Question

write an equation for a rational function with:
vertical asymptotes at $x = -3$ and $x = 3$
x-intercepts at $x = 4$ and $x = -2$
horizontal asymptote at $y = 10$
$y = \square$

Explanation:

Step1: Set denominator for vertical asymptotes

For vertical asymptotes at $x=-3$ and $x=3$, the denominator has factors $(x+3)$ and $(x-3)$.
Denominator: $(x+3)(x-3) = x^2 - 9$

Step2: Set numerator for x-intercepts

For x-intercepts at $x=4$ and $x=-2$, the numerator has factors $(x-4)$ and $(x+2)$.
Numerator (before horizontal asymptote adjustment): $(x-4)(x+2) = x^2 - 2x - 8$

Step3: Adjust for horizontal asymptote

A horizontal asymptote $y=10$ means the ratio of leading coefficients of numerator and denominator is 10. Multiply the numerator by 10.
Adjusted numerator: $10(x^2 - 2x - 8) = 10x^2 - 20x - 80$

Answer:

$y = \frac{10(x-4)(x+2)}{(x-3)(x+3)}$ or $y = \frac{10x^2 - 20x - 80}{x^2 - 9}$