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To solve for \( h \) and \( k \) in the exponential function \( y = 2^{x - h} + k \), let's analyze the given information.

### Step 1: Identify \( k \)
From the graph, the horizontal asymptote is \( y = 2 \). For exponential functions of the form \( y = 2^{x - h} + k \), the horizontal asymptote is at \( y = k \). Therefore:
\[
k = 2
\]

### Step 2: Use the Point (1, 3) to Solve for \( h \)
We know that when \( x = 1 \), \( y = 3 \). Substitute these values along with \( k = 2 \) into the equation:
\[
3 = 2^{1 - h} + 2
\]

Subtract 2 from both sides:
\[
1 = 2^{1 - h}
\]

Since \( 2^0 = 1 \), we have:
\[
1 - h = 0
\]

Solving for \( h \), we find:
\[
h = 1
\]

### Final Answer
\[
h = 1, \quad k = 2
\]

Would you like more details on any of these steps?

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### Related Questions
1. How does changing \( h \) shift an exponential function on the x-axis?
2. What effect does \( k \) have on the vertical position of the exponential curve?
3. How do we determine the horizontal asymptote of an exponential function?
4. What is the general form of a transformed exponential function?
5. How would the function change if we used a base other than 2?

### Tip:
In exponential functions, \( h \) affects horizontal shifts, while \( k \) adjusts vertical position or asymptote level.

Find the values of h and k for the following exponential function y = 2^(x - h) + k given the point (1, 3) and the horizontal asymptote y = 2.

Learn how to find the values of h and k for the exponential function y = 2^(x - h) + k using a given point and horizontal asymptote. This problem covers transformations of exponential functions, suitable for students in Grades 9-12.

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