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.