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 use an exponential decay function to find a, the amount at the beginning of the time period.
Exponential Decay
Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time
Here's an exponential decay function:
y = a(1-b)^{x}
- y: Final amount remaining after the decay over a period of time
- a: The original amount
- x: Time
- The decay factor is (1-b).
- The variable, b, is percent decrease in decimal form.
Purpose of Finding the Original Amount
If you are reading this article, then you are probably ambitious. 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 rate
- 6: The number of years for the investment to grow
- a: 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 doesn't look like a linear equation (6a = $120,000), but it's solvable. Stick with it!
a(1 +.08)^{6} = 120,000
Be careful: Do not solve this exponential equation by dividing 120,000 by 6. It's 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 to invest is approximately $75,620.36.
3. Freeze -you're 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)
Answers and Explanations to the Questions
Woodforest, Texas, a suburb of Houston, is determined to close the digital divide in its community.
A few years ago, community leaders discovered that their citizens were computer illiterate: they did not have access to the Internet and were shut out of the information superhighway. The leaders established World Wide Web on Wheels, a set of mobile computer stations.
World Wide Web on Wheels has achieved its goal of only 100 computer illiterate citizens in Woodforest. Community leaders studied the monthly progress of World Wide Web on Wheels. According to the data, the decline of computer illiterate citizens can be described by the following function:
100 = a(1 - .12)^{10}
1. How many people are computer illiterate 10 months after the inception of World Wide Web on Wheels? 100 people
Compare this function to the original exponential growth function:
100 = a(1 - .12)^{10}
y = a(1 + b)^{x}
The variable, y, represents the amount of computer illiterate people at the end of 10 months, so 100 people are still computer illiterate after World Wide Web on Wheels began to work in the community.
2. Does this function represent exponential decay or exponential growth? This function represents exponential decay because a negative sign sits in front of the percent change, .12.
3. What is the monthly rate of change? 12%
4. How many people were computer illiterate 10 months ago, at the inception of World Wide Web on Wheels? 359 people
Use Order of Operations to simplify.
100 = a(1 - .12)^{10}
100 = a(.88)^{10} (Parenthesis)
100 = a(.278500976) (Exponent)
Divide to solve.
100(.278500976) = a(.278500976)/(.278500976)
359.0651689 = 1a
359.0651689 = a
Use Order of Operations to check your answer.
100 = 359.0651689(1 - .12)^{10}
100 = 359.0651689(.88)^{10} (Parenthesis)
100 = 359.0651689(.278500976) (Exponent)
100 = 100 (Okay, 99.9999999…It's just a bit of a rounding error.) (Multiply)
5. If these trends continue, how many people will be computer illiterate 15 months after the inception of World Wide Web on Wheels? 52 people
Plug in what you know about the function.
y = 359.0651689(1 - .12) ^{x}
y = 359.0651689(1 - .12) ^{15}
Use Order of Operations to find y.
y = 359.0651689(.88)^{15} (Parenthesis)
y = 359.0651689(.146973854) (Exponent)
y = 52.77319167 (Multiply)