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