The given equation for the graph is:

$y = A \cdot 2^x + k$

Looking at the image, we can observe that the curve seems to be a transformed exponential function. To solve for $A$ and confirm $k$:

1. Analyze the horizontal asymptote: The graph appears to approach $y = 1$ as $x \to \infty$, which suggests that $k = 1$. This matches the given value of $k$.

2. Check for points on the curve: By inspecting the graph, we can choose specific points to substitute into the equation $y = A \cdot 2^x + k$.

   • When $x = 0$, the graph seems to pass through approximately $y = 0$.
     • Substituting $x = 0$ and $y = 0$ into the equation: $0 = A \cdot 2^0 + 1 \implies 0 = A + 1 \implies A = -1$

Thus, the value of $A$ is $-1$.

Final Equation:

$y = -1 \cdot 2^x + 1$

Let me know if you'd like more details or have any questions!

Related Questions:

1. How does the horizontal asymptote relate to the parameter $k$?
2. Why is the base of the exponential function $2^x$ significant?
3. How can you verify the value of $A$ using other points on the graph?
4. What happens to the graph if $A$ is positive instead of negative?
5. How does changing $k$ affect the overall position of the graph?

Tip:

To better understand exponential functions, always check key points like $x = 0$ and look for asymptotes, which provide valuable information about $A$ and $k$.