Question

How do you graph `f(x)=8/(x(x+2))` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

holes: a value that causes both the numerator and denominator to equal zero. there are no holes in this rational function.

vertical asymptotes: it's a line `->` set the denominator of the rational function equal to `0`:

`x(x+2) = 0`
vertical asymptotes: `x = 0, x = -2`

horizontal asymptotes:
the following are the rules for solving horizontal asymptotes:
let m be the degree of the numerator
let n be the degree of the denominator

if m > n, then there is no horizontal asymptote

if m = n, then the horizontal asymptote is dividing the coefficients of the numerator and denominator

if m < n, then the horizontal asymptote is `y = 0`.

As we can see in our rational function, the denominator has a larger degree of `x`. So the horizontal asymptote is `y = 0`.

x-ints: x-intercepts are the top of the rational function. Since the numerator just says `8`, that means that there are no x-ints.

y-ints: y-intercepts are when you plug in `0` to the function:
`8/(0(0+2))`
`8/0 ->` undefined, so there are no y-ints.

Hope this helps!