Notes 9 4 Finding Exponential Equations

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1 Notes 9 4 Finding Exponential Equations Dec 22 10:39 AM 1

2 y = ab x We need the initial condition (a) and the growth factor (b) Solving method #1 Need two points one of which is the y intercept. y intercept is known 1. Re write general equation, substitute the y intercept/initial condition in for "a". 2. Substitute ANOTHER ordered pair in for x and y. 3. Solve for b 4. Write final equation in terms of x with your new a and b values Solving method #2 Need two point neither of which is the y intercept. y intercept is not known 1. Choose two data points & substitute into y = ab x (write 2 equations put largest "x" as first equation.) 2. Divide the two eq. WHY? 3. Solve for b 4. Use one equation from earlier & substitute "b" into it; solve for "a" 5. Write final equation in terms of x with your new "a" and "b" values Feb 25 9:02 AM 2

3 Solving method #1 Need two points one of which is the y intercept. y intercept is known 1. Re write general equation, substitute the y intercept/initial condition in for "a". 2. Substitute ANOTHER ordered pair in for x and y. 3. Solve for b 4. Write final equation in terms of x with your new a and b values Solving method #2 Need two point neither of which is the y intercept. y intercept is not known 1. Choose two data points & substitute into y = ab x (write 2 equations put largest "x" as first equation.) 2. Divide the two eq. WHY? 3. Solve for b 4. Use one equation from earlier & substitute "b" into it; solve for "a" 5. Write final equation in terms of x with your new "a" and "b" values Feb 16 4:33 PM 3

4 Solving method #1 Need two points one of which is the y intercept. y intercept is known 1. Re write general equation, substitute the y intercept/initial condition in for "a". 2. Substitute ANOTHER ordered pair in for x and y. 3. Solve for b 4. Write final equation in terms of x with your new a and b values Solving method #2 Need two point neither of which is the y intercept. y intercept is not known 1. Choose two data points & substitute into y = ab x (write 2 equations put largest "x" as first equation.) 2. Divide the two eq. WHY? 3. Solve for b 4. Use one equation from earlier & substitute "b" into it; solve for "a" 5. Write final equation in terms of x with your new "a" and "b" values Feb 25 9:03 AM 4

5 AdvAlg9.4FittingExponentialModelsToData.notebook Determine the growth factor between many(all) the values of the dependent variable. If these are all the same (a constant) or very near the same an exponential model is approprieate. One thing that MUST be true when you find these growth factors is that the time interval between each value must be the same. Feb 25 9:15 AM 5

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7 Annual growth rate Feb 25 9:15 AM 7

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Population in 1840 was 17,070,000. Using this as the starting point makes the 1840 year zero and 17.07 million the initial value so y = 17.07 b x Population in 1860 was 31,440,000. Population of 31.

9 Population in 1840 was 17,070,000. Using this as the starting point makes the 1840 year zero and million the initial value so y = b x Population in 1860 was 31,440,000. Population of million 20 years after 1840 so = b 20 x is the number of years after 1840 This is the annual growth factor y = 17.07(1.0315) (x 1840) here x is the year y = 17.07(1.0315) ( ) y = million Feb 25 9:15 AM 9

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18 Notes 9 4 Finding Exponential Equations Use Data from Lesson Master 9 4A and general equation y = ab x Solving method #1 y intercept is known (0, 14) 1. Re write general equation, substitute y intercept in for "a" value. 2. Substitute ANOTHER x and f(x) from table ( 3, 4.21) 3. Solve for b 4. Write final equation in terms of x with your new a and b values Dec 22 10:39 AM 18

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20 Solving method #2 y intercept is not known 1. Choose two data points & substitute into y = ab x (write 2 equations put largest "x" as first equation.) 2. Divide the two eq. WHY? 3. Solve for b 4. Use one equation from earlier & substitute "b" into it; solve for "a" 5. Write final equation in terms of x with your new "a" and "b" values y = ab x (2, 6.28) ( 4, 2.82) Dec 22 10:54 AM 20

21 Ratio: Population of specific year Population 10 years earlier If the ratio is constant, an exponential model is appropriate Annual growth factor between 1840 and 1850?? **This data needs two different exponential models Dec 22 10:58 AM 21

22 Solving method #1 y intercept is known y = ab x Dec 22 11:04 AM 22

23 Solving method #2 y intercept is NOT known y = ab x Dec 22 11:04 AM 23

24 In 20 days, how much of the substance will be present? Dec 22 11:21 AM 24

25 Population Time (hours) Dec 22 11:22 AM 25

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8.6. Write and Graph Exponential Decay Functions. Warm Up Lesson Presentation Lesson Quiz

8.6. Write and Graph Exponential Decay Functions. Warm Up Lesson Presentation Lesson Quiz 8.6 Write and Graph Exponential Decay Functions Warm Up Lesson Presentation Lesson Quiz 8.6 Warm-Up. Evaluate. 3 ANSWER 8. Evaluate. 4 ANSWER 6 3. The table shows how much money Tess owes after w weeks.