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 using exponential growth functions to make predictions.
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
Exponential Growth Example: Shopping at Thrift Stores
I regret that I was too uppity and ignorant to shop at thrift stores when I was a college student. Eighteen-year-old me thought that second hand stores were cedar chests of musty, old clothes from a deceased person's closet. Since I was a "big time" resident advisor earning $80 a month, I just had to purchase new clothes at the mall. At step shows and talent shows and parties, the other "big time" girls were mirror images of me. Although I wasn't wearing a dead woman's dress, my festive spirit died right there on the dance floor.
After I graduated and started shopping at Edloe and Co., a thrift store, I discovered high quality, unique clothes at affordable prices. Ever since the start of the Great Recession, shoppers have become more budget conscious; thrift stores are more popular than ever.
Exponential Growth in Retail
Edloe and Co. relies on word of mouth advertising, the original social network. Fifty shoppers each told 5 people, and then each of those new shoppers told 5 more people and so on. The manager recorded the growth of store shoppers.
- Week 0: 50 shoppers
- Week 1: 250 shoppers
- Week 2: 1,250 shoppers
- Week 3: 6,250 shoppers
- Week 4: 31,250 shoppers
First, how do you know that this data represents exponential growth? Ask yourself 2 questions.
- Are the values increasing? Yes
- Do the values demonstrate a consistent percent increase? Yes.
How to Calculate Percent Increase
Percentage increase: (Newer - Older)/(Older) = (250 - 50) / 50 = 200/50 = 4.00 = 400%
Verify that the percentage increase persists throughout the month:
Percentage increase: (Newer - Older)/(Older) = (1,250 - 250)/250 = 4.00 = 400%
Percentage increase: (Newer - Older)/(Older) = (6,250 - 1,250)/1,250 = 4.00 = 400%
Careful - do not confuse exponential and linear growth.
The following represents linear growth:
- Week 1: 50 shoppers
- Week 2: 50 shoppers
- Week 3: 50 shoppers
- Week 4: 50 shoppers
Note: Linear growth means a consistent number of customers (50 shoppers a week); exponential growth means a consistent percent increase (400%) of customers.
How to Write an Exponential Growth Function
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.
Fill in the blanks:
- a = 50 shoppers
- b = 4.00
y = 50(1 + 4)^{x}
Note: Don't fill in values for x and y. The values of x and y will change throughout the function, but the original amount and percent change will remain constant.
Use the Exponential Growth Function to Make Predictions
Assume that the recession, the primary driver of shoppers to the store, persists for 24 weeks. How many weekly shoppers will the store have during the 8^{th} week?
Careful, do not double the number of shoppers in week 4 (31,250 *2 = 62,500) and believe it's the correct answer. Remember, this article is about is exponential growth, not linear growth.
Use Order of Operations to simplify.
y = 50(1 + 4)^{x}
y = 50(1 + 4)^{8}
y = 50(5)^{8} (Parenthesis)
y = 50(390,625) (Exponent)
y = 19,531,250 (Multiply)
19,531,250 shoppers
Exponential Growth in Retail Revenues
Prior to the start of the recession, the store's monthly revenue hovered around $800,000.
A store's revenue is the total dollar amount that customers spend in the store on goods and services.
Edloe and Co. Revenues
- Prior to recession: $800,000
- 1 month after recession: $880,000
- 2 months after recession: $968,000
- 3 months after recession: $1,171,280
- 4 months after recession: $1,288,408
Exercises
Use the information about Edloe and Co's revenues to complete 1 -7.
- What are the original revenues?
- What's the growth factor?
- How does this data model exponential growth?
- Write an exponential function that describes this data.
- Write a function to predict revenues in the 5^{th} month after the start of the recession.
- What are the revenues in the 5^{th} month after the start of the recession?
- Assume that the domain of this exponential function is 16 months. In other words, assume that the recession will last for 16 months. At what point will revenues surpass 3 million dollars?