8.2 exploring exponential models
TRANSCRIPT
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8.2 Writing Exponential Models
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What is an exponential equation?
An exponential equation has the general form
y=abx
1b and 0b ,0 where a
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Growth Factor, Decay Factor
Given the general form
y=abx
When b > 1, b is the growth factor
When 0 < b < 1, b is the decay factor
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Growth or Decay???
x)2.1(10y
x)9(.5y x)54.1(50y
x)70(.2.5y xy )2(4
x)07(.100y
Growth
Decay
Growth
Decay
Growth
Decay
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Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5). xaby
ab
3
20
13
205 b
b
31205 b
320 ab
Start with the general form.
Choose a point. Substitute for x and y using (3, 20)
Solve for a
Substitute x and y using (1, 5) and a using ab
3
2
Division property of exponents
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Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5).
2
45
20
205
205
2
2
2
b
b
b
b
Simplify
Solve for b
2
5
8
20
2
202033
b
aGo back to the equation for a; substitute in b and solve for a
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Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5).
xaby
xy )2(2
5
Going back to the general form, substitute in a and b
The exponential equation passing through the points (3,20) and (1,5) is xy )2(
2
5
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Let’s Try One Find the exponential equation passing through
the points (2,4) and (3,16). xaby
ab
2
4
32
416 b
b
23416 b
24 ab
Start with the general form.
Choose a point. Substitute for x and y using (2, 4)
Solve for a
Substitute x and y using (3, 16) and a using ab
2
4
Division property of exponents
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Writing Exponential Equations
1416 b4b
Simplify
Solve for b
25.04
1
4
42
aGo back to the equation for a; substitute in b and solve for a
xaby
xy )4(25.0 Going back to the general form, substitute in a and b
The exponential equation passing through the points (2,4) and (3,16) is
xy )4(25.0
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Putting it all together . . .
Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.
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Modeling Growth with an Exponential Equation
The growth factor can be found in word problems using b = 1 + r
where r = rate or amount of increase. You can substitute your new b into your general equation to find the exponential function.
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EX- a guy puts $1000 into a simple 3% interest account. What is the exponential equation?
xy )03.1( 1000
r = rate 3% (write as 0.03)
b = 1 + r = 1.03
x = time
a = amount put into the account ($1,000)
xaby
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EX – a colony of 1000 bacteria cells doubles every hour. What is the exponential equation?
r = 1 (why not 2?)
b = r + 1 = 2
x = time (in hours)
a = the original number in the colony (1,000 bacteria )
xy )2( 1000
xaby
b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1
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EX- a $15000 car depreciates at 10% a year. What is the exponential equation?
r = - 10% (the car is worth 10% less each year)
b = 1 - r = 1 – 0.1 = 0.9
x = time (in years)
a = amount put into the account ($15,000)
xy )9.0( 15000
xaby
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Compound Interest
The formula for compound interest:
( ) 1
ntr
A t Pn
Where n is the number of times per year interest is being compounded and r is the annual rate.
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Compound Interest - Example
Which plan yields the most interest? Plan A: A $1.00 investment with a 7.5% annual
rate compounded monthly for 4 years Plan B: A $1.00 investment with a 7.2% annual
rate compounded daily for 4 years
A:
B:
12(4)0.075
1 1 1.348612
365(4)0.072
1 1 1.3337365
$1.35
$1.34