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 $k$.