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 word problems 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
Uses of Exponential Growth in Real Life:
- Values of home prices
- Values of investments
- Increased membership of a popular social networking site
Here’s an exponential growth function:
y = a(1 + b)^{x}
- y: Final amount remaining over a period of time
- a: The original amount
- x: Time
- The growth factor is (1 + b).
- The variable, b, is percent change 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)
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
Original Worksheet
Farmer and Friends
Use the information about the farmer's social networking site to answer questions 1-5.
A farmer started a social networking site, farmerandfriends.org, that shares backyard gardening tips. When farmerandfriends.org enabled members to post photos and videos, the website's membership grew exponentially. Here’s a function that describes that exponential growth.
120,000 = a(1 + .40)^{6}
- How many people belong to farmerandfriends.org 6 months after it enabled photo-sharing and video-sharing? 120,000 people
Compare this function to the original exponential growth function:
120,000 = a(1 + .40)^{6}y = a(1 +b)^{x}
The original amount, y, is 120,000 in this function about social networking. - Does this function represent exponential growth or decay? This function represents exponential growth for two reasons. Reason 1: The information paragraph reveals that "the website membership grew exponentially." Reason 2: A positive sign is right before b, the monthly percentage change.
- What is the monthly percent increase or decrease? The monthly percent increase is 40%, .40 written as a percent.
- How many members belonged to farmerandfriends.org 6 months ago, right before photo-sharing and video-sharing were introduced? About 15,937 members
Use Order of Operations to simplify.
120,000 = a(1.40)^{6}
120,000 = a(7.529536)
Divide to solve.
120,000/7.529536 = a(7.529536)/7.529536
15,937.23704 = 1a
15,937.23704 = a
Use Order of Operations to check your answer.
120,000 = 15,937.23704(1 + .40)^{6}120,000 = 15,937.23704(1.40)^{6}
120,000 = 15,937.23704(7.529536)
120,000 = 120,000 - If these trends continue, how many members will belong to the website 12 months after the introduction of photo-sharing and video-sharing? About 903,544 members
Plug in what you know about the function. Remember, this time you have a, the original amount. You are solving for y, the amount remaining at the end of a time period.
y = a(1 + .40)^{x}y = 15,937.23704(1+.40)^{12}
Use Order of Operations to find y.
y = 15,937.23704(1.40)^{12}
y = 15,937.23704(56.69391238)
y = 903,544.3203