From the graph you provided and the base function $f(x) = e^x$, it appears that the transformed function $g$ involves a vertical stretch and possibly a horizontal shift.

Analyzing the Graph:

• The graph passes through the point (0, 3), indicating a vertical stretch.
• The original function $f(x) = e^x$ passes through (0, 1).

Transformation Observation:

• Given $g(x) = a \cdot e^x$ and $g(0) = 3$, substituting $x = 0$ into the transformation, we have: $a \cdot e^0 = 3$ $a \cdot 1 = 3$ So, $a = 3$.

Concluding the Equation: The equation of the transformed function $g$ in terms of the function $f$ is: $g(x) = 3f(x) = 3e^x$

Would you like to know more about how transformations affect the graph of exponential functions, or do you have any other questions on this topic? Here are some follow-up queries you might find helpful:

1. What would the graph look like if we also shifted $g(x)$ horizontally?
2. How do reflections across the x-axis or y-axis affect the graph of $f(x) = e^x$?
3. Can you explain the effect of changing the base of the exponential function from $e$ to another base like 2 or 10?
4. How would adding a vertical shift modify the equation and graph of $g(x)$?
5. What is the effect of a horizontal compression on the graph of an exponential function?

Tip: When analyzing transformations of functions, always consider how each transformation—shifts, stretches, compressions, and reflections—affects the original graph in both shape and position.