What are Exponential Growth Functions

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. The following focuses on using exponential growth functions to make predictions. Exponential Growth Exponential growth is 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 pricesValues of investmentsIncreased membership of a popular social networking site Exponential Growth in Retail Edloe and Co. relies on word of mouth advertising, the original social network. Fifty shoppers each told five people, and then each of those new shoppers told five more people, and so on. The manager recorded the growth of store shoppers. Week 0: 50 shoppersWeek 1: 250 shoppersWeek 2: 1,250 shoppersWeek 3: 6,250 shoppersWeek 4: 31,250 shoppers First, how do you know that this data represents exponential growth? Ask yourself two questions. Are the values increasing? YesDo 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 shoppersWeek 2: 50 shoppersWeek 3: 50 shoppersWeek 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 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. Fill in the blanks: a 50 shoppersb 4.00 y 50(1 4)x Note: Dont 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 8th week? Careful, do not double the number of shoppers in week 4 (31,250 *2 62,500) and believe its 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 stores monthly revenue hovered around $800,000. A stores revenue is the total dollar amount that customers spend in the store on goods and services. Edloe and Co. Revenues Prior to recession: $800,0001 month after recession: $880,0002 months after recession: $968,0003 months after recession: $1,171,2804 months after recession: $1,288,408 Exercises Use the information about Edloe and Cos revenues to complete 1 through 7. What are the original revenues?Whats 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 fifth  month after the start of the recession.What are the revenues in the fifth  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?