –0.005)t Substitute 41 for a and 0.005 for r.

= 41(0.995)t Simplify.

a. Let P be the number of acres (in millions), and let t be the time (in years) since 1963. The initial value is 41, and the decay rate is 0.005.

b. To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 2002?


EXAMPLE 5

P = 41(0.995)39 33.7 Substitute 39 for t. Use a calculator.

ANSWER

There were about 33.7 million acres of Ponderosa pine forests in 2002.

b. To the nearest tenth, about how many million acres of Ponderosa pine forests were therein 2002?


WHAT IF? In Example 5, suppose the decay rate of the forests remains the same beyond 2002. About how many acres will be left in 2010?

5.

SOLUTION

P = a(1 – r)t Write exponential decay model.

= 41(1 –0.005)t Substitute 41 for a and 0.005 for r.

= 41(0.995)t Simplify.

a. Let P be the number of acres (in millions), and let t be the time (in years) since 1963. The initial value is 41, and the decay rate is 0.005.


To find the number of acres will be left in 2010, 47 years after 1968,substitute 47 for t

P = 41(0.995)47 Substitute 47 for t.

= 32.4

There will be about 32.4 million acres of Ponderosa pine forest in 2010.

ANSWER

Review – Ch. 8 – PUT HW QUIZZES HERE


Simplify the expression. Write your answer using exponents.

1. 145 142

ANSWER 147

3. [(m –3)6]4

2. [(–8)4]3

ANSWER (–8)12


ANSWER (m – 3)24


4. –(2s)3

ANSWER -8s3

A website had about 102 hits after a week. After a year,it had about 103 times the number of hits of the first week. About how many hits did it have at the end of the year ?

5.

ANSWER About 10,000 hits


ANSWER 64

1. Simplify . 63 65

64

ANSWER 103

ANSWER s24

27r3

3. Simplify s8

3r3.

–110

2. Simplify 107 4.


ANSWER 103

The order of magnitude of the power output of a nuclear-powered aircraft carrier is about 106 watts. The order of magnitude of peak power at Hoover Dam is about 109 watts. How many times as great is the power output of Hoover Dam as the power output of a nuclear-powered aircraft carrier?

4.


1. Evaluate 52

– 3

ANSWER 8125

4–7 432. Evaluate

ANSWER 1256

3. Simplify 6a –4 b 0.

ANSWER 6 a4


4. Simplify 8x3y –4

12x2y –3 .

ANSWER 2x3y

A human cell uses on average about 10–12 watts of power. The laser in a CD-R drive uses 109 times as many watts. About how many watts of power does the laser in a CD-R drive use?

5.

ANSWER About 10–3


Write the number in scientific notation.1. 100,500

2. 0.0203

ANSWER 30,600,000

ANSWER 105 1.005

ANSWER 2.03 10–2

3. Write 3.06 107 in standard form.


The diameter of Mercury is about 4.9 103 kilometers. The diameter of Venus is about 1.2 104 kilometers. Find the ratio of the diameter of Venus to that of Mercury. Round to the nearest hundredth.

4.

ANSWER about 2.5


Write the number in scientific notation.1. 100,500

2. 0.0203

ANSWER 30,600,000

ANSWER 105 1.005

ANSWER 2.03 10–2

3. Write 3.06 107 in standard form.


The diameter of Mercury is about 4.9 103 kilometers. The diameter of Venus is about 1.2 104 kilometers. Find the ratio of the diameter of Venus to that of Mercury. Round to the nearest hundredth.

4.

ANSWER about 2.5


1. Graph y = 1.4x.

ANSWER

Your family bought a house for $150,000 in 2000. The value of the house increases at an annual rate of 8%. What is the value of the house after 5 years?

2.

ANSWER About $220,399


2. The population in a town has been declining at a rate of 2% per year since 2001. The population was 84,223 in 2001.What was the population in 2006?

ANSWER

ANSWER about 76,130

13

1. Graph y =( )x

Warm-Up – X.X

Vocabulary – X.X• Holder

• Holder 2

• Holder 3

• Holder 4

Notes – X.X – LESSON TITLE.• Holder•Holder•Holder•Holder•Holder

Examples X.X

unit 6 – chapter 8

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