7-6 & 7-7 Exponential Functions Evaluate and graph exponential functions.
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Transcript of 7-6 & 7-7 Exponential Functions Evaluate and graph exponential functions.
Exponential Growth, modeled by the following y = a
𝑦=𝑎 ∙𝑏𝑥
Initial amount
The base & when b>1, called the Growth factor(1 + the percent rate written as a decimal)
exponent
(this is when x = 0)
Exponential Decay
𝑦=𝑎 ∙𝑏𝑥
Initial amount
The base is the decay factor (1 – percent rate written as a decimal)
exponent
(this is when x = 0)
What is the graph of y = ?
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1
0
1
2
3
4
5
6
7
8
9
10x y = (x, y)
-2
-1
0
1
2
y =
y =
y =
y =
y =
(-2, )
(-1, 1 )
(0, 3)
(1, 6)
cc
(2, 12)
Does the table or rule represent a linear or an exponential function?
ANSWER: EXPONENTIAL FUNCTION.
B. y = 3x
A.
ANSWER: LINEAR FUNCTION.
Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. The function f(x) = gives the population after x weeks. How many beetles will there be after 56 days?
f(x) =
=
=
= Answer: after 56 days, there will be 7,680 beetles.
What does x represent?
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?
𝑦=𝑎 ∙𝑏𝑥
Let y = The annual amount spent in restaurants (in billions of dollars)
Let a = The initial amount:
Let b = The growth factor:
Let x =
360
(1 + %) or 1 + .07 = 1.07
The number of years since 2005: 10
𝑦=360 ∙1.0710
𝑦=708.174488𝑦=$708billion
Compound interest: When a bank pays interest on both the principal and the interest an account has earned. (it uses the following formula)
A=P(1+ rn)n t
A = The balance
P = the principal (the initial deposit)
r = the annual interest rate----convert from % to a decimal—(move 2 places to the left)
n = the number of times interest is compounded per year
t= the time in years
A=P(1+ rn)n t
Find the balance in the account after the given period:
$12,000 principal earning 4.8% compounded annually, after 7 years
P =r =n =t =
12,000.04817
A=12 ,000(1+ .0481
)1(7)
A = $16,661.35
A=P(1+ rn)n t
Find the balance in the account after the given period:
$20,000 principal earning 3.5% compounded monthly, after 10 years
P =r =n =t =
20,000.0351210
A=20 ,000 (1+ .03512
)12(10)
A = $28,366.90
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 ∙𝑏𝑥
Let y = The atmospheric pressure (in kilopascals)Let a = The initial amount:
Let b = The decay factor:
Let x =
101
(1 - %) or 1 - .115 = .885
The altitude (in thousands of meters) 3
𝑦=101 ∙.8853
𝑦=70.0085𝑦=70 kilopascals