Transcript 7-6 Exponential Functions

7-6 & 7-7 Exponential Functions
Evaluate and graph exponential
functions
Exponential function
A function in the form of
y = 𝑎 ∙ 𝑏𝑥
Examples:
Exponential Growth, modeled by the following
y = a∙ 𝑏 𝑥
Initial amount (this is when x = 0)
𝑦 =𝑎∙𝑏
𝑥
exponent
The base & when b>1,
called the Growth factor
(1 + the percent rate written as a
decimal)
Exponential Decay
Initial amount (this is when x = 0)
𝑦 =𝑎∙𝑏
𝑥
exponent
The base is the decay factor
(1 – percent rate written as a decimal)
What is the graph of y = 3 ∙ 2𝑥 ?
x
y = 3 ∙ 2𝑥
-2
2−2
y=3∙
(x, y)
10
(-2,
3
)
4
9
8
7
-1
0
y=3∙
2−1
y = 3 ∙ 20
(-1, 1
1
)
2
c
c
6
5
4
(0, 3)
3
2
1
y = 3 ∙ 21
1
(1, 6)
0
-1
2
y = 3 ∙ 22
(2, 12)
-2
-5
-4
-3
-2
-1
0
1
2
3
4
5
Does the table or rule represent a linear or an exponential function?
A.
ANSWER: EXPONENTIAL FUNCTION.
B. y = 3x
ANSWER:
LINEAR FUNCTION.
Suppose 30 flour beetles are left undisturbed in a warehouse bin.
The beetle population doubles each week. The function
f(x) = 30 ∙ 2𝑥 gives the population after x weeks. How many beetles
will there be after 56 days?
f(x) = 30 ∙ 2𝑥
What does x
represent?
= 30 ∙ 28
= 30 ∙ 256
= 7680
Answer: after 56
days, there will be
7,680 beetles.
Evaluate the function for the given value.
𝑦 = 3 ∙ 4𝑥 for x = 3
𝑦 =3∙4
(3 )
𝑦 = 3 ∙ 64
𝑦 = 192
Since 2005, the amount of money spent at restaurants in the US has increased
about 7% each year. In 2005, about $360 billion was spent at restaurants. If the
trend continues, about how much will be spent at restaurants in 2015?
𝑥
𝑏
𝑦=𝑎∙
Let y = The annual amount spent in restaurants (in billions of dollars)
Let a = The initial amount: 360
Let b = The growth factor:
(1 + %) or
Let x = The number of years since 2005:
𝑦 = 360 ∙
10
1.07
𝑦 = 708.174488
𝑦 = $708 billion
1 + .07 = 1.07
10
Compound interest:
When a bank pays interest on both the principal and
the interest an account has earned. (it uses the
following formula)
A = The balance
r = the annual interest rate---convert from % to a
decimal—(move 2 places to
the left)
r nt
A = P( 1 + )
n
P = the principal
(the initial deposit)
t= the time in years
n = the number of
times interest is
compounded per year
Find the balance in the account after the given period:
$12,000 principal earning 4.8% compounded
annually, after 7 years
r nt
A = P( 1 + )
n
.048 1(7)
A = 12,000( 1 +
)
1
A = $16,661.35
P = 12,000
r = .048
n=1
t= 7
Find the balance in the account after the given period:
$20,000 principal earning 3.5% compounded
monthly, after 10 years
r nt
A = P( 1 + )
n
.035 12(10)
A = 20,000( 1 +
)
12
A = $28,366.90
P = 20,000
r = .035
n = 12
t = 10
The kilopascal is unit of measure for atmospheric pressure. The atmospheric
pressure at sea level is about 101 kilopascals. For every 1000-m increase in
altitude, the pressure decreases about 11.5%. What is the approximate pressure
at an altitude of 3000 m?
𝑦=𝑎∙
𝑥
𝑏
Let y = The atmospheric pressure (in kilopascals)
Let a = The initial amount: 101
Let b = The decay factor:
(1 - %) or
1 - .115 = .885
Let x = The altitude (in thousands of meters)
𝑦 = 101
3
∙.885
𝑦 = 70.0085
𝑦 = 70 kilopascals
3
Pg 457: 9-21 odd & 20
pg 464: 9-21 odd (skip 13)