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The question asks for an equation in the form \( y = A \cdot 2^x + k \) that describes the transformation of the base function \( y = 2^x \).

### Step 1: Analyze the Transformation
1. **Vertical Reflection**: The graph opens downward, which suggests that the coefficient \( A \) is negative.
2. **Vertical Shift**: The graph's horizontal asymptote appears to be around \( y = -3 \), indicating a vertical shift of \( k = -3 \).

### Step 2: Determine the Value of \( A \)
The original function \( y = 2^x \) has been reflected and shifted. Let's use a point from the graph to determine the exact value of \( A \). 

From inspection, it looks like the graph passes through \( (0, -1) \).

Using the point \( (0, -1) \):
\[
y = A \cdot 2^0 + k
\]
\[
-1 = A \cdot 1 - 3
\]
\[
A - 3 = -1
\]
\[
A = 2
\]

Since the graph is reflected, we take \( A = -2 \).

### Final Equation
Thus, the equation of the transformed graph is:
\[
y = -2 \cdot 2^x - 3
\]

Would you like further details or have any questions about this solution?

### Related Questions:
1. How would the equation change if the vertical shift was different?
2. What if the reflection were along the x-axis instead of the y-axis?
3. How does changing \( A \) affect the steepness of the graph?
4. What transformations occur if \( k \) is positive?
5. How do we derive transformations of exponential functions in general?

**Tip:** Always check the new asymptote of transformed exponential graphs to determine the vertical shift parameter \( k \).

The graph below shows a transformation of y = 2^x. Write an equation of the form y = A * 2^x + k for the graph shown.

Explore how to determine the equation for a transformation of the exponential function y = 2^x. This problem involves analyzing transformations, including vertical reflections and shifts, to form the equation y = A * 2^x + k. Ideal for students in Grades 9-11.

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