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The problem asks for the value of \( k \) when the function \( f(x) = \frac{1}{2}(x + 3)^2 \) is rewritten in standard form as:

\[
f(x) = \frac{1}{2}x^2 + kx + \frac{9}{2}
\]

To solve this, we need to expand the given expression and match it to the standard form.

### Step 1: Expand \( \frac{1}{2}(x + 3)^2 \)

We start by expanding \( (x + 3)^2 \):

\[
(x + 3)^2 = x^2 + 6x + 9
\]

Now multiply by \( \frac{1}{2} \):

\[
f(x) = \frac{1}{2}(x^2 + 6x + 9)
\]

Distribute the \( \frac{1}{2} \):

\[
f(x) = \frac{1}{2}x^2 + 3x + \frac{9}{2}
\]

### Step 2: Compare with the standard form

Now, compare this result with the standard form:

\[
f(x) = \frac{1}{2}x^2 + kx + \frac{9}{2}
\]

We can see that the value of \( k \) is \( 3 \).

Thus, the value of \( k \) is \( \boxed{3} \).

---

Would you like more details on how to expand or any other concepts clarified? Here are a few related questions:

1. How do you expand a binomial square like \( (x + 3)^2 \)?
2. What is the general method for converting a quadratic function into standard form?
3. How can you use completing the square to convert a quadratic into standard form?
4. What is the difference between vertex form and standard form of a quadratic function?
5. How would the value of \( k \) change if the expression was \( \frac{1}{3}(x + 3)^2 \)?

Tip: When expanding binomials, always remember to square both terms and apply the distributive property carefully.

The function f(x) = (1/2)(x + 3)^2 can be written in standard form as f(x) = (1/2)x^2 + kx + 9/2. What is the value of k?

Learn how to find the value of k when rewriting the quadratic function f(x) = (1/2)(x + 3)^2 in standard form. This problem covers expanding binomials and matching terms in quadratic equations, suitable for Grades 8-10.

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