8.1 Exploring Exponential Models

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8.1 Exploring Exponential Models
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What is an exponential equation?

• An exponential equation has the general form
  
• yabx

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Growth Factor, Decay Factor

• Given the general form yabx
• When b gt 1, b is the growth factor

• When 0 lt b lt 1, b is the decay factor

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Growth or Decay???

Growth

Decay
Growth
Decay
Growth
Decay
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Graphs of Exponential Functions
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What is an asymptote?
Walking halfway to the wall
An Asymptote is a line that a graph approaches as
x or y increases in absolute value.
In this example, the asymptote is the x axis.
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Graphing Exponential Functions
X .5x Y100(.5)x
-3
-2
-1
0
1
2
3
Complete the table using the integers -3 through
3 for x.
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Lets graph one together
X .5x Y100(.5)x
-3 .5-3 800
-2 .5-2 400
-1 .5-1 200
0 .50 100
1 .51 50
2 .52 25
3 .53 12.5
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Lets try one
X .5x Y2(.5)x
-3
-2
-1
0
1
2
3
Complete the table using the integers -3 through
3 for x. Then graph the function.
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Lets try one
X .5x Y2(.5)x
-3
-2
-1
0
1
2
3
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Lets try one
X 10x Y5(10)x
-3
-2
-1
0
1
2
3
Complete the table using the integers -3 through
3 for x. Then graph the function.
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Lets try one
X 10x Y5(10)x
-3
-2
-1
0
1
2
3
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Writing Exponential Equations

• Find the exponential equation passing through the
  points (3,20) and (1,5).

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
Division property of exponents
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Writing Exponential Equations

• Find the exponential equation passing through the
  points (3,20) and (1,5).

Simplify
Solve for b
Go 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).

Going back to the general form, substitute in a
and b
The exponential equation passing through the
points (3,20) and (1,5) is
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Lets Try One

• Find the exponential equation passing through the
  points (2,4) and (3,16).

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
Division property of exponents
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Writing Exponential Equations
Simplify
Solve for b
Go back to the equation for a substitute in b
and solve for a
Going back to the general form, substitute in a
and b
The exponential equation passing through the
points (2,4) and (3,16) is
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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?

r rate 3 (write as 0.03) b 1 r 1.03 x
time a amount put into the account (1,000)
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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 )
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)