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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.

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

Learn how to write the equation of a transformed exponential function, based on a reflection and a vertical shift. This graph shows y = 2^x reflected over the x-axis and shifted downward. Suitable for students in Grades 9-12.

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