The given graph is a transformation of the exponential function $y = 2^x$, and you are tasked with writing an equation of the form $y = A \cdot 2^x + k$.

Let's analyze the graph:

1. Reflection: The graph is a reflection of $y = 2^x$ over the x-axis because the curve slopes downward as $x$ increases. This indicates that the coefficient $A$ is negative.

2. Vertical Shift: The graph appears to be shifted downward. The horizontal asymptote of the original function $y = 2^x$ is at $y = 0$, but for this graph, the asymptote is at $y = -3$. Therefore, the vertical shift $k$ is $-3$.

3. Equation form: The reflection and vertical shift suggest the equation will take the form: $y = -A \cdot 2^x - 3$ where $A$ is positive.

To determine $A$, we use a point on the graph. Let's take the point $(0, -1)$, which lies on the graph.

Substitute $x = 0$ and $y = -1$ into the equation: $-1 = -A \cdot 2^0 - 3$ $-1 = -A - 3$ Solve for $A$: $-1 + 3 = -A$ $A = 2$

Thus, the equation of the graph is: $y = -2 \cdot 2^x - 3$

Final Answer:

$y = -2 \cdot 2^x - 3$

Would you like further clarification or details?

Here are 5 related questions you could explore:

1. How does the transformation change if $k$ is positive?
2. What if the reflection was over the y-axis instead of the x-axis?
3. How would a horizontal shift affect the equation?
4. What would the graph look like if $A$ were a fraction?
5. How does the base of the exponential function (e.g., using 3 instead of 2) affect the transformation?

Tip: To identify reflections, check whether the graph is flipped relative to the original function about an axis.