Exponential Growth/Decay

The Exponential Growth/Decay formula is one of the most common equation forms to see on the ACT. It can be used to express a multitude of different situations, which means it can show up theoretically as a variable function or practically within a word problem. The formula looks like this:

Final = Initial (1±growth/decay rate)^time

Key Takeaways

• If it is a growth function, then the number inside of the parenthesis should be greater than 1
• If it is a decay function, then the number inside of the parenthesis should be less than 1
• Final is the output of the equation. It is like the y-value in a linear equation. It is the thing that is being calculated
• Initial is the starting point of the equation. It is like the intercept in a linear equation. It says what the starting point is for the equation.
• T is usually the main variable in the equation. It says how long the equation needs to be run for (how many cycles it goes through), as well as how often it will happen.

How It Shows Up In Problems

Exponential growth and decay problems are often word problems. The problem will describe an example of a real-life situation that utilizes this formula. The problem will then either:

• Present the correct version of the formula, and then ask students to interpret one element of the formula

OR

• Give several different possible iterations of the formula, from which the student must choose the correct version of the formula as described in the problem

Common Exponential Growth/Decay Formulas or Questions

This formula can be applied to a wide variety of situations, but there are a few variations of it that have shown up on multiple exams and are therefore more likely to show up on any given exam.

Financial/Compound Interest

The compound interest formula shows how an investment growths with time. The exponent is normally equal to the number of years that the investment grows for or the number of months divided by 12.

Final = Initial (1±interest rate)^years or 

Final = Initial (1±interest\ rate)months12

Half-Life

The “half-life” says that after X number of years, the amount of a given element remaining will be half of what was initially there. This is why the rate inside of the parenthesis is 

12and the exponent will be equal to

number\ of\ years\ elapsedthe half–life (in years).

Final = Initial (12)# yearshalf–life years

Doubling

The doubling formula is the same as the half-life formula, but with a two inside of the parenthesis and where the denominator of the exponent is the time it takes to double rather than the time it takes to halve. This can be expressed in years, but is often expressed in smaller units like hours.

Final = Initial (2)timetime\ to\ double