Question

How do you graph `f(x)=(x^2-1)/x` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

You have a vertical asymptote at x=0 because that would make the denominator equal zero. There's a slant asymptote at `y=x` because `x^2/x=x`

Explanation:

You know this graph can't exist at `x=0`, since that would make the denominator equal `0`. Because the polynomial on the top is of a bigger degree, there is a slant asymptote. Dividing the initial terms, we get `x^2/x=x`, so there is a slant asymptote at `y=x`.
Since we can factor the top into `(x-1)(x+1)`, we know the function has two solutions at `x=+-1`.

Plotting these solutions and following the asymptotes makes this a straightforward graph to sketch: graph{(x^2-1)/x [-10, 10, -5, 5]}

Also, not all rational functions are so easy to predict the behavior of, so creating a table of x and y values is always a good idea! And if you need more information about how to find the asymptotes, look here.