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The question is asking for a description of the transformation applied to the function \( f(x) = 3x^5 - 6x^2 + 21x \) to form the new function \( g(x) = -9(3(2x)^5 - 6(2x)^2 + 21(2x)) \).

Let's break down the transformation step by step:

1. **Horizontal Dilation (Stretch/Compression)**: The term \( 2x \) inside the function means a horizontal dilation. Since \( 2x \) compresses the function horizontally by a factor of \( \frac{1}{2} \). This is because multiplying by a number greater than 1 compresses the function.

2. **Vertical Dilation (Stretch/Compression)**: The factor of \( -9 \) outside the entire function affects the vertical scaling. Since \( 9 > 1 \), it stretches the function vertically by a factor of 9. The negative sign indicates a reflection across the x-axis.

3. **Reflection**: The negative sign in front of the factor \( -9 \) causes a reflection over the x-axis.

Now, putting these pieces together:
- The factor of \( 2x \) indicates a **horizontal compression** by a factor of \( \frac{1}{2} \).
- The factor of \( -9 \) indicates a **vertical dilation** by a factor of 9 and a **reflection across the x-axis**.

So, the correct description of the transformation is:

- **A horizontal dilation of a factor of \( \frac{1}{2} \), a vertical dilation of a factor of 9, and reflection across the x-axis**.

This matches the third option in the choices provided.

Do you have any questions, or would you like more details? Here are some related questions for further practice:

1. What happens if the horizontal dilation factor is negative?
2. How does changing the coefficient inside the function affect the horizontal transformation?
3. What would happen if the vertical dilation factor was less than 1?
4. Can a transformation involve both horizontal and vertical reflections?
5. How can you recognize vertical translations from the function's form?

**Tip**: Always remember that coefficients inside the function affect the x-axis (horizontal transformations), while coefficients outside the function affect the y-axis (vertical transformations).

The function f(x) is given by f(x) = 3x^5 - 6x^2 + 21x. A function g is created as a result of a transformation of f. If g(x) = -9(3(2x)^5 - 6(2x)^2 + 21(2x)), which of the following describes the transformation?

Understand the transformation of the polynomial function f(x) = 3x^5 - 6x^2 + 21x into g(x) = -9(3(2x)^5 - 6(2x)^2 + 21(2x)). This solution covers horizontal and vertical dilations as well as reflections over the x-axis, ideal for high school students studying function transformations.

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