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

Question

Write an equation for a rational function with:

Vertical asymptotes at x = 4 and x = -2

x intercepts at x = -3 and x = 6

Horizontal asymptote at y = 6

Answer 1

Hello Ashley,

Vertical asymptotes are found from setting the denominator equal to zero. So x = 4 and x = -2 as vertical asymptotes mean you at least have (x-4)(x+2) at the denominator

x-intercepts are found from setting your numerator equal to zero. So x = -3 and x = 6 mean you at least have (x+3)(x-6) at the numerator

Horizontal asymptote of y = 6 tells you that:

1. the degree (a.k.a the exponent of x) of the numerator will be the same as the degree of the denominator. So, if you have x² at the top, you'll have x² at the bottom.
2. The ratio of the leading coefficients at the top and bottom has to equal 6. So if you have a leading coefficient of 12 at the top, you gotta have 2 at the bottom since 12/2 = 6.

Combining all previous points, one rational equation would be:

f(x) = 6*(x+3)(x-6)/((x-4)(x+2) )