Rachel has been watching the number of alligators that live in her neighborhood. The number of alligators changes each week.

n	f(n)
1	48
2	24
3	12
4	6

Which function best shows the relationship between n and f(n)? (1 point)

f(n) = 48(0.5)n − 1

f(n) = 48(0.5)n

f(n) = 24(0.5)n

f(n) = 96(0.5)n − 1

To determine which function best describes the relationship between \( n \) and \( f(n) \), let's analyze the data and the given options.

The data provided shows the number of alligators \( f(n) \) for each week \( n \):

\[
\begin{array}{c|c}
n & f(n) \\
\hline
1 & 48 \\
2 & 24 \\
3 & 12 \\
4 & 6 \\
\end{array}
\]

### Observing the Pattern:
Notice that \( f(n) \) is halving each week:
- \( f(1) = 48 \)
- \( f(2) = 24 \) (which is \( \frac{48}{2} \))
- \( f(3) = 12 \) (which is \( \frac{24}{2} \))
- \( f(4) = 6 \) (which is \( \frac{12}{2} \))

This suggests an exponential decay where each week the number of alligators is halved.

### General Form of the Exponential Decay Function:
The general form of the exponential function is:

\[
f(n) = a \cdot b^n
\]

Where:
- \( a \) is the initial amount (at \( n = 0 \))
- \( b \) is the base of the exponent (in this case \( 0.5 \) since the number is halving each week)

### Applying to the Given Options:
Let's examine the function \( f(n) = 48(0.5)^n \):

- At \( n = 1 \):
  \[
  f(1) = 48 \cdot (0.5)^1 = 48 \cdot 0.5 = 24 \quad \text{(But we need 48)}
  \]
  This does not match \( f(1) = 48 \).

Let's examine the function \( f(n) = 48(0.5)^{n-1} \):

- At \( n = 1 \):
  \[
  f(1) = 48 \cdot (0.5)^{1-1} = 48 \cdot (0.5)^0 = 48 \cdot 1 = 48
  \]
  This matches \( f(1) = 48 \).

- At \( n = 2 \):
  \[
  f(2) = 48 \cdot (0.5)^{2-1} = 48 \cdot (0.5)^1 = 48 \cdot 0.5 = 24
  \]
  This matches \( f(2) = 24 \).

- At \( n = 3 \):
  \[
  f(3) = 48 \cdot (0.5)^{3-1} = 48 \cdot (0.5)^2 = 48 \cdot 0.25 = 12
  \]
  This matches \( f(3) = 12 \).

- At \( n = 4 \):
  \[
  f(4) = 48 \cdot (0.5)^{4-1} = 48 \cdot (0.5)^3 = 48 \cdot 0.125 = 6
  \]
  This matches \( f(4) = 6 \).

### Conclusion:
The correct function that describes the relationship is:

\[
\boxed{f(n) = 48(0.5)^{n-1}}
\]

This function correctly models the given data.

Would you like more details on how exponential decay functions work, or do you have any other questions?

### Related Questions:
1. How would the function change if the initial number of alligators was 96?
2. What is the value of \( f(5) \) using the correct function?
3. How does the function change if the number of alligators decreases by a third each week instead of a half?
4. Can you express the function using a different base, such as \( 2^{-n} \)?
5. How does the function's graph look, and what are its key features?

### Tip:
When analyzing a pattern of exponential decay, pay attention to the factor by which the quantity decreases to identify the base of the exponent.

Explore the relationship between n and f(n) in an exponential decay scenario. Analyze the data of alligator population change over weeks and determine the correct exponential decay function. Suitable for high school students studying exponential functions.

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