To construct a rational function with the given properties, let's analyze each condition and incorporate it into the equation step-by-step:

1. Vertical Asymptotes:

   • Vertical asymptotes occur where the denominator of a rational function is zero.
   • To have vertical asymptotes at $x = 2$ and $x = -4$, the denominator must include the factors $(x - 2)$ and $(x + 4)$.
   • Thus, the denominator can be written as $(x - 2)(x + 4)$.

2. x-intercepts:

   • x-intercepts occur where the numerator of a rational function is zero.
   • To have x-intercepts at $(6, 0)$ and $(-3, 0)$, the numerator must include the factors $(x - 6)$ and $(x + 3)$.
   • Thus, the numerator can be written as $(x - 6)(x + 3)$.

3. Horizontal Asymptote:

   • A horizontal asymptote at $y = 9$ suggests that the degrees of the numerator and denominator are equal, and the leading coefficient of the numerator should be 9.
   • Therefore, we need to multiply the numerator by 9.

Putting it all together, the rational function can be written as: $f(x) = \frac{9(x - 6)(x + 3)}{(x - 2)(x + 4)}$

Final Answer:

$f(x) = \frac{9(x - 6)(x + 3)}{(x - 2)(x + 4)}$

Would you like more details on how each property is integrated?

Related Questions:

1. How can we verify the horizontal asymptote of a rational function?
2. How would additional x-intercepts change the form of this function?
3. What changes if the horizontal asymptote was $y = 0$ instead of $y = 9$?
4. How can we find the y-intercept of this rational function?
5. What is the end behavior of this function as $x \to \pm \infty$?

Tip:

For rational functions, the horizontal asymptote depends on the degree of the polynomial in the numerator and the denominator.