Let's use the form \( f(x) = \left(\frac{1}{2}\right)^{k(x-d)} + c \) to determine the possible equation.

### Analyzing the Given Form
We know from the graph:
- The horizontal asymptote is \( y = 3 \), so \( c = 3 \).
- The function passes through the point \((0, 4)\) and \((1, 6)\).
- The function appears to increase exponentially, which means \( k > 0 \).

### Step 1: Plug in the Point \((0, 4)\)

Using the point \((0, 4)\):

\[
4 = \left(\frac{1}{2}\right)^{k(0-d)} + 3
\]

Simplifying:

\[
1 = \left(\frac{1}{2}\right)^{-kd}
\]

\[
\left(\frac{1}{2}\right)^{-kd} = 1 \quad \Rightarrow \quad -kd = 0 \quad \Rightarrow \quad d = 0
\]

So, the function simplifies to:

\[
f(x) = \left(\frac{1}{2}\right)^{kx} + 3
\]

### Step 2: Plug in the Point \((1, 6)\)

Now, using the point \((1, 6)\):

\[
6 = \left(\frac{1}{2}\right)^{k(1)} + 3
\]

Simplifying:

\[
3 = \left(\frac{1}{2}\right)^k
\]

To solve for \( k \), take the natural logarithm on both sides:

\[
\ln(3) = k \ln\left(\frac{1}{2}\right)
\]

\[
k = \frac{\ln(3)}{\ln\left(\frac{1}{2}\right)} \quad \Rightarrow \quad k = -\frac{\ln(3)}{\ln(2)}
\]

So the equation becomes:

\[
f(x) = \left(\frac{1}{2}\right)^{-\frac{\ln(3)}{\ln(2)} \cdot x} + 3
\]

### Final Answer

The possible equation for the exponential function in the required form is:

\[
f(x) = \left(\frac{1}{2}\right)^{-\frac{\ln(3)}{\ln(2)} \cdot x} + 3
\]

This equation correctly reflects the behavior of the graph, increasing as \(x\) increases, with an asymptote at \(y = 3\).

Would you like further clarification on this, or do you have any other questions?

---

**Relative Questions:**

1. How does the value of \(k\) influence the steepness of the graph?
2. What effect does changing \(d\) have on the graph of the function?
3. Can the value of \(c\) ever be negative, and what would that mean for the graph?
4. How would the graph change if \(k\) were positive instead of negative?
5. What is the significance of the base \(\frac{1}{2}\) in the function?
6. How would the function behave as \( x \) approaches infinity?
7. What would happen if \( d \) were a non-zero constant?
8. Can this form be converted back into the \( f(x) = ab^{(x-h)} + c \) form?

**Tip:** When dealing with exponential functions, the base and exponent determine the rate and direction of growth.

Explore the exponential function f(x) = (1/2)^k(x-d) + c with detailed analysis and step-by-step solutions. Learn about the impact of parameters such as k, d, and c on the graph's behavior. Suitable for advanced high school students and learners interested in exponential functions.

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