Let's solve this step-by-step based on the given transformations of the function $f(x) = 2x^4 - x^2 + x + 4$.

The transformations required are:

1. Reflection in the y-axis: This changes $f(x)$ to $f(-x)$, flipping the graph horizontally.
2. Vertical stretch by a factor of 3: This multiplies the entire function by 3, giving us $3f(-x)$.
3. Translation 1 unit down: This shifts the graph down by subtracting 1 from the function, resulting in $3f(-x) - 1$.

Step-by-step process:

1. Reflection in the y-axis: Replace $x$ with $-x$ in the original function $f(x)$: $f(-x) = 2(-x)^4 - (-x)^2 + (-x) + 4 = 2x^4 - x^2 - x + 4$ So after reflection, $f(-x) = 2x^4 - x^2 - x + 4$.

2. Vertical stretch by a factor of 3: Multiply the entire function by 3: $3f(-x) = 3(2x^4 - x^2 - x + 4) = 6x^4 - 3x^2 - 3x + 12$

3. Translation 1 unit down: Subtract 1 from the function: $g(x) = 6x^4 - 3x^2 - 3x + 12 - 1 = 6x^4 - 3x^2 - 3x + 11$

Final answer:

$g(x) = 6x^4 - 3x^2 - 3x + 11$

Would you like further details or clarifications? Here are some related questions to deepen your understanding:

1. How does reflecting in the y-axis affect the graph of a polynomial function?
2. What happens to the function if we reflect it in the x-axis instead?
3. How does a vertical stretch compare to a horizontal stretch in terms of effect on the graph?
4. Can you describe how horizontal shifts differ from vertical shifts in a function?
5. What if the translation was 1 unit up instead of down?

Tip: When working with transformations, always apply them in the correct order: reflection, stretch/compression, and translation.