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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?

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### 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.

Identify the initial amount and the growth or decay factor, b, for the exponential function y = 450 * (0.75)^t.

Learn how to identify the initial amount and the decay factor in the exponential function y = 450 * (0.75)^t. This problem covers exponential functions, decay rates, and initial values. Suitable for high school students in Grades 9-10.

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