The function *f* is given by *f* (*x*) = 3*x*5 – 6*x*2 + 21*x*. A function *g* is created as a result of a transformation of *f*. If *g*(*x*) = –4(3(6*x*)5 – 6(6*x*)2 + 21(6*x*)), which of the following describes the transformation?

A horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the y-axis A horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the x-axis A horizontal dilation of a factor of 6, a vertical dilation of a factor of 4, and reflection across the x-axis A horizontal dilation of a factor of 6, a vertical dilation of a factor of 4, and reflection across the y-axis

We are given two functions:

1. \( f(x) = 3x^5 - 6x^2 + 21x \)
2. \( g(x) = -4\left( 3(6x)^5 - 6(6x)^2 + 21(6x) \right) \)

### Step 1: Identifying transformations

#### Horizontal dilation:
The expression \( g(x) = -4\left( 3(6x)^5 - 6(6x)^2 + 21(6x) \right) \) shows that in each term, \( x \) has been replaced by \( 6x \). This indicates a horizontal dilation by a factor of \( 1/6 \). When \( x \) is replaced by \( 6x \), it compresses the function horizontally by a factor of \( 6 \), which corresponds to a dilation of \( 1/6 \).

#### Vertical dilation:
The function \( g(x) \) has an overall factor of \( -4 \) in front of the entire expression. This factor affects the vertical scaling. Specifically, multiplying the function by \( -4 \) applies a vertical dilation by a factor of \( 4 \).

#### Reflection:
The negative sign in front of \( g(x) = -4\left( 3(6x)^5 - 6(6x)^2 + 21(6x) \right) \) implies a reflection across the **x-axis**.

### Step 2: Verifying the correct transformation
Now let's compare these findings with the options provided:

- **Horizontal dilation**: The dilation factor is \( 1/6 \), which is due to replacing \( x \) with \( 6x \).
- **Vertical dilation**: The factor \( -4 \) means there is a vertical dilation by a factor of 4.
- **Reflection**: The negative sign indicates a reflection across the x-axis.

### Conclusion:
The correct description is **a horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the x-axis**.

Thus, the correct answer is:
**A horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the x-axis**.

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Would you like more details or have any further questions? 

Here are five related questions to expand on the concept:
1. How does a vertical dilation affect the graph of a function?
2. What is the difference between a reflection across the x-axis and the y-axis?
3. Can you explain the effect of replacing \( x \) with \( kx \) for any value of \( k \)?
4. How would the graph of the function change with only a horizontal dilation?
5. How can you combine different transformations to achieve a desired effect on a graph?

**Tip**: To identify horizontal transformations, focus on how the input \( x \) is modified inside the function—replacing \( x \) with \( kx \) indicates a horizontal dilation by \( 1/k \).

The function f 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) = –4(3(6x)^5 – 6(6x)^2 + 21(6x)), which of the following describes the transformation? A horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the y-axis A horizontal dilation of a factor of 1/6, a vertical dilation of a factor of 4, and reflection across the x-axis A horizontal dilation of a factor of 6, a vertical dilation of a factor of 4, and reflection across the x-axis A horizontal dilation of a factor of 6, a vertical dilation of a factor of 4, and reflection across the y-axis.

Explore how function transformations work with a detailed breakdown of the transformation of f(x) = 3x^5 – 6x^2 + 21x into g(x) = -4(3(6x)^5 – 6(6x)^2 + 21(6x)). This problem covers horizontal dilation by a factor of 1/6, vertical dilation by a factor of 4, and reflection across the x-axis. Suitable for Grades 10-12.

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