To determine which function rule matches the graph provided, let’s analyze the graph's characteristics:

The graph shows an exponential growth function that:

1. Starts near zero as $x$ approaches negative values.
2. Increases rapidly as $x$ becomes positive.

Each function provided is of the form $y = c \cdot 4^x$, where $4^x$ is the base exponential function. The value of $c$ affects the vertical stretch or compression of the graph.

Analysis of Options

1. $y = 2 \cdot 4^x$: This would result in a function that grows relatively quickly, doubling the base $4^x$.
2. $y = \frac{1}{2} \cdot 4^x$: This function would grow, but at a slower rate compared to $y = 4^x$ due to the $\frac{1}{2}$ factor.
3. $y = \frac{1}{4} \cdot 4^x$: This function would also show exponential growth but at an even slower rate.
4. $y = 4^x$: This represents pure exponential growth without any scaling factor, making it the standard exponential curve.

Matching with the Graph

The graph matches $y = 4^x$ best. The function is an unmodified exponential growth without additional scaling, consistent with the rapid increase in the graph at positive $x$ values.

Final Answer

The function rule that matches the graph is: $y = 4^x$

Would you like further details or have any questions?

Here are some related questions:

1. What is the effect of different values of $c$ in the function $y = c \cdot 4^x$?
2. How does the base of the exponential function (4 in this case) impact the graph's growth rate?
3. What would happen if the base was a fraction instead, like $y = 4^{-\frac{x}{2}}$?
4. How can you determine the horizontal asymptote for exponential functions?
5. What would the graph look like if the function was $y = -4^x$?

Tip: Exponential growth functions with a base greater than 1 increase as $x$ increases, while functions with a base between 0 and 1 decrease as $x$ increases.