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 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

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.

### Exponential Decay

Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time

Exponential Decay in Real Life:

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

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 (6*a* = $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)

1*a* = 75,620.35523

*a* = 75,620.35523

The original amount, or the amount that your family should 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)

### Practice Exercises: Answers and Explanations

Here are examples of how to solve for the original amount, given the exponential function:

**84 =***a*(1+.31)^{7}

Use Order of Operations to simplify.

84 =*a*(1.31)^{7}(Parenthesis)^{ }

84 =*a*(6.620626219) (Exponent)

Divide to solve.

84/6.620626219 =*a*(6.620626219)/6.620626219

12.68762157 = 1*a*

12.68762157 =*a*

Use Order of Operations to check your answer.

84 = 12.68762157(1.31)^{7}(Parenthesis)

84 = 12.68762157(6.620626219) (Exponent)

84 = 84 (Multiplication)*a*(1 -.65)^{3}= 56

Use Order of Operations to simplify.*a*(.35)^{3}= 56 (Parenthesis)*a*(.042875) = 56 (Exponent)

Divide to solve.*a*(.042875)/.042875 = 56/.042875*a*= 1,306.122449

Use Order of Operations to check your answer.*a*(1 -.65)^{3}= 56

1,306.122449(.35)^{3}= 56 (Parenthesis)

1,306.122449(.042875) = 56 (Exponent)

56 = 56 (Multiply)*a*(1 + .10)^{5}= 100,000

Use Order of Operations to simplify.*a*(1.10)^{5}= 100,000 (Parenthesis)*a*(1.61051) = 100,000 (Exponent)

Divide to solve.*a*(1.61051)/1.61051 = 100,000/1.61051*a*= 62,092.13231

Use Order of Operations to check your answer.

62,092.13231(1 + .10)^{5}= 100,000

62,092.13231(1.10)^{5}= 100,000 (Parenthesis)

62,092.13231(1.61051) = 100,000 (Exponent)

100,000 = 100,000 (Multiply)

**8,200 =***a*(1.20)^{15}

Use Order of Operations to simplify.

8,200 =*a*(1.20)^{15}(Exponent)

8,200 =*a*(15.40702157)

Divide to solve.

8,200/15.40702157 =*a*(15.40702157)/15.40702157

532.2248665 = 1*a*

532.2248665 =*a*

Use Order of Operations to check your answer.

8,200 = 532.2248665(1.20)^{15}

8,200 = 532.2248665(15.40702157) (Exponent)

8,200 = 8200 (Well, 8,199.9999...Just a bit of a rounding error.) (Multiply.)*a*(1 -.33)^{2}= 1,000

Use Order of Operations to simplify.*a*(.67)^{2}= 1,000 (Parenthesis)*a*(.4489) = 1,000 (Exponent)

Divide to solve.*a*(.4489)/.4489 = 1,000/.4489

1*a*= 2,227.667632*a*= 2,227.667632

Use Order of Operations to check your answer.

2,227.667632(1 -.33)^{2}= 1,000

2,227.667632(.67)^{2}= 1,000 (Parenthesis)

2,227.667632(.4489) = 1,000 (Exponent)

1,000 = 1,000 (Multiply)*a*(.25)^{4}= 750

Use Order of Operations to simplify.*a*(.00390625)= 750 (Exponent)

Divide to solve.*a*(.00390625)/00390625= 750/.00390625

1a = 192,000

a = 192,000

Use Order of Operations to check your answer.

192,000(.25)^{4}= 750

192,000(.00390625) = 750

750 = 750

Edited by Anne Marie Helmenstine, Ph.D.