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?

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.