Intro%20to%20Exponential%20Functions

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Intro to Exponential Functions
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Contrast
View differences using spreadsheet
LinearFunctions Change at a constant rate Rate of change (slope) is a constant
ExponentialFunctions Change at a changing rate Change at a constant percent rate
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Contrast

• Suppose you have a choice of two different jobs
  at graduation
• Start at 30,000 with a 6 per year increase
• Start at 40,000 with 1200 per year raise
• Which should you choose?
• One is linear growth
• One is exponential growth

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Which Job?
Year Option A Option B
1 30,000 40,000
2 31,800 41,200
3 33,708 42,400
4 35,730 43,600
5 37,874 44,800
6 40,147 46,000
7 42,556 47,200
8 45,109 48,400
9 47,815 49,600
10 50,684 50,800
11 53,725 52,000
12 56,949 53,200
13 60,366 54,400
14 63,988 55,600

• How do we get each nextvalue for Option A?
• When is Option A better?
• When is Option B better?
• Rate of increase a constant 1200
• Rate of increase changing
• Percent of increase is a constant
• Ratio of successive years is 1.06

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Example

• Consider a savings account with compounded yearly
  income
• You have 100 in the account
• You receive 5 annual interest

At end of year Amount of interest earned New balance in account
1 100 0.05 5.00 105.00
2 105 0.05 5.25 110.25
3 110.25 0.05 5.51 115.76
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View completed table
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Compounded Interest

• Completed table

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Compounded Interest

• Table of results from calculator
• Set y screen y1(x)1001.05x
• Choose Table (Diamond Y)
• Graph of results

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Exponential Modeling

• Population growth often modeled by exponential
  function
• Half life of radioactive materials modeled by
  exponential function

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Growth Factor

• Recall formula new balance old balance 0.05
  old balance
• Another way of writing the formula new balance
  1.05 old balance
• Why equivalent?
• Growth factor 1 interest rate as a fraction

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Decreasing Exponentials

• Consider a medication
• Patient takes 100 mg
• Once it is taken, body filters medication out
  over period of time
• Suppose it removes 15 of what is present in the
  blood stream every hour

At end of hour Amount remaining
1 100 0.15 100 85
2 85 0.15 85 72.25
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Fill in the rest of the table
What is the growth factor?
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Decreasing Exponentials

• Completed chart
• Graph

Growth Factor 0.85 Note when growth factor lt
1, exponential is a decreasing function
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Solving Exponential Equations Graphically

• For our medication example when does the amount
  of medication amount to less than 5 mg
• Graph the functionfor 0 lt t lt 25
• Use the graph todetermine when

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General Formula

• All exponential functions have the general
  format
• Where
• A initial value
• B growth factor
• t number of time periods

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Typical Exponential Graphs

• When B gt 1
• When B lt 1

View results of Bgt1, Blt1 with spreadsheet
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Assignment

• Lesson 3.1A
• Page 112
• Exercises1 23 odd
• Lesson 3.1B
• Pg 113
• Exercises25 37 odd