Write an equation for a rational function with: Vertical asymptotes at x = 2 and x = − 4 x -intercepts at ( 6 , 0 ) and ( − 3 , 0 ) Horizontal asymptote at y = 9

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.

Write an equation for a rational function with: Vertical asymptotes at x = 2 and x = −4, x-intercepts at (6, 0) and (−3, 0), Horizontal asymptote at y = 9

Construct a rational function equation with vertical asymptotes at x = 2 and x = -4, x-intercepts at (6, 0) and (-3, 0), and a horizontal asymptote at y = 9. Understand the role of numerator and denominator factors in forming rational functions with specified asymptotic and intercept properties, suitable for advanced algebra students in Grades 11-12.

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