Let's break this down step-by-step! The form asks questions about exponential growth functions and their components. Here's how to fill it in:

1. What is the role of $a$? Restrictions for $a$:

• Role of $a$:
  $a$ represents the initial value (or starting amount) of the function. It is the value of $f(x)$ when $x = 0$.
• Restrictions for $a$:
  $a$ must be greater than 0 ($a > 0$) in exponential growth functions because a negative or zero value would not make sense for modeling growth scenarios.

2. What is the role of $b$? Restrictions for $b$:

• Role of $b$:
  $b$ is the base of the exponential expression. It determines the growth or decay rate of the function. In an exponential growth function, $b > 1$ (growth factor). For decay, $0 < b < 1$.
• Restrictions for $b$:
  $b$ must be positive ($b > 0$) because a negative base would create undefined results for certain $x$-values. $b \neq 1$ because if $b = 1$, the function becomes constant.

3. What is the difference between the growth rate and the growth factor?

• Growth Rate ($r$):
  The percentage by which the quantity increases over each unit interval.
• Growth Factor ($1 + r$):
  The multiplier applied to the initial value during each interval of growth. For example, if $r = 0.05$ (5%), the growth factor is $1.05$.

The growth rate $r$ is the rate of change, while the growth factor $1 + r$ is the total multiplier.

4. Annual Compound Interest Formula:

• Formula:
  $A = P(1 + r)^t$
  Where:
  $A$ = final amount,
  $P$ = principal (initial amount),
  $r$ = annual interest rate (in decimal form),
  $t$ = time in years.

5. Compound Interest Formula (compounding $n$ periods):

• Formula:
  $A = P\left(1 + \frac{r}{n}\right)^{nt}$
  Where:
  $n$ = number of compounding periods per year.

Would you like detailed examples for any of these formulas or terms? 😊

Follow-up questions to expand understanding:

1. How do $a$, $b$, and $r$ relate to real-life applications like population growth?
2. How does compound interest differ when $n = 1$ versus $n > 1$?
3. Can exponential decay functions also use these formulas?
4. Why must $a > 0$ for exponential models?
5. What happens to $A$ as $n \to \infty$ in the compound interest formula?

Tip: Always convert percentages into decimals (e.g., 5% = 0.05) when using these formulas.