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