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The problem in the image asks about transforming the function \( f(x) = 3x^2 + 2x + 1 \) through a **vertical dilation by a factor of 2**.

### Step-by-Step Explanation:
- **Vertical dilation** means multiplying the entire function by a constant factor.
- If we dilate \( f(x) \) by a factor of 2, we multiply the entire function by 2:

\[
f(x) = 3x^2 + 2x + 1
\]
\[
2f(x) = 2(3x^2 + 2x + 1) = 6x^2 + 4x + 2
\]

So, we are looking for an equation where the right-hand side matches \( 6x^2 + 4x + 2 \).

### Analyzing the Options:
- **Option A:** \( m(x) = 12x^2 + 4x + 1 \) – This does not match because the coefficient of \( x^2 \) is incorrect (it should be 6, not 12).
- **Option B:** \( k(x) = 6x^2 + 4x + 2 \) – This matches the transformation \( 2f(x) \).
- **Option C:** This expression represents an additive transformation, not a multiplicative one.
- **Option D:** This adds 2 to \( f(x) \), which is not a vertical dilation.

### Correct Answer:
The correct answer is **B**: \( k(x) = 6x^2 + 4x + 2 \), which is the result of multiplying \( f(x) \) by 2.

Would you like further details or have any other questions?

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### Related Questions:
1. How do you distinguish between vertical and horizontal transformations in functions?
2. What is the difference between a vertical dilation and a vertical shift?
3. How would the graph of \( f(x) = 3x^2 + 2x + 1 \) look after a vertical compression by a factor of 1/2?
4. What happens if you apply a negative factor for vertical dilation?
5. How do transformations affect the vertex of a quadratic function?

#### Tip:
Always remember, for vertical transformations, multiplying affects the "height" of the graph, while adding or subtracting affects its "position" vertically.

The function f is given by f(x) = 3x^2 + 2x + 1. The graph of which of the following functions is the image of the graph of f after a vertical dilation of the graph of f by a factor of 2?

Explore how to perform a vertical dilation on the quadratic function f(x) = 3x^2 + 2x + 1 by a factor of 2. This problem covers the concept of transformations in algebra and demonstrates how to apply vertical scaling to functions, suitable for students in Grades 9-12.

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