Question

How do you graph `y=(x^2-4)/(x^2+3)` using asymptotes, intercepts, end behavior?

Answer 1

Only a horizontal asymptote `y=1`
Intercepts are `0,-4/3` , (2,0) and (-2,0)
minimum at `(0,-4/3)`
And limit `x->+-oo` is `y->1`

Explanation:

We start by finding the Domain
Here it is `RR` as the denominator is always positive
`x^2+3>0`
So there is no horizontal asymptote
The intercepts are `(0,-4/3)` on the y axis
and `(2,0) and (-2,0)` on the x axis
The we look for symmetry, `f(-x)=f(x)`
So thre is symmetry about the y axis
limit `y=1`
`x->-oo`

limit `y=1`
`x->oo`

So `y=1` is a horizontal asymptote
We can calculate the first derivative `y'=(2x(x^2+3)-2x(x^2-4))`
`=(2x^3+6x-2x^3+8x)/(x^2+3)=(14x)/(x^2+3)^2`
So there is a minimum at `(0,-4/3)`

graph{(x^2-4)/(x^2+3) [-8.89, 8.89, -4.444, 4.445]}