exponential functions chapter 4. 4.1 properties of exponents know the meaning of exponent, zero...
TRANSCRIPT
![Page 1: Exponential Functions Chapter 4. 4.1 Properties of Exponents Know the meaning of exponent, zero exponent and negative exponent. Know the properties of](https://reader036.vdocument.in/reader036/viewer/2022062621/551bf1f6550346b4588b666e/html5/thumbnails/1.jpg)
Exponential Functions
Chapter 4
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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.
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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
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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
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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,
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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
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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
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Order of Operations
• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction
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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
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Examples
95
4523
4253
24
64
64
yx
yyxx
yxyx
1218
34363
34363
346
216
6
6
6
vu
vu
vu
vu
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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
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Zero Exponent
• For b ≠ 0,
• Examples,10 b
0,1
124
15
0
0
0
xyxy
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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
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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
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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
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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 )(
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Exponential vs Linear Functions
• x is a exponent • x is a base
xxf 2)( 12)( xxf
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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
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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
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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
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Group Exploration
• If time,– p173
nb1
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4.2 Rational Exponents
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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
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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
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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
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Examples
27
1
)3(
1
)81(
1
81
181
16)2())32(()32(
9)3()27(27
33414343
445154
223132
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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,
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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
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4.3 Graphing Exponential Functions
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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)(
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Graph of an exponential function is called an exponential curve
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x
xg
2
14)(
x y
-1 8
0 4
1 2
2 1
3 1/2
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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
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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)(
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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
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Intercepts
• y-intercept for the form:
is (0,a)• y-intercept for the form:
is (0,1)
xaby
xby
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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
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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