exponential function and compound interest

1. The Exponential Functions

2. Let’s review the basics of exponential notations.
The Exponential Functions

3. The quantity A multiplied to itself N times is written as AN.
Let’s review the basics of exponential notations.
The Exponential Functions

4. The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
The Exponential Functions
N times

5. base
exponent
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
The Exponential Functions
N times

6. base
exponent
Rules of Exponents
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
The Exponential Functions
N times

7. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
The Exponential Functions
N times

8. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
For example x9x5 = x14
The Exponential Functions
N times

9. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
Division Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
For example x9x5 = x14
The Exponential Functions
N times

10. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
Division Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notations.
For example x9x5 = x14
For example x9
x5 = x9–5 = x4
The Exponential Functions
N times

11. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
Division Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power Rule: (AN)K = ANK
Let’s review the basics of exponential notations.
For example x9x5 = x14
For example x9
x5 = x9–5 = x4
The Exponential Functions
N times

12. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
Division Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power Rule: (AN)K = ANK
Let’s review the basics of exponential notations.
For example x9x5 = x14
For example x9
x5 = x9–5 = x4
For example (x9)5 = x45
The Exponential Functions
N times

13. base
exponent
Multiplication Rule: ANAK =AN+K
Rules of Exponents
Division Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power Rule: (AN)K = ANK
Let’s review the basics of exponential notations.
For example x9x5 = x14
For example x9
x5 = x9–5 = x4
For example (x9)5 = x45
The Exponential Functions
We extends the definition of exponents to negative and
fractional exponents in the following manner.
N times

14. Since = 1
A1
A1
The Exponential Functions

15. Since = 1 = A1 – 1 = A0A1
A1
The Exponential Functions

16. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
The Exponential Functions

17. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
The Exponential Functions

18. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since =
1
AK
A0
AK
The Exponential Functions

19. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K,
1
AK
A0
AK
The Exponential Functions

20. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
The Exponential Functions

21. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Since (A )k = A = (A1/k )k,
k
The Exponential Functions

22. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
The Exponential Functions

23. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Fractional Powers: A1/k = A.
k
The Exponential Functions
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

24. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

25. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
b. 91/2 =
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

26. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 =
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

27. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

28. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k

29. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
pull the numerator
outside

30. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3 = 3–3
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
pull the numerator
outside

31. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
1
33
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3 = 3–3 = =
1
27
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
pull the numerator
outside

32. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
The Exponential Functions

33. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10
122
100
The Exponential Functions

34. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122
122
100 100
The Exponential Functions

35. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
The Exponential Functions

36. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
The Exponential Functions

37. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159..
10 ..
Example B.
The Exponential Functions

38. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1
10 10
Example B.
31
10
The Exponential Functions

39. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14
10 10 10
Example B.
31
10
314
100
The Exponential Functions

40. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141
10 10 10 10
Example B.
31
10
314
100
3141
1000
The Exponential Functions

41. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10
Example B.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions

42. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10
Example B.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions

43. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 1385.45..
Example B.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions

44. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 1385.45..
Example B.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Exponential Functions

45. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 1385.45..
Example B.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Exponential Functions
Exponential functions are functions of the form f(x) = bx
where b>0 and b  0.

46. For decimal-exponent, we write the decimal as a fraction and
use the radical definition.
For example, 101.22 = 10 =( 10 )122  16.59586….
122
100 100
For real-number-exponent, we approximate the real number
with fractions and use the fractional power to approximate the
exact answer.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 1385.45..
Example B.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Exponential Functions
Exponential functions are functions of the form f(x) = bx
where b>0 and b  0. Exponential functions show up in
finanance, biology and many other scientific disciplines.

47. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
The Exponential Functions

48. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
The Exponential Functions

49. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1
The Exponential Functions

50. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2
The Exponential Functions

51. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4
The Exponential Functions

52. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16
The Exponential Functions

53. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
The Exponential Functions

54. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions

55. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).

56. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),

57. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x),

58. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".

59. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions

60. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:

61. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:
t -4 -3 -2 -1 0 1 2 3 4
y=2t

62. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1

63. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1 2 4 8 16

64. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/2 1 2 4 8 16

65. Example C. A germ splits into two germs once every day.
How many germs will there be after one day? Two days?
Three days? Four days? t days?
No. of
days
0 1 2 3 4 t
No. of
gems
1 2 4 8 16 2t
Hence P(t) = 2t gives the number of germs after t days.
The Exponential Functions
The exponential function bx is also written as expb(x).
For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"
may be expressed as "exp4(x) = 25".
Graph of Exponential Functions
To graph f(t) = 2t = y we make a table:
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

66. The Exponential Functions
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

67. (0,1)
The Exponential Functions
Graph of y = 2x
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

68. (0,1)
(1,2)
The Exponential Functions
Graph of y = 2x
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

69. (0,1)
(1,2)
(2,4)
(3,8)
The Exponential Functions
Graph of y = 2x
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

70. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
y=2t
The Exponential Functions
Graph of y = 2x
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

71. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
y=2t
The Exponential Functions
Graph of y = 2x
t -4 -3 -2 -1 0 1 2 3 4
y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16

72. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
Graph of y = bx where b>1
y=2t
The Exponential Functions
Graph of y = 2x

73. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
Graph of y = bx where b>1
y=2t
The Exponential Functions
To graph f(t) = (½)t, we make a table.
Graph of y = 2x

74. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
Graph of y = bx where b>1
y=2t
The Exponential Functions
To graph f(t) = (½)t, we make a table.
t -4 -3 -2 -1 0 1 2 3 4
y= (½)t
Graph of y = 2x

75. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
Graph of y = bx where b>1
y=2t
The Exponential Functions
To graph f(t) = (½)t, we make a table.
t -4 -3 -2 -1 0 1 2 3 4
y= (½)t 1 1/2 1/4 1/8 1/16
Graph of y = 2x

76. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
Graph of y = bx where b>1
y=2t
The Exponential Functions
To graph f(t) = (½)t, we make a table.
t -4 -3 -2 -1 0 1 2 3 4
y= (½)t 16 8 4 2 1 1/2 1/4 1/8 1/16
Graph of y = 2x

77. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)t
The Exponential Functions
Graph of y = (½)x

78. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)t
Graph of y = bx where 0<b<1
The Exponential Functions
Graph of y = (½)x

79. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)t
Graph of y = bx where 0<b<1
The Exponential Functions
Graph of y = (½)x
Compound Interest
When an account offer interest on top of previously
accumulated interest, it is called compound interest.

80. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
The Exponential Functions

81. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
The Exponential Functions

82. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01
The Exponential Functions

83. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
The Exponential Functions

84. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01
The Exponential Functions

85. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
The Exponential Functions

86. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions

87. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula

88. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)

89. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)
i = periodic rate (interest rate for a period)

90. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)
i = periodic rate (interest rate for a period)
N = number of periods

91. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)
i = periodic rate (interest rate for a period)
N = number of periods
A = total accumulated value after N periods

92. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)
i = periodic rate (interest rate for a period)
N = number of periods
A = total accumulated value after N periods
Compound Interest Formula: A = P (1 + i )N

93. Example D. We deposit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 1 month? 2 month? 3 month? and after 4 month?
After the 1st month: 1000 + 1000*0.01 = 1010 $
After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
The Exponential Functions
Compound Interest Formula
Let P = principal (the money deposited)
i = periodic rate (interest rate for a period)
N = number of periods
A = total accumulated value after N periods
Compound Interest Formula: A = P (1 + i )N
With this formula, we may compute the return after N
periods directly.

94. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
The Exponential Functions

95. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
The Exponential Functions

96. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
The Exponential Functions

97. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3
The Exponential Functions

98. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
The Exponential Functions

99. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions

100. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4

101. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $

102. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
Often for compound interest, the annual (yearly) rate r is
given, and the number of periods in one year K is given.

103. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
Often for compound interest, the annual (yearly) rate r is
given, and the number of periods in one year K is given.
In this case, the periodic rate is i = r
K

104. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
Often for compound interest, the annual (yearly) rate r is
given, and the number of periods in one year K is given.
In this case, the periodic rate is i = r
K
For example, if the annual compound interest rate is
r = 8% and compounded 4 times a year,

105. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
Often for compound interest, the annual (yearly) rate r is
given, and the number of periods in one year K is given.
In this case, the periodic rate is i = r
K
For example, if the annual compound interest rate is
r = 8% and compounded 4 times a year, then the perioic rate is
i = = 0.02.4
0.08

106. Example D. We depsit $1,000 in an account that gives 1%
interest compounded monthly. How much money are there
after 3 month? and after 4 month?
Use the formula A = P (1 + i )N
P = 1000, i = 0.01, after 3 month, N = 3,
A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
After 4 month, N = 4,
The Exponential Functions
A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
Often for compound interest, the annual (yearly) rate r is
given, and the number of periods in one year K is given.
In this case, the periodic rate is i = r
K
For example, if the annual compound interest rate is
r = 8% and compounded 4 times a year, then the perioic rate is
i = = 0.02. If it's compunded 12 times a year, then i = .4
0.08
12
0.08

107. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
The Exponential Functions

108. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
The Exponential Functions

109. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
N = (20 years)(4 times per years) = 80
The Exponential Functions

110. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
N = (20 years)(4 times per years) = 80
Hence A = 1000(1 + 0.02 )80
The Exponential Functions

111. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
N = (20 years)(4 times per years) = 80
Hence A = 1000(1 + 0.02 )80
= 1000(1.02)80  4875.44 $
The Exponential Functions

112. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
N = (20 years)(4 times per years) = 80
Hence A = 1000(1 + 0.02 )80
= 1000(1.02)80  4875.44 $
The Exponential Functions
If K is the number of times compounded in a year and t is the
number of years, then N = Kt is the total number of periods.

113. Example: We deposited $1000 in an account with annual
compound interest rater = 8%, compounded 4 times a year.
How much will be there after 20 years?
We have
4
0.08
P = 1000, i = = 0.02,
N = (20 years)(4 times per years) = 80
Hence A = 1000(1 + 0.02 )80
= 1000(1.02)80  4875.44 $
The Exponential Functions
If K is the number of times compounded in a year and t is the
number of years, then N = Kt is the total number of periods.
Together with i = r/k, the formula A = P (1 + i )N is
A = P (1 + )Ktr
K
where P = principal, r = annual rate,
K = number of periods in one year, t = number of years.

114. The Exponential Functions