Introduction to Exponential Growth and Decay Models

Exponential Growth ($b > 1$)

When the base $b$ is greater than $1$, the function represents exponential growth. Every time $x$ increases by $1$, the previous value is multiplied by a number larger than $1$, causing the total to increase at an accelerating rate.

For example, if $b = 2$, the values will double every step ($2, 4, 8, 16 \dots$). If $b = 1.05$, the values increase by $5%$ every step.

Exponential Decay ($0 < b < 1$)

When the base $b$ is a fraction between $0$ and $1$, the function represents exponential decay. Multiplying a number by a fraction less than $1$ results in a smaller number. As $x$ increases, the value of $f(x)$ gets closer and closer to zero but theoretically never reaches it.

For example, if $b = 0.5$, the value is halved every step ($100, 50, 25, 12.5 \dots$). If $b = 0.9$, the value decreases by $10%$ every step.

Exam Tip: If you are given a percentage rate, be careful when converting it to the base $b$.

• For growth of $r%$: $b = 1 + r$ (e.g., $5%$ growth $\rightarrow b = 1 + 0.05 = 1.05$)
• For decay of $r%$: $b = 1 - r$ (e.g., $5%$decay$\rightarrow b = 1 - 0.05 = 0.95$)

Worked Example: Modeling a Population

Let's put these concepts into practice. Suppose a biologist starts an experiment with $500$ bacteria cells. The population is observed to triple every hour.

1. Identify the components:

• The starting amount ($a$) is $500$.
• The growth factor ($b$) is $3$ (since it triples).

2. Construct the model: $f(x) = 500 \cdot 3^x$

3. Predict a future value: How many bacteria will there be after $4$ hours? We substitute $x = 4$ into our equation:

$$\begin{aligned} f(4) &= 500 \cdot 3^4 \\ f(4) &= 500 \cdot 81 \\ f(4) &= 40,500 \end{aligned}$$

After 4 hours, the population has grown to $40,500$ cells.

Summary Table

To keep these rules straight, remember this quick reference:

By identifying the starting value and the multiplier, you can model almost any phenomenon that changes proportionally to its current size.