To determine $h$ in the function $f(x) = a(x - h)^2 + k$, we need to identify the vertex of the quadratic function, since the vertex form of a parabola $f(x) = a(x - h)^2 + k$ has $(h, k)$ as the vertex.

From the graph:

1. The vertex of the parabola is at the point $(1, 4)$.
2. Therefore, $h = 1$ (the $x$-coordinate of the vertex).
3. We can also see that $k = 4$ (the $y$-coordinate of the vertex).

So, $h = 1$.

Would you like further details on this problem, or do you have additional questions?

Here are some related questions to deepen your understanding:

1. How would you determine the value of $a$ for this function?
2. How does the vertex form $f(x) = a(x - h)^2 + k$ help in identifying transformations of the parent function $y = x^2$?
3. If $h$ changes, how does it affect the graph of the parabola?
4. What does the parameter $k$ represent in the vertex form of a quadratic function?
5. How can you rewrite this function in standard form $f(x) = ax^2 + bx + c$ using the values of $h$, $k$, and $a$?

Tip: In the vertex form $f(x) = a(x - h)^2 + k$, $h$ and $k$ directly represent the horizontal and vertical shifts of the basic parabola $y = x^2$.