porfiru1nm

2022-12-31

Write an equation for a rational function with: Vertical asymptotes at x = 5 and x = -4 x intercepts at x = -6 and x = 4 Horizontal asymptote at y = 9?

funjisnickenskdg

Expert

(1) Given that there are vertical asymptotes at x=5 and x=-4, the factors in the denominator are (x-5) and (x+4).
(2) As we have x-intercepts at x=-6 and x=4, we have (x+6) and (x-4) in numerator as factors
(3) As we have horizontal asymptote at y=9, the highest power of numerator and denominator are same and coefficient of numerator is 9 times that of denominator
Hence, rational function is $\frac{9\left(x+6\right)\left(x-4\right)}{\left(x-5\right)\left(x+4\right)}$ or $\frac{9{x}^{2}+18x-216}{{x}^{2}-x-20}$

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