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# Exponential Growth & Decay

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### Exponential Growth & Decay

1. 1. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8.2 & 8.3
2. 2. Growth Formula <ul><li>y= a b x </li></ul><ul><li>a > 0 & b = 1 + rate </li></ul><ul><li>a is the starting amount </li></ul><ul><li>b is the base (growth factor, increase) </li></ul><ul><li>We know this is a growth problem because b is always greater than one </li></ul>06/01/09 Bitsy Griffin PH 8.2
3. 3. Growth Formula <ul><li>Don’t forget to convert the rate from a % to a decimal BEFORE you add it to the 1 </li></ul><ul><li>BUT in conversation, you use the % </li></ul><ul><li>You have to be comfortable moving back and forth between the two. </li></ul><ul><li>x is the number of increases (usually years) </li></ul>06/01/09 Bitsy Griffin PH 8.2
4. 4. Examples 1 & 2 <ul><li>Reminders </li></ul><ul><li>Money has two decimal places. </li></ul><ul><li>Round people to whole numbers </li></ul>06/01/09 Bitsy Griffin PH 8.2
5. 5. Examples 1 & 2 <ul><li>The population of a city is 450,000 people increases 2.5% per year. Determine what the population of that city would be after each of the following years: </li></ul><ul><li>A. 1 year </li></ul><ul><li>B. 3 years </li></ul><ul><li>C. 6 years </li></ul><ul><li>D 10 years </li></ul>06/01/09 Bitsy Griffin PH 8.2
6. 6. Show your work: <ul><li>y= a b x </li></ul><ul><li>a = 450,000 </li></ul><ul><li>b = 1 + .025 </li></ul><ul><li>y =450,000(1.025) x </li></ul><ul><li>A. 1 year => y =450,000(1.025) 1 </li></ul><ul><li>B. 3 years => y =450,000(1.025) 3 </li></ul><ul><li>C. 6 years => y =450,000(1.025) 6 </li></ul><ul><li>D 10 years => y =450,000(1.025) 10 </li></ul>06/01/09 Bitsy Griffin PH 8.2
7. 7. Use your calculator <ul><li>After you find the equation, put it in the Function Editor and use your table of values to find the various increases asked for. </li></ul><ul><li>Remember to tab over to the Y column so that you can see the complete value </li></ul>06/01/09 Bitsy Griffin PH 8.2
8. 8. Compound Interest <ul><li>Your parents deposited \$500 in an account paying 6.5% interest, compounded annually 20 years ago. </li></ul><ul><li>y= a ● b x </li></ul><ul><li>a = 500 </li></ul><ul><li>b = 1.065 </li></ul><ul><li>y= 500 ● 1.065 x </li></ul><ul><li>This is simple interest </li></ul>06/01/09 Bitsy Griffin PH 8.2
9. 9. Compound Interest <ul><li>There are other ways the interest can be figured: </li></ul><ul><li>Semi-annually (2x yearly) </li></ul><ul><li>Quarterly (4x yearly) </li></ul><ul><li>Monthly (12x yearly) </li></ul><ul><li>Daily (365x yearly) </li></ul><ul><li>In each case, a little more interest will be earned. </li></ul>06/01/09 Bitsy Griffin PH 8.2
10. 10. Compound Interest <ul><li>Your parents deposited \$500 in an account paying 6.5% interest, compounded annually 20 years ago. </li></ul><ul><li>y= a ● b x </li></ul><ul><li>a = 500 </li></ul><ul><li>b = 1.065 </li></ul><ul><li>y= 500 ● 1.065 x </li></ul><ul><li>This is simple interest </li></ul>06/01/09 Bitsy Griffin PH 8.2
11. 11. Compound Interest Basic Set-up <ul><li>Semi-annual – 2x yearly </li></ul><ul><li>y= a ● b x </li></ul><ul><li>a = 500 </li></ul><ul><li>b = 1 + .065/2 – this is because you the 6.5% is divided out over the year. You have to leave the + in. It’s only the rate that’s divided. </li></ul><ul><li>x = (2●20) – This must be in () or the entire equation will be raised to 2 and then multiplied by 20 (oops!) </li></ul>06/01/09 Bitsy Griffin PH 8.2
12. 12. Calculator tip: <ul><li>Make the interest periods </li></ul><ul><li>2 = 002 </li></ul><ul><li>4 = 004 </li></ul><ul><li>12 = 012 </li></ul><ul><li>365 = 365 </li></ul><ul><li>You are working on the main screen of your calculator. When you hit 2 nd , Enter, this will save you some work. </li></ul>06/01/09 Bitsy Griffin PH 8.2
13. 13. Compound Interest Compounded Differently <ul><li>y = 500(1 + .065/002) (00 2 ● 20 ) </li></ul><ul><li>\$ </li></ul><ul><li>y = 500(1 + .065/004) (00 4 ● 20 ) </li></ul><ul><li>\$ </li></ul><ul><li>y = 500(1 + .065/012) (012 ● 20 ) </li></ul><ul><li>\$ </li></ul><ul><li>y = 500(1 + .065/365) (365 ● 20 ) </li></ul><ul><li>\$ </li></ul>06/01/09 Bitsy Griffin PH 8.2
14. 14. 8.3 Decay Formula <ul><li>y= a b x </li></ul><ul><li>a > 0 & b = 1 - rate </li></ul><ul><li>a is the starting amount </li></ul><ul><li>b is the base (decay factor, decrease) </li></ul><ul><li>We know this is a decay problem because b is always less than one </li></ul>06/01/09 Bitsy Griffin PH 8.2
15. 15. Decay Formula <ul><li>A city of 140,000 has a 1% annual decrease in population. Determine the city’s population after each of the follow years. </li></ul><ul><li>A. 2 years </li></ul><ul><li>B. 5 years </li></ul><ul><li>C. 10 years </li></ul><ul><li>D. 20 years </li></ul>06/01/09 Bitsy Griffin PH 8.2
16. 16. Show your work: <ul><li>y= a b x </li></ul><ul><li>a = 140,000 </li></ul><ul><li>b = 1 - .01 </li></ul><ul><li>y =450,000(.99) x </li></ul><ul><li>A. 1 year => y =450,000(.99) 2 </li></ul><ul><li>B. 3 years => y =450,000(.99) 5 </li></ul><ul><li>C. 6 years => y =450,000(.99) 10 </li></ul><ul><li>D 10 years => y =450,000(.99) 20 </li></ul>06/01/09 Bitsy Griffin PH 8.2
17. 17. Decay Formula <ul><li>How do you know the difference between growth and decay when you see the formulas? </li></ul><ul><li>How do you know the difference between growth and decay when you see formulas with figures in them? </li></ul>06/01/09 Bitsy Griffin PH 8.2
18. 18. Growth or Decay? <ul><li>500(1.035)^x </li></ul><ul><li>2502(0.98)^x </li></ul><ul><li>850(0.65)^x </li></ul><ul><li>200(1.05)^x </li></ul>06/01/09 Bitsy Griffin PH 8.2