The problem is to determine the equation of the transformed exponential function based on the graph provided. The general form is $y = A \cdot 2^x + k$, where $A$ is the vertical stretch/compression factor and $k$ is the vertical shift.

Step 1: Identify the horizontal asymptote

From the graph, the horizontal asymptote is at $y = -4$. This indicates the vertical shift $k = -4$.

Step 2: Find a point on the graph

From observation of the graph, the point $(0, -2)$ lies on the curve. Substituting $x = 0$ and $y = -2$ into the equation:

$-2 = A \cdot 2^0 - 4$

Simplify:

$-2 = A \cdot 1 - 4$

$A = 2$

Step 3: Write the equation

Using $A = 2$ and $k = -4$, the equation of the graph is:

$y = 2 \cdot 2^x - 4$

Final Answer:

The equation is:

$y = 2 \cdot 2^x - 4$

Do you have any questions, or would you like a more detailed explanation?

Related Questions:

1. How do you determine the horizontal asymptote of an exponential graph?
2. What happens to the graph when the value of $A$ is negative?
3. How does changing $k$ affect the graph of $y = 2^x$?
4. How do you find the equation of an exponential function given two points?
5. What are the key features of the graph of $y = A \cdot b^x + k$?

Tip:

Always check for the horizontal asymptote and a specific point on the graph to determine transformations of exponential functions accurately.