Question

How do you write a rational function that has the following properties: a zero at x= 4, a hole at x= 7, a vertical asymptote at x= -3, a horizontal asymptote at y= 2/5?

Answer 1

Rational function is `(2x^2-22x+56)/(5x^2-20x-105)`

Explanation:

A zero at `x=4` means we have `(x-4)` as a factor in numerator;

a hole at `x=7` means, we have `x-7` a factor both in numerator as well as denominator;

a vertical asymptote at `x=-3` means `x+3` a factor in denominator only

a horizontal asymptote at `y=2/5` means highesr degrees in both numerator and denominator are equal and their coefficients are in ratio of `2:5`

Hence desired rational function is `(2(x-4)(x-7))/(5(x-7)(x+3))`

i.e. `(2x^2-22x+56)/(5x^2-20x-105)`

See its graph down below. Observe vertical asymptote `x=-3` and horizontal asymptote `y=2/5`. We have a zero at `x=4` as functtion passes through `(4,0)`. Hole is not seen as `x-7` cancels out, but we know, the function is not definedat this point.

graph{(2x^2-22x+56)/(5x^2-20x-105) [-10.67, 9.33, -4.4, 5.6]} 1