Transcript Document

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
2.
For use after Lesson 7.1
Solve the linear system by graphing.
2x + y = – 3
– 6x + 3y = 3
ANSWER
(–1, –1)
Daily Homework Quiz
3.
For use after Lesson 7.1
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?
ANSWER
2 angel fish and 6 clown loaches
Daily Homework Quiz
For use after Lesson 7.2
Solve the linear system using substitution
1.
–5x – y = 12
3x – 5y = 4
(–2, –2)
ANSWER
2. 2x + 9y = –4
x – 2y = 11
ANSWER
(7, –2 )
Daily Homework Quiz
For use after Lesson 7.3
Solve the linear system using elimination.
1.
–5x +y = 18
3x – y = –10
ANSWER
2.
(–4, –2)
4x + 2y = 14
4x – 3y = –11
ANSWER
3.
(1, 5)
– 7x – 3y = 11
4x – 2y = 16
ANSWER
(1, – 6)
Daily Homework Quiz
For use after Lesson 7.4
4. 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.
ANSWER
9 times
Daily Homework Quiz
For use after Lesson 7.3
3. 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.
ANSWER
$3.50
Daily Homework Quiz
1.
For use after Lesson 7.6
Write a system of inequalities for the shaded
region.
ANSWER
x < 2,
y > x +1
Prerequisite Skills
1.
VOCABULARY CHECK
Identify the exponent and the base in the
expression 138.
ANSWER
Exponent: 8, base: 13
2.
Copy and complete: An expression that
represents repeated multiplication of the same
factor is called a(n) ? .
ANSWER
Power
Prerequisite Skills
SKILLS CHECK
Evaluate the expression.
3.
x2 when x = 10
4.
ANSWER
ANSWER
100
5. r2 when r = 5
6
ANSWER
25
36
a3 when a = 3
27
6.
z3 when z = 1
2
ANSWER
1
8
Prerequisite Skills
SKILLS CHECK
Order the numbers from least to greatest.
7.
6.12, 6.2, 6.01
8.
ANSWER
0.073, 0.101, 0.0098
ANSWER
6.01, 6.12, 6.2
0.0098, 0.073, 0.101
Write the percent as a decimal.
9.
4%
ANSWER
0.04
10.
0.5%
ANSWER
0.005
11. 13.8%
ANSWER
0.138
12.
145%
ANSWER
1.45
Prerequisite Skills
13.
SKILLS CHECK
Write a rule for the function.
ANSWER
f (x) = x + 2
Warm-Up – 8.1
Lesson 8.1, For use with pages 488-494
Evaluate the expression.
1.
x4 when x = 3
3.
ANSWER
2.
a2
81
when a = –6
ANSWER
-a2 when a = –6
36
ANSWER
-36
Lesson 8.1, For use with pages 488-494
Evaluate the expression.
3.
m3 when m = –5
ANSWER
–125
4. A food storage container is in the shape of a cube.
What is the volume of the container if one side is
4 inches long? Use V = s3.
ANSWER
64 in.3
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.
1.
32 37 = 32 + 7 = 39
2.
5 59 = 51 59 = 51 + 9 = 510
3.
(– 7)2(– 7) = (– 7)2 (– 7)1
= (– 7)2+1
= (–7)3
4.
x2 x6 x
= x2 x6 x1
= x2 + 6 + 1
= x9
EXAMPLE 2
a.
Use the power of a power property
(25)3 = 25
3
b.
= 215
c.
(x2)4 = x2
= x8
[(–6)2]5
= (–6)2
5
= (–6)10
4
d.
[(y + 2)6]2 = (y+ 2)6
= (y + 2)12
2
GUIDED PRACTICE
for Example 2
Simplify the expression.
5.
(42)7 = 42
7
6.
= 414
7.
(n3)6 = n3
= n18
[(–2)4]5
= (–2)4
5
= (–2)20
6
8.
[(m + 1)5]4 = (m + 1)5
4
= (m + 1)20
EXAMPLE 3
b.
b.
Use the power of a product property
a.
(24
13)8 = ?
a.
248
138
(9xy)2 = (9 x
y)2 =
92
x2 y2 = 81x2y2
EXAMPLE 3
c.
(–4z)2 = (–4 z)2
(–4)2 z2 = 16z2
c.
d.
d.
d.
d.
d.
d.
Use the power of a product property
– (4z)2 = – (4 z)2 =
– (42 z2) = –16z2
What is the order of magnitude of 90,000?
90,000 is closest to 100,000 = 105, so Order of magnitude is 5.
What is the order of magnitude of 99?
99 is closest to 100 = 102, so Order of magnitude is 2.
EXAMPLE 4
Use all three properties
Simplify
(2x3)2 x4
(2x3)2 x4 = 22 (x3)2 x4
Power of a product property
= 4 x6 x4
Power of a power property
= 4x10
Product of powers property
for Examples 3, 4 and 5
GUIDED PRACTICE
Simplify the expression.
9.
10.
11.
(42
12)2 = 422 122
(–3n)2 = (–3 n)2 = (–3)2 n2 = 9n2
(9m3n)4 = 94
(m3) 4 n 4
= 6561
m12 n4
= 6561 m12n4
Power of a product property
Product of powers property
Power of a power property
GUIDED PRACTICE
12.
5 (5x2)4
for Examples 3, 4 and 5
=5
54 (x2)4
Power of a product property
= 5 625 x8
Power of a power property
= 3125x8
Product of powers property
EXAMPLE 5
Solve a real-world problem
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?
EXAMPLE 5
Solve a real-world problem
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
ANSWER
The USDA studied about 107, or 10,000,000, bees.
Warm-Up – 8.2
Lesson 8.2, For use with pages 495-501
Evaluate the expression.
1
1. q3 when q =
4
ANSWER
1
64
2. c2 when c =
ANSWER
9
25
3
5
Lesson 8.2, For use with pages 495-501
Evaluate the expression.
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
10.
about 106 or 1,000,000
(–5n2)2 = (–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
a.
Use the quotient of powers property
810 = 810 – 4
84
= 86
b.
(– 3)9 = (– 3)9 – 3
(– 3)3
= (– 3)6
c.
12
54 58 = 5
57
57
= 512 – 7
= 55
EXAMPLE 1
d.
Use the quotient of powers property
x6
1
6
x = 4
x
x4
= x6 – 4
= x2
GUIDED PRACTICE
for Example 1
Simplify the expression.
1.
611 11 – 5
=6
65
= 66
2.
(– 4)9
9–2
=
(–
4)
2
(– 4)
= (– 4)7
3.
94 93 = 97
92
92
= 97 – 2
= 95
GUIDED PRACTICE
4.
y8
1
8
y = 5
y
y5
= y8 – 5
= y3
for Example 1
EXAMPLE 2
a.
b.
Use the power of quotient property
x 3 x3
y = y3
–7
x
2
=
–7
x
2
(– 7)2
49
=
=
x2
x2
EXAMPLE 3
4x2
5y
a.
b.
5
2
a
b
3
Use properties of exponents
(4x2)3
=
(5y)3
43 (x2)3
= 3 3
5y
64x6
=
125y3
(a2)5
1
2a2 = b5
a10
= 5
b
a10
=
2a2b5
8
= a
2b5
1
2a2
1
2a2
Power of a quotient property
Power of a product property
Power of a power property
Power of a quotient property
Power of a power property
Multiply fractions.
Quotient of powers property
GUIDED PRACTICE
for Examples 2 and 3
Simplify the expression.
5.
a 2 a2
= 2
b
b
5
6. = – y
7.
x2
4y
3
–5
=
y
2
3
2)2
(
x
=
(4y)2
x4
= 2 2
4y
x4
=
16y2
(– 5)3
125
= 3
=–
y
y3
Power of a quotient property
Power of a product property
Power of a power property
GUIDED PRACTICE
8.
2s
3t
3
t5
16
23 s3
= 3 3
3 t
for Examples 2 and 3
t5
16
8 s3 t 5
=
27 t3 16
8 s3 t5
=
27 16t3
s3 t 2
=
54
Power of a quotient property
Power of a power property
Multiply fractions.
EXAMPLE 4
Solve a multi-step problem
Fractal Tree
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 1 unit
2
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.
EXAMPLE 4
Solve a multi-step problem
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?
EXAMPLE 4
Solve a multi-step problem
SOLUTION
a. Step Number of new branches
1
3 = 31
b.
2
9 = 32
3
27 = 33
4
81 = 34
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
5
Step 5 is 3 = 33 = 27 times the number of new
32
branches added at Step 2.
GUIDED PRACTICE
for Example 4
9. 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 1 . What is the length of a new branch
2
added at
Step 9?
SOLUTION
1
1 9
( 9 ) = 512 units
EXAMPLE 5
Solve a real-world problem
ASTRONOMY
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
SOLUTION
Luminosity of Canopus (watts)
1030
30 - 26
4
10
10
=
=
=
Luminosity of the sun (watts)
1026
ANSWER
Canopus is about 104 times as luminous as the sun.
GUIDED PRACTICE
for Example 5
9. WHAT IF? Sirius is considered the brightest star in the
sky. Sirius is less luminous than Canopus, but Sirius
1
appears to be brighter because it is much closer
to the
2
Earth. The order of magnitude of the luminosity of
Sirius is 1028 watts. How many times more luminous is
Canopus than Sirius?
SOLUTION
Luminosity of Canopus (watts)
1030
30 - 28
2
10
10
=
=
=
Luminosity of the sun (watts)
1028
ANSWER
Canopus is about 102 times as luminous as Sirius
Warm-Up – 8.3
Lesson 8.3, For use with pages 502-508
1. Simplify (– 3x)2.
ANSWER
9x2
a3
2. Simplify 2b
ANSWER
5
.
a15
32b5
Lesson 8.3, For use with pages 502-508
3. The order of magnitude of Earth’s mass is about
1027 grams. The order of magnitude of the sun’s
mass 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
Notes – 8.3 – Zero and Negative Exponents
Summary of Exponent Rules – on page 504 of your textbook
Examples 8.3
EXAMPLE 1
3 –2= 1
32
1
=
9
b. (–7 )0 = 1
a.
Use definition of zero and negative exponents
Definition of negative exponents
Evaluate exponent.
Definition of zero exponent
EXAMPLE 1
1
5
c.
–2
1
1 2
5
1
=
1
25
=
= 25
d.
Use definition of zero and negative exponents
Definition of negative exponents
Evaluate exponent.
Simplify by multiplying numerator
and denominator by 25.
0 – 5 = 1 (Undefined) a – n is defined only for a nonzero
05
number a.
GUIDED PRACTICE
for Example 1
Evaluate the expression.
1.
2
3
0
=1
Definition of zero exponents
Evaluate the expression.
2.
(–8)
–2
1
=
(–8) 2
1
=
64
Definition of negative exponents
Evaluate exponent.
GUIDED PRACTICE
for Example 1
Evaluate the expression.
3.
1
13
=
–3
1
2
2
Definition of negative exponents
1
=
1
8
=8
Simplify by multiplying numerator
and denominator by 8.
GUIDED PRACTICE
for Example 1
Evaluate the expression.
4.
(–1 )0 = 1
Definition of zero exponent
EXAMPLE 2
a.
Evaluate exponential expressions
6– 4 64 = 6– 4 + 4
Product of a power property
= 60
Add exponents.
=1
Definition of zero exponent
EXAMPLE 2
b.
Evaluate exponential expressions
(4– 2)2 = 4– 2
Power of a product property
= 4– 4
Multiply exponents.
1
= 4
4
Definition of negative exponents
1
=
256
c.
2
1
4
=
3
3– 4
= 81
Evaluate power.
Definition of zero exponent
Evaluate power.
EXAMPLE 2
d.
Evaluate exponential expressions
5– 1
–1 –2
=
5
52
Quotient of powers property
= 5– 3
Subtract exponents.
1
= 3
5
Definition of negative exponents
=
1
125
Evaluate power.
GUIDED PRACTICE
for Example 2
Evaluate the expression.
5.
1
3
=
4
4– 3
Definition of negative exponent
= 64
Evaluate power.
Evaluate the expression.
6.
(5– 3) – 1 = 5– 3
–1
Power of a product property
= 53
Multiply exponents.
= 125
Evaluate power.
GUIDED PRACTICE
for Example 2
Evaluate the expression.
7.
(– 3 ) 5 (– 3 ) – 5 = (– 3 ) 5 + ( – 5) Product of a power property
= (– 3)0
Add exponents.
=1
Definition of zero exponent
GUIDED PRACTICE
for Example 2
Evaluate the expression.
8.
6– 2
–2 –2
=
6
62
= 6– 4
=
1
64
1
=
1296
Quotient of powers property
Subtract exponents.
Definition of negative
exponents
Evaluate power.
EXAMPLE 3
Use properties of exponents
Simplify the expression. Write your answer using only
positive exponents.
a.
(2xy – 5)3 = 23 x3 (y – 5)3
Power of a product property
= 8 x3 y – 15
Power of a power property
8x3
= 15
y
Definition of negative exponents
EXAMPLE 3
b.
Use properties of exponents
y5
(2x)– 2y5 =
(2x)2(– 4x2y2)
– 4x2y2
Definition of negative exponents
y5
=
(4x)2(– 4x2y2)
Power of a product property
y5
=
–16x4y2
Product of powers property
= –
y3
16x4
Quotient of powers property
EXAMPLE 1
Fractional Exponents
Warm-Up – 8.4
Lesson 8.4, For use with pages 512-519
1. Simplify using only positive exponents:
x2y-3
ANSWER
x2/y3
1. Simplify using only positive exponents:
(6x-2y3)-3
ANSWER
x6 / (216y9)
2. Simplify using positive exponents
2
8x-2y-6
ANSWER
x2y6
4
Lesson 8.4, For use with pages 512-519
1. Order the numbers 0.014, 0.1, 0.01 from least to
greatest.
ANSWER
0.01, 0.014, 0.1
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
about
1
1011
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. – cont
1. 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
a.
42,590,000 = 4.259 107 Move decimal point 7 places to
b.
0.0000574 = 5.74 10-5
the left.
Exponent is 7.
Move decimal point 5 places to
the right.
Exponent is – 5.
EXAMPLE 2
a.
Write numbers in standard form
2.0075 106 = 2,007,500 Exponent is 6.
Move decimal point 6 places to
the right.
b.
1.685 10-4 = 0.0001685 Exponent is – 4.
Move decimal point 4 places to
the left.
GUIDED PRACTICE
for Examples 1 and 2
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
4.5
10 – 4 = 0.00045
Move decimal point 5 places to
the left.
Exponent is 5.
Exponent is – 4.
Move decimal point 4 places to
the left.
EXAMPLE 3
Order numbers in scientific notation
Order 103,400,000 , 7.8
to greatest.
8
10 , and 80,760,000 from least
SOLUTION
STEP 1
Write each number in scientific notation, if necessary.
103,400,000 = 1.034
108
80,760,000 = 8.076
107
EXAMPLE 3
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.
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 than
7.8 108.
So, 8.076
107 < 1.034
108 < 7.8
108.
EXAMPLE 3
Order numbers in scientific notation
STEP 3
Write the original numbers in order from least to
greatest.
80,760,000; 103,400,000; 7.8
108
EXAMPLE 4
Compute with numbers in scientific notation
Evaluate the expression. Write your answer in scientific
notation.
a. (8.5 102)(1.7 106)
= (8.5 • 1.7) (102•106) Commutative property and
= 14.45
= 1.445
108
109
associative property
Product of powers property
Product of powers property
EXAMPLE 4
b. (1.5
Compute with numbers in scientific notation
–3 2
10 ) =
1.52
–3 2
(10 )
Power of a product property
= 2.25 (10 – 6)
c.
(1.2 10 4)
1.2
=
1.6
1.6 (10 – 3)
Power of a power property
10 4
10 – 3
= 0.75 (10 7)
Product rule for fractions
Quotient of powers property
= (7.5
10 – 1)
10 7 Write 0.75 in scientific notation.
= 7.5
(10 – 1
10 7) Associative property
= 7.5
(10 6)
Product of powers property
GUIDED PRACTICE
2.
for Examples 3 and 4
Order 2.7 × 10 5, 3.401 × 10 4, and 27,500 from least to
greatest.
SOLUTION
STEP 1
Write each number in scientific notation, if necessary.
27,500 = 2.75 × 104
for Examples
3 andnotation
4
Order numbers
in scientific
GUIDED PRACTICE
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 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
So, 2.7
104 < 2.7
105 < 3.401
104
EXAMPLE 3
Order numbers in scientific notation
STEP 3
Write the original numbers in order from least to
greatest.
27,500; 3.401 × 104, and 2.7 × 105
for Examples 3 and 4
GUIDED PRACTICE
Evaluate the expression. Write your answer in
scientific notation.
3. (1.3
–5 2
10 ) =
1.32
–5 2
(10 )
= 1.69 (10 –10)
4.
4.5 10 5
4.5
=
–2
1.5
10
1.5
=3
10 7
10 5
10 – 2
Power of a product property
Power of a power property
Product rule for fractions
Quotient of powers property
for Examples 3 and 4
GUIDED PRACTICE
Evaluate the expression. Write your answer in
scientific notation.
5. (1.1
107) (1.7
= (1.1
= 4.62
102)
1.7) (102 107) Commutative property and
109
associative property
Product of powers property
Warm-Up – 8.5
Lesson 8.5, For use with pages 520-527
1. Simplify – (– 4x2)3.
ANSWER
64x6
2. Simplify
ANSWER
2. Simplify
ANSWER
2a3
3b
2
4a6
9b2
2a-3b5
3b-1
2b6
3a3
Lesson 8.5, For use with pages 520-527
3. The table shows the cost of tickets for a matinee.
Write a rule for the function.
Tickets, t 2
5
Cost, c
ANSWER
c = 2t + 1
4 6 8
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
Write a function rule
+1
+1
+1
+1
X
–2
–1
0
1
2
y
2
4
8
16
32
2
2
2
2
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.
EXAMPLE 1
Write a function rule
STEP 2
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
8, so a = 8.
STEP 3
Write the function rule. A rule for the function is y = 8 2x.
GUIDED PRACTICE
1.
for Example 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.
for Example 1
GUIDED PRACTICE
+1
+1
+1
+1
X
–2
–1
0
1
2
y
3
9
27
81
343
3
3
3
3
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.
GUIDED PRACTICE
for Example 1
STEP 2
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.
STEP 3
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.
EXAMPLE 2
Graph an exponential function
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
all positive real numbers.
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.
EXAMPLE 3
Compare graphs of exponential functions
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.
GUIDED PRACTICE
2.
for Examples 2 and 3
Graph y = 5x and identify its domain and range.
SOLUTION
Domain: all real numbers,
range: all positive real numbers
GUIDED PRACTICE
for Examples 2 and 3
1
3. Graph y = 3
2x. Compare the graph with the
graph of y = 2x.
SOLUTION
The graph is a vertical shrink
of the graph of y = 2x.
GUIDED PRACTICE
4.
for Examples 2 and 3
Graph y = – 13 2x. Compare the graph with the
graph of y = 2x.
SOLUTION
The graph is a vertical
shrink with a reflection in
the x–axis of the graph of
y = 2x.
EXAMPLE 4
Solve a multi-step problem
Collector Car
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.
EXAMPLE 4
Solve a multi-step problem
a. Write a function that models the value of the car over
time.
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.
SOLUTION
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.
EXAMPLE 4
Solve a multi-step problem
C = a(1 + r)t
Write exponential growth model.
= 11,000(1 + 0.069)t
Substitute 11,000 for a and 0.069 fo
= 11,000(1.069)t
Simplify.
b. To find the value of the car in 2004, 20 years after
1984, substitute 20 for t.
C = 11,000(1.069)20
≈ 41,778
Substitute 20 for t.
Use a calculator.
EXAMPLE 4
Solve a multi-step problem
ANSWER
In 2004 the value of the car was about $41,778.
EXAMPLE 5
Standardized Test Practice
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?
A $300
B $304.16
C $1344.56
D $781,250
SOLUTION
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 fo
Simplify.
Use a calculator.
You will have $304.16 in 5 years.
EXAMPLE 5
Standardized Test Practice
ANSWER
The correct answer is B.
A
B
C
D
GUIDED PRACTICE
5.
for Examples 4 and 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.
GUIDED PRACTICE
C = a(1 + r)t
for Examples 4 and 5
Write exponential growth model.
= 11,000(1 + 0.069)t
Substitute 11,000 for a and 0.069 fo
= 11,000(1.069)t
Simplify.
To find the value of the car in 1994, 30 years after
1984, substitute 30 for t.
C = 11,000(1.069)30
≈ 21,437
ANSWER
Substitute 30 for t.
Use a calculator.
In 2004 the value of the car was about $21,437.
GUIDED PRACTICE
6.
for Examples 4 and 5
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
Write exponential growth model.
= 250(1 + 0.035)
Substitute 250 for a, 0.035 for r, and 5
= 250(1.035)5
Simplify.
296.92
ANSWER
Use a calculator.
You will have $296.92 in 5 years.
Warm-Up – 8.6
Lesson 8.6, For use with pages 530-538
1. Evaluate
ANSWER
2. Evaluate
ANSWER
1 3
.
2
1
8
1 –2.
4
16
2. Evaluate
x . –2
2y
ANSWER
4y2/x2
Lesson 8.6, For use with pages 530-538
3. The table shows how much money Tess owes after
w weeks. Write a rule for the function.
Week, w
Owes, m
ANSWER
0 1 2 3
50 45 40 35
m = 50 – 5w
3. Given the following points, write the linear equation in
Slope intercept form:
(1,4) and (2,6)
ANSWER
y = 2x + 2
EXAMPLE 5
Standardized Test Practice
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?
SOLUTION
y = a(1 + r)t
y = 250(1 + .05)t
= 250(1.05)5
319.07
= 250(1.05)20
663.32
You will have $319.07 in 5 years and 663.32 in 20 years.
EXAMPLE 5
Standardized Test Practice
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?
SOLUTION
y = a(1 + r)t
y = 100000(1 + .045)t
= 100000(1.045)30
374,532
You will pay almost $375,000 for your $100,000 house!
EXAMPLE 5
Standardized Test Practice
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?
SOLUTION
y = a(1 + r)t
y = 5000(1 + 0.18/12)12t
= 5000(1.015)12
5978.09
You will pay $5978.09 after 1 year and $7,147.51 after
two years.
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
EXAMPLE 2
Graph an exponential function
1
Graph the function y = 2
range.
x
and identify its domain and
SOLUTION
STEP 1
Make a table of values. The domain is all real numbers.
x
–2
–1
0
1
2
y
4
2
1
1
2
1
4
EXAMPLE 2
Graph an exponential function
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 3
Compare graphs of exponential functions
Graph the functions y = 3
1
1x
–
and y = 3
2
1x
2
x
1
Compare each graph with the graph of y =. 2
Compare graphs of exponential functions
EXAMPLE 3
SOLUTION
x
1
y=
2
–2
4
12
–1
2
6
0
1
3
1
1
2
1
4
3
2
3
4
2
x
y =3
1
2
x
1 1
y= –
3 2
–4
3
–2
3
–1
3
–1
6
– 1
12
x
EXAMPLE 3
Compare graphs of exponential functions
1 x
Because the y-values for y = 3 2 are 3 times the
1 x
corresponding y-values for y =
, the graph of
2
1 x
1 x
y=3
is a vertical stretch of the graph of y =
.
2
2
1 1 x
– 1 times the
Because the y-values for y = –
are
3 2
3
1 x
corresponding y-values for y = 2 , the graph of
1 1 x
y = – 3 2 is a vertical shrink with reflection in the
1 x
x-axis of the graph of y = 2 .
GUIDED PRACTICE
for Examples 2 and 3
2. Graph the function y =(0.4)
and range.
x
and identify its domain
SOLUTION
STEP 1
Make a table of values. The domain is all real numbers.
x
–2
–1
y
0.16 –0.4
0
1
2
1
0.4
0.16
GUIDED PRACTICE
for Examples 2 and 3
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
Tell whether the graph represents exponential growth
or exponential decay.Then write a rule for the function.
SOLUTION
The graph represents exponential
growth (y = abx where b > 1). The yintercept 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.
EXAMPLE 4
Classify and write rules for functions
Tell whether the graph represents exponential growth
or exponential decay.Then write a rule for the function.
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.
y = abx
Write function.
4 = 8 . b1
Substitute.
0.5 = b
Solve.
A function rule is y = 8 (0.5)x.
GUIDED PRACTICE
for Example 4
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.
GUIDED PRACTICE
for Example 4
SOLUTION
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.
y = abx
Write function.
8 = 10 . b1
Substitute.
0.8 = b
Solve.
A function rule is y = 10 (0.8)x.
EXAMPLE 5
Solve a multi-step problem
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.
EXAMPLE 5
b.
Solve a multi-step problem
To the nearest tenth, about how many million
acres of Ponderosa pine forests were there in 2002?
SOLUTION
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.
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.
EXAMPLE 5
b.
Solve a multi-step problem
To the nearest tenth, about how many million acres
of Ponderosa pine forests were therein 2002?
Substitute 39 for t. Use a calculator.
P = 41(0.995)39 33.7
ANSWER
There were about 33.7 million acres of
Ponderosa pine forests in 2002.
GUIDED PRACTICE
5.
for Example 5
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?
SOLUTION
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.
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.
GUIDED PRACTICE
for Example 5
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
ANSWER
There will be about 32.4 million acres of
Ponderosa pine forest in 2010.
Review – Ch. 8 – PUT
HW QUIZZES HERE
Daily Homework Quiz
For use after Lesson 8.1
Simplify the expression. Write your answer using
exponents.
1. 145 142
ANSWER
2.
147
[(–8)4]3
ANSWER
(–8)12
Simplify the expression.
3.
[(m –3)6]4
ANSWER
(m – 3)24
Daily Homework Quiz
4.
–(2s)3
ANSWER
5.
For use after Lesson 8.1
-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 ?
ANSWER
About 10,000 hits
Daily Homework Quiz
3 65
6
1. Simplify
.
64
ANSWER 64
2. Simplify
107
ANSWER
103
–1 4.
10
8 3.
s
3. Simplify
3r
ANSWER
s24
27r3
For use after Lesson 8.2
Daily Homework Quiz
For use after Lesson 8.2
4. 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?
ANSWER
103
Daily Homework Quiz
1. Evaluate 5
2
ANSWER
2.
3.
8
125
Evaluate 4–7
ANSWER
–3
43
1
256
Simplify 6a –4 b 0.
ANSWER
6
a4
For use after Lesson 8.3
Daily Homework Quiz
For use after Lesson 8.3
8x3y –4
4. Simplify
.
12x2y –3
ANSWER
2x
3y
5. 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?
ANSWER
About 10–3
Daily Homework Quiz
For use after Lesson 8.4
Write the number in scientific notation.
1. 100,500
ANSWER
2.
1.005
105
0.0203
ANSWER
2.03
3. Write 3.06
ANSWER
10–2
107 in standard form.
30,600,000
Daily Homework Quiz
For use after Lesson 8.4
4. 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.
ANSWER
about 2.5
Daily Homework Quiz
For use after Lesson 8.4
Write the number in scientific notation.
1. 100,500
ANSWER
2.
1.005
105
0.0203
ANSWER
2.03
3. Write 3.06
ANSWER
10–2
107 in standard form.
30,600,000
Daily Homework Quiz
For use after Lesson 8.4
4. 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.
ANSWER
about 2.5
Daily Homework Quiz
1.
For use after Lesson 8.5
Graph y = 1.4x.
ANSWER
2. 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?
ANSWER
About $220,399
Daily Homework Quiz
For use after Lesson 8.6
x
1
1. Graph y = ( )
3
ANSWER
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
about 76,130
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