https://file.aiquickdraw.com/mathbot/file/1730741858584h2tgfljc.jpg

To determine which graph represents the function \( f(x) = 4 \cdot 5^{-x} \), let's analyze the properties of this function:

1. **Form of the Function**:
   - The function \( f(x) = 4 \cdot 5^{-x} \) is an exponential decay function because the base \( 5^{-x} = \frac{1}{5^x} \) indicates a negative exponent, causing the function to decay as \( x \) increases.
   
2. **Behavior of \( f(x) \)**:
   - As \( x \to \infty \) (as \( x \) increases positively), \( f(x) \) approaches \( 0 \), since \( 5^{-x} \) will get closer and closer to zero.
   - As \( x \to -\infty \) (as \( x \) decreases negatively), \( f(x) \) grows very large, because \( 5^{-x} \) will become very large for negative \( x \).

3. **Key Points**:
   - At \( x = 0 \): \( f(0) = 4 \cdot 5^{0} = 4 \cdot 1 = 4 \).
   - At \( x = 1 \): \( f(1) = 4 \cdot 5^{-1} = 4 \cdot \frac{1}{5} = 0.8 \).
   - This confirms that \( f(x) \) decreases as \( x \) increases.

4. **Identification of the Correct Graph**:
   - The correct graph should show a rapid decay from a high value on the left side (negative \( x \)) toward zero on the right side (positive \( x \)).
   - The graph should pass through \( (0, 4) \) and have a horizontal asymptote at \( y = 0 \).

Looking at the provided options:
- The correct graph is the **top-left** one, which shows exponential decay behavior, decreasing from left to right and approaching zero as \( x \) becomes positive.

Would you like more details or have any questions?

### Related Questions:
1. How would the graph change if the function were \( f(x) = 4 \cdot 5^x \) instead of \( f(x) = 4 \cdot 5^{-x} \)?
2. What would happen to the graph if we changed the coefficient from 4 to a different number?
3. How does the base (5 in this case) impact the shape and steepness of the decay?
4. What if the exponent were \( -2x \) instead of \( -x \)? How would that affect the graph?
5. How does this graph compare to a function with a base between 0 and 1, such as \( f(x) = 4 \cdot 0.5^x \)?

### Tip:
For exponential functions, a negative exponent typically indicates decay, while a positive exponent results in growth.

Which graph represents \( f(x) = 4 \cdot 5^{-x} \)?

Analyze the exponential decay function f(x) = 4 * 5^(-x) to determine its corresponding graph. This problem focuses on identifying the characteristics of exponential functions, including decay behavior and horizontal asymptotes, suitable for students in Grades 9-11.

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