To determine the equation of the form $y = A \cdot 2^x + k$ for the transformed graph, let’s analyze the graph step-by-step.

Step 1: Identify the transformations

1. Reflection: The graph of $y = 2^x$ usually increases as $x$ increases. Here, however, the graph decreases, suggesting a reflection across the x-axis. This means $A$ will be negative.
2. Vertical Scaling: The curve seems stretched, suggesting $A$ might not just be -1.
3. Vertical Shift: The graph does not appear to approach zero as $x \to -\infty$, indicating a vertical shift. This will determine $k$.

Step 2: Choose points to solve for $A$ and $k$

From the graph:

• At $x = 0$, $y = 2$.
• At $x = 1$, $y = -1$.

Step 3: Set up equations based on these points

Using the general form $y = A \cdot 2^x + k$:

1. For $x = 0$: $2 = A \cdot 2^0 + k \Rightarrow 2 = A + k$

2. For $x = 1$: $-1 = A \cdot 2^1 + k \Rightarrow -1 = 2A + k$

Step 4: Solve the system of equations

From the first equation: $k = 2 - A$

Substitute $k = 2 - A$ into the second equation: $-1 = 2A + (2 - A)$ $-1 = A + 2$ $A = -3$

Then substitute $A = -3$ back into $k = 2 - A$: $k = 2 - (-3) = 5$

Solution

The equation for the transformed graph is: $y = -3 \cdot 2^x + 5$

Would you like further explanation on any of these steps, or have additional questions?

Follow-up Questions

1. What effect does changing the value of $A$ have on the graph?
2. How does a vertical shift (the $k$ value) affect the asymptote of an exponential graph?
3. How would the equation change if the graph were reflected over the y-axis instead?
4. What transformations are needed to turn $y = 2^x$ into $y = 4 \cdot 2^x - 7$?
5. How does the base of the exponent (here, 2) affect the steepness of the exponential curve?

Tip

When analyzing transformations, always check the reflection, scaling, and shifts in order; they each have a distinct impact on the shape and position of the graph.