The graph shows an exponential function that increases rapidly as $x$ becomes positive and approaches zero as $x$ becomes negative. Based on the vertical stretch and shape, let's examine each option:

1. $y = 10 \cdot 4^x$: This would have a large initial value and a steep rise, as the factor of 10 stretches it vertically.
2. $y = 2 \cdot 4^x$: This also has an exponential growth pattern but with a smaller vertical stretch compared to the first option.
3. $y = 3 \cdot 10^x$: This would grow extremely quickly because of the base 10, indicating a very steep exponential increase.
4. $y = 1 \cdot 4^x$: This function has a base of 4, which matches the observed growth rate closely without any additional stretch factor.

From the graph, it appears to be an exponential function with a base of 4 and no additional vertical stretch factor, so the most fitting option is: $y = 1 \cdot 4^x$

Would you like further details on why this choice fits the graph?

Here are some related questions to deepen understanding:

1. How does changing the base of an exponential function affect its growth rate?
2. What role does the coefficient (like 10, 2, or 1 in these options) play in shaping the graph?
3. How would the graph change if the base was less than 1, such as $y = 4^{-x}$?
4. What are some real-world phenomena that exhibit exponential growth like this?
5. How can we find the equation of an exponential function given only its graph?

Tip: Exponential functions with larger bases grow faster as $x$ increases, and adding a coefficient in front of the base affects the initial height but not the growth rate.