Exponential Functions - How to Find the Starting Value

Exponential Functions - How to Find the Starting Value Exponential functions tell the stories of explosive change. The two types of exponential functions are exponential growth and exponential decay. Four variables - percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period - play roles in exponential functions. This article focuses on how to find the amount at the beginning of the time period, a. Exponential Growth Exponential growth: the change that occurs when an original amount is increased by a consistent rate over a period of time Exponential Growth in Real Life: Values of home pricesValues of investmentsIncreased membership of a popular social networking site Heres an exponential growth function: y a(1 b)x y: Final amount remaining over a period of timea: The original amountx: TimeThe growth factor is (1 b).The variable, b, is percent change in decimal form. Exponential Decay Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time Exponential Decay in Real Life: Decline of Newspaper ReadershipDecline of strokes in the U.S.Number of people remaining in a hurricane-stricken city Heres an exponential decay function: y a(1-b)x y: Final amount remaining after the decay over a period of timea: The original amountx: TimeThe decay factor is (1-b).The variable, b, is percent decrease in decimal form. Purpose of Finding the Original Amount Six years from now, perhaps you want to pursue an undergraduate degree at Dream University. With a $120,000 price tag, Dream University evokes financial night terrors. After sleepless nights, you, Mom, and Dad meet with a financial planner. Your parents bloodshot eyes clear up when the planner reveals an investment with an 8% growth rate that can help your family reach the $120,000 target. Study hard. If you and your parents invest $75,620.36 today, then Dream University will become your reality. How to Solve for the Original Amount of an Exponential Function This function describes the exponential growth of the investment: 120,000 a(1 .08)6 120,000: Final amount remaining after 6 years.08: Yearly growth rate6: The number of years for the investment to growa: The initial amount that your family invested Hint: Thanks to the symmetric property of equality, 120,000 a(1 .08)6 is the same as a(1 .08)6 120,000. (Symmetric property of equality: If 10 5 15, then 15 10 5.) If you prefer to rewrite the equation with the constant, 120,000, on the right of the equation, then do so. a(1 .08)6 120,000 Granted, the equation doesnt look like a linear equation (6a $120,000), but its solvable. Stick with it! a(1 .08)6 120,000 Be careful: Do not solve this exponential equation by dividing 120,000 by 6. Its a tempting math no-no. 1. Use Order of Operations to simplify. a(1 .08)6 120,000 a(1.08)6 120,000 (Parenthesis) a(1.586874323) 120,000 (Exponent) 2. Solve by Dividing a(1.586874323) 120,000 a(1.586874323)/(1.586874323) 120,000/(1.586874323) 1a 75,620.35523 a 75,620.35523 The original amount, or the amount that your family should invest, is approximately $75,620.36. 3. Freeze -youre not done yet. Use order of operations to check your answer. 120,000 a(1 .08)6 120,000 75,620.35523(1 .08)6 120,000 75,620.35523(1.08)6 (Parenthesis) 120,000 75,620.35523(1.586874323) (Exponent) 120,000 120,000 (Multiplication) Practice Exercises: Answers and Explanations Here are examples of how to solve for the original amount, given the exponential function: 84 a(1.31)7Use Order of Operations to simplify.84 a(1.31)7 (Parenthesis) 84 a(6.620626219) (Exponent)Divide to solve.84/6.620626219 a(6.620626219)/6.62062621912.68762157 1a12.68762157 aUse Order of Operations to check your answer.84 12.68762157(1.31)7 (Parenthesis)84 12.68762157(6.620626219) (Exponent)84 84 (Multiplication)a(1 -.65)3 56Use Order of Operations to simplify.a(.35)3 56 (Parenthesis)a(.042875) 56 (Exponent)Divide to solve.a(.042875)/.042875 56/.042875a 1,306.122449Use Order of Operations to check your answer.a(1 -.65)3 561,306.122449(.35)3 56 (Parenthesis)1,306.122449(.042875) 56 (Exponent)56 56 (Multiply)a(1 .10)5 100,000Use Order of Operations to simplify.a(1.10)5 100,000 (Parenthesis)a(1.61051) 100,000 (Exponent)Divide to solve.a(1.61051)/1.61051 100,000/1.61051a 62,092.13231Use Order of Operations to check your answer.62,092.13231(1 .10)5 100,00062,092.13231(1.10)5 100,000 (Parenthesis)62,092.13231(1.61051) 100,000 (Exponent)100,000 100,00 0 (Multiply) 8,200 a(1.20)15Use Order of Operations to simplify.8,200 a(1.20)15 (Exponent)8,200 a(15.40702157)Divide to solve.8,200/15.40702157 a(15.40702157)/15.40702157532.2248665 1a532.2248665 aUse Order of Operations to check your answer.8,200 532.2248665(1.20)158,200 532.2248665(15.40702157) (Exponent)8,200 8200 (Well, 8,199.9999...Just a bit of a rounding error.) (Multiply.)a(1 -.33)2 1,000Use Order of Operations to simplify.a(.67)2 1,000 (Parenthesis)a(.4489) 1,000 (Exponent)Divide to solve.a(.4489)/.4489 1,000/.44891a 2,227.667632a 2,227.667632Use Order of Operations to check your answer.2,227.667632(1 -.33)2 1,0002,227.667632(.67)2 1,000 (Parenthesis)2,227.667632(.4489) 1,000 (Exponent)1,000 1,000 (Multiply)a(.25)4 750Use Order of Operations to simplify.a(.00390625) 750 (Exponent)Divide to solve.a(.00390625)/00390625 750/.003906251a 192,000a 192,000Use Order of Operations to check your answer.192,000(.25)4 750192,000(.00390625) 750750 750

May Have vs. Might Have

May Have vs. Might Have May Have vs. Might Have May Have vs. Might Have By Maeve Maddox Speaking of a murderer who was apprehended in 1998, a law enforcement officer was quoted as saying: When all this happened, if I wasnt there, he may have gotten away with it. As the speaker was there in the past and the murderer did not get away, standard usage calls for this construction: When all this happened, if I hadn’t been there, he might have gotten away with it. Might is the past tense of may. Ideally, may is the form to use when talking about a current situation, and might is the form to use in referring to an event from the past. In practice, the two forms are used interchangeably, as demonstrated by these headlines from different Web sites: 10 Civilizations That Might Have Beaten Columbus To America Polynesians  may have beaten Columbus  to South America. US-bound passengers may have to switch on mobile phones for security [Cellphone] owners might have to undergo extra screening before boarding Researchers May Have Discovered The Consciousness On/Off Switch Scientists might have just found the brain’s â€Å"off switch† 6 Signs That You Might Be Psychic Signs You May be Psychic 7 Mistakes You Might Make Before Your Job Interview 5 Money Mistakes Even Good Savers May Make Fans might have to wait weeks before Dodgers games come to their TVs Apple Fans May Have to Wait Longer for Larger iPhone Most of the time, the interchange of may and might does not present a problem. The Oxford Dictionaries site declares that if the truth of a situation isn’t known at the time of use, then either is acceptable. The one context in which might is always the better choice is one in which the event mentioned did not in fact occur: If JFK had not been assassinated, civil rights legislation might have been delayed. If the English had defeated the Normans at Hastings, we might have inherited fewer spelling problems. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Grammar category, check our popular posts, or choose a related post below:Types of RhymeIn Search of a 4-Dot EllipsisMood vs. Tense