# Solving Exponential Growth Functions: Social Networking

## Algebra Solutions: Answers and Explanations

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.

### 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

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
1. 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.
2. 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.
3. What is the monthly percent increase or decrease? The monthly percent increase is 40%, .40 written as a percent.
4. 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