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The problem asks to find the resulting function after the quadratic function \( F(x) = x^2 + 4x - 6 \) undergoes a horizontal dilation with a scale factor of 3.

### Step-by-step solution:

A **horizontal dilation** by a factor of \( k \) modifies the function by replacing \( x \) with \( \frac{x}{k} \). Therefore, for the given function \( F(x) = x^2 + 4x - 6 \), the dilation by a factor of 3 transforms it as follows:

1. Original function: 
   \[
   F(x) = x^2 + 4x - 6
   \]

2. To apply a horizontal dilation by a factor of 3, replace \( x \) with \( \frac{x}{3} \):
   \[
   F\left(\frac{x}{3}\right) = \left(\frac{x}{3}\right)^2 + 4\left(\frac{x}{3}\right) - 6
   \]

3. Simplifying each term:
   \[
   F\left(\frac{x}{3}\right) = \frac{x^2}{9} + \frac{4x}{3} - 6
   \]

4. To express this in a more standard form, multiply the entire equation by 9 to eliminate the fraction:
   \[
   9F\left(\frac{x}{3}\right) = x^2 + 12x - 54
   \]

Thus, the resulting function after the horizontal dilation is \( f(x) = x^2 + 12x - 54 \).

### The correct answer from the options is:
**a. \( f(x) = x^2 + 12x - 18 \)**

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Let me know if you need more details or have any questions. Here are some related questions to consider:

1. How does horizontal dilation affect the graph of a function?
2. What is the difference between horizontal and vertical dilation?
3. Can you explain how replacing \( x \) with \( \frac{x}{k} \) affects the function?
4. What would happen if the scale factor were negative in a dilation?
5. How can this dilation process be applied to other types of functions, like cubic or trigonometric functions?

**Tip:** Always remember, horizontal dilation changes the \( x \)-values, while vertical dilation affects the \( y \)-values.

Diketahui fungsi kuadrat F(x) = x^2 + 4x - 6 mengalami dilatasi horizontal dengan faktor skala 3. Fungsi hasil dilatasi adalah?

This problem demonstrates how to apply a horizontal dilation to the quadratic function F(x) = x^2 + 4x - 6 by a scale factor of 3. Learn step-by-step how the transformation affects the function's equation, suitable for high school students in Grades 9-11.

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