Exponential Functions

Graphing Functions

Geometric Sequence

• A geometric sequence has a common ratio.

• Are these geometric sequences?

...64,32,16,8,4)

321,

161,

81,

41,

21)

...30,265,20,15,10,5)

c

b

a

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x 1 2 3 4 …Y 3 5 7 9 …

x 1 2 3 4 …

Y 3 6 12 24 …

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x 1 3 5 7 …

Y 3 …43

23

x 1 3 5 7 …Y 3 5 8 13 …

83

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X 2 4 6 8 …

y 3 -1 -5 -9 …

X 1 2 5 7 …

Y 131

91

271

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X 2 5 8 11 …

y 8 12 18 27 …

X 2 5 3 4

y 2 -7 -1 -4

X 0 -1 2 1

Y 8 40 0.32 1.6

Exponential Function Characteristics

• Exponential functions model Geometric sequences.

• In a growth graph the pattern shows that the y values increase slowly at first as x increases and then it grows rapidly as x gets larger.

• In a decay graph the pattern shows that the y values decreases slowly at first as x increases and then it decays rapidly as x gets larger.

Graphs xy 3

X y0 1

1 3

2 9

3 27

4 81

1 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

10

10

20

20

30

30

40

40

50

50

60

60

70

70

80

80

90

90

100

100

Graphing xy )5.0(

x y0 11 0.52 0.253 0.1254 0.0625

1 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

Which is growing faster?xy 2 xy 5

x y0 11 22 43 84 16

x y0 11 52 253 1254 625

Which graph is steeper?xy 2

xy 51 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

5

5

10

10

15

15

20

20

25

25

30

30

1 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

5

5

10

10

15

15

20

20

25

25

30

30

Do these graphs show growth or decay? What point do these graphs have in common(Focal Point)? As x gets smaller what happens to y?

Which is steeper?x

xy

21)5.0(

xxy

41)25.0(

1 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

1 2 3 4 5 6

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

Do these graphs show growth or decay? What point do these graphs have in common(Focal Point)? As x gets very large what happens to y?

Conclusion about Growth and Decay functions

• In the function • If b>1, the function

will show growth.

• In the function • If 0<b<1, the function

will show decay

xby

Conclusions about common points

• All functions of the form have a common point at (0,1)

undefinedisexceptb 00 01

xby

Focus B

• Simplest exponential function

onentxandbasebwhereby x exp

Writing exponential functions

• On Jan 1 Donnie borrowed a dollar from his brother, Mike, who charged him 5% interest compounded daily. The function describing the amount of money he owes,y, in terms of days,x,is

xy 05.1

Table

X Y0 11 1.052 1.103 1.164 1.22… …25 3.39

What is the common ratio?

What is the connection between the base of the equation and the interest rate.

xy 05.1

Connecting the situation to the equation

• If Donnie had borrowed $100 instead of $1, what would the equation be?

• If Mike had charged him 7% interest on his $100 loan, what would the equation be?

Page 118 #30 like this• Leo buys a minivan

for $18 000. The vehicle depreciates by 15% each year.

• Set up a table of values to show the van’s value for the first four years.

• What is the common ratio?

x 0 1 2 3 4

y 18 000

Writing an exponential function

• If Bradford bought a rare stamp for $800. It appreciates 5% per year. What exponential function represents the value,y, of the stamp after x years?