exponential functions functions that have the exponent as the variable
TRANSCRIPT
Exponential Functions
Functions that have the exponent as the variable.
Exponential Functions
Functions that have the exponent as the variable.
xaxf )( - “a” is our base raised to some exponent “ x “ that varies
- if a > 1, the graph shows exponential growth
Exponential Functions
Functions that have the exponent as the variable.
xaxf )(
xxf 2)(
- “a” is our base raised to some exponent “ x “ that varies
- if a > 1, the graph shows exponential growth
- these functions explode as “x” gets larger
Exponential Functions
Functions that have the exponent as the variable.
xaxf )(
xxf 2)(
xxf 10)(
- “a” is our base raised to some exponent “ x “ that varies
- if a > 1, the graph shows exponential growth
- these functions explode as “x” gets larger
Exponential Functions
Functions that have the exponent as the variable.
xaxf )( - “a” is our base raised to some exponent “ x “ that varies
- if 0 < a < 1, the graph shows exponential decay
- the graph approaches zero as “x” gets larger
x
xf
2
1)(
Exponential Functions
Functions that have the exponent as the variable.
xaxf )( - “a” is our base raised to some exponent “ x “ that varies
- if 0 < a < 1, the graph shows exponential decay
- the graph approaches zero as “x” gets larger
x
xf
2
1)( xxf 2.0)(
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE : Graph xxf 2)(
x y
3 8
2
1
0
-1
-2
-3
82)3( 3 f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE : Graph xxf 2)(
x y
3 8
2 4
1
0
-1
-2
-3
42)2(
82)3(2
3
f
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE : Graph xxf 2)(
x y
3 8
2 4
1 2
0
-1
-2
-3
221
42)2(
82)3(
1
2
3
f
f
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE : Graph xxf 2)(
x y
3 8
2 4
1 2
0 1
-1 .5
-2 .25
-3 .125
125.023
25.022
5.021
12)0(
221
42)2(
82)3(
3
2
1
0
1
2
3
f
f
f
f
f
f
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE # 2 : Graph x
xf
4
1)(
x y
3 0.02
2
1
0
-1
-2
-3
02.04
1)3(
3
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE # 2 : Graph x
xf
4
1)(
x y
3 0.02
2 0.06
1
0
-1
-2
-3
06.04
12
02.04
1)3(
2
3
f
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE # 2 : Graph x
xf
4
1)(
x y
3 0.02
2 0.06
1 0.25
0
-1
-2
-3
25.04
1)1(
06.04
12
02.04
1)3(
1
2
3
f
f
f
Exponential Functionsxaxf )(
To graph these functions we only need to complete an x / y table…
EXAMPLE # 2 : Graph x
xf
4
1)(
x y
3 0.02
2 0.06
1 0.25
0 1
-1 4
-2 16
-3 64
644
13
164
12
44
11
14
10
25.04
1)1(
06.04
12
02.04
1)3(
3
2
1
0
1
2
3
f
f
f
f
f
f
f
Exponential Functions
Applications : Compound Interest
Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods.
Exponential Functions
Applications : Compound Interest
Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods.
trPA 1 Compound Interest equation
Exponential Functions
Applications : Compound Interest
Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods.
trPA 1
EXAMPLE # 1 : If $7,500 is invested at 12% interest compounded yearly, how much would be in the account after 5 years ?
Exponential Functions
Applications : Compound Interest
Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods.
trPA 1
EXAMPLE # 1 : If $7,500 is invested at 12% interest compounded yearly, how much would be in the account after 5 years ?
56.217,13
7623.17500
12.17500
12.017500
5 ,12.0 7500,P
5
5
A
A
A
A
tr
Exponential Functions
trPA 1
EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ?
Exponential Functions
trPA 1
EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ?
A = 9000 , r = 0.0067 ( 0.08/12 …divide your interest by 12 )
t = 72 ( 12 months x 6 years )
Exponential Functions
trPA 1
EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ?
A = 9000 , r = 0.0067 ( 0.08/12 …divide your interest by 12 )
t = 72 ( 12 months x 6 years )
18.14556
617.19000
0067.19000
0067.01900072
72
A
A
A
A
Exponential Functions
trPA 1
EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ?
Exponential Functions
trPA 1
EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ?
A = 15,000 r = 0.0075 ( 0.09 / 12 )
T = 60 ( 5 x 12 )
Exponential Functions
trPA 1
EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ?
A = 15,000 r = 0.0075 ( 0.09 / 12 )
T = 60 ( 5 x 12 )
P
P
P
P
P
38.95805657.1
15000
5657.115000
0075.115000
0075.011500060
60
Exponential Functions
ht
cc
2
1M
The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass.
Where : c = original mass
t = time
h = half life
Exponential Functions
ht
cc
2
1M
The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass.
Where : c = original mass
t = time
h = half life
Example : Plutonium has a half-life of 24,360 years. How much of a 2 kg sample would be left after 50,000 years ?
Exponential Functions
ht
cc
2
1M
The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass.
Where : c = original mass
t = time
h = half life
Example : Plutonium has a half-life of 24,360 years. How much of a 2 kg sample would be left after 50,000 years ?
kg 482.02M
241.022M
2
122M
2
122M
053.2
2436050000