The problem provided involves identifying the initial amount and the growth or decay factor in an exponential function. Here’s a breakdown of the solution:

The exponential function is: $y = 450 \cdot (0.75)^t$

1. Initial Amount (Part b):

   • In an exponential function of the form $y = a \cdot b^t$, the initial amount $a$ is the coefficient before the base $b$.
   • Here, $a = 450$.
   • So, the initial amount is 450.

2. Growth or Decay Factor (Part a):

   • The base $b$ in the equation is 0.75.
   • Since $b = 0.75$, which is less than 1, this indicates a decay process.
   • To find the decay rate, we calculate $1 - b = 1 - 0.75 = 0.25$, which represents a 25% decay rate per year.

Summary of Answers:

• Part a:
  • $b = 0.75$, so the quantity is shrinking.
  • The decay rate is 25% per year.
• Part b:
  • The initial amount is 450.

Would you like further details on how exponential decay works or have any questions?

Additional Practice Questions:

1. How would the function change if the decay rate was 30% instead?
2. What would the function be if the initial amount was doubled?
3. How would the rate change if $b$ were 0.85 instead of 0.75?
4. What is the value of $y$ after 3 years?
5. If the base $b$ were greater than 1, what kind of growth would it represent?

Tip: In exponential functions, the base $b$ determines if the quantity grows or shrinks—values of $b$ greater than 1 indicate growth, while values between 0 and 1 indicate decay.