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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.

Write a rule for g(x) that represents the indicated transformations of the graph of f(x) = 2x^4 - x^2 + x + 4. The transformations are a reflection in the y-axis, a vertical stretch by a factor of 3, and a translation 1 unit down.

Solve a problem involving transformations of the function f(x) = 2x^4 - x^2 + x + 4. Learn about reflections, vertical stretches, and translations in polynomial functions. Suitable for students in Grades 10-12.

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