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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 \( y = x^2 \).

Use the graph of the quadratic function f to write its formula as f(x) = a(x − h)^2 + k. Determine h in f(x) = a(x − h)^2 + k.

Explore how to determine the value of h in the quadratic function f(x) = a(x − h)^2 + k by analyzing the graph and identifying the vertex. This problem covers key algebraic concepts in quadratic functions, focusing on the vertex form.

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