Applications of Exponential Models

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Applications of Exponential Models

• Sections 7.1 7.2 continued. . .

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What am I going to learn?

• Relationship between percent increase and
  exponential growth
• Relationship between percent decrease and
  exponential decay
• Translating exponential functions
• Exponential decay and half-life
• Compounded interest

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Investigate

• Do the investigation in your note templates with
  a partner.
• Sample Answers
• 50, since 100 is all of a number or 100 1
  and
• 1 x 50 50.
• 50, since 75 is 25 more than 50 and 25 is half
  of 50 or
• .5 x 50 25 and 25 50 75
• 50(1.5) 75. This is a good formula. One, it
  works. Two it makes sense since you need 50 and
  half of 50 more.

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Percent Increase and Exponential Growth

• If the percent increase is r, then the growth
  factor b 1 r.
• r can be found by

• Ex If the annual percent increase of a
  population is 1.5, then the growth factor is 1
  .015 1.015 b.

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Model for Percent Increase or Decrease

• The exponential growth model is used when a
  quantity is increased by a fixed percent over a
  given time period.
• The exponential decay model is used when a
  quantity is decreased by a fixed percent over a
  given time period.
• The model is
• a
• r
• t
• y

the initial quantity
the percent of increase or decrease
the time period
the final quantity
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Example 1

• Refer to the graph from this link. In 2000, the
  annual rate of increase in the U.S. population
  was 1.24.
• Find the growth factor for the U.S. population.
• Suppose the rate of increase continues to be
  1.24. Write a function to model exponential
  growth.

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Example 5

• Growth factor
• b 1 r 1 .0124 1.0124
• Writing an exponential function
• Use the general form
• y is the US population
• t is the number of years after 2000.
• a about 281 million, since a is initial value
  and 2000 is the year of the rate of increase.
• b 1 r 1.0124.
• Substitute a and b to get y 281(1.0124)t

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Example 6

• The exponential decay graph shows the expected
  depreciation for a car over four years. Estimate
  the value of the car after 6 years.
• What do we need to do?
• Find the decay factor.
• Find an exponential function.
• Plug x 6 into the function to find the solution.

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Example 6

• Decay factor
• r (12,000 20,000) / 20,000 -0.4
• -.04/4 to get rate of increase per year.
• b 1 r 1 (-0.1) 0.9
• Writing an exponential function
• Use the general form y abt
• Let t number of years after purchase
• Let y value of car
• a is the initial value of the car, 20,000 b
  0.9
• Substitute a and b to get y 20,000(0.9)t
• Value after 6 years Plug in x 6 to equation.
• y 20,000(0.9)6 10,628.82

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Try It Out!

• Suppose that you want to buy a used car that
  costs 11,800. The expected depreciation of the
  car is 20 per year. Estimate the depreciated
  value of the car after 6 years.
• About 3090

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Transforming the Exponential Function
Note A negative sign in front of the a reflects
the graph across the x-axis. If , then
the curve rises or falls faster. If ,
then the curve rises or falls slower.
Similar to transforming the other six families!
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Quick Review

• How are the following graphs transformed?
• y 4(1/2)x 3
• Up 3 units Rises faster Key point starts at (0,
  4)
• y -4(1/2)x-2
• Reflected over the x-axis Right 2 units Rises
  faster Key point starts at (0, 4)
• y (1/2)x-2 3
• Right 2 units Down 3 units
• y 1/4(1/2)x1 2
• Left 1 unit Down 2 units Rises slower Key
  point (0, ¼)