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The problem given is:

\[ f(x) = 2x^2 - 10x + 5 \]

The function \( g(x) \) is defined as \( g(x) = f(x+3) \). We need to find the value of \( x \) for which \( g(x) \) reaches its minimum.

### Step 1: Express \( g(x) \) in terms of \( x \)
To do this, substitute \( x + 3 \) into the function \( f(x) \):

\[
g(x) = f(x + 3) = 2(x + 3)^2 - 10(x + 3) + 5
\]

### Step 2: Expand and simplify the expression for \( g(x) \)
Expand \( (x + 3)^2 \):

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

Now substitute this back into the equation for \( g(x) \):

\[
g(x) = 2(x^2 + 6x + 9) - 10(x + 3) + 5
\]

Distribute the terms:

\[
g(x) = 2x^2 + 12x + 18 - 10x - 30 + 5
\]

Combine like terms:

\[
g(x) = 2x^2 + 2x - 7
\]

### Step 3: Find the minimum value of \( g(x) \)
The minimum value of a quadratic function \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = 2 \):

\[
x_{\text{min}} = -\frac{2}{2 \times 2} = -\frac{2}{4} = -\frac{1}{2}
\]

So, the minimum value of \( g(x) \) occurs at \( x = -\frac{1}{2} \).

### Step 4: Verify with the answer choices
The answer choices provided were integers, but our calculated minimum point is \( x = -\frac{1}{2} \), which does not correspond directly to any of the choices.

This discrepancy suggests a mistake might have been made in expanding the function or interpreting the problem. Let's check it again to ensure correctness:

- We correctly applied the formula for the vertex of the parabola.
- However, the exact minimum value might be needed to compare against the options, or it's possible the problem's structure demands an adjustment in interpretation.

### Final Answer
Given the above, it seems the options might represent the y-value (the minimum point) rather than the x-value, or there might be a misinterpretation of the question. I would recommend reconsidering the problem statement to match the answer choices provided.

Let me know if you need further details or have any questions!

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1. How do you determine the vertex of a quadratic function?
2. Can the minimum value be found using derivatives?
3. How does completing the square help in finding the vertex?
4. What are the implications of translating a function horizontally?
5. How do you confirm if a function opens upwards or downwards?

**Tip:** When dealing with quadratic functions, always check if you're solving for the vertex (x-coordinate) or the actual minimum/maximum value (y-coordinate).

Discover how to find the minimum value of the quadratic function g(x) = 2x^2 + 2x - 7. This problem explores quadratic functions, the vertex formula, and finding the minimum point. Suitable for Grades 10-12 students studying advanced algebra.

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