The given equation is $y = 2 \cdot 5^x$. Let us break it down to identify the requested components:

1. Original amount ($a$): This is the initial value of the function when $x = 0$. In this case, $a = 2$.

2. Growth factor ($b$): This is the base of the exponential term. Here, $b = 5$.

3. Rate of increase ($r$): The growth factor is related to the rate of increase using the formula: $b = 1 + r$ Solving for $r$: r = b - 1 = 5 - 1 = 4 \quad \text{(or 400% as a percentage)}.

Final Answer:

• Original amount ($a$): $2$
• Growth factor ($b$): $5$
• Rate of increase ($r$): $4$ (or $400\%$).

Would you like me to explain these steps further or provide examples?

Relative Questions:

1. How do you determine the growth factor in exponential functions?
2. What is the difference between the growth factor and the rate of increase?
3. How do exponential growth and decay differ in terms of their formulas?
4. Can the original amount ($a$) ever be negative? What does that imply?
5. How would the function change if the growth factor was less than 1?

Tip:

Remember that in exponential functions, $b > 1$ indicates growth, while $0 < b < 1$ indicates decay!