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Intro to Exponential Functions
Lesson 3.1
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Exponential Functions
Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions Change at a changing rate Change at a constant percent rate
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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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Contrast Suppose you have a choice of two different jobs at graduation
Start at $30,000 with a 10% per year increase Start at $40,000 with $1000 per year raise Which should you choose? One is linear growth One is exponential growth
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Which Job? How do we get each next value for Option A?
1 30,000 40,000 2 33,000 41,000 3 36,300 42,000 4 39,930 43,000 5 43,923 44,000 6 48,315 45,000 7 53,147 46,000 8 58,462 47,000 9 64,308 48,000 10 70,738 49,000 11 77,812 50,000 12 85,594 51,000 13 94,153 52,000 14 103,568 53,000 How do we get each next value for Option A? When is Option A better? When is Option B better? Rate of increase a constant $1000 Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.10
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Amount of interest earned
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 * 0.05 = $5.51 $115.76 4 5 View completed table
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Compounded Interest Completed table
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Compounded Interest Table of results from calculator Graph of results
Set y= screen y1(x)=100*1.05^x Choose Table (Diamond Y/F5) 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 * old balance Another way of writing the formula new balance = 1.05 * old balance Why equivalent? Growth factor: interest rate as a fraction
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Assignment Lesson 3.1A Page 112 Exercises 1 – 23 odd
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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 3 4 5 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 < 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 function for 0 < t < 25 Use the graph to determine when
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Typical Exponential Graphs
When B > 1 When B < 1 View results of B>1, B<1 with Excel
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Assignment Lesson 3.1B Pg 113 Exercises 25 – 37 odd
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