Write an equation for a rational function with:

Question

Vertical asymptotes at x = -6 and x = -2

x intercepts at x = 2 and x = -3

Horizontal asymptote at y = 10

y=

Answer 1

In order to have two vertical asymptotes at x=-6 and x=-2 the function must tend to infinity at those points.

so you can start by building the denominator as (x+6)(x+2)

Regarding the x intercepts at x=2 and x=-3, it implies that the function is zero at those points, so you can construct the numerator as (x-2)(x+3).

So at this stage the function reads f(x)=(x-2)*(x+3)/(x+6)(x+2)

However, when you take the limit of this function as x tends to infinity, you will obtain the value 1. thus, the horizontal asymptote is y=1. In order to get y=10, you can simply multiply the numerator by the factor 10.

Finally, a function which satisfies all those requirements is given by f(x)=10*(x-2)*(x+3)/(x+6)(x+2)

Best,

Davide