exponential functions chapter 4. 4.1 properties of exponents know the meaning of exponent, zero...
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Exponential Functions
Chapter 4
4.1 Properties of Exponents
• Know the meaning of exponent, zero exponent and negative exponent.
• Know the properties of exponents.
• Simplify expressions involving exponents
• Know the meaning of exponential function.
• Use scientific notation.
Exponent
• For any counting number n,
• We refer to as the power, the nth power of b, or b raised to the nth power.
• We call b the base and n the exponent.
nb
boforsfan
n bbbbbcot
Examples
32222222
8133333
1024444444
5
4
5
When taking a power of a negative number,
if the exponent is even the answer will be positive
if the exponent is odd the answer will be negative
Properties of Exponents
Product property of exponents
Quotient property of exponents
Raising a product to a power
Raising a quotient to a power
Raising a power to a power
mnnm
n
nn
nnn
nmn
m
nmnm
bb
cc
b
c
b
cbbc
nmandbbb
b
bbb
0,
0,
Meaning of the Properties
532
32
32
532
bbb
bbbbbbb
bbbbbbb
bbb
nm
factorsnmfactorsnfactorsm
nm
nmnm
bbbbbbbbbbbbbbb
bbb
n
n
factorsn
factorsn
factorsn
n
n
nn
c
b
cccc
bbbb
c
b
c
b
c
b
c
b
c
b
cc
b
c
b
0,
Product property of exponents Raising a quotient to a power
Simplifying Expressions with Exponents
• An expression is simplified if:– It included no parenthesis– All similar bases are combined
– All numerical expressions are calculated
– All numerical fractions are simplified– All exponents are positive
53xx xx66
57 08
Order of Operations
• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction
Warning
• Note: When using a calculator to equate powers of negative numbers always put the negative number in parenthesis.
• Note: Always be careful with parenthesis
162].....162[]162[ 444
22 55 xx
Examples
95
4523
4253
24
64
64
yx
yyxx
yxyx
1218
34363
34363
346
216
6
6
6
vu
vu
vu
vu
Examples (Cont.)
6
6
30
5
44
5937
53
97
hg
hg
hg
hg
3
7
35
3
7
5849
3
754
89
4
3
4
3
36
27
r
qp
r
qp
rqp
qp
21
915
373
33353
37
335
64
9
4
3
4
3
r
qp
r
qp
r
qp
Zero Exponent
• For b ≠ 0,
• Examples,10 b
0,1
124
15
0
0
0
xyxy
Negative Exponent
• If b ≠ 0 and n is a counting number, then
• To find , take its reciprocal and switch the sign of the exponent
• Examples,
nn
bb
1
nb
44
22
149
1
7
17
xx
Negative Exponent (Denominator)
• If b ≠ 0 and n is a counting number, then
• To find , take its reciprocal and switch the sign of the exponent
• Examples,
nnb
b
1
nb
1
44
22
1
8199
1
xx
Simplifying Negative Exponents
2433
13
333
55
805810805810
275757
5
2575
7
7
5
xxxx
x
xxx
x
x
x
5
12125)8(4)7(12
87
412
5
1212584127
12
847
87
412
5
8
5
8
5
8
25
40
5
8
5
8
5
8
5
8
25
40
x
yyxyx
yx
yx
x
yyxyx
x
yyx
yx
yx
Exponential Functions
• An exponential function is a function whose equation can be put into the form:
– Where a ≠ 0, b > 0, and b ≠ 1.
– The constant b is called the base.
16
5,4
16
52
5
)2(5)4(
)4(
)2(5)(
4
4f
ffind
xf x
)384,3(
384
646
)4(6
)3(
)4(6)(
3
ffind
xf x
xabxf )(
Exponential vs Linear Functions
• x is a exponent • x is a base
xxf 2)( 12)( xxf
Scientific Notation
• A number written in the form:
where k is an integer and-10 < N ≤ -1 or
1 ≤ N < 10
• Examples
5108.4
kN 10
23108905.5 531036.4
Scientific to Standard Notation
• When k is negative move the decimal to the left
0.3255
1000325.5
10325.5 3
move the decimal 3 places to the right
56300008.0
00001.0100000
15.8
10
1563.8
10563.8
5
5
move the decimal 5 places to the left
• When k is positive move the decimal to the right
Standard to Scientific Notation
• if you move the decimal to the right, then k is positive
• if you move the decimal to the left, then k is negative
910938.2
.9380000002
000,000,938,2
410039.2
039.00020
0002039.0
move the decimal 9 places to the right
move the decimal 4 places to the left
Group Exploration
• If time,– p173
nb1
4.2 Rational Exponents
Rational Exponents ( )
• For the counting number n, where n ≠ 1,– If n is odd, then is the number whose nth
power is b, and we call the nth root of b– If n is even and b ≥ 0, then is the
nonnegative number whose nth power is b, and we call the principal nth root of b.
– If n is even and b < 0, then is not a real number.
• may be represented as .
nb1
nb1
nb1
nb1
nb1
nb1
n b
nb1
Examples
½ power = square root
⅓ power = cube root
not a real number since the 4th power of any real number is non-negative
41
4141
331
331
221
)81(
3)81(81
27)3(,3)27(
82,28
366,636
Rational Exponents
• For the counting numbers m and n, where n ≠ 1 and b is any real number for which is a real number,
• A power of the form or is said to have a rational exponent.
0.
1
11
bb
b
bbb
nmnm
nmmnnm
nb1
nmb nmb
Examples
27
1
)3(
1
)81(
1
81
181
16)2())32(()32(
9)3()27(27
33414343
445154
223132
Properties of Rational Exponents
Product property of exponents
Quotient property of exponents
Raising a product to a power
Raising a quotient to a power
Raising a power to a power
mnnm
n
nn
nnn
nmn
m
nmnm
bb
cc
b
c
b
cbbc
nmandbbb
b
bbb
0,
0,
Examples
4/54
2
4
3
2
1
4
32/14/3
575
4
5
3
54
53
101053
5
1
653135635356 322)8()(8)8(
yyyyy
xxx
x
xxxxx
66
3
4
3
1
8
34/1
4/38
4/3
4/3
8
4/384/3113
4/3
11
3
2738181
818181
81
vvvv
vvv
v
v
4.3 Graphing Exponential Functions
Graphing Exponential Functions by hand
x y
-3 1/8
-2 1/4
-1 1/2
0 1
1 2
2 4
3 8
xxf 2)(
Graph of an exponential function is called an exponential curve
x
xg
2
14)(
x y
-1 8
0 4
1 2
2 1
3 1/2
Base Multiplier Property
• For an exponential function of the form
• If the value of the independent variable increases by 1, then the value of the dependent variable is multiplied by b.
xaby
x increases by 1, y increases by b
x y
-3 1/8
-2 1/4
-1 1/2
0 1
1 2
2 4
3 8
x y
-1 8
0 4
1 2
2 1
3 1/2
x
xg
2
14)(xxf 2)(
Increasing or Decreasing Property
• Let , where a > 0.
• If b > 1, then the function is increasing– grows exponentially
• If 0 < b < 1, then the function is decreasing– decays exponentially
Intercepts
• y-intercept for the form:
is (0,a)• y-intercept for the form:
is (0,1)
xaby
xby
Intercepts
• Find the x and y intercepts:• y-intercept
• x-intercept– as x increases by 1, y is multiplied by 1/3.– infinitely multiplying by 1/3 will never equal 0– as x increases, y approaches but never equals 0– no x-intercept exists, instead the x-axis is called the
horizontal asymptote
x
xf
3
16)(
6)1(63
16)(
0
xf
Reflection Property
• The graphs
• are reflections of each other across the x-axis
x
x
abxg
abxf
)(
)(
a > 0 a > 0
a < 0 a < 0
4.4 Finding Equations of Exponential Functions
Finding an Equation Using a Table
• Refer to the Base Multiplier Property– as x increases by 1, y
is multiplied by the base
• Find the y-intercept– (0, a) = (0, 2)
• Find the constant– multiplying by 4
• Write the equation
x F(x)
0 2
1 8
2 32
3 148
4
x
x
y
aby
)4(2
Linear vs Exponential
x F(x)
0 243
1 81
2 27
3 9
4 3
x F(x)
0 81
1 70
2 59
3 48
4 37x
xf
3
1243)( 8111)( xxf
Solving for b
3
2727
27
)6(66
3636
36
3/13/133 3
3
2/12/12
2
b
bb
b
orb
bb
b
2
)32(32
32
381
)81(81
81
5/15/155 5
5
4
4/14/144 4
4
b
bb
b
bb
b
bb
b
No Real Solutions
When Solving For b
• For ,– n is even, always have a positive and
negative answer– n is odd, always have one positive answer
• For ,– n is even, always no real solutions– n is odd, always have one negative answer
kbn
kbn
Solving for b (cont)
• Remember to always simplify both sides of the equations first.
5
9
25
81
25
8125
8125
81
2
2
24
2
4
b
b
b
b
b
5
3
25
81
25
81
4
2
4
2
4
b
b
b
b
Finding an Equation of Exponential Curves Using Two Points
• Given (0,4) and (5,128)• We know (0,4) is y-intercept (0,a) so a = 4
• Substitute (5, 128) into the equation
xby 4
2
32
32
4
4
4
128
4128
5 55
5
5
5
b
b
b
b
b
xy )4(2
Example
• (2,1) (5,7)
• Plug both points into the standard equation
• Divide to cancel a term
5
2
7
1
ab
ab
91.1
7
7
1
7
3 33
325
2
5
a
b
bb
ab
ab
Example (cont)
• Now we have b
• We can substitute one of the points and solve for a …..(2,1)
xay )91.1(
27.65.3
1
65.31
)91.1(1 2
a
a
axy )91.1(27.
4.5 Using Exponential Functions to Model Data
Exponential Models
• Exponential model – exponential function, or its graph, that describes the relationship between two quantities in an authentic situation
• Exponentially related – If all the data points for a situation lie on an exponential curve
• Approximately exponentially related – If all the data points lie on or close to an exponential curve
• Suppose there are 5 million bacteria on a banana at 8am on Monday. Every bacterium divides into 2 every hour, on average.
• Give a function for the situation
• Predict the number of bacteria at 8pm on Monday.
• 20,480 million bacteria
xxf )2(5)(
480,20
)096,4(5
)2(5 12
y
y
y
Exponential Functions of Time
• y is the amount
• t amount of time
• a is the initial amount when t =0
• b is constant by which a grows or decays over time
taby
• A person invests 8,000 in an account and interest is compounded 5% annually.
• Write an equation to model the data
– Value is 100% of the original deposit plus 6% after the first year so we use 1.06 as our base
• What is the value after 10 years?
• The value would be $14,326.78 after 10 years
xxf )06.1(8000)(
78.326,14
)06.1(8000 10
y
y
Half-life
• If a quantity decays exponentially, the half-life is the amount of time it takes for that quantity to be reduced to half
2
a
a
half life
• Californium-251 is a radioactive element with a half-life of 900 years.
• Give an equation to model the data.
• What percent would be left after 600 years?
• about 63%
900/
2
1100)(
t
tg
99.62
)6299(.100
)5(.100
)5(.100)(3/2
900/600
tg
Base of Exponential Model
• b > 1, grows exponentially by a rate of
b – 1
• 0 < b < 1, decays exponentially by a rate of 1 – b
• Notice: b ≠ 1, always equals one therefore the equation would be a horizontal line at the initial amount (a)
n1
• A person’s heart attack risk can be estimated by using Framingham point scores. If men’s risk of a heart attack is 1% at a Fram-score of 0, and 20% at a Fram-score of 15. Given they are exponentially linearly related, find an equation to model the data.
• (0,1) and (15, 20)• a = 1
• Predict the percent risk for a person with a score of 20
x
x
xf
b
b
bxf
)22.1()(
22.1
20
1)(15
%53
36.53
)22.1()( 20
xf
Exponential Functions from Tables
• A person’s heart attack risk can be estimated by using Framingham point scores. Men’s risk of having a heart attack in the next 10 years is shown in the table.
Framingham Score
Risk)
(percent
0 1
5 2
10 6
15 20
17 30
Plot a Scattergram
• Chose two points – (5,2) (15,20)
• Divide the equations
• Find a
5
15
2
20
ab
ab
26.1
10
2
20
10
5
15
b
b
ab
ab
62.
03.3220
)26.1(20 15
a
a
a
b represents the percent risk grows exponentially by 26% for each score
point
a represents the approximate initial
percent risk at a score of 0 points
Graph line over scattergram to check
xsf )26.1(62.)(