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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.

If function f is the parent exponential function f(x) = e^x, what is the equation of the transformed function g in terms of function f? Replace the value of a to complete the equation.

Learn how to transform the parent exponential function f(x) = e^x into g(x) by finding the correct value of a. This involves analyzing graph transformations and vertical stretches, suitable for Grades 9-11.

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